study of active structural vibration control strategies for chatter

Research Collection

Doctoral Thesis

Active structural methods for chatter mitigation in millingprocess

Author(s): Monnin, Jérémie

Publication Date: 2013

Permanent Link: https://doi.org/10.3929/ethz-a-009920936

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Diss. ETH No. 21015

Active Structural Methods for ChatterMitigation in Milling Process

A dissertation submitted to the

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

JÉRÉMIE MONNIN

M.Eng.

born February 2nd 1980

citizen of Nidau - Switzerland

accepted on the recommendation of

Prof. Dr. Konrad Wegener, examiner

Prof. Dr. John Lygeros, co-examiner

Dr. Guillaume Ducard, co-examiner

2013

Acknowledgements III

Acknowledgements

This thesis attempts to present the major outcomes achieved in a work spreads overapproximately six and a half years realized in the framework of the IDEFIX project atinspire AG. Over such a long time period, many peoples contributed in one way or anotherto the results presented here.

First of all, the expression of my deepest gratitude goes to Prof. Konrad Wegener, super-visor of my thesis and head of the Institute for Machine Tools and Manufacturing (IWF)of the ETH Zurich, for giving me the opportunity to work on this project through which somany exciting topics, such as manufacturing technology, rotor dynamics, control theory,mechatronic system design and experimental mechanics, could have been explored.I am specially grateful to Dr. Fredy Kuster, director of the Processes’ group at the IWF,for his great scientific but also moral support during my whole time at inspire AG.The idefix project would not have been possible without the collaboration of the indus-trial partners: Step-Tec AG and Mikron Agie Charmilles AG. In particular, I would liketo thank Kurt Schneider, Edwin Reinhard, Adrian Wittwer, Markus Bauder for their con-tribution in the development of the prototype spindle, and Jean-Philippe Besuchet, DavidSchranz, Thomas Hüpscher and Philippe Keller for their support in the realization of theexperiments.This work was supported by the Commission for Technology and Innovation (CTI) ofthe Federal Department of Economic Affairs of Switzerland and I particularly express mygratitude to Dr. Pierre Pahud, the delegated expert.I also want to thank the co-examiners of this thesis: Prof. John Lygeros, for the precioustime he gave to the examination of this work, and Dr. Guillaume Ducard, for his alwaysvaluable advices and his kindness.

Several inspire/IWF colleagues collaborated on this project and must be thanked. I espe-cially think of Dr. Harald Wild, Dr. Roger Margot, Albert Weber, Roman Glauser andKarl Ruhm.The author needs to thank the IFMS FH Burgdorf for its partnership on the actuatingsystem design. In particular, my gratitude goes to Florian Fässler, Sandro Schnegg and

IV Acknowledgements

Markus Zimmermann.I also want to thank Prof. Yusuf Altintas who kindly gave me advices on chatter stabilitypredictions, Dr. Saurabh Aggarwal for the interesting discussions on the chatter detection,Prof. Marc Bodson for his hints on the applicability of adaptive control, Prof. Roy Smithfor his explanations on robust control theory and Dr. Stan Pietrzko for the discussion onactive vibration control.

Special thanks go to some other inspire/IWF colleagues who contributed to make my timein Zurich a wonderful experience. I first think of my office mates: Dr. Sherline Wun-der, Dr. Sebastian Buhl, Michal Kuffa, Dr. Pascal Maglie, Uygar Pala, Dr. MohammadRabey, Darko Smolenicki, Dr. Markus Steinlin, Stefan Thoma, Dr. Fabio Wagner Pintoand more than a colleague, my friend Dr. Eduardo Weingärtner.I also want to thank Dr. Marije van der Klis, Ewa Grob, Katalin Stutz, Marianne Kästli,Mansur Akbari, Felipe Tadeu Barata De Macedo, Dr. Angelo Gil Boeira, Jens Boos,Dr. Sergio Bossoni, Dr. Bernhard Bringmann, Rossano Carbini, Karl-Robert Deibel,Dr. Markus Ess, Michael Gebhardt, Adam Gontarz, Günter Graf, Michael Gull, Dr.Michael Hadorn, Marcel Henerichs, Dr. Sascha Jaumann, Dr. Christian Jäger, Dr. NicolasJochum, Dr. Wolfang Knapp, Ricardo Knoblauch, Dr. Thomas Liebrich, Umang Mara-dia, Dr. Josef Mayr, Josef Meile, Willi Müller, Hop Nguyên, Raoul Roth, Dr. NiklausRüttimann, Dr. Zoltan Sarosi, Nikolas Schaal, Florian Sellmann, Daniel Spescha, JosefStirnimann, Dr. Martin Stöckli, Dr. Martin Suter, Robert Transchel, Dr. Guilherme Var-gas, Christian Walter, Dr. Sascha Weikert, Lukas Weiss, Dr. Carl Wien, Sandro Wiggerand probably many more.

At the end of this thesis, my thoughts go to my former Professor of vibrations at theUniversity of Applied Sciences of St-Imier, Dr. Olivier Bernasconi, who made me aware ofthe beauty and interest of structural vibrations and encouraged me to pursue my studies.

Of course, all this work might not have been possible without the unconditional supportof my parents and brothers.

I dedicate this thesis to my son Gaël Kouadio, who came to birth during this work andgave me more motivation to come to an end, and to my wonderful wife Amlan Éléonorefor her patience and without whom I most probably couldn’t find the strength to finalizethis task.

Zurich, February 2013Jérémie Monnin

Contents V

Contents

Acknowledgements III

Nomenclature and Abbreviations X

Abstract XVII

Kurzfassung XIX

Résumé XXI

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 General Objective and Scope of the Thesis . . . . . . . . . . . . . . . . . . 2

1.3 Proposed Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Literature Survey 9

2.1 Chatter in Manufacturing Processes . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Methods for Chatter Mitigation . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Process Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Off-Line Process Methods . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 On-Line Process Methods . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Regeneration Disturbing Methods . . . . . . . . . . . . . . . . . . . . . . . 22

VI Contents

2.4.1 Process Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Structural Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.1 Passive Structural Methods . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2 Active Structural Methods . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.2.1 Active Bearing Support . . . . . . . . . . . . . . . . . . . 25

2.5.2.2 Active Bearing . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.2.3 Active Ancillary Bearing . . . . . . . . . . . . . . . . . . . 30

2.5.2.4 Active Spindle Housing Support . . . . . . . . . . . . . . . 31

2.5.2.5 Active Tool . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2.6 Active Workpiece . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2.7 Active Mass Actuator . . . . . . . . . . . . . . . . . . . . 33

2.5.2.8 Active Structural Control Strategies for Chatter Mitigation 34

2.6 Specific Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Simulation Tools 40

3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Passive Mechatronic Structure . . . . . . . . . . . . . . . . . . . . . 42

3.1.1.1 Active System Design . . . . . . . . . . . . . . . . . . . . 43

3.1.1.2 Spindle Mechanical Model . . . . . . . . . . . . . . . . . . 43

3.1.1.3 Coupling with Machine Structure . . . . . . . . . . . . . . 45

3.1.1.4 Coupling with Actuating System . . . . . . . . . . . . . . 46

3.1.1.5 Second Order Model . . . . . . . . . . . . . . . . . . . . . 48

3.1.1.6 First Order Model . . . . . . . . . . . . . . . . . . . . . . 49

3.1.2 Milling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.3 Active Control System . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.4 Passive Structure coupled to Milling Process . . . . . . . . . . . . . 58

3.1.5 Active Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.6 Global System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Contents VII

3.2 Approximated Models for Control Design and Stability Analysis . . . . . . 62

3.2.1 Approximated Plant-Process System . . . . . . . . . . . . . . . . . 62

3.2.2 Approximated Global System . . . . . . . . . . . . . . . . . . . . . 64

3.2.3 Full and Perturbation Dynamics . . . . . . . . . . . . . . . . . . . . 65

3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.1 Time-Domain Simulation . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.2 Semi-Discretization Method . . . . . . . . . . . . . . . . . . . . . . 70

3.3.3 Zeroth Order Approximation Method . . . . . . . . . . . . . . . . . 70

3.3.4 Linear Time-Invariant Approximation Method . . . . . . . . . . . . 73

3.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Active Structural Control for Process Stability Improvement 75

4.1 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.1 Disturbance Rejection Scheme . . . . . . . . . . . . . . . . . . . . . 75

4.1.2 Stabilization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Observability and Controllability Aspects . . . . . . . . . . . . . . . . . . . 77

4.3 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.1 System Norms Definition . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.2 Generalized Plant Definition . . . . . . . . . . . . . . . . . . . . . . 81

4.3.3 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.4 Optimal Controller Resolution . . . . . . . . . . . . . . . . . . . . . 83

4.3.5 Disturbance Rejection Scheme . . . . . . . . . . . . . . . . . . . . . 84

4.3.6 Stabilization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.7 Computation of Control Effort . . . . . . . . . . . . . . . . . . . . . 88

4.3.8 Choice of Weighting Functions . . . . . . . . . . . . . . . . . . . . . 89

4.4 Design of Active System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Actuating System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4.2 Sensing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Experiments using Prototype Spindle 96

VIII Contents

5.1 Preliminary Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Selected Machining Parameters . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Monitoring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Chatter Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.1 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5 Frequency Responses at Tool Tip . . . . . . . . . . . . . . . . . . . . . . . 106

5.6 Identification of Average Cutting Force Model Coefficients . . . . . . . . . 111

5.7 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.7.1 Plant Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.7.2 Optimal Controllers Design . . . . . . . . . . . . . . . . . . . . . . 121

5.7.2.1 Disturbance Rejecting Controller . . . . . . . . . . . . . . 123

5.7.2.2 Stabilizing Controller . . . . . . . . . . . . . . . . . . . . . 123

5.7.2.3 Controllers Comparison . . . . . . . . . . . . . . . . . . . 128

5.7.3 Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . 130

5.8 Closed-Loop Frequency Responses at Tool Tip . . . . . . . . . . . . . . . . 131

5.8.1 Comparison between Predictions and Experiments . . . . . . . . . . 131

5.8.2 Influence of Sensing and Actuating Point on Control Performance . 132

5.9 Stability Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.9.1 Process Stability in Passive Configuration . . . . . . . . . . . . . . 134

5.9.1.1 Comparison between Stability Prediction Methods . . . . 135

5.9.1.2 Stability Chart Uncertainty . . . . . . . . . . . . . . . . . 135

5.9.2 Process Stability in Active Configuration . . . . . . . . . . . . . . . 140

5.9.2.1 Disturbance Rejection Scheme . . . . . . . . . . . . . . . . 140

5.9.2.2 Stabilization Scheme . . . . . . . . . . . . . . . . . . . . . 142

5.9.3 Intermediate Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 151

6 Main Performance Limitations of Active System 152

6.1 Sensing Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.2 Filtering of Forced Vibration Component . . . . . . . . . . . . . . . . . . . 154

Contents IX

7 Conclusion and Outlook 162

Bibliography 165

X Nomenclature and Abbreviations

Nomenclature and Abbreviations

Some efforts have been made to determine a consistent notation that would cover thewhole document and where the nomenclature accepted by the community of the differentscientific fields present in this work, such as manufacturing technology, structural androtordynamics as well as mechatronics and control theory, would be as much as possiblerespected.

Medium letters are generally employed for scalar or non-specified quantities. Bold lower-case letters explicitly refer to vectors and bold upper-case letters to matrix entities. Trans-fer and frequency response functions are denoted by upper-case letters.

Symbols

General Parameters

f Frequency (Hz)

i i2 = −1

i, j,k Unit vectors of coordinate systemI Identity matrixn Vector sizes Laplace variable (s−1)

t Time (s)

T Time period (s)

X, Y, Z Machine coordinate system directionsω Angular frequency (rad/s)

Ω Spin pulsation or self-rotational speed of the shaft (rad/s)

Nomenclature and Abbreviations XI

Structural Parameters

c Viscous damping coefficient (Ns/m)

D Viscous damping matrixf External nodal forces vectorfn Undamped natural frequency (Hz)

G Gyroscopic matrixk Static stiffness (N/m)

ku Actuator factor (N/V)

kL Structural stiffness of actuator (N/m)

K Stiffness matrixKσ Stress stiffness matrixm Mass (kg)

M Mass matrixMr Rotation mass matrixq Nodal displacement DOFs vectorR Modal gainum Input modal vectorym Output modal vectorωd Damped natural angular frequency (rad/s)

ωn Undamped natural angular frequency (rad/s)

ζ Damping ratio (−)

Milling Process Parameters

ae Radial depth of cut (m)

ap Axial depth of cut (m)

b Undeformed chip width (m)

blim Critical chip width (m)

bmin Unconditionally stable chip width (m)

db Differential chip width (m)

df Differential cutting force (N)

dS Infinitesimal cutting edge length (m)

dz Infinitesimal increment of z coordinate (m)

fc Chatter frequency (Hz)

fP Cutting forces generated by the kinematics of milling processfSR Spindle revolution frequency (Hz)

XII Nomenclature and Abbreviations

fTP Tooth passing frequency (Hz)

fz Feed rate per tooth (m/tooth)

g Immersion function (−)

h Dynamic undeformed chip thickness (m)

h0 Nominal undeformed chip thickness (m)

kf,1.1 Specific feed force (N/mm2)

k fSR Harmonics of spindle revolution frequencyk fTP Harmonics of tooth passing frequencyK Cutting force coefficientKf Cutting pressure in feed direction (N/m2)

KP Cutting stiffnessmf Kienzle exponent (−)

n Rotational speed of the spindle (rpm)

Nt Number of cutter teethq Number of complete undulations between two subsequent cuts and SLD

lobe numberQ Material removal rate (cm3/min)

R Cutter radius (m)kTj Homogenous transformation matrix from jth coordinate reference system

to kth coordinate reference systemTn Time period of one spindle revolution (s)

Tz Time period between two successive cuts (s)

vc Cutting speed (m/s)

vf Feed velocity (m/s)

wR Regenerative cutting force (N)

z Axial coordinate along the cutting edge from the end tip (TCP) to the top(m)

ǫ Phase shift between two subsequent cut undulations (rad)

φ Immersion angle (rad)

φ0 Initial angle of the tool (rad)φin Entry angle of cut (rad)

φout Exit angle of cut (rad)

κ Cutter entering angle (rad)

λ Cutter helix angle (rad)

ωc Chatter angular frequency (rad/s)

σP Variance of the reduced Poincaré map for chatter detection

Nomenclature and Abbreviations XIII

θ Counterclockwise rotation angle between machine and tool coordinate sys-tems (rad)

Control Parameters

A System matrixB Input matrixC Transfer function of the controllerC Output matrixD Feedthrough matrixE State estimatorF Closed-loop transfer function of the plant with the controllerG Open-loop transfer function of the plant or the passive mechatronic ma-

chine tool structureJ Cost functionalK Feedback regulatorKc Controller gainL Loop transfer functionp PoleP Transfer function of the machining processR Regenerative effectS Output sensitivity functionT Complementary sensitivity functionu Controller output or actuator inputv Measurement noisew Exogenous input or process disturbanceW Weighting functiony Controller input or sensor outputz Exogenous output or relative deviation between tool and workpiece from

nominal tool path

Subscripts

(.)a Related to the axial direction of tooth coordinate system(.)c Related to the shear cutting force or to the coupling force and element(.)C Related to the controller(.)e Related to the edge cutting force

XIV Nomenclature and Abbreviations

(.)f Related to the forced vibrations(.)g Related to the overall system(.)GC Related to the coupled system formed by the plant and the controller(.)GP Related to the coupled system formed by the plant and the process(.)GR Related to the coupled system formed by the plant and the regenerative

effect(.)lim At the stability limit(.)m Related to the machine coordinate system(.)q Related to the nodal displacement vector(.)r Related to the radial direction of tooth coordinate system(.)s Related to the spindle coordinate system or to the self-excited vibrations(.)t Related to the tool coordinate system or to the tangential direction of tooth

coordinate system(.)u Related to the plant input u

(.)w Related to the plant input w

(.)X Related to the X-direction(.)y Related to the plant output y

(.)Y Related to the Y -direction(.)z Related to the plant output z

Superscripts

(.)B Related to beam FE elements(.)D Related to lumped disk FE elements(.)S Related to spring-damper FE elements(.)P Related to the generalized plant

Operators

(.)T Transpose(.)∗ Complex conjugate(.)H Conjugate transpose‖.‖2 H2 norm‖.‖∞ H∞ normarg(.) Argument of a complex quantityarg min(.) Argument of the minimumdiag(.) Diagonal matrix

Nomenclature and Abbreviations XV

E . Expectation operatorfloor(.) Round down functionmag(.) Modulus of a complex quantitymin(.) Min functionRe(.) Real part of a complex quantitytr(.) Matrix traceλj(.) jth eigenvalueσj(.) jth singular valueσ(.) Maximum singular value

Abbreviations

ADD Active damping deviceAMB Active magnetic bearingAVC Active vibration controlAWH Active workpiece holderDOC Depth of cutDOF Degree of freedomER ElectrorheologicalFE Finite elementFRF Frequency response functionHSC High speed cuttingHSM High speed machining or high speed millingLHP Left-half planeLQG Linear-quadratic-GaussianLQR Linear-quadratic regulatorLTI Linear time-invariantMIMO Multiple inputs, multiple outputsMRR Material removal rateMSV Maximum singular valueN4SID Numerical algorithm for subspace state space system identificationNCDT Non-contact displacement transducerNLMS Normalized least mean squaresPI Proportional-integralPID Proportional-integral-derivativeRHP Right-half planeRKF Recursive Kalman filter

XVI Nomenclature and Abbreviations

RT Real-timeSD Semi-discretizationSDOF Single degree of freedomSISO Single input, single outputSLD Stability lobes diagramTCP Tool center pointTDS Time-domain simulationTF Transfer functionTMD Tuned mass damperZOA Zeroth order approximation

Abstract XVII

Abstract

The demands for productivity, quality and efficiency from the production industry tendto exacerbate the apparition of vibrational problems during machining operations. Morespecifically, the self-excited vibrations between the tool and the workpiece, commonlycalled chatter, constitute one of the most detrimental phenomena for the production qual-ity. Different types of chatter can occur during the cutting process. In milling, regenerativechatter corresponds to the most critical effect.The increasing occurrence of vibrational problems requires the development of innovativesolutions. Among them, the use of active structural control probably represents one ofthe most promising ways to deal with this problem mainly due to their high adaptivityand their great development potential. However, the applications of such methods remainmarginal in the machine tool industry essentially because of their important developmentcosts. This work aims to provide recommendations able to help the implementation ofsuch techniques in order to encourage their transfer from the academic field to the ma-chine tool industry.The proposed concept consists in integrating the active system inside an existing motorspindle. The active system is composed of piezoelectric actuators driving the support ofthe front bearings in the radial plane and accelerometers located near the actuators. Acontroller delivers reference signals to the actuating system based on the information re-ceived by the sensing system and according to a predefined strategy. This work proposesthe use of optimal control techniques to elaborate strategies suited for the mitigation ofchatter.

From the deficiencies in the state of the art on this topic, the objectives of this work aremore precisely formulated.A physical model, helping the design and the dimensioning of the active system as wellas the study of different control strategies, is developed and validated. It must be ableto predict the behavior of the global system composed of the dynamics of the machinestructure, the mechatronic system and their interaction with the milling process.The design of the active control system respects the considered industrial constraints and

XVIII Abstract

has been patented. From the observations made during machining tests, several recom-mendations on the design of the mechatronic system susceptible to enhance its performanceare suggested.Two different model-based approaches susceptible to increase the productivity in chatterfree conditions are formulated. The first one, called disturbance rejection, attempts tominimize the influence of the disturbances coming from the cutting process on the tooltip vibrations. The second one, named stabilization, explicitly considers the machiningparameters in the control design and modifies the dynamics of the spindle so that thestability of the process is guaranteed.The performance of the proposed concept is verified on a broad range of representativemilling operations. The obtained results demonstrate its capability to improve the pro-ductivity in stable machining conditions.

In this work, the lack of reliability of the proposed stability prediction methods is alsopointed out. This makes the use of model-based control techniques challenging and con-stitutes the main aspect limiting the industrial application of the concept. Therefore,an improvement of the prediction methods or a combination with adaptive schemes isrecommended for further developments.

Keywords

Milling, Regenerative Chatter, Motor Spindle, Mechatronic System, Optimal Control

Kurzfassung XIX

Kurzfassung

Die Anforderungen bezüglich Produktivität, Qualität und Effizienz, die an die spanen-de Fertigung gestellt werden, führen zu einer Verschärfung der Schwingungsprobleme imZerspanungsprozess. Im Besonderen besitzen die selbst-erregten Schwingungen zwischenWerkzeug und Werkstück, das so genannte Rattern, einen wesentlichen Einfluss auf dieQualität der produzierten Bauteile. Unterschiedliche Typen des Ratterns können währenddes Zerspanungsprozesses auftreten. Beim Fräsprozess ist das regenerative Rattern daskritischste Phänomen.Die Verschärfung der Schwingungsproblematik verlangt innovative Lösungen. Aufgrundihrer hohen Flexibilität und des enormen Entwicklungspotentials ist die Verwendung vonaktiven strukturellen Schwingungskontroll-Techniken wahrscheinlich eine der am meistenerfolgsversprechenden Wege. Die industriellen Anwendungen solcher Methoden bleibenim Werkzeugmaschinen-Bereich aber marginal, hauptsächlich wegen ihren Entwicklungs-kosten. Diese Arbeit hat zum Ziel, Richtlinien für den Einsatz solcher Techniken vorzu-schlagen, um den Transfer von der Wissenschaft in die Werkzeugmaschinenindustrie zuunterstützen.Das vorgeschlagene Konzept basiert darauf, ein aktives System in eine existierende Motor-Spindel zu integrieren. Das aktive System besteht aus piezoelektrischen Aktuatoren, welchedie vordere Lagerung der Spindel in einer radialen Ebene ansteuern, sowie aus Beschleu-nigungssensoren, die nahe bei den Aktuatoren positioniert sind. Ein Regler liefert Signalezum Aktuator, welche auf den Messsignalen der Sensoren sowie einer vorgegebenen Regel-strategie basieren. Diese Arbeit präsentiert optimale Regelungstechniken, um das Ratternzu minimieren.

Ausgehend von den Schwachpunkten des aktuellen Stands der Technik werden die Zielendieser Arbeit detailliert erläutert.Ein physikalisches Modell, das für die Dimensionierung des aktiven Systems und die Stu-die der unterschiedlichen Regelstrategien hilfreich ist, wird entwickelt und validiert. DiesesModell muss das Verhalten des globalen Systems, welches sich aus der Interaktion derStrukturdynamik der Maschine sowie des mechatronischen Systems mit dem Fräsprozess

XX Kurzfassung

zusammensetzt, prognostizieren können.Das Design des aktiven Systems berücksichtigt die Anforderungen für die industrielle An-wendung und wurde patentiert. Basierend auf Erfahrungen während der Fräsbearbeitungwerden mehrere Empfehlungen bezüglich der Auslegung des mechatronischen Systems vor-geschlagen, um dessen Leistung zu erhöhen.Zwei unterschiedliche modell-basierte Ansätze werden formuliert, welche beide die Produk-tivität bei stabilen Fräsbedingungen ohne regeneratives Rattern erhöhen. Der erste An-satz, der „disturbance rejection“ genannt wird, versucht den Einfluss der Prozessstörungenauf die Schwingungen des Werkzeugs zu minimieren. Der zweite Ansatz, die so genannte„stabilization“, betrachtet explizit die Zerspanungsparameter im Aufbau des Reglers undverändert der Dynamik der Spindel, so dass die Prozessstabilität garantiert wird.Die Effizienz des ausgewählten Konzepts wird auf einem breiten Spektrum repräsentativerFräsversuche verifiziert. Die Ergebnisse beweisen seine Fähigkeit, die Produktivität unterstabilen Fräsbedingungen zu erhöhen.

In dieser Arbeit wird ebenfalls auf die mangelnde Verlässlichkeit der Stabilitätsprogno-sen hingewiesen. Dies macht die Verwendung von modell-basierten Regelungstechnikenanspruchsvoll und stellt den hauptsächlich limitierenden Aspekt für eine industrielle An-wendung des Konzepts dar. Deshalb wird vor einer Weiterentwicklung der in dieser Arbeitvorgestellten Methoden eine Verbesserung der bisher verwendeten Stabilitätsprognosenoder eine Kombination dieser mit einer adaptiven Regelung empfohlen.

Stichwörter

Fräsprozess, Rattern, Motor-Spindel, mechatronisches System, optimale Regelung

Résumé XXI

Résumé

Les exigences imposées par l’industrie de production en termes de productivité, de qualitéet d’efficacité tendent à exacerber l’apparition de problèmes vibratoires durant l’usinage.Plus particulièrement, les vibrations auto-entretenues entre l’outil de coupe et la pièce àusiner, plus communément appelées : broutement d’usinage ou broutage, constituent l’undes principaux phénomènes à éradiquer. Différents types de broutement peuvent survenirdurant le processus d’usinage. En fraisage, le broutement régénératif constitue le phéno-mène le plus critique.L’apparition croissante de problèmeds vibratoires nécessite la mise en place de solutionsinnovantes. Parmi elles, l’utilisation de contrôle actif de vibrations structurelles repré-sente certainement l’une des voies les plus prometteuses, notamment en raison du degréde flexibilité qu’elles offrent et de leur fort potentiel de développement. Cependant, lesapplications industrielles de ce type de méthodes restent marginales dans le domaine dela machine-outil, principalement à cause de leur coût de développement élevé. Ce travailvise à fournir des recommandations pouvant aider la mise en place de ce genre de tech-niques afin d’encourager leur transfert du domaine académique à celui de l’industrie de lamachine-outil.Le concept proposé consiste à intégrer le système actif à l’intérieur d’une électro-brocheexistante. Le système actif se compose d’actuateurs piézoélectriques agissant radialementsur le support des paliers avant ainsi que d’accéléromètres disposés sur le pourtour de cemême support. Un contrôleur transmet aux actuateurs des signaux de consigne sur la basedes informations délivrées par les capteurs en fonction d’une stratégie de contrôle prédé-finie. Ce travail propose l’utilisation de techniques de contrôle optimal pour l’élaborationde stratégies adaptées à la réduction du broutement d’usinage.

À partir des lacunes dans l’état de l’art sur le sujet, les objectifs de ce travail sont plusprécisément définis.Un modèle physique, pouvant aider le design et le dimensionnement du système actif ainsique l’étude de différentes stratégies de contrôle, est développé et validé. Il doit pouvoirprédire le comportement du système à l’étude, composé de la dynamique de la structure

XXII Résumé

de la machine et du système mécatronique ainsi que de son interaction avec le processusde fraisage.Le design du system de contrôle actif respecte les contraintes industrielles prises en consi-dération et a été breveté. Les observations faites durant les tests d’usinage permettentde suggérer plusieurs recommandations pouvant améliorer les performances du systèmemécatronique.Deux différentes approches reposant sur un modèle et susceptibles d’augmenter la produc-tivité dans des conditions d’usinage exemptes de broutement sont formulées. La première,appelée : « disturbance rejection », tente de minimiser l’influence des perturbations pro-venant du processus de coupe sur les vibrations de la pointe de l’outil. La seconde, nom-mée : « stabilization », considère explicitement les paramètres d’usinage lors du designdu contrôleur et modifie la dynamique de la broche de manière à ce que la stabilité duprocessus soit garantie.La performance du concept proposé est vérifiée sur une large gamme d’opérations représen-tatives de fraisage. Les résultats obtenus démontrent sa capacité à améliorer la productivitédans des conditions d’usinage stables.

Dans ce travail, le manque de fiabilité des méthodes utilisées pour la prédiction de l’ap-parition du broutement est également mis en évidence. Ceci rend délicat l’utilisation detechniques de contrôle basées sur un modèle et constitue le principal aspect pouvant limi-ter l’application industriel du concept. C’est la raison pour laquelle une amélioration desméthodes de prédiction ou une combinaison avec des procédures de type adaptatives estrecommandée la poursuite du développement.

Mots-clés

Fraisage, broutement d’usinage, broutage, électro-broche, système mécatronique, contrôleoptimal

1

Chapter 1

Introduction

1.1 Motivation

Several aspects of the current trends in production technique exacerbate mechanical vi-bration problems coming from the interaction between the manufacturing process and themachine tool structure. The increasing demand from the market for productivity, accuracyand efficiency leads to contradictory requirements in the design as well as in the use ofstate of the art machines tools. In particularly, the dynamical loads generated by the pro-cess tend to increase in order to fulfill the demand for productivity. At the same time, theefficiency requirements lead to the production of high dynamical machine tools with lowenergy consumption. This implies the use of lightweight structures and a rationalizationof the moving parts inducing low damping properties and thus low dynamical stiffness.Also, the development of novel materials requiring extreme machining conditions as wellas the rising complexity of tool or workpiece geometries contribute to the propensity ofvibrations to occur. On the other hand, the demand for an increasing accuracy makessuch vibrational problems always less tolerable.

Under severe machining conditions, mainly two types of detrimental structural vibrationsare susceptible to arise between the tool and the workpiece out of the interaction betweenthe process and the machine tool structure dynamics. These two types are the resonancesand the instabilities. Generally, the instabilities, commonly called: chatter, are morecritical for the production than resonances, due to the fact that much more energy isinduced by the closed-loop formed between the machine and the process.

Different phenomena may give birth to process instabilities. In milling operations, themost dominant cause of chatter comes from a regenerative effect induced by the phaseof the current oscillations between the tool and the workpiece with the undulations left

2 1. Introduction

by the previous cuts on the machined surface. This type of chatter is called: regenerative

chatter.

The apparition of chatter between the tool and the workpiece is detrimental for the qualityof the surface finish and the respect of the workpiece tolerances but also leads to anexcessive tool wear and may highly decrease the lifetime of some machine elements, likespindle bearings for instance. It is characterized by an excessive acoustic noise emissionand represents a waste of energy. So, generally speaking, the chatter occurrence increasesthe production costs and it is now well established by the manufacturing community thatit constitutes one of the main limiting factors for the productivity and must be mitigated.

An intuitive way to suppress chatter is to reduce the process load by decreasing the materialremoval rate and thus the productivity. Since this effect has started being investigated,several more efficient ways have been proposed. However, in order to deal with the risingdiversity of vibration problems resulting from the new challenges in production techniques,some innovative solutions remain to be developed.

Active structural control techniques probably present the greatest potential of develop-ment. They are characterized by the use of a sensing system delivering information to acontroller. Based on a predefined control strategy, this controller transmits instructionsto an actuating system in order to mitigate the detected vibrations. The main attractive-ness of these methods comes from their high degree of adaptivity due to the fact that thecontroller can be easily reprogrammed. This constitutes a key aspect in modern manufac-turing techniques, as a great amount of production costs can be saved using one flexiblemachine tool instead of several specialized ones. Thanks to the latest developments inelectronics and information technology, such solutions become always more attractive forindustrial applications. Nevertheless, some efforts still need to be made for the reduc-tion of the engineering costs required by the implementation of such solutions though theelaboration of methods, guidelines and benchmarks.

1.2 General Objective and Scope of the Thesis

Through the elaboration of an original concept dedicated to the improvement of processstability in metal cutting and more specifically, in milling operations, the general objectiveof this thesis is to provide useful information and recommendations susceptible to help thedesign, the implementation and the optimization of active structural control systems. Inorder to encourage their integration into industrial applications, special emphasis must bebrought on their adaptability as well as on their industrial transferability.

1.2 General Objective and Scope of the Thesis 3

The system considered in the present study lies on three interacting elements, namelythe machine tool structure, the milling process and the active control loop. A simplifieddiagram of the overall system is represented in figure 1.1, where the transfer function of themechatronic structure of the machine tool including the integrated sensors and actuatorsof the active system corresponds to G. In control theory, this subsystem is often calledthe plant. The milling process is represented by P and the controller of the active systemby C. Different variables carry the information between each of these subsystems. Theprocess acts on the machine structure through the cutting forces w acting between thetool tip and the workpiece. This load combined with the finite compliance of the structuregenerates relative deviations z from the programmed tool path. On the other side, thesensors integrated in the machine tool structure provide information y on the currentstate of the plant to the controller. Based on this information and the strategy used bythe controller, instruction signals u are delivered to the actuating system. This overallsystem can be represented by the flowchart depicted in figure 1.2.

Figure 1.1: Simplified diagram of overall considered system. TCP: tool center point; G: transfer functions

of mechatronic machine tool structure; P : milling process transfer function; C: controller transfer function;

u: actuator input; y: sensor output.

This work is focused on the mitigation of regenerative chatter in milling process but canbe transferred, to some extent, to other machining processes, such as turning or grindingoperations, for example. Furthermore, no specific type of machining center is a priori

considered here but milling operations involving peripheral milling are taken into account.

Chatter may be caused by structural modes from different parts of the machine structure

4 1. Introduction

P

w z

G

C

yu

Figure 1.2: Flowchart of overall system. w: process disturbance; z: relative deviations between tool and

workpiece from nominal tool path.

and thus different corresponding frequency ranges. The most dominant natural modesof the whole machine usually lie between 50 and 200 Hz. Critical modes due to a localdeformation of the spindle, the tool or the workpiece generate higher chatter frequencies.In the present study, the focus gets on chatter phenomena primarily caused by the firstbending modes of the spindle shaft assembly, typically lying in the frequency range from500 Hz to 3 kHz.

1.3 Proposed Concept

The experience gained over the last century by the designers allows to achieve machinetools with high dynamic capabilities. However, on the other hand, the manufacturersmust cope with a market demand for the production of workpieces involving an increasingcomplexity and quality requirements. This leads to the machining of novel materialsrequiring extreme cutting conditions or to exotic tools or workpiece geometries. Suchtrend exacerbates the apparition of vibrational problems coming from the natural modeslocal to the spindle shaft, the tool or the workpiece more than the global modes of thewhole machine structure.

The frequency range of such vibrations is often out of the bandwidth of the machineaxes steering system. Consequently, some additional low-authority subsystems presentinghigher dynamics must be integrated into the machine structure. The implementation ofsuch complementary mechatronic subsystems, a priori independent of the axes controller,can take different forms. In order to guide its integration, the weakest part of the structuralchain of the machine must be identified. The main spindle, or more specifically its spindleshaft coupled with the tool assembly, generally constitutes the most critical element dueto the low damping induced by the rolling bearings. In the great majority of the cases,

1.4 Outline of the Thesis 5

chatter problems arise due to vibrations of the tool tip in the radial plane of the spindlerotational axis corresponding to the Z-direction of the machine. These radial deviationscome from the first bending modes of the spindle shaft assembly.

It is decided to restrict the integration of the complementary active system to the spindleunit. Its proximity with the tool tip provides a good compromise between adaptivity,as it is independent of tool change, and performance. Acting close from the end of thestructural chain, formed by the machine structure coupled to the tool and the workpiece,implies an action on lighter inertia and thus less power or higher dynamics of the actuatingsystem. Limiting the sensing system to the spindle stage also means that the disturbancescoming from the process cannot directly be detected but only the resulting deviations.This limits the control scheme to feedback action.

The active system is integrated into an existing spindle type so that it could be pro-posed as an option to the customer. The type chosen for the integration is the Step-TecHPC170CC-X-36-9.5/30-1FD-HSK-A63 motor spindle with cool-core option and hydraulictool clamping, shown in figure 1.3. This spindle is specially dedicated to the mold anddie industry, where five-axes high performance machining requirements must be fulfilled,which means high feed rates with large depths of cut leading to important cutting loadsand great material removal rate. The spindle must thereby guarantee sufficient torqueand stiffness. It must also fulfill the high speed machining requirements imposed by themanufacturing of aluminium parts for the aerospace industry, where high cutting speedsand large feed rates with low depths of cut are used. This need for high stiffness andhigh rotational speed leads to the use of hybrid ceramic-steel rolling bearings to supportthe spindle shaft in rotation. Three angular contact ball bearings in O-arrangement aremounted at the front of the spindle, constraining the radial and axial degrees of freedom ofthe spindle shaft. One roller bearing supports the rear of the shaft, enabling the constraintof the two radial rotational degrees of freedom. The front bearing assembly is constantlypreloaded using elastic elements. Between the front and rear bearings, an asynchronousmotor drives the rotation of the shaft. Some complementary specifications of this spindleare listed in table 1.1.

1.4 Outline of the Thesis

In chapter 2, the chatter phenomenon encountered in milling process is more deeply de-scribed. The different possible ways for its mitigation are then discussed. Active structuralmethods are especially emphasized and some existing solutions are presented. From theidentified deficiencies in the state of the art, the specific objectives of this thesis are for-

6 1. Introduction

Figure 1.3: Step-Tec motor spindle type HPC170CC-X-36-9.5/30-1FD-HSK-A63. Courtesy of Step-Tec.

Main specifications Values

External diameter 170 mmRated power, S6/40% 36 kWRated torque, S6/40% 36.2 Nm

Base speed 9’500 rpmMaximal speed 28’000 rpm

Tool clamping device HSK-A63Clamping force 18 kN

Lubrication Oil-airCooling Water

Integrated motor Asynchronous (4 poles)Weight approx. 75 kg

Table 1.1: Specifications of Step-Tec motor spindle type HPC170CC-X-36-9.5/30-1FD-HSK-A63.

mulated.

The different tools developed for the prediction and the simulation of the system studiedhere are described in chapter 3. The models used to describe the physics of the passivemechatronic structure, the milling process and the controller as well as the resulting dif-ferent combinations of the coupled systems are defined. Some approximated models usedfor the control design are presented. They are also used to analyze the process stabilitywith the help of different techniques. The validation of the different models is described.

The next chapter deals with the study of active structural methods dedicated to the

1.5 Conventions 7

mitigation of chatter. The selected control strategies are first presented. Two differentmodel-based approaches are formulated. Some observability and controllability aspectsare then discussed. The control strategies are more deeply investigated. The last sectiondescribes the design of the mechatronic system integrated into the prototype spindle. Thetechnological choices leading to the elaboration of the actuating and sensing systems areexposed.

The different experimental investigations realized with the prototype spindle are reviewedin chapter 5. Some preliminary experiments, for the concept validation and the spindlecommissioning, are briefly discussed. The performances of the prototype spindle are thentested in real machining conditions. The selected machining conditions are defined and theprocedure for the chatter detection during the experiments is described. The frequencyresponses at the tool tip of the selected tool assemblies are measured. The experimentaldetermination of the cutting force coefficients used by the process model is presented inthe next section. The design of the selected control strategy is detailed and the influenceof its implementation on the tool tip dynamics is discussed in the following part. Finally,the stability improvements provided by the active system in representative machiningconditions are presented.

Based on the results of the machining tests, the main limitations of the proposed conceptare evaluated and some recommendations are formulated in the next chapter.

The last chapter concludes the thesis by a summary of the main obtained results, someconclusive remarks and the perspectives provided by this work.

1.5 Conventions

In order to prevent any confusion, some lexical conventions must be specified. Here is a listof several terms used in this document referring to different, but closely related, objects:

Actuator Piezoelectric multilayer stackSensor Accelerometer or displacement transducerActuating system System formed by all the actuators connected to the power

amplifiersSensing system System formed by all the sensors connected to the signal con-

ditionersControl strategy Instructions provided by the controller to the actuating sys-

tem according to the information delivered by the sensingsystem (cognitive element)

8 1. Introduction

Controller Digital real-time (RT) controller connected to the sensor sig-nal conditioners and the power amplifiers; System formedby the anti-aliasing filters, analog-to-digital converters, dig-ital real-time controller with implemented control strat-egy, digital-to-analog converters, multiplexers and connectingboards

Control system or Ac-

tive system

System formed by the sensing system, the controller and theactuating system

Active Synonym for closed-loop, i.e. under the action of the activesystem

Plant or passive mecha-

tronic structure

System formed by the prototype spindle connected to theactuating and sensing systems

Passive Synonym of open-loop, i.e. free from the influence from thecontrol system

Passive spindle Prototype spindle with the active system set on its workingpoint (preloaded actuators) but without generating any cog-nitive action (open-loop configuration)

Active spindle Prototype spindle including the active system, i.e. systemformed by the plant connected to the controller (closed-loopor feedback configuration)

Conventional spindle Standard Step-Tec HPC170CC-X-36-9.5/30-1FD-HSK-A63spindle without integrated active system

Spindle unit System formed by the spindle housing, spindle shaft and itsbearings

Spindle assembly System formed by the spindle unit, tool-holder and toolShaft assembly System formed by the spindle shaft, tool-holder and tool

9

Chapter 2

Literature Survey

Regenerative chatter in milling process, mechatronic system design and active structuralvibration control are the main topics addressed in this work. All these themes, with theirlatest developments, are reviewed in this chapter. From the inferred research gap, thethesis objectives are then more precisely formulated.

2.1 Chatter in Manufacturing Processes

On a machine tool structure presenting low dynamic stiffness, four different types of me-chanical vibrations are susceptible to occur in milling operation: Free, forced, parametricand self-excited vibrations.Free vibrations arise when the structure is submitted to a sudden deviation from its equi-librium state position and then left freely vibrating. Such vibrations can be generated byimpulse forces from the moving axis inertia due to a inappropriate tool path programing.Forced vibrations are caused by an external periodic excitation acting on the structureat specific frequencies, such as unbalance forces coming from the spindle rotation or thekinematic variation of the chip thickness. If the excitation frequency becomes close froma natural frequency of the structure, a resonance problem may occur leading to large butbounded amplitude oscillations.In milling, parametric vibrations can also occur due to the variation at the tooth passingfrequency of the cutting stiffness matrix. This latter is defined as the coefficients puttingin relation the deviations between the tool tip and the workpiece with the correspondingvariations of the cutting forces.Finally, self-excited vibrations may arise due to the closed-loop interaction between thecut and the machine structure dynamics. This interaction can become unstable due to

10 2. Literature Survey

the fact that the oscillating movements of the structure themselves are sustaining the pe-riodic excitation force leading to an exponential increase of vibration amplitudes. Thisphenomenon is called chatter and constitutes, among these four types of vibration, themost detrimental effect for the cut.

Chatter is often referred as a vibrational phenomenon between the tool and the workpiecefrom which undesired wavy surface finish is left. The surface resulting from unstablemachining condition presents distinctive chatter marks as illustrated in figure 2.1.

Figure 2.1: Resulting surface finish in stable and unstable machining conditions.

A more precise definition has been accepted by the manufacturing community. It is agreedthat chatter represents unstable self-excited vibrations between the tool and the workpieceoccurring during machining processes and characterized by a specific frequency, calledchatter frequency.

Different effects have been recognized as possible source for the occurrence of such insta-bility. As suggested in [1, 2, 3], chatter can be classified in two main categories: primaryand secondary chatters. The former is related to some effects induced by the dynamicsof the cut itself. Several principles have been identified as primary chatter sources. Thefrictional chatter, described in [4, 5], explains the apparition of unstable self-excited vi-bration by the relation between the cutting forces and the cutting speed susceptible toinduce negative damping into the process due to a rubbing effect on the clearance face.Machine structures presenting dynamics with strong coupling between the thrust and cut-ting directions are susceptible to generate unstable vibrations due to the non conservativenature of the cutting force model, as presented in [6, 7, 8, 9]. This phenomenon is similarto the aerodynamic flutter and is named mode coupling principle.The last identified principle occurs during the formation of segmented chip through thermo-

mechanical self-excited relaxation vibrations. This is explained by Davis et al. in [10]. Thecorresponding chatter frequency of this later case is much higher than the other primarytypes.

Secondary chatter refers to a regenerative effect between the current and previous cutting

2.1 Chatter in Manufacturing Processes 11

paths. Depending on the phase between the current vibrations of the cutter and the un-dulations of the wavy surface left by the previous cuts, a variable chip thickness may becreated. In this case, as the cutting force is directly dependent on the chip thickness, aresulting oscillatory tool load arises. From the phase relation between this varying forceand the cutter vibrations, a positive work may be generated. This energy is supplied tothe oscillatory system leading to an exponential increase of the cutter vibration ampli-tudes in the workpiece material until some nonlinear effects, as jump out of the cut or toolbreakage, occur.This phenomenon thus requires an overlapping of the cut to occur. Since this is the casein the great majority of metal removal operations and as its stability limit is often muchlower than the primary types, this secondary or regenerative chatter constitutes the mostdominant type of chatter encountered in practice, especially in milling operations. In thefollowing text, if no specification is made, the term chatter refers to the regenerative type.Figure 2.2 illustrates the dynamics of the interaction between the machine structure dy-namics and the milling process with the regenerative phenomenon. The regenerative effectinduces time delay terms in the dynamics of the whole system and the particularity inmilling is the rotating kinematics of the tool leading to periodic directional effects. Theresulting coupled system can be described by a set of inhomogeneous delay differentialequations with periodic coefficients and in terms of transfer functions can be depicted bythe flowchart in figure 2.3. In this figure, the dynamics of the machine structure is repre-sented by the block G, which generates the tool center point (TCP) deviations z inducedby the cutting forces excitation w. These cutting forces, generated by the milling processP , correspond to the sum of the loads fP , coming from the kinematic variation of the chipthickness, and the cutting forces wR, resulting from the regenerative effect R.

The regenerative effect is nothing but the difference between the current and the pre-vious TCP deviations, multiplied by the cutting stiffness KP . In the Laplace domain,the delayed term z(t − Tz) becomes z(s) e−sTz , where the tooth passing time periodcorresponds to Tz and the Laplace variable to s.

As previously mentioned, the stability of the process depends on the phase differencebetween the current oscillations of the tool in the workpiece and the wavy surface leftby the previous cuts. This phase difference is determined by the relation between thevibrational frequency of the cutter, i.e. the chatter frequency, and the cutting speed,given by the spindle rotational speed. Thus, a convenient way to represent the stabilityof the milling process is to plot the stability limit in the parameter plane: depth of cut(DOC) vs spindle rotational speed. Two different DOCs are used to define the milling


n

jt

it

kt

vf

kY

cY

kX cX

WorkpieceCutter

Figure 2.2: Regenerative effect in peripheral milling process. n: spindle rotational speed; vf : feed rate;

kj , cj : static stiffness, resp., viscous damping coefficient in the jth direction, j = X,Y ; it, jt,kt: unit

vectors of non-rotating tool coordinate system.

Gw z

KP+

+

fPP

wR

e-sTz

+

-

R

Figure 2.3: Flowchart of regenerative effect in milling process. G: transfer function (TF) of machine

structure dynamics; P : TF of milling process; R: regenerative effect TF; KP : cutting stiffness; w: cutting

forces; z: tool center point deviations; fP : kinematic cutting forces; wR: regenerative cutting forces;

Tz: tooth passing time period; s: Laplace variable.

operation condition, namely the radial and axial DOCs. Both are represented in figure2.4.Due to the repetitive nature of the phase difference, the pattern of the stability limitpresents a series of lobes. Such stability chart is known as stability lobes diagram (SLD).

Single point operation, such as orthogonal plunge turning operation, with a single degreeof freedom (SDOF) oscillating system in feed direction to represent the relative motionsbetween tool and workpiece corresponds to a simple case for the modeling of regenerative

2.1 Chatter in Manufacturing Processes 13

ae

Workpiece

ap

Cutter

n

vf

it

jt

kt

Figure 2.4: Axial and radial depths of cut in peripheral milling operation. ap: axial depth of cut; ae: radial

depth of cut.

chatter. Figure 2.5 represents the considered system. The tool tip deviation around thenominal feed movement given by the feed rate vf is z. The rotational speed of the spindle in(rpm) is n. The mass of the vibrating system is m, c is its viscous damping coefficient and k,its static stiffness. Its corresponding undamped natural angular frequency is ωn =

√

k/m.

n

z

m

k

c

vf

Workpiece

Cutter

Figure 2.5: SDOF model in plunge turning. n: spindle rotational speed; vf : feed rate; z: tool tip deviation;

m: mass of oscillating system; c: viscous damping coefficient; k: static stiffness.

The relation between the chatter frequency, the spindle speed and the resulting numberof undulations over one revolution is given by

60

nωc = ǫ+ 2π q (2.1)

where ωc is the chatter angular frequency, q, the number of complete undulations over onerevolution, i.e. q ∈ N, and ǫ, the phase shift between the current and previous oscillations.


The dynamics of the tool tip deviations z is given by

m z(t) + c z(t) + k z(t) = w(t) (2.2)

where w is the cutting force in feed direction. According to Kienzle and Victor [11], for agiven cutting speed, the cutting force is a power function of the chip thickness. For thestability analysis, w can be linearized around the nominal chip thickness, such that

w(t) = Kf b h(t). (2.3)

The undeformed chip width corresponds to b, h is the dynamic undeformed chip thicknessand Kf is the cutting pressure in feed direction in (N/m2), given by Kienzle and Victor as

Kf = kf,1.1 · 106(

h0 · 103)−mf (2.4)

where h0 is the nominal chip thickness in (m) given by the feed rate vf . The Kienzleexponent is mf and kf,1.1 is the specific feed force in (N/mm2) such that kf,1.1 is equal tothe feed force for an undeformed chip area of 1 by 1 mm2. These coefficients are derivedfrom cutting tests.The dynamic chip thickness h is given by

h(t) = h0 + z(t− Tn)− z(t) (2.5)

with Tn corresponding to the time period of one spindle revolution, given by Tn = 60/n.The stability analysis in frequency domain can be used to determine the correspondingstability lobe diagram, as described in [12]. Under the assumption that the whole systemis linear, the characteristic equation is derived from the ratio between the dynamic andnominal chip thicknesses in Laplace domain, expressed as

h(s)

h0

=1

1 + (1− e−sTn) b Kf Gzw(s)(2.6)

where s ∈ C is the Laplace variable and G, the receptance function. The receptancetransfer function corresponds to the dynamic compliance as suggested in [13] and is definedas

G(s) =z(s)

w(s)=

1

m s2 + c s+ k. (2.7)

After the transformation in frequency domain by replacing s by iωc, where i2 = −1, andsome manipulations of the conditions given by the characteristic equation, the stabilitycondition can be formulated as following.

2.2 Methods for Chatter Mitigation 15

The cutting process is asymptotically stable if and only if, for a spindle speed equal to

n =60 ωc

(2 q + 1)π + 2 arg(

G(ωc)) , (2.8)

the undeformed chip width b is strictly smaller than the critical value blim given by

blim =−1

2Kf Re(

G(ωc)) . (2.9)

It is to note that, in the above expressions and in the following text, in order to simplifythe notation, the function G(ω) corresponds to the frequency response function (FRF),strictly defined as G(iω).From the above condition, it is possible to plot the critical value blim in function of n fordifferent values of q. For a given spindle speed, the lowest stability limit gives the SLD, asrepresented in figure 2.6. On the left-hand side of this figure, the magnitude and real partof the receptance function at the tool tip are represented. The corresponding stabilitylobes diagram and chatter frequency are plotted on the right-hand side. As visible fromequation (2.9), the minimal critical chip width bmin, also called unconditionally stable chipwidth, corresponds to the minimum of the receptance real part, min

(

Re(G))

. In thisfigure, the influence of the structural parameters c and k is also represented. The increaseof the viscous damping coefficient c produces a vertical shifting of the SLD in larger chipwidths. An increase of the static stiffness k tends to move the SLD in the right-handside, i.e. in higher spindle speeds, and increase the chatter frequencies. Also, due to thestiffening effect induced by the process interaction, the chatter frequency is always higherthan the natural frequency of the SDOF system.

2.2 Methods for Chatter Mitigation

Chatter phenomenon results from the interaction of two complex systems, namely thedynamics of the machine structure and the cutting process, where a lot of parameters fromdifferent natures are involved. An appropriate tuning of these parameters can stabilizethe process. The objective is to modify the parameters in such a way that the processstability is guaranteed for the desired productivity requirements. Depending on the processconditions, some parameters may be more appropriate than others. In particular, someare more efficient in the sense that they require less effort to achieve the same stabilization


ωc

n

Stability lobes diagrams

b lim

Re(G(ω

))

ω

m, c, 1.5 k

m, 5 c, k

m, c, k

Tool tip receptances

mag(

G(ω

))

arg min(

Re (G))

ωn

ωn

bmin

min(

Re (G))

0

Chatter

Stable

Figure 2.6: Influence of the variation of the structural parameters c and k on the receptance functions

and their corresponding stability lobes diagrams. arg min(.): argument of the minimum; blim: critical chip

width; bmin: unconditionally stable chip width; min (.): min function; ω: angular frequency; ωc: chatter

angular frequency; ωn: undamped natural angular frequency.

performance.

Many different methods have been proposed to prevent or suppress chatter. These methodscan be classified in three main categories regarding the parameters they are influencing.The methods based on the process parameters, named here process methods, try to cor-rectly select or adjust the parameters related to the process in order to prevent, respectivelysuppress chatter. These methods take advantage of the stability lobes pattern by findingthe maxima of stability. In this category, two subdivisions can be identified: off-line andon-line methods. The off-line or passive methods select suitable machining parameters,i.e. spindle speed and depths of cut, based on the predicted stability lobes diagrams. Theythus require accurate models of the machine structure and process dynamics. The on-lineor active methods monitor the process and as soon as chatter is detected, the processparameters are adjusted in order to stabilize the cut. So, they need the integration inthe machine environment of a monitoring system transmitting information to a decision-making device providing new set of machining conditions.

2.2 Methods for Chatter Mitigation 17

Another class of methods influences the structural parameters in order to modify the sta-bility lobes pattern so that the process stability is guaranteed for the selected machiningconditions. These methods are called structural methods. Among them, two subcategoriescan be identified as passive and active methods. Passive methods do not require anyenergy supply and try to transfer the energy from the primary system, the unstable oscil-lating structure, to a secondary system, e.g. an oscillator or a dissipative device. Activestructural methods lie on a sensing system delivering information to a controller which,according to a predefined strategy, delivers instructions to an actuating system. Such sys-tems require external energy supply. Their main advantage over passive systems consistsin their high adaptivity degree due to a wide bandwidth of the actuating system com-bined with the fact that the control strategy is often implemented into a controller easilyreprogrammable. Also, the integrated sensing system automatically provides monitoringcapabilities.The energy induced by the regenerative effect leading to chatter lies on periodic phenom-ena and requires a certain amount of time to fully develop. A third category of methodstries to disturb the periodicity of the feedback loop between the machine structure and theprocess. These methods are called here regeneration disturbing methods. Among them, adistinction can be made between those employing parameters related to the process andthose using structural parameters.Figure 2.7 represents the proposed classification of the methods dedicated to the mitigationof chatter. All these methods are more deeply described in the following text.

Process

methods

Structural

methods

Regeneration

disturbing

methods

Off-line On-line Passive Active Process Structural

Planning based

on SLD

-Chatter detec-

tion and

characterization

-Machining

parameters

adjustment

Energy transfer

to passive

auxiliary system

Improvement of

structural

properties using

active loop

Alteration of

process

parameters

periodicity

Alteration of

structural

parameters

periodicity

Figure 2.7: Classification of the methods for chatter mitigation.


2.3 Process Methods

As previously mentioned, process methods take advantage of the lobe pattern of the stabil-ity charts to determine suitable machining conditions. They are thus particularly efficientin high spindle speed domain where larger stability pockets are reachable.These methods can be divided into off-line and on-line categories. Several examples ofboth are given in the following subsections.

2.3.1 Off-Line Process Methods

The off-line process methods consist in adapting the machining parameters of influence,such as spindle speed and depths of cut, in order to guarantee stable cutting conditions forthe productivity requirements. In machining operation, the productivity can be expressedin terms of rate of material removed from the workpiece. In milling, the material removalrate (MRR) Q is given by

Q = ae ap fz Nt n (2.10)

where ap and ae are the axial, respectively the radial depths of cut, fz is the feed per tooth,Nt, the number of teeth and n, the spindle speed. The depths of cut exert a dominantand almost monotonically influence on the process stability. This is why the stabilizationof the process using these parameters leads to an unwanted loss of productivity. Anotherparameter of great influence is the rotational speed of the main spindle, as it determinesthe phase shift between the current structural vibrations and the undulations left by theprevious cuts, which is determinant for the regenerative effect. The feed rate has almostno influence on the process stability and the cutter geometry plays only a second orderrole.

The great potential to increase chatter free productivity through an appropriate choice ofaxial depth of cut and spindle speed based on the stability lobes diagram has been putinto light by Tlusty and Polacek [14] and Tobias and Fishwick [15]. Budak and Tekeli [16]proposed to not only consider the axial but also the radial depth of cut in the stabilityanalysis in order to maximize the chatter free MRR.

These methods require reliable stability predictions based on representative models of theinteraction between the machine structure and the process as well as an accurate identi-fication of the modeling parameters. The structural dynamics can be identified via FRFmeasurements at the tool tip and on the workpiece to get the relative tool-workpiece recep-tance functions. Some commercial solutions, such as CutProTM [17] or MetalMAXTM [18],provide software and hardware kits to measure the tool tip receptances using impulse ham-

2.3 Process Methods 19

mer and accelerometer and derive the stability lobes. In some cases, the use of impulsehammer and accelerometer is not adequate. For instance, the mass of the accelerometermight disturb the measurement, as discussed in [19]. In [20, 21], the influence of the spin-dle speed must be considered and measurements with the rotating spindle are performed.In this latter case, non-contact transducers, such as laser, capacitive or eddy current sen-sors, are required. Non-contact excitation may also be applied using magnetic bearing, aspresented in [22].The receptance functions can also be estimated using modeling methods. In [23, 24, 25, 26],the finite element method is used. However, as the stability predictions are very sensitiveto the modal properties of the critical modes of the machine, especially the modal damp-ing, only qualitative predictions are possible to be obtained. A combined approach canbe employed using measured FRFs on the spindle nose coupled to a finite element (FE)model of the tool through receptance coupling methods, as described in [27, 28, 29, 30].Depending on the operating conditions, different models may be used to describe theinteraction between the machining parameters, the tool tip deviations from the desirednominal path and the cutting forces generated by the process and acting on the machinetool structure. For chatter modeling, analytical empirical models, such as those presentedin [12, 31], are used. These models can be derived by performing cutting tests in differentmachining conditions and recording the cutting forces using dynamometer. Orthogonalcutting tests may be led with different rake angles, feed rates and cutting speeds. In thiscase, the collected data are only dependent on workpiece material. They can be appliedto the milling case using some geometrical and kinematic transformations as long as thecutting edge geometry satisfies the assumptions of oblique cutting. For more complexcutting edge geometries, a mechanistic approach is used where cutting tests at differentfeed rates and cutting speeds must be performed. The obtained model data are then onlyvalid for the considered combination of workpiece material and tool.Once both structural and process parameters are determined, different methods, such asfrequency-domain [32, 33, 34] or time-domain approaches [35, 36, 37, 38, 39, 40], may beemployed to predict the stability of the machining process. Depending on the tool speci-fications and the main spindle power, stable machining conditions satisfying productivityrequirements can be defined.

2.3.2 On-Line Process Methods

In some cases, reliable stability predictions may be very challenging to get and the use ofoff-line methods hazardous. An alternative way is to monitor the process and as soon aschatter problems arise, modify the machining parameters to reach stable conditions. The


most elementary method consists in a monitoring from the operator itself. In practice,chatter occurrence is usually quite obvious due to a characteristic loud noise emissionand poor surface finish. The operator can thus opt for different strategies. A possiblechoice is a reduction of the depths of cut. However, this leads to a proportional decreaseof the productivity. A smarter strategy is to try to increase the spindle speed to fallinto larger stability pockets which, by keeping the feed per tooth constant, improves atthe same time the productivity. If the chatter phenomenon can be characterized and itsfrequency identified, the spindle speed can be tuned such that a harmonic of the toothpassing frequency matches the chatter frequency. The reason is that, in this case, thephase between the undulations from the previous cut and the current vibrations of thecutter is almost equal to zero, which implies a variation of the chip thickness only due tothe process kinematic and thus minimizes the occurrence of self-excitation. Such spindlespeeds correspond to the peaks of stability between two lobes.

The phase shift between two subsequent cuts ǫ, expressed in function of the chatter fre-quency fc and the tooth passing frequency fTP , is given by

ǫ

2π+ q =

fcfTP

(2.11)

wherefTP =

Nt n

60. (2.12)

The characterization of the chatter phenomenon requires the integration of sensors in theenvironment of the machine delivering information to the monitoring system. Differenttypes of sensor can be used. Microphones may record the sound emission. Accelerometersor displacement sensors located as close as possible from the oscillating part of the ma-chine can also be employed. The measurement of the cutting forces may also be used todetect chatter. They can be directly measured using a dynamometer table for instance,or indirectly via the power requirements of the main spindle motor.Microphone is very easy to integrate in the machine and its acoustic signal can be di-rectly transmitted to the sound card of every standard computer. However, in shop floorenvironment, its signal is susceptible to be disturbed by the sounds coming from other ap-plications. Accelerometers or displacement sensors are less sensitive to noise coming fromthe environment than microphones but they are more expensive and their placement on themachine structure requires more effort. The use of dynamometers has several limitations.First, they are generally more expensive compared to microphones and accelerometers.Also, their measurement bandwidth is much smaller due to their low internal dynamics.

In order to improve the reaction time between the chatter detection and the decidedaction, automated procedures replace the operator. The objective and the main challenge

2.3 Process Methods 21

of such methods is to apply corrections sufficiently early so that the workpiece or toolcan be preserved from chatter damages. The key aspects are thus the chatter detectionand identification, and the strategy to correct the process parameters. In milling, themain difficulty for the detection of chatter comes from the superposition of forced andself-excited vibrations. The self-excited vibrations can be distinguished from the forcedvibrations using time-domain methods, such as demodulation or parametric modeling tofilter the signal provided by the sensors. Frequency-domain methods, based on the Fourieror wavelet transforms, may also be used.

In [41], based on the work of Faassen [2], van Dijk et al. demonstrate that accelerometersare particularly suitable for chatter detection. They use parametric modeling with normal-ized least mean squares (NLMS) adaptive filter to determine the model parameters and aKalman filter to identify the chatter frequency in the build-up phase of the phenomenon.As soon as the onset of chatter is detected, a corrected spindle speed is computed andsent to the machine command using an override function. The feed rate is adapted toguarantee the same feed per tooth. Two different strategies are presented for the choiceof the new spindle speed. The first one sets the spindle speed such that the estimateddominant chatter frequency corresponds to a higher harmonic of the new tooth passingfrequency. From relation (2.11), setting the phase shift ǫ to zero, the new selected spindlespeed nk is given by

nk =60 fc,kNt qk

(2.13)

with

qk = floor

(

60 fc,kNt nk

)

(2.14)

where floor (.) is the round down function and k represents the iteration subscript.However, as the chatter frequency varies with the spindle speed, these new machiningparameters may not exactly lie in the middle of the stable peak and may still cross thestability boundary. Based on the new detected chatter frequency fc,k+1, a second spindlespeed nk+1 is chosen and so on until a convergence towards a stable region can be iterativelyreached. If not, the depths of cut must then be reduced.The second strategy uses extremum seeking control in order to minimize the perturbationvibrations.

The Harmonizer R© [42] is a commercial software using sound signal for the detection andidentification of chatter conditions and makes suggestion for the selection of chatter-freespindle speeds. It is now also available as an application for smartphones.


2.4 Regeneration Disturbing Methods

A second class of methods attempts to break the regenerative feedback loop between thestructure and the process in order to prevent the apparition of self-excited vibrations. Ituses the fact that the regenerative effect needs periodicity between each cut to build up.Some methods propose to break this periodicity using process parameters and some othersusing structural parameters.

2.4.1 Process Parameters

Slavicek [43] first proposed to use tools with variable pitch angle to prevent the apparitionof self-excited vibrations. Since then, the use of such tools has been intensively investi-gated. Some more recent studies on this topic can be found in [44, 45, 46]. If properlydesigned, this method can be applied in both high and low spindle speeds but it requiresthe possibility of grinding its own cutters.As demonstrated in [47, 48], tools with variable helix angle may also be successfully usedto prevent the regenerative effect to arise.End mill with serrated cutting edge have also been developed in order to decrease theresulting cutting forces and stabilize roughing operations [49].Serrated cutting edge as well as variable pitch or helix angles are now common features ofroughing milling cutters proposed by the tool manufacturers.

Finally, continuous modulation of the spindle speed may be used to disturb the regen-erative feedback loop, as presented in [50, 51, 52, 53]. This method is known as spindlespeed variation. The amplitude and frequency of the modulation are based on the detectedchatter frequency.A limitation of this method comes from the modulation frequency restricted by the band-width and power of the main spindle motor. Successful stability improvement of the sta-bility have been demonstrated in low spindle speed range but, as shown in [54], for highspindle speeds, this method seems to be limited to the stabilization of flip bifurcationswhich are usually not dominant in milling. The different types of bifurcation encounteredin milling process are discussed in section 3.3.

2.4.2 Structural Parameters

The regenerative feedback loop may also be disturbed using a variation of the structuralproperties of the machine.

2.5 Structural Methods 23

Segalman and Redmond [55] propose to vary the structural impedance, defined as theratio of force over velocity, of a milling cutter using electrorheological (ER) fluid locatedat the interface between the tool sleeve and the tool. Applying electric field to the ER fluidmodifies the natural frequencies of the tool tip. Changing the modal properties betweentwo subsequent cuts disrupts the regenerative effect and stabilizes the process. Best resultswere obtained using sinusoidal variation of the electric field at a frequency equal to thehalf of the tooth passing frequency.Wang and Fei [56] developed a similar system dedicated to the stabilization of chatterin boring operation. ER fluid is integrated in the boring bar. Artificial neural networkis used to filter the sensor signal and detect chatter occurrence. The strategy here is touse the nonlinear viscoelastic behavior of the ER fluid and its sensitivity to the vibrationamplitudes to dissipate a maximum of energy and stabilize the system.

Such nonlinear strategies have been more formally studied. In [57, 58], Stépán and In-sperger suggest the use of time-varying delayed feedback control force to produce act-and-wait control. The delay term induced by the regenerative effect leads to infinite dimen-sional system. The controller is periodically switched on and off for a longer time periodthan the delay coming from the machining process in order to eliminate the memory effectand obtain finite dimensional system.In [59], Butcher and Mann describe the use of optimal quadratic controller with finitehorizon as well as delay state feedback control.

2.5 Structural Methods

Another approach for chatter mitigation consists in adapting the structural parametersso that the process stability can be guaranteed for the specified machining conditions.In comparison with the process methods, such structural methods have the advantage ofbeing well suited over the whole spindle speed range. Their main drawback comes fromthe fact that their implementation is complex, as it generally requires the integrationof a subsystem into the machine structure which also induces a weakening of the overallstructure dynamics that must be compensated by a specific action of the dedicated device.

A differentiation can be made between structural methods using passive or active systems.

2.5.1 Passive Structural Methods

Several passive devices have been proposed to stabilize the cutting process. Their greatmajority tries to increase the structural damping of the machine, shifting the stability limit


to larger depths of cut. Different types of damper have been used. In [60], Semercigil andChen propose to use an impact damper integrated into a slender tool holder to dissipatethe energy of vibration and stabilize the milling process.

Ancillary tuned mass dampers (TMD) have also been extensively used to modify thedynamics of the critical natural mode of the structure in order to maximize its stabil-ity against chatter. Tarng et al. [61] investigated the use of a tuned vibration absorberbased on a piezoelectric inertia actuator in turning operation. Different approaches havebeen proposed for the selection of the right structural parameters for the design of theTMD. Sims [62] proposes an analytical method specially dedicated to chatter suppression.Yang et al. [63] suggest the use of multiple TMDs for the stabilization of turning processand demonstrated the possibility to improve the performances over single TMD withoutsignificantly increasing the total additional mass.

Passive structural solutions have been commercialized. For instance, SECO TOOLS ABproposes the series SteadylineTM [64] of shell mill holders for long milling applicationsintegrating passive damping system. SANDIVK TOOLING Coromant also offers the seriesSilent Tools R© [65] for boring, turning and milling operations. The Smart DamperTM series[66] from BIG Daishowa Seiki CO LTD are slender tool holders with built-in dampingmechanism for deep hole boring or milling operations.

In [67, 68], semi-active methods have also been tested where the structural parametersof the system passively dissipating the energy from the primary system can be adjustedusing active materials such as magnetorheological fluids.

2.5.2 Active Structural Methods

Numerous studies explore the mitigation of chatter phenomenon by means of active struc-tural methods. The proposed solutions differ by their implementation and control strategy.Several reviews have already been published on this topic [69, 70].

As previously mentioned, chatter is a highly dynamic phenomenon. Its correspondingfrequency range usually lies well above the bandwidth of the steering system of the machineand thus cannot be directly compensated by the main axes. Its control requires theintegration of additional subsystems with higher dynamics. In order to minimize the inertiamoved by the actuating system and thus reduce its required power and dimensions, thegreat majority of the proposed concepts are integrated as close as possible to the process,either on the tool side or on the workpiece side. In milling or grinding operations, themain spindle unit constitutes a well suited element for the integration of the active system.There are basically two types of solution. In the first one, the actuating system lies within


the force path of the spindle structure, e.g. the actuators can drive the support of aspindle bearing. Such solution is called active bearing support and corresponds to theconcept proposed in this study. Instead of conventional passive rolling bearing, active

bearing, such as electromagnetic bearing for instance, may also be used to radially drivethe rotational axis of the spindle shaft. The second solution is to place the actuatingsystem parallel to the force path. In this case, the system represents an active ancillary

bearing on the spindle shaft.Other solutions propose the integration of the actuating system outside of the spindleunit. The use of the interface between the spindle unit and the machine tool structure iscalled here active spindle support. Also, the active system can be integrated in the toolor the tool holder itself. Such solutions are named active tools. Some active workpieces

or workpiece holders have been designed. Finally, Active mass actuators have also beenused to damp the vibrations of a specific part of the machine structure. Several examplesof each of these implementations are described in the following sections.

For a given implementation of the active system, different strategies may be used by thecontroller in order to mitigate the chatter apparition. Several control strategies are alsolisted below.

2.5.2.1 Active Bearing Support

Active bearing support solutions propose the integration of the actuating system withinthe force path between the tool tip and the machine tool structure. The actuating systemsteers the radial displacements of the outer ring of the rolling bearing. A guiding systemis required to guarantee the movements in the radial plane and prevent any displacementin the other degrees of freedom (DOFs) that cannot be constrained by the actuators. Thestiffness of the actuators is serial with the structural stiffness of the spindle. The staticstiffness of the actuators must thus be sufficient to guarantee the accuracy requirementsat tool tip. This is given by the passive structural stiffness of solid state actuators or usingcontrol system with displacement feedback.

Probably the most similar implementation to the one presented here has been developedby the consulting engineering office Wölfel GmbH in collaboration with Weiss Spindeltech-nologie GmbH and presented in 2005 at the international trade fair EMO in Hannover.This solution consists in an active motor milling spindle designed for high speed machiningoperations and presenting a maximal spindle speed equal to 30’000 rpm and a power of80 kW. The front bearing is actively supported in both radial directions by two orthogonalpreloaded piezoelectric stack actuators. One pair of accelerometers and two pairs of non-contact displacement transducers (NCDTs) are used to sense the radial vibrations in the


vicinity of the actuators and to deliver the corresponding signals to a digital controller. Adecentralized controller, i.e. each radial directions are independently regulated, attemptsto increase the structural damping of the spindle shaft support using collocated controlstrategies, like direct velocity feedback and second order filter, to reduce the propensityto chatter. Collocated control is a single variable control system where the sensor andthe corresponding actuator are located at the same place and the multiplication of thesensed variable with the actuating variable represents a power flow. Lead compensator isalso employed to compensate eventual phase shift between the sensors and the actuators.Figure 2.8 represents the global system.In [71], the active system shows the ability to reduce about 30% of the amplitudes at thetool tip of the first bending mode of the shaft located around 510 Hz. The correspondingprojections on the process stability indicate a 50% increase of the unconditionally stableaxial depth of cut, i.e. the limit under which the predictions guarantee a stable process forevery spindle speeds corresponding to the lowest bound of the stability lobes diagram alsocalled absolute stability limit. Some machining tests described in [72] confirm an increaseof the critical axial depth of cut of 40%.More details can be found in the doctoral thesis of Ries [73]. Unfortunately only fewexperimental machining conditions are presented to confirm the previous stability pro-jections and sketch the limitations of the system. Also, no comparison of the proposedconcept with an equivalent standard spindle is presented. To the best knowledge of theauthor, until now, no commercial product integrating this system has been offered byWeiss Spindeltechnologie GmbH to the market.

Another similar solution, called smart spindle unit, has been developed by the SandiaNational Laboratories and first presented in [74]. In the smart spindle unit, the frontbearing is actively supported in the radial plane by two orthogonal pairs of electrostrictivestack actuators working in push-pull configuration and guided by hydrostatic bearings, asshown in figure 2.9. This spindle unit has been designed to be integrated into an octa-hedral hexapod milling machine, presented in [75, 76]. The information on the currentstate of the tool tip deviations is provided by strain gauges located at the root of therotating tool itself. The signals from the strain gages are transmitted to the controller via

a telemetry system placed at the spindle nose. The angular position of the spindle shaftmeasured by a decoder at the end of the spindle allows to determine the deformations ofthe tool in the inertial coordinate reference system of the machine.A linear-quadratic-Gaussian (LQG) control, minimizing the influence of the process dis-

turbance on the tool tip, has been implemented and found to provide sufficient trade-offbetween robustness and performance. An optimal H∞ controller has also been tested in[78].


YX

Z

Figure 2.8: Active motor milling spindle concept. Source: [71].

The proposed system is designed to operate at moderate spindle speeds where the influenceof the process damping — originated by frictional forces induced by an effective negativeclearance angle — becomes negligible and where the stability limit is almost independentof the spindle speed, i.e. where the stability lobes are not visible. More information onthe process damping effect can be found in [31, 79, 80].In full immersion, the maximum stable axial DOC can be improved by an order of mag-nitude. With quarter and half immersions, the results obtained in [77] show the ability toincrease the stable axial DOC by factors 4 to 5.The sensing system used in this solution makes difficult its transposition into a commercialproduct where different types of tool are used and must be quickly replaced. Moreover,some problems of observability and controllability of the critical mode have been observedand an additional mass has been added close to the end of the tool such that the criticalmode of the tool becomes more visible by the control system. This modification shifts thefirst critical mode from 800 Hz to 453 Hz that makes the dynamics of the spindle assemblymuch more sensitive to chatter, due to a higher compliance, and thus less representativeof the machining conditions encountered in production.

The IFAS institute for fluid power drives and controls of the RWTH Aachen elaboratedthe construction of a milling spindle, presented in [81], where the outer ring of the frontbearing is actively supported by three cylindrical pistons hydraulically actuated in bothradial directions and guided with six hydrostatic bearings. Each actuating piston is con-


Figure 2.9: Smart Spindle Unit. Source: [77].

trolled by a high dynamic piezo-servovalve.The use of hydraulic actuators allows larger strokes (up to ±0.5 mm) than solid stateactuators, such as piezoelectric or electrostrictive actuators, for a same available imple-mentation volume. It is thus conceivable to not only compensate the vibrations of theshaft but also the static or lower dynamic deformations from the rest of machine struc-ture. Collocated non-contact eddy current displacement sensors are used to capture theshaft deviations at the front bearing. The controller consists in a proportional-integral(PI) control.


The system shows the ability to modify the dynamics at the tool tip. More specifically,the phase of the receptance function of the critical mode around 300 Hz is modified suchthat it does not cross -90 deg. However, no machining tests are presented. The achievedbandwidth of the proposed active system is quite limited due to the low dynamics of theactuating system presenting a cutoff frequency around 120 Hz.

2.5.2.2 Active Bearing

Mainly due to their interesting cost-effectiveness ratio, rolling bearings remain the mostwidespread solution for the rotational guiding of the main spindle shaft of modern machinetools. Using the principle of electromagnetic suspension, electromagnetic bearings can alsobe used to fully support in rotation the spindle shaft with the main advantage of being freeof friction. They are active by nature and, thus, their stiffness and damping properties aregiven by a control loop. In comparison with rolling bearings, higher static stiffness can beachieved but their maximum loading rate is significantly lower, as discussed in [82]. Activemagnetic bearings (AMBs) are then well suited for high speed cutting (HSC) applicationscharacterized by high cutting speeds and low cutting forces. Their active nature can beused for self-balancing or to improve process stability. However, due to their very highcosts, their use remains limited to special applications.

Kyung and Lee [83] developed a milling spindle shaft suspended by five-axes AMBs whereeach AMB is independently controlled using a proportional-derivative (PD) controller. Aparametric study showed that the control gains of the rear AMB do not influence thestability of the process. The derivative control gain of the front bearing plays a significantrole on the milling stability and must be large enough to prevent chatter occurrence.

In [84, 85, 86], the development of a high speed AMB spindle prototype presenting threeradial magnetic bearings and one single axial bearing is described. Optical sensors are usedto measure the spindle shaft displacements at the radial bearings location. The prototypespindle is able to turn up to 32’000 rpm and susceptible to deliver 67.1 kW. In [87], Knospecompared different types of controller in order to minimize the tool tip deviations. Optimalcontroller based on H∞ design and robust µ-synthesis controllers are compared with aconventional proportional-integral-derivative (PID) controller. The results show that areduction of 40% of the tool compliance is achievable using multivariable controllers.

The ETH Zürich developed a series of high speed spindles involving active magnetic bear-ings. More information of that can be found in [88, 89]. In [90], it is mentioned that suchspindle is susceptible to rotate twice faster than ball bearing spindles with greater power,longer life and less maintenance.


Recently, Gourc et al. [91] performed some experiments using a high speed IBAG HF400-M spindle. They point out the difficulty to obtain accurate chatter stability predictionsin high speed milling using such spindle and propose an improved model.

2.5.2.3 Active Ancillary Bearing

Several systems using an active ancillary bearing have also been proposed. In comparisonwith active support solutions previously described, such systems place the actuating sys-tem in parallel to the force path transmitted by the spindle bearings. The bearing loadingcan thus be reduced but the integration of an ancillary bearing requires space leading toan increase of the shaft length inducing a reduction of its structural stiffness.

In the frame of the AdHyMo (Adaptronische, hybridgelagerte Motorspindel zur prozess-sicheren und ratterfreien HPC-Fräsbearbeitung) research project, an active motor spindlesupported by conventional ball bearings and integrating an active ancillary electromag-netic bearing located between the front ball bearings and the tool has been developed.The main objective of the project is the detection and mitigation of the chatter vibrationsusing active damping approach. The ability of the hybrid supported spindle to monitorits main functionalities, like the loading of the ball bearings, and the process, such as toolwear, is also an issue. The results achieved in this project are summarized in [92].The deviation of the spindle shaft are sensed by one pair of eddy-current sensors locatedat the front of the prototype spindle and almost collocated with the stators of the magnetbearing. In the presented configuration, the natural frequency of the first bending modeof spindle shaft, which is critical for the process stability, is located at 520 Hz. The band-width achieved by the active system makes possible the control of this first natural mode.For a given spindle speed, a robust approach based on µ-synthesis procedure is used todesign a controller that actively increases the damping of the first bending mode of spin-dle shaft and improve the process stability. The spindle shows a spindle speed-dependentdynamics which induces a sever decrease of the performance provided by a unique robustcontroller. Several linear time-invariant (LTI) robust controllers are designed for differentspindle speeds and an adaptive scheme is used for the transition ranges. Three differentgain-scheduling approaches are presented in [93], namely bumpless transfer, fuzzy gainscheduling and gain scheduling of controller poles and zeros. All seem to provide satisfy-ing active damping performances in the transition zones but differ by their implementationwhich can be more or less suitable depending on the application.In order to avoid saturation problems of the electromagnetic actuator, the synchronouscomponent of the signals generated by the controller is filtered using adaptive feedforwardscheme, as described in [94].


Some other strategies, for instance optimal linear-quadratic control, have also been tested.In [95], a lead-lag controller is manually tuned in order to damp the first bending modeand stabilize the poles of the closed-loop system considering the process interaction basedon the semi-discretization method. Some machining tests demonstrate an increase equalto 100% of the productivity in stable conditions.The active system is also used to identify the spindle dynamics during the milling processand detect the chatter occurrence using recursive identification of a second order modelcorresponding to the critical mode of the spindle. Chatter appears when the roots of theidentified second order system become unstable.

In [96], another concept using an ancillary electromagnetic bearing located between thefront and the rear rolling bearings of a HSC motor spindle is presented. The rear rollingbearing is supported by a squeeze film damper. The objective is to adapt the propertiesof the spindle to the selected machining conditions in order to cope with a wider range ofoperations by finding an optimal working point between high static or dynamic stiffnessand self-balancing capabilities. This concept has been patented [97].

2.5.2.4 Active Spindle Housing Support

Some solutions propose to place the active system around the spindle housing. Due toheavier inertia driven by the actuators, the control bandwidth of such system is generallystrongly limited to low frequencies and thus, they are suitable for quasistatic and lowdynamic fine positioning of the TCP or chatter mitigation of low frequency critical modes,as for instance natural modes coming from the global machine structure.

In [98], a tripod adaptronic spindle system for milling machine is described. Threepreloaded piezo-actuators, mounted with strain gages, work in parallel kinematics andthree eddy current sensors are used to measure the tool tip deflection and the spindledisplacement in Z-direction.The system shows the potential to compensate the static deflection of slender tools due toprocess loading using 3D-dynamometer. A reduction of 50 up to 90% of the tool deflectionerror can be achieved during cutting tests. Harmonic counter vibrations, generated by theactuating system, are used to counteract the chatter vibrations by using a compensationphase-opposite to the tool deflections. No improvement of the process stability could bedemonstrated but a reduction of the chatter vibration amplitudes of 10 to 15% is obtained.

A hexapod kinematic structure using three pairs of piezoelectric actuators controlled byPID controllers is presented in [99]. Five capacitive displacement sensors are used for theestimation of the TCP position and the control loop. The controller bandwidth seems


to be limited to 130 Hz. The system is used to create an irregular surface in a boringoperation.

In [100], it is proposed to integrate an active system along the Z-slider of a portal machineto improve the static and dynamic compliance at the tool tip. Two compensation modules,integrating piezoelectric actuators with strain gages and a collocated force cell, are actingalong the spindle axis in order to compensate the bending modes of the Z-slider. Thedynamic displacements are compensated using a notch controller for each active moduledelivering current reference signal to the actuator amplifier and receiving the velocitysignal of the TCP derived from accelerometers. The static deformations are controlledusing a PI controller delivering the voltage reference signal to the amplifier based on theTCP displacements measured by displacement sensors.The system demonstrates the ability to completely compensate the static TCP deviationsconsidering a process force of 350 N. The resonance amplitude of the second bending modeof the Z-slider located at 49 Hz can also be damped by a factor 10 which should improveits stability against chatter by almost the same order.

2.5.2.5 Active Tool

The kinematics of the milling process as well as the tool change requirements make theintegration of active systems at the tool or tool holder stage inconvenient. This is whymost of such application concern turning or boring operations. As this work focuses onmilling operation, only one example is presented here.

Rojas et al. [101] studied the possibility to actively improve boring process using magne-tostrictive actuators and ER elastomer dampers. The actuator is mounted between theboring bar and the tool holder. A triaxial accelerometer is used to measure the tool tipdeviations and a force cell is placed between the actuator and the boring bar. A velocityfeedback with a PI controller is compared with a least mean squares adaptive filter.The experiments using the magnetostrictive actuator delivered mitigated results mainlydue to a limited bandwidth of the control system and an undersized actuator. The employof ER elastomer damper instead of the actuator shows the ability to reduce high frequencyvibrations leading to a better surface finish.

2.5.2.6 Active Workpiece

As chatter or, more generally, vibration problems in manufacturing process occur betweentool and workpiece, active compensation may also be integrated on the workpiece side, inthe workpiece holder or in the machine table.


Zhang and Sims [102] proposed to integrate piezoelectric actuators on the structure ofthin-walled and flexible workpieces in order to improve process stability. Two piezoelectricceramic patches are bound on both sides of the workpiece structure submitted to bendingdeformations. One plate acts as strain sensor and the other as actuator. They usedpositive position feedback as collocated control strategy to increase the damping of thefirst bending mode.Experimental tests demonstrated a possible increase of the critical axial DOC up to seventimes using the active device.

Haase [103] developed an active workpiece holder (AWH) for vertical machining centersintegrating two preloaded piezoelectric actuators. The AWH possesses two DOFs in theradial plane. Strain gages, force sensors and accelerometers are used to deliver the requiredinformation to the controller. Adaptive control using fxLMS (filtered-x least mean squares)algorithm is employed to estimate the model of the machine structure dynamics and tocontrol the resulting vibrations between the workpiece and the tool.The proposed system demonstrates the ability to reduce the vibrations in stable conditionsby approximately 85% and also to stabilize chatter vibrations.Abele et al. [104] continued the development of this system to make it more user friendly,robust and suitable for industrial integration.

Brecher et al. [105] presented similar AWH using two piezoelectric actuators enablingplanar movements in the radial plane of the spindle. The control strategy consists inmoving the workpiece to follow the spindle vibrations so that the critical mode becomesunobservable by the process. The tool tip deviations are estimated using an accelerometerattached to the spindle. The system is used to suppress the influence of the first criticalmode of the machine coming from the tool side and located at 60 Hz. A productivityincrease in stable conditions of 50% can be achieved by the proposed device.

2.5.2.7 Active Mass Actuator

Active mass actuators may also be used to damp the critical modes of the machine struc-ture. They consist in a reaction mass connected to the vibrating structure via an actuatingsystem moving the mass. Inertial forces resulting by the movements of the mass caused bythe actuator can be transmitted to the structure and influence its dynamics. They havethe advantage of being easily implementable on the machine but add mass and requireoften more space than the previously mentioned solutions.

The company Micromega Dynamics SA proposes a compact active damping device (ADD)[106] susceptible to increase structural damping of any mode of the machine tool observable


by the integrated sensor. The ADD consists in a vibration sensor collocated with an inertialelectromagnetic actuator and a simple velocity feedback controller.Ganguli [107] verified the ability of the ADD to suppress chatter in turning and millingoperations.

Brecher et al. [108] used an electrohydraulic actuator to drive the reaction mass. Thedamping of a critical mode of the machine located around 50 Hz is demonstrated leadingto an increase of 180% of the critical axial DOC.

2.5.2.8 Active Structural Control Strategies for Chatter Mitigation

Basically, the control strategies susceptible to improve the machining process stability canbe classified into two categories: disturbance rejection and stabilization schemes. Referringto figure 2.10, the former does not explicitly consider the process interaction in the controldesign step but intends to minimize the influence of the process excitation w on the TCPdeviations z. The process is considered here as an exogenous disturbance with generalcharacteristics. The control objective is thus the minimization of a certain norm of theclosed-loop transfer function Fzw between w and z,

z(s) = Fzw(s) w(s). (2.15)

The transfer function Fzw is given by

Fzw(s) = Gzw(s) +Gzu(s) C(s) S(s)Gyw(s) (2.16)

where S is the output sensitivity given by

S(s) =(

I −Gyu(s) C(s))−1

(2.17)

with I being the identity matrix.Disturbance rejection schemes lower the magnitude of the critical resonance peaks at thetool tip. This tends to cause a general increase of the unconditionally stable axial DOC,as previously seen in figure 2.6. They are thus especially interesting at low spindle speedswhere no higher stability pocket can be found.

In all the aforementioned active structural solutions, the control strategies correspondto disturbance rejection schemes where the process stability is checked a posteriori ofthe design step. However, some authors attempted to explicitly integrate the processdynamics P into the control design procedure. The control objective is then to guaranteethe stability of the overall closed-loop system of figure 2.10 for the considered machiningconditions. Such strategies are called here stabilization schemes.


C

Gzw

Gyw

Gzu

Gyu

y

w

u

+

+

+

+

zG

P

Figure 2.10: Flowchart of overall system with detailed plant structure. Gkj : transfer function of the plant

between the jth input and the kth output with j = w, u and k = z, y.

Among these strategies, Mei et al. [109] designed an optimal linear-quadratic controllerwhere the control force takes the time delay induced by the regenerative effect into account.Simulation results of a single point orthogonal cut show a substantial increase of the processstability limit.

In [110], Shiraishi et al. described the implementation of an optimal linear-quadraticintegral controller using a Luenberger state observer for the chatter stabilization and theminimization of the tracking error of the cutter on a bench type lathe. A stepping motordrives the tool position and an eddy current displacement transducer delivers its actualposition to the observer. A Padé approximation is used to model the time delay termcoming from the regenerative effect.Some experiments in turning showed the ability of the controller to stabilize some chatterconditions and to set the tool position tracking error to zero.

Chen and Knospe presented in [111] a robust stabilization design using µ-synthesis withDK-iteration exploring the chatter mitigation through three different approaches for theconsideration of the spindle speed in the control design process, named: speed-independent,speed-specified and speed-interval. The former tries to maximize the unconditional sta-bility limit guarantying a chatter-free operation for all spindle speeds. The second oneconsiders a specific spindle speed where the stable depth of cut is maximized and the


latter one takes into account a spindle speed interval.Using robust stabilization synthesis techniques, it is possible to derive controller guaran-tying a stable process over a certain range of values for some varying parameters, suchas the depth of cut or the spindle speed. These ranges are thus considered as parametricuncertainties. The first strategy only considers an uncertainty over the cutting stiffness,or equivalently, the depth of cut, in order to guarantee a stability for a depth of cut goingfrom zero to a specific maximum. The second approach approximates the time delay via

a Padé approximation and also considers the uncertainty related to the cutting stiffness.Beside this uncertainty, the third controller takes into account a certain range of valuesfor the time delay related to the variation of the spindle speed. A stable filter of finite-dimension coupled to a specific uncertainty is used to cover the range of time delays andto conservatively guarantee the stability over the range of considered spindle speeds.The final end of this study is the implementation of such techniques for the control of ac-tive magnetic bearings supporting a high speed machining (HSM) spindle. However, thearticle presents an application of the proposed approaches to the case of turning operationusing an AMB system.All three strategies demonstrate a great potential to increase the productivity in chatter-free conditions over a standard optimized PID controller. Figure 2.11 shows the predictedstability lobes derived from the measured tool frequency responses for the different con-trollers. The spindle speed considered for the two speed-dependent controllers is 2’800 rpm.As visible in figure 2.11, these two controllers are susceptible to tailor the tool dynamicsin order to maximize the stable cutting stiffness at the specified spindle speed.

The robust stabilization based on the speed-interval approach presented in [111] is appliedby van Dijk and co-workers in [112, 113] to the case of a HSM spindle fully supported byAMBs. Figure 2.12 shows the stability lobes obtained using the proposed controller fortwo different spindle speed ranges. An important increase of the predicted stable depth ofcut in these selected spindle speed ranges is visible.

2.6 Specific Objectives of the Thesis

The demands imposed by the market to the production techniques industry tend to ex-acerbate the occurrence of structural vibration problems in manufacturing processes. Inmilling operations, regenerative chatter constitutes one of the main limiting factors forthe productivity. The great diversity of the machining conditions encountered in practicecombined to the complexity of the phenomenon makes its mitigation difficult.From the last century, chatter phenomenon has been extensively studied to predict its ap-

2.6 Specific Objectives of the Thesis 37

Figure 2.11: Stability lobes as calculated from experimentally obtained tool compliance with: 1) PID

controller; 2) speed-independent controller; 3) speed-specified controller, n = 2’800 rpm; and 4) speed-

interval controller, n ∈ [2’727, 2’857] rpm. Areas below each curve are predicted to be stable operating

conditions. Source: [111].

parition and determine appropriate machining conditions guarantying chatter-free cut. Inparallel, different methods susceptible to counteract this phenomenon have been developed.Among them, active structural methods present a high degree of adaptivity susceptible tocope with the great diversity of the encountered machining conditions. They present theadvantage of providing efficiency from low to high spindle speed ranges and to cope witha wide scope of different cutting tools and workpieces dynamics due to a broad bandwidthof the actuating system. They are based on two elements, namely, the hardware imple-mentation and the strategy used by the controller. The challenge consists in finding theappropriate combination between both.The continuous decrease of the costs/effectiveness ratio of mechatronic hardware observedsince several decades in conjunction with their performance increase and the recent devel-opments in the control theory have made this kind of methods always more attractive forindustrial purpose. Combined with their great adaptivity, it makes them one of the mostpromising approaches to deal with chatter problem.

The aforementioned literature review points out the great potential of development of suchmethods but only very few solutions have been transferred into a commercial product sofar. This means that more efforts must be provided by the research community in order


Figure 2.12: Stability lobes diagrams, determined using the linearized nonautonomous milling model, for

reduced-order controllers designed for two different range of spindle speeds, n ∈ [24’990, 25’010] rpm and

n ∈ [30’000, 32’000] rpm. Source: [112].

to fulfill the industrial requirements.The conjunction of different reasons is responsible for this state of facts. The first onecomes from an economical point of view. Indeed, these solutions remain more expensivethan the process methods, for instance. Another major economical limiting factor comesfrom the engineering costs involved for the integration and the control design steps. In thisperspective, the author thinks that a possible way to decrease these costs is to optimizeand standardize the design procedures. This constitutes the background aim of this work.

The present work focuses on the following identified deficiencies in the state of the art. Oneof the major advantages of active structural systems is their ability to be efficient in lowas well as in high spindle speed ranges. In low spindle speed range, disturbance rejectionschemes are usually sufficient to improve the stability limit. In high speed range, greaterstability increase may be achieved using stabilization schemes. Both approaches are thuscomplementary. However, the active bearing support solutions presented in subsection2.5.2.1 have only been used in conjunction with disturbance rejection schemes but neverwith stabilization. Finally, stabilization schemes applied to milling process and discussedin subsection 2.5.2.8 have not been experimentally validated in machining conditions.

2.6 Specific Objectives of the Thesis 39

The first specific objective of this thesis is the development of a motor spindle integratinga mechatronic system dedicated to the stabilization of regenerative chatter in millingprocess. An active bearing support solution is selected for the integration of this system.The transferability of the proposed concept into a commercial product must determinethe design of the active system by respecting the industrial constraints.A representative physical model of the mechatronic system, susceptible to help the designand the elaboration of control strategies, must be defined and validated.Emphasis must be brought on the elaboration of an adequate control strategy. A model-based procedure using optimal control theory is used to design the controller. It mustsatisfy the performance expectations and must be experimentally validated over a range ofrepresentative machining conditions. In particular, the formulation of the control objectivemust consider the possibility to choose between disturbance rejection and stabilizationschemes.In the end, the proposed concept must be evaluated and some recommendations for itseventual improvement, formulated.

40 3. Simulation Tools

Chapter 3

Simulation Tools

As seen in the previous chapters, the overall system considered here lies on three fun-damental interconnected subsystems, namely: the passive mechatronic structure G, themilling process P and the controller C corresponding to the cognitive element. Physi-cally, the mechatronic structure is composed of the spindle with the integrated sensorsand actuators. It also includes the dynamics of the whole machine structure up to theworkpiece as well as the signal conditioners and amplifiers required by the control loop.It constitutes the central element connected to both other subsystems through input andoutput variables. The process is coupled to the structure by the cutting loads w. Theresulting structural deviations from the nominal cutting path z influence the process loadsvia the regenerative effect R. This closed-loop between G and P makes possible the ap-parition of self-excited vibrations leading to chatter. In addition to the cutting forces wR

induced by the regenerative effect, some cutting forces fP , generated by the variation ofthe chip thickness due to the process kinematics, excite the machine structure. In parallel,the plant G is also coupled to a controller delivering reference signals u to the actuatoramplifiers based on the information y provided by the couplers connected to the integratedsensors. The flowchart representing this global system is shown in figure 3.1.

To design the active system, a physical model, able to simulate the dynamics of the spin-dle assembly coupled to the actuators and sensors, must be constructed. In particular,this model must guide the positioning and the dimensioning of the actuating and sensingsystems. The influence of the control system on the spindle assembly dynamics and onthe process stability must also be predicted. One thus requires analysis tools, such asmodal and harmonic response analysis of the mechatronic system, as well as time-domainsimulations and stability analysis considering the coupling with the milling process.Commercial software exists to model the structural behavior, such as ANSYSR© [114], orthe process-structure interaction, like CutProTM [17], but none can simultaneously con-

3.1 Modeling 41

C

Gzw

Gyw

Gzu

Gyu

y

w

u

+

+

+

+

zG

R+

+

wR

Figure 3.1: Detailed flowchart of overall system.

sider the interaction between all three subsystems. This is why, during this work, an eigencode has been developed on the Matlab/SimulinkR© [115] platform, as it offers an in-teresting flexibility/complexity compromise for numerical computation, existing toolboxesfor control design, the resolution of differential equations and to display the results. Italso presents interesting memory allocation properties for large arrays manipulation andthe possibility to generate standalone executable files.

In the following sections, the modeling of the global system, depicted in figure 3.1, isdescribed. Some approximated models, used for the stability analysis and the controldesign, are derived. The key issue of the process stability analysis is presented and,finally, the validation of the developed simulation tools is described.

3.1 Modeling

In this section, the modeling of the three subsystems: the plant G, the milling process P

and the controller C, is detailed and different combinations of them are formulated.


3.1.1 Passive Mechatronic Structure

Two different types of model are used in this work to represent the behavior of the passivemechatronic structure G. The first one is derived from physical principles using the finiteelement method for the structure dynamics and analytical models for the mechatronicelements. Such models are specially helpful for the design step of the prototype spindle.The second type is based on experimental data and the parameter values of a predefinedmathematical model are estimated generally using regression methods. This second typeis usually more precise than the first one and may be used for model-based schemes, suchas control design.In this chapter, the construction of the physical model is presented. The derivation of thesecond model is discussed in chapter 5.

The element G, also called plant in control theory, corresponds to the assembly composedby the machine tool, the spindle unit, the tool-holder, the tool and the workpiece. It alsotakes into account the integrated sensors and actuators used by the active system. As thefocus here is made on problems originated by the spindle shaft assembly, it is decided tolimit the boundaries of the physical model to the system formed by the spindle housing,the bearings, the spindle shaft, the tool-holder and the tool. Indeed, as mentioned in theintroduction, one primarily looks at chatter cases caused by the first bending modes fromthe spindle assembly, which typically occur in the frequency range between 500 Hz and3 kHz. Within this frequency range, the first bending modes of the whole spindle shaft aswell as those local to the tool end are present. These modes are normally well uncoupledfrom the most critical modes of the machine tool, that are generally located between 50and 200 Hz. The rest of the whole machine tool structure is not explicitly considered hereas the modeling of the higher modes of the machine is generally hazardous. This limitationof the system boundaries is also motivated by the fact that this model is essentially usedfor the design of the prototype spindle and thus does not require the consideration of thewhole machine tool. However, assuming that a sufficiently precise model of the wholemachine structure could be derived, its coupling to the physical model of the spindleassembly could be easily achieved, as discussed in section 3.1.1.3.

The modeling is based on the following main assumptions. In order to be able to applystandard analysis tools such as modal and harmonic analysis or control design procedures,the passive mechatronic structure of the active spindle is assumed to be stable and lineartime-invariant for a given spindle speed and machine axes position. The mechanical be-havior of angular contact ball bearings is highly dependent on the rotational speed of theshaft, the temperature and the axial preload, depending on the cutting forces applied attool tip, as demonstrated in [23, 82]. This preload dependency leads to a nonlinear model

3.1 Modeling 43

for the bearing stiffness, which is here linearized around its nominal working conditionbased on the data provided by the bearing manufacturer. Thermomechanical couplingeffects are also neglected by considering a behavior at thermal equilibrium. The intrinsicdynamics of the electromechanical and electrical devices, like the sensors, actuators, signalconditioners and amplifiers, is considered as ideal. This means that the effects inherent totheir own dynamics and disturbing their expected functioning are located wide outside ofthe system bandwidth.

3.1.1.1 Active System Design

Before going deeper into modeling, the design of the active system integrated into theprototype spindle is briefly described. Only the selected working principle and the ar-rangement of the final solution are presented here in order to help the understanding ofthe model construction. More details on the active system design are given in section 4.4.

The main challenge of the design consists in integrating an actuating system susceptibleto satisfy the performance expectations with respect to the tight industrial constraints interms of basic specifications maintaining, available implementation space and costs. Theactuating system is constituted of two pairs of piezoelectric stack actuators working inpush-pull configuration and orthogonally arranged in the radial plane of the spindle. Theysteer the support of the front bearing assembly of the spindle shaft. Each piezoelectricactuator is driven by a power amplifier connected to the controller.The sensing system is composed of two uniaxial accelerometers orthogonally located onthe front bearing support in the same axial plane as the actuating system so that, interms of axial position, they can be considered as collocated with the actuators. This canbe realized by an angular shifting of the accelerometers around spindle axis. AdditionalNCDTs can be used to sense the radial deviations of the spindle shaft relative to thehousing at different axial positions. The signals generated by the sensors are processed bysome signal conditioners before being delivered to the controller.The concept of the active system is represented in figure 3.2.

3.1.1.2 Spindle Mechanical Model

In order to simulate the behavior of the global system, one needs a model able to predict thedeviations from the nominal tool trajectory between the tool and the workpiece resultingfrom the process excitation, the information provided by the integrated sensors as well asthe influence of the instructions delivered to the actuating system. A mechanical modelof the spindle assembly dynamics and mechatronic models for the sensing and actuating


w

z

y

u

is

Tool piezoelectric stack

TCP

NCDT

Accelerometer

Power amplifier

Signal conditioners

Spindle shaft

n

js

ksis

js

ks

Figure 3.2: Concept of active spindle. NCDT: non-contact displacement transducer; is, js,ks: unit vectors

of spindle coordinate system.

systems are thus necessary.

Finite elements are used to model the structural behavior of the mechanical model. Dueto the axial symmetry of the spindle assembly, Timoshenko rotating beam finite elementswith translational and rotational DOFs are well suited as they can save a large amountof computational efforts in comparison with conventional tetrahedral or quadrilateral ele-ments with only translational DOFs. They can also easily take into account the influenceof the rotation on the structural behavior using rotordynamics theory.The global mechanical spindle assembly model thus considers rotating beam elements forthe modeling of the spindle shaft structure, standard non-rotating beam elements for thespindle housing, linear spring-damper elements for the modeling of bearings, the tool in-terface or the coupling with the rest of the machine tool structure, as well as rotatingor non-rotating lumped disk elements for the modeling of elements presenting negligibledeformation but important inertial influence. After the assembling of all elementary ma-trices and the removal of constrained DOFs, the resulting global system, expressed in theinertial reference frame of the machine, can be described by the following linear ordinaryset of second order differential equations

Ms qs(t) +Ds qs(t) +Ks qs(t) = fs(t) (3.1)

where qs ∈ Rnqs are the unknown nodal displacement DOFs and nqs is the total number

of unconstrained DOFs. The subscript (.)s refers to the spindle unit. The global massmatrix is Ms and Ds is the global viscous damping matrix, Ks, the global stiffness matrixand fs, the external force vector acting on the structure and coming from unbalance, the

3.1 Modeling 45

machining process or the actuators. These global matrices are given by

Ms = MB +MD, Ds = DB +DS − Ω(

GB +GD)

,

Ks = KB +KBσ +KS − Ω2

(

MBr +MD

r

)

, fs(t) = fB(t) + fD(t). (3.2)

The superscript (.)B is used for beam elements, (.)D, for lumped disk elements and (.)S, forspring-damper elements. Ω is the negative angular velocity around the Z-axis of the spindleshaft in (rad/s) (see figure 3.2). The rolling bearings of the spindle are modeled usingspring-damper elements so that KS = KS(Ω, fap), with fap being the axial preload forceof the angular contact ball bearings. The G matrices are the gyroscopic matrices. The KB

σ

matrix is the stress stiffness matrix, considering the stiffening effects due to axial preloadfa applied on beam elements. The Mr matrices are the rotation mass matrices involvedby spin softening effects originated by the centrifugal forces. As proportional damping isconsidered for the structural damping, the influence of the damping of rotating elementson the stiffness is not taken into account here. Moreover, due to the slender geometry ofthe spindle shaft assembly, the centrifugal stiffening effect is neglected.

The procedure to derive these different matrices can be found in [23, 116].

3.1.1.3 Coupling with Machine Structure

The spindle assembly FE model can be coupled to the rest of the machine using spring-damper elements. If the machine structure dynamics is given by the following set ofdifferential equations, where the time dependency is omitted for more convenience,

Mm qm +Dm qm +Km qm = fm + fmc. (3.3)

The subscript (.)m refers to the machine structure. The nodal displacement DOFs belong-ing to the machine structure model are qm ∈ R

nqm and nqm is the number of correspondingfree DOFs. fm represents the external nodal forces acting on the machine structure and fmc

corresponds to the nodal reaction forces generated by the coupling elements characterizedby the stiffness and viscous damping matrices Kc and Dc, respectively.

fmc = −Bq,mc

(

Kc (Cq,mc qm −Cq,sc qs) +Dc (Cq,mc qm −Cq,sc qs))

(3.4)

where Bq,mc is the input matrix spreading the coupling forces to the global DOFs of themachine structure qm and Cq,jc, j = m, s, are the output matrices extracting the couplingDOFs from the global DOF vectors of the machine and the spindle assembly, respectively.

Similarly, for the spindle assembly structure,

Ms qs +Ds qs +Ks qs = fs + fsc (3.5)


where fsc corresponds to the nodal reaction forces generated by the coupling elements.

fsc = −Bq,sc

(

Kc (Cq,sc qs −Cq,mc qm) +Dc (Cq,sc qs −Cq,mc qm))

(3.6)

where Bq,sc is the input matrix spreading the coupling forces to the global DOFs of thespindle assembly qs.

The resulting global coupled system becomes

M q+D q+K q = f (3.7)

with

q =

[

qs

qm

]

, M =

[

Ms 0

0 Mm

]

, f =

[

fs

fm

]

,

D =

[

Ds +Bq,sc Dc Cq,sc −Bq,sc Dc Cq,mc

−Bq,mc Dc Cq,sc Dm +Bq,mc Dc Cq,mc

]

,

K =

[

Ks +Bq,sc Kc Cq,sc −Bq,sc Kc Cq,mc

−Bq,mc Kc Cq,sc Km +Bq,mc Kc Cq,mc

]

(3.8)

and q ∈ Rnq where nq = nqs + nqm.

Here, it is assumed that the main critical modes are coming from the spindle assembly andare uncoupled from the most flexible modes of the machine structure so that the dynamicsat tool tip can be studied by the FE model of the spindle assembly considering a simplespring-damper system at the boundaries of the spindle housing representing the stiffnessof the machine structure.

3.1.1.4 Coupling with Actuating System

The actuating system is composed of two pairs of piezoelectric stack actuators orthogonallyoriented in the radial plane and working in push-pull configuration (see figure 3.2). Specialhigh dynamic charge amplifiers are used to drive the actuators. The amplifier controls theelectrical charge on the piezoceramic stack and, by doing so, allows a reduction of thehysteresis effect, deteriorating the linear behavior of the actuator. This is specially truein low frequency domain. It also leads to a higher passive structural stiffness. Moreinformation concerning the actuating system is given in section 4.4.The resulting electromechanical system can be represented by the following relation wherethe force generated by the jth actuator is given by

fj = kuj uj − kLj (qe − qb) (3.9)

3.1 Modeling 47

where uj is the driving reference signal delivered by the controller to the charge amplifier,qb, the displacement of the node at the bottom of piezoelectric stack along the actuatormain deformation axis and qe, the displacement at the actuator end, parallel with qb andacting on the support of the front bearing. kuj and kLj are the actuator factor and itsstructural stiffness, respectively.

qb = cq,b q, qe = cq,e q (3.10)

where cq,k, k = b, e, are the output vectors extracting the displacement DOFs, qb, re-spectively qe, from the global nodal displacement DOFs q, defined in relation (3.8). Thesubscript (.)q refers to the nodal displacement DOFs vector q. The corresponding globalforce vector becomes

fj = bq,pj

(

kuj uj − kLj cq,pj q)

. (3.11)

cq,pj = cq,e − cq,b and bq,pj = bq,e − bq,b, where bq,e is the input vector spreading thegenerated force fj to the global nodal displacement DOFs vectors q and bq,b, the inputvector spreading the reaction force at the bottom of the actuator to the global DOFs vectorq.

With na actuators, the resulting sum of the global force vectors fp is given by

fp =na∑

j=1

fj

= Bq,p

(

Ku u−KL Cq,p q)

(3.12)

with

u =

u1

u2

...una

, Ku = diag (ku1, ku2, . . . , kuna) , KL = diag (kL1, kL2, . . . , kLna

)

(3.13)and

Bq,p =[

bq,p1 bq,p2 · · · bq,pna

]

, Cq,p =

cq,p1

cq,p2...

cq,pna

. (3.14)

Coupled to the overall machine structural model (3.7) and considering only external dis-turbance forces fw coming from the process, this gives the following set of equations forthe description of the mechatronic model.

Mq q+Dq q+Kq q = fw +Bq,u u (3.15)


with

Mq = M, Dq = D, Kq = K+Bq,p KL Cq,p, Bq,u = Bq,p Ku. (3.16)

Considering the nodal process forces vector w,

fw = Bq,w w (3.17)

where Bq,w is the input matrix spreading the process forces to the global nodal displace-ment vector q.

The actuator constants are experimentally derived. For the use of the power amplifier incharge control mode, the constants ku,Q and kD

L , corresponding to the actuator factor incharge mode and the structural stiffness at constant electric charge density displacement,respectively, must be considered.Two pairs of identical actuators in charge mode working in push-pull arrangement composethe actuating system. This means that na = 4, kL1 = kL2 = . . . = kL4 = kD

L andku1 = ku2 = . . . = ku4 = ku,Q. So,

KL = diag(

kDL , k

DL , k

DL , k

DL

)

. (3.18)

Also, considering the actuators 1 and 2 working in X-direction and the actuators 3 and 4,in Y -direction (see figure 3.2), u1 = −u2 = uX and u3 = −u4 = uY where uX and uY arethe driving signals in X, respectively in Y -direction. Defining

u =

[

uX

uY

]

, (3.19)

this leads to

Ku =

ku,Q 0

−ku,Q 0

0 ku,Q0 −ku,Q

. (3.20)

3.1.1.5 Second Order Model

From (3.15), the behavior of the passive mechatronic structure is described by the set ofsecond order ordinary differential equations

Mq q+Dq q+Kq q = Bq,w w +Bq,u u (3.21)

where Bq,w is the process input matrix and Bq,u the control input matrix.

3.1 Modeling 49

Two types of output must be derived: the deviations of the TCP z and the output signalsfrom the sensing system y. z corresponds to the DOFs at the TCP and y, in its mostgeneral expression and considering ideal measurement systems, represents an image of thedisplacements, velocities or accelerations, depending on the type of sensor used, at specificlocations on the structure. So these outputs can be written as

z = Cq,z q, y =

yq

yq

yq

(3.22)

withyq = Cq,y q, yq = Cq,y q, yq = Cq,y q (3.23)

where Cq,z is the process output matrix and Cq,y, Cq,y and Cq,y, the control outputmatrices.

3.1.1.6 First Order Model

In control applications, the first order formulation of the system dynamics is usually pre-ferred.

Defining the state vector

x =

[

q

q

]

, (3.24)

equations (3.21) and (3.22) can be rewritten as

x = A x+B

[

w

u

]

(3.25)

[

z

y

]

= C x+D

[

w

u

]

(3.26)

with

B =[

Bw Bu

]

, C =

[

Cz

Cy

]

, D =

[

Dzw Dzu

Dyw Dyu

]

(3.27)

and x ∈ Rn, w ∈ R

nw , z ∈ Rnz , u ∈ R

nu , y ∈ Rny .

The exogenous forces and torques generated by the milling process and acting on the TCPcorrespond to w. So, considering a concentrated application point of these loads, nw = 6


andw =

[

wX wY wZ wθX wθY wθZ

]T

(3.28)

where wj, j = X, Y, Z, represent the translational forces and wθj , the torques. The result-ing TCP deviations are represented by z. Assuming that only the radial deviations at thetool tip influence the chip thickness, the three translational displacements are considered,so that nz = 3 and

z =

zXzYzZ

. (3.29)

The controller outputs are u, i.e the signals delivered to the power amplifiers of theactuators. As two pairs of actuators are working in push-pull configuration (see section3.1.1.4), nu = 2 and u is given by (3.19).

The outputs from the sensors, which correspond to the signals delivered by the sensorconditioners, are given by y. At this point, the configuration of the sensing system is leftgeneral.

From the second order system, the different matrices of the first order system are derivedas

A =

[

0 I

−M−1q Kq −M−1

q Dq

]

, Bw =

[

0

M−1q Bq,w

]

, Bu =

[

0

M−1q Bq,u

]

,

Cz =[

Cq,z 0

]

, Cy =

Cq,y 0

0 Cq,y

−Cq,y M−1q Kq −Cq,y M

−1q Cq

,

Dzw = 0, Dzu = 0, Dyw =

0

0

Cq,y Bq,w

, Dyu =

0

0

Cq,y Bq,u

(3.30)

where I is the identity matrix.

In the Laplace domain, the transfer functions of the resulting system can be written as[

z(s)

y(s)

]

= G(s)

[

w(s)

u(s)

]

(3.31)

where

G(s) =

[

Gzw(s) Gzu(s)

Gyw(s) Gyu(s)

]

=

[

Cz

Cy

]

(sI−A)−1[

Bw Bu

]

+

[

Dzw Dzu

Dyw Dyu

]

. (3.32)

3.1 Modeling 51

The state-space realization of G over x ∈ Rn can be rewritten in compact form as

Gs=

A Bw Bu

Cz Dzw Dzu

Cy Dyw Dyu

. (3.33)

3.1.2 Milling Process

The model of the milling process P must predict the resulting loads generated betweenthe tool and the workpiece based on the chosen machining parameters as well as thecurrent and past structural deviations of the TCP from its nominal trajectory. In millingoperation, the process model is based on two parts: the kinematic model and the cuttingforce law. The kinematic model describes the resulting chip geometry from the movementsbetween the tool and the workpiece. The cutting force law describes, from the differentmachining parameters and chip geometry, the resulting cutting forces.

The combination of the feed movement with the rotation of the cutter leads to a trochoidalpath of the cutting edges relative to the workpiece. Depending on the ratio between thefeed rate and the spindle speed, this trochoidal path may be satisfactorily approximatedby a circular function. This is specially true for low feed rates with high spindle speedsand, except for high feed machining, the resulting variation of the nominal chip thicknesscan be approximated by a sine function.

Basically, two types of model are used to predict the cutting forces in machining processes.The first one does not require any specific experimental data from machining tests butdiscretizes the chip formation area so that fundamental thermomechanical principles canbe applied. The accuracy of the results highly depends on the discretization density andrequires the use of numerical methods. Finite element method, as proposed in [117, 118,119, 120], as well as meshfree methods, like smoothed particle hydrodynamics presented in[121], have been applied. Due to the complexity of the chip formation phenomenon, suchmethods require high computational efforts to get representative results. The differenceof time scales between the chip formation process and the machine structure dynamicsmakes them oversized and not well adapted when the focus is the stability of the process-structure interaction. In this case, macroscopic analytical models are better suited for thecutting force prediction. Such models require the determination of coefficients based onmachining tests where the cutting forces are monitored. Different analytical models canbe found in the literature [11, 12, 122, 20, 123, 31]. Here, the linear average cutting force

model is used. This model has been proposed by Altintas and Lee [124] and takes intoaccount some shear and edge cutting forces. The former forces result from shear stress


depending on shear angle and from the friction between the chip and the rake face. Theedge cutting forces are due to the friction of the cutting edge in contact with the generatedsurface. It allows a simple identification of the cutting force coefficients and usually leads toaccurate predictions even for complex tool geometries. The value of the coefficients can bedetermined from orthogonal cutting tests or identified using a mechanistic approach. Themechanistic approach requires a calibration for every different tool and workpiece pair butcan cope with complex cutting edge geometry. The approach based on orthogonal cuttingtests only needs experiments for a given workpiece material but remains valid only forsimple cutting edge geometry.

Peripheral milling is considered here, i.e. the machined surface is essentially generatedby the periphery of the cutter. The process is defined by the axial and radial cuttingdepths, ap, respectively ae, the feed rate vf and the spindle rotational speed n. Figure 2.4illustrates these parameters.In this work, three different reference coordinate systems are used. The first one is thecoordinate system of the spindle which is represented in figure 3.2. The second one is thetool coordinate system shown in figure 2.4. The spindle coordinate system correspondsto the machine axes and, by convention, the X-direction of the tool coordinate system(indicated by the unit vector it) corresponds to the feed direction, the Y -direction (unitvector jt) to the normal direction and Z (unit vector kt) is parallel to the spindle rotationalaxis. The positive direction of the angular velocity of the cutter relatively to the workpieceis opposite to the Z-direction of the tool coordinate system. The last considered coordinatesystem is rotating with the tool and is attached to the cutting tooth, as presented in figure3.4.To describe the tool geometry, the cutter is discretized along its rotational Z-axis, so thatthe geometrical parameters, such as the radius R as well as the helix and entering angles,λ and κ respectively, can be expressed as a function of the z coordinate. The definitionof these angles is shown in figure 3.3. Moderate helix angles and entering angles close to90 deg are assumed to derive the following relations.

The three components of the differential cutting force dfj generated on an infinitesimalcutting edge of length dS on the jth tooth are given by the average cutting force lawas

dfq,j(t, z) =(

Kqc hj(t, z) db(z) +Kqe dS(z))

gj(t, z) (3.34)

where q = r, t, a and j = 1, . . . , Nt, with Nt being the total number of teeth of thecutter.

The subscript (.)r corresponds to the radial component, (.)t, to the tangential and (.)a,

3.1 Modeling 53

z it

jt

kt

n

R

Figure 3.3: Geometrical parameters of milling cutter. R: tool radius; z: coordinate along axial direction;

κ: tool entering angle; λ: tool helix angle; it, jt,kt: unit vectors of non-rotating tool coordinate system.

to the axial. The corresponding coordinate system local to the cutting tooth is shown infigure 3.4.

The shear and the edge cutting force coefficients obtained from experimental tests are Kqc

and Kqe respectively.

The infinitesimal uncut chip width db, represented in figure 3.4, is given by

db(z) =dz

sinκ(z)(3.35)

where dz is the corresponding infinitesimal increment of the z coordinate. Similarly andconsidering moderate helix angle and entering angle close to 90 deg, dS can be approxi-mated by

dS(z) =db(z)

cosλ(z). (3.36)

The function gj determines whether the jth tooth is cutting or not, based on its angularposition φj and the entry and exit immersion angles, φin, respectively φout. This functionis defined as

gj(t, z) = g(

φj(t, z))

with

g(φj) =

1 if ∃k ∈ Z such that φin ≤ φj + k2π ≤ φout,

0 otherwise.(3.37)

Also, assuming regular pitch angles between two teeth, a constant spindle speed andneglecting the influence of the TCP deviations,

φj(t, z) = φ(t)− (j − 1)2π

Nt

−tanλ(z)

Rj(z)z (3.38)

where Rj is the radius of the jth tooth and φ(t) = φ0+Ωt, with φ0 being the initial angularposition of the reference tooth cutting edge and Ω = 2πn/60.


Figure 3.4: Tool and tooth coordinate systems with infinitesimal undeformed chip geometry. ij , jj ,kj : unit

vectors of jth tooth coordinate system.

In steady machining conditions and using circular approximation for the jth toothpath, the instantaneous undeformed chip thickness hj including the regenerative effectis given by

hj(t, z) =(

fz sinφj sinκ

+[

sinφj sinκ cosφj sinκ − cosκ]

(

δx(t)− δx(t− Tz))

)

gj (3.39)

where Tz is the time delay between the passage of the tooth j and the previous tooth(j − 1).

For convenience, the time and z dependencies are not explicitly specified. The vector δx

represents the translational deviations of the TCP from its nominal trajectory due to the

3.1 Modeling 55

vibrations between the tool and the workpiece, expressed in the tool coordinate systemand defined as

δx =

δxX

δxY

δxZ

. (3.40)

This vector is given from the deviations z, expressed in the spindle coordinate system anddefined in (3.29), by

δx(t) = tTz,s z(t) (3.41)

where tTz,s is the transformation matrix from the spindle coordinate system to the toolcoordinate system. This transformation is required in the case where the feed direction ofthe cutter into the workpiece is different from the X-axis of the machine. This transfor-mation corresponds to a simple rotation matrix around the spindle rotational axis, suchthat

tTz,s =

cos θ sin θ 0

− sin θ cos θ 0

0 0 1

(3.42)

where the angle θ corresponds to a counterclockwise rotation of the unit vector is up tothe vector it around the Z-axis.The chip thickness is the most important parameter for the study of chatter, as it in-troduces the regenerative effect into the process loop. Some other models of the chipthickness can be considered. For instance, under high amplitudes chatter vibrations, thecutting edge may jump out of the cut, leading to multiple regenerative effects. This mayeasily be taken into account using time-domain simulations.

The resulting differential cutting forces and torques, acting on the TCP and expressed inthe tool coordinate system, are given by

dft,j(t, z) =tTj(t, z) dfj(t, z) (3.43)

where

dfj =[

dfr,j dft,j dfa,j

]T

, dft,j =[

dftX,j dftY,j dftZ,j dftθX ,j dftθY ,j dftθZ ,j

]T

,


tTj(t, z) =

1 0 0 0

0 1 0 0

0 0 1 0

0 −z 0 0

z 0 0 0

0 0 0 1

− sinφj(t, z) sinκ(z) − cosφj(t, z) − sinφj(t, z) cosκ(z)

− cosφj(t, z) sinκ(z) sinφj(t, z) − cosφj(t, z) cosκ(z)

cosκ(z) 0 − sinκ(z)

0 Rj(z) 0

. (3.44)

The subscripts (.)X , (.)Y and (.)Z correspond to the feed, the normal and the axial directions,

respectively, as represented in figure 3.4.

By integrating these differential loads along the Z-axis and after summing them, the total loads

acting on the TCP, expressed in the tool coordinate system, are given by

ft(t) =

Nt∑

j=1

∫ z2

z1

dft,j(t, z) (3.45)

where, usually, z1 = 0 and z2 = ap.

The resulting cutting forces acting on the TCP but expressed in the machine coordinate system,

previously defined in (3.28), are

w(t) = w,sTt ft(t). (3.46)

The transformation matrix w,sTt from the tool to the spindle coordinate system is expressed as

w,sTt =

cos θ − sin θ 0 0 0 0

sin θ cos θ 0 0 0 0

0 0 1 0 0 0

0 0 0 cos θ − sin θ 0

0 0 0 sin θ cos θ 0

0 0 0 0 0 1

. (3.47)

The resulting forces vector can be rewritten as

w(t) = fP (t) +wR(t) (3.48)

with

fP (t) =w,sTt

Nt∑

j=1

∫ z2

z1

tTj

(

Kc fz sinφj +Ke1

sinκ cosλ

)

gj dz (3.49)

3.1 Modeling 57

and

wR(t) = KP (t)(

z(t− Tz)− z(t))

. (3.50)

The cutting stiffness matrix KP is given by

KP (t) = −

(

w,sTt

Nt∑

j=1

∫ z2

z1

tTj Kc

[

sinφj cosφj − cotκ]

tTz,s gj dz

)

(3.51)

where the time and z dependencies are not explicitly specified for convenience. Also, the shear

and edge cutting force coefficient vectors are defined as

Kc =[

Krc Ktc Kac

]T, Ke =

[

Kre Kte Kae

]T. (3.52)

The function fP corresponds to the cutting forces resulting from the kinematic variation of the

chip thickness and the edges cutting forces.

Also,

KP (t) = KP (t+ T ), fP (t) = fP (t+ T ) (3.53)

where T = Tz when considering an ideal tool without runout. If a tool with runout is considered,

T = Tn, where Tn is the time period between two spindle revolutions.

The transfer function of the regenerative effect R between the TCP deviations z and the corre-

sponding cutting forces and torques wR, represented in figure 3.1, can thus be written

wR(s) = R(s) z(s) (3.54)

with

R(s) = KP (s)(

e−sTz − I)

. (3.55)

3.1.3 Active Control System

A general form for the control transfer function is formulated which is able to represent all types

of control strategy susceptible to be considered, such as single variable, multivariable state-space

controllers or delayed state feedback.

A linear proper controller with delayed terms and time-varying coefficients of the following

general form is defined here.

u(t) = KC(t) xC(t) +KC,0(t) y(t) +KC,T (t) y(t− Tz) (3.56)

xC(t) = AC(t) xC(t) +BC,0(t) y(t) +BC,T (t) y(t− Tz) (3.57)

where xC ∈ RnC is the state vector of the controller.


The subscripts (.)0 and (.)T respectively refer to the current and delayed time signals.

The transfer function of the controller between the signals y delivered by the sensing system and

the corresponding reference signals u transmitted to the actuating system can be written as

u(s) = C(s) y(s) (3.58)

where

C(s) = KC(s)(

sI−AC(s))−1(

BC,0(s) +BC,T (s) e−sTz

)

+KC,0(s) +KC,T (s) e−sTz . (3.59)

3.1.4 Passive Structure coupled to Milling Process

For simulation or analysis matters, it may be interesting to use models of different configurations

between the three subsystems previously described. Here, the milling process coupled to the plant

without considering the controller is described. Figure 3.5 represents the coupling between the

plant and the process.

P

y

w

u

z

G

Figure 3.5: System formed by the plant G and the process P .

The coupling of the process model formulated by (3.48) to the plant G, expressed by (3.25) and

(3.26), gives the following system.

x(t) = AGP,0(t) x(t) +AGP,T (t) x(t− Tz)

+BGPu,0(t) u(t) +BGPu,T (t) u(t− Tz)

+BGPw,T (t)w(t− Tz) +BGPw(t) fP (t) (3.60)

y(t) = CGP,0(t) x(t) +CGP,T (t) x(t− Tz)

+DGPu,0(t) u(t) +DGPu,T (t) u(t− Tz)

+DGPw,T (t)w(t− Tz) +DGPw(t) fP (t) (3.61)

with

AGP,0(t) = A−Bw SGP (t)KP (t)Cz, AGP,T (t) = Bw SGP (t)KP (t)Cz,

BGPu,0(t) = Bu −Bw SGP (t)KP (t)Dzu, BGPu,T (t) = Bw SGP (t)KP (t)Dzu,

BGPw,T (t) = Bw SGP (t)KP (t)Dzw, BGPw(t) = Bw SGP (t),

CGP,0(t) = Cy −Dyw SGP (t)KP (t)Cz, CGP,T (t) = Dyw SGP (t)KP (t)Cz,

3.1 Modeling 59

DGPu,0(t) = Dyu −Dyw SGP (t)KP (t)Dzu, DGPu,T (t) = Dyw SGP (t)KP (t)Dzu,

DGPw,T (t) = Dyw SGP (t)KP (t)Dzw, DGPw(t) = Dyw SGP (t) (3.62)

where

SGP (t) =(

I+KP (t)Dzw

)−1. (3.63)

To solve these equations, the following initial conditions must be defined.

x(t) = ξx(t), u(t) = ξu(t) for − Tz + t0 ≤ t ≤ t0 (3.64)

and

w(t) = ξw(t) for − Tz + t0 ≤ t < t0. (3.65)

3.1.5 Active Structure

For the control design, it is necessary to represent the system with the process excitation not

explicitly described. This corresponds to the so-called active structure that is represented in

figure 3.6.

C

y

w

u

zG

Figure 3.6: System formed by the plant G and the controller C.

Considering the particular form of the control feedback (3.56), the coupled system can be ex-

pressed by[

x(t)

xC(t)

]

= AGC,0(t)

[

x(t)

xC(t)

]

+AGC,T (t)

[

x(t− Tz)

xC(t− Tz)

]

+BGCw,0(t)w(t) +BGCw,T (t)w(t− Tz)

+BGCu,T (t) u(t− Tz) (3.66)

z(t) = CGC,0(t)

[

x(t)

xC(t)

]

+CGC,T (t)

[

x(t− Tz)

xC(t− Tz)

]

+DGCw,0(t)w(t) +DGCw,T (t)w(t− Tz)

+DGCu,T (t) u(t− Tz) (3.67)

where

AGC,0(t) =

[

A+BuSGC(t)KC,0(t)Cy BuSGC(t)KC(t)

BC,0(t)Cy +BC,0(t)DyuSGC(t)KC,0(t)Cy AC +BC,0(t)DyuSGC(t)KC(t)

]

,


AGC,T (t) =

[

Bu SGC(t)KC,T (t)Cy 0

BC,T (t)Cy +BC,0(t)Dyu SGC(t)KC,T (t)Cy 0

]

,

BGCw,0(t) =

[

Bw +Bu SGC(t)KC,0(t)Dyw

BC,0(t)Dyw +BC,0(t)Dyu SGC(t)KC,0(t)Dyw

]

,

BGCw,T (t) =

[

Bu SGC(t)KC,T (t)Dyw

BC,T (t)Dyw +BC,0(t)Dyu SGC(t)KC,T (t)Dyw

]

,

BGCu,T (t) =

[

Bu SGC(t)KC,T (t)Dyu

BC,T (t)Dyu +BC,0(t)Dyu SGC(t)KC,T (t)Dyu

]

,

CGC,0(t) =[

Cz +Dzu SGC(t)KC,0(t)Cy Dzu SGC(t)KC(t)]

,

CGC,T (t) =[

Dzu SGC(t)KC,T (t)Cy 0

]

,

DGCw,0(t) = Dzw +Dzu SGC(t)KC,0(t)Dyw

DGCw,T (t) = Dzu SGC(t)KC,T (t)Dyw, DGCu,T (t) = Dzu SGC(t)KC,T (t)Dyu (3.68)

with

SGC(t) =(

I−KC,0(t)Dyu

)−1. (3.69)

In order to solve these equations, the following initial conditions must be defined.

x(t) = ξx(t), xC(t) = ξxC(t), w(t) = ξw(t) for − Tz + t0 ≤ t ≤ t0 (3.70)

and

u(t) = ξu(t) for − Tz + t0 ≤ t < t0. (3.71)

3.1.6 Global System

The global system is represented in figure 3.1 and is described by the following set of (n+nC)

differential equations.

xg(t) = Ag,0(t) xg(t) +Ag,T (t) xg(t− Tz)

+Bgy,T (t) y(t− Tz) +Bgy,2T (t) y(t− 2Tz)

+Bgw,T (t)w(t− Tz) +Bgw(t) fP (t) (3.72)

3.1 Modeling 61

with

xg(t) =

[

x(t)

xC(t)

]

,

Ag,0(t) =

[

A0 +Bu,0 KC,0 Sg C0 Bu,0 (I+KC,0 Sg Du,0)KC

BC,0 Sg C0 AC +BC,0 Sg Du,0 KC

]

,

Ag,T (t) =

[

AT +Bu,0KC,0SgCT Bu,TKC(t− Tz) +Bu,0KC,0SgDu,TKC(t− Tz)

BC,0SgCT BC,0SgDu,TKC(t− Tz)

]

,

Bgy,T (t) =

[

Bu,0 (KC,T +KC,0Sg (Du,0KC,T +Du,TKC,0(t− Tz))) +Bu,TKC,0(t− Tz)

BC,T +BC,0Sg (Du,0KC,T +Du,TKC,0(t− Tz))

]

,

Bgy,2T (t) =

[

Bu,0 KC,0 Sg Du,T KC,T (t− Tz) +Bu,T KC,T (t− Tz)

BC,0 Sg Du,T KC,T (t− Tz)

]

,

Bgw,T (t) =

[

Bw,T +Bu,0 KC,0 Sg Dw,T

BC,0 Sg Dw,T

]

,

Bgw(t) =

[

Bw +Bu,0 KC,0 Sg Dw

BC,0 Sg Dw

]

, (3.73)

with

Sg(t) =(

I−Du,0 KC,0

)−1(3.74)

and where

A0 = AGP,0(t), AT = AGP,T (t), Bu,0 = BGPu,0(t), Bu,T = BGPu,T (t),

C0 = CGP,0(t), CT = CGP,T (t), Du,0 = DGPu,0(t), Du,T = DGPu,T (t),

Bw,T = BGPw,T (t), Dw,T = DGPw,T (t), Bw = BGPw(t), Dw = DGPw(t). (3.75)

The following initial conditions must be defined

x(t) = ξx(t), xC(t) = ξxC(t) for − Tz + t0 ≤ t ≤ t0 (3.76)

and

y(t) = ξy(t) for − 2Tz + t0 ≤ t < t0, w(t) = ξw(t) for − Tz + t0 ≤ t < t0. (3.77)

In the above relations, the explicit dependency of time has been sometimes omitted for conve-

nience.


3.2 Approximated Models for Control Design and Sta-

bility Analysis

3.2.1 Approximated Plant-Process System

In order to be able to apply standard tools for the control design, the coupled system of the

passive plant with the process, described by relations (3.60) and (3.61), must be approximated

by a LTI system. In a first step, the feedthrough terms of the plant, Dzw and Dzu, are set to

zero. This can be physically interpreted when looking at the second order model of the passive

structure as shown by (3.22). The resulting coupled system can be simplified to

x(t) =(

A−Bw KP (t)Cz

)

x(t) +Bw KP (t)Cz x(t− Tz)

+Bw fP (t) +Bu u(t) (3.78)

z(t) = Cz x(t) (3.79)

y(t) =(

Cy −Dyw KP (t)Cz

)

x(t) +Dyw KP (t)Cz x(t− Tz)

+Dyw fP (t) +Dyu u(t). (3.80)

The periodic time-varying coefficients of the differential equations can be approximated by their

average values. This approximation is especially acceptable for large radial immersions of the

cutter into workpiece material and tools with important number of teeth. The average value of

the periodic coefficients is given by

KP =1

T

∫ t0+T

t0

KP (t) dt (3.81)

where T = Tz if an ideal tool without runout is considered and T = Tn if a runout is modeled.

The delay terms must be approximated by rational functions using Padé approximation. Here,

the delay term is approximated by a proper system of ndth order. For

xT (t) ∼= x(t− T ), (3.82)

the Laplace transform gives

xT (s) ∼= e−sTx(s).

One defines the following LTI state-space system,

xd(t) = Ad xd(t) + bd x(t) (3.83)

xT (t) = cd xd(t) + dd x(t), (3.84)

such that

cd (sI−Ad)−1

bd + dd ∼= e−sT (3.85)

with xd ∈ Rnd .

3.2 Approximated Models for Control Design and Stability Analysis 63

Let xk(t) being the kth element of the state vector x(t) and xT,k(t), the approximation of the

kth element of the delayed state vector x(t − T ). If one considers a constant time delay and an

approximation based on the same order nd for all the elements of the state vector, the approximate

delayed state vector can be written

xT (t) = CD xD(t) +DD x(t) (3.86)

where

xT =

xT,1

xT,2...

xT,n

, xT ∈ Rn, xD =

xd,1

xd,2

...

xd,n

, xD ∈ Rndn, (3.87)

and

CD = diag (cd, cd, · · · , cd) , DD = diag (dd, dd, · · · , dd) . (3.88)

The dynamics of the state vector xD is thus given by

xD(t) = AD xD(t) +BD x(t) (3.89)

with

AD = diag (Ad,Ad, · · · ,Ad) , BD = diag (bd,bd, · · · ,bd) . (3.90)

The global state vector

x′ =

[

x

xD

]

(3.91)

can be defined, where x is an approximation of x.

By integrating (3.86) and (3.89) into the system described from the equations (3.78) to (3.80)

and by approximating the periodic coefficients by their average value, the following LTI

system is obtained.

x′(t) = A′GP x′(t) +B′

GPw fP (t) +B′GPu u(t) (3.92)

z(t) ∼= C′GPz x

′(t) (3.93)

y(t) ∼= C′GPy x

′(t) +Dyw fP (t) +Dyu u(t) (3.94)

with

A′GP =

[

A+Bw KP Cz (DD − I) Bw KP Cz CD

BD AD

]

,

B′GPw =

[

Bw

0

]

, B′GPu =

[

Bu

0

]

, C′GPz =

[

Cz 0

]

,


C′GPy =

[

Cy +Dyw KP Cz (DD − I) Dyw KP Cz CD

]

. (3.95)

The stability of the process-machine structure interaction can be studied by the eigenvalues of

the matrix A′GP and, as expected, the forcing term fP does not play any role on the process

stability.

It is to note that the size of the approximated system increases with the order nd of the Padé

approximation by a factor n and the size of the resulting system is equal to (nd + 1)n instead of n

for the original system. This size increase must be considered in the design step of the controller.

A smaller size system, equal to (nd/2 + 1)n, can be obtained using a model based on the second

order structural system and where the nodal displacement DOFs vector q(t−T ) is approximated

instead of the state vector x(t− T ).

3.2.2 Approximated Global System

A similar procedure can be applied to the overall system including the controller. To do so,

a LTI controller without delay term, of the form

xC(t) = AC xC(t) +BC,0 y(t) (3.96)

u(t) = KC xC(t) +KC,0 y(t), (3.97)

is considered. This controller, coupled to the system given by expressions (3.78) to (3.80),

leads to the following global system.

xg(t) = Ag,0(t) xg(t) +Ag,T (t) xg(t− Tz) +Bgw fP (t) (3.98)

with

Ag,0(t) =

[

A+BuKC,0Sg (Cy −DywKPCz)−BwKPCz Bu (I+KC,0SgDyu)KC

BC,0Sg (Cy −DywKPCz) AC +BC,0SgDyuKC

]

,

Ag,T (t) =

[

Bu KC,0 Sg Dyw KP Cz +Bw KP Cz 0

BC,0 Sg Dyw KP Cz 0

]

,

Bgw =

[

Bw +Bu KC,0 Sg Dyw

BC,0 Sg Dyw

]

(3.99)

and

Sg = (I−Dyu KC,0)−1 . (3.100)

3.2 Approximated Models for Control Design and Stability Analysis 65

After approximation of the periodic coefficients and the time delay terms, the following LTI

global system is derived.

x′g(t) = A′

g x′g(t) +B′

gw fP (t) (3.101)

where

x′g =

[

x′

xC

]

, x′g ∈ R

(nd+1)n+nC ,

A′g =

[

A′GP +B′

GPu (I−KC,0Dyu)−1

KC,0C′GPy B′

GPu (I−KC,0Dyu)−1

KC

BC,0 (I−DyuKC,0)−1

C′GPy AC +BC,0 (I−DyuKC,0)

−1DyuKC

]

,

B′gw =

[

B′GPw +B′

GPu (I−KC,0 Dyu)−1

KC,0 Dyw

BC,0 (I−Dyu KC,0)−1

Dyw

]

. (3.102)

This model can be used to study the stability of the overall system.

3.2.3 Full and Perturbation Dynamics

As observed in section 3.2.1, the stability of the milling process does not depend on the term fP .

For chatter control, a way to improve the control efficiency is to focus the controller action on

the terms involved in the process stability, which means induced by the regenerative effect, as

suggested by van Dijk et al. in [112].

The exogenous disturbances generated by the process w can be dissociated into one synchronous

part wf , generating forced vibrations, and one asynchronous part ws susceptible to lead to self-

excited vibrations and chatter instability, such that

w(t) = wf (t) +ws(t) (3.103)

where

wf (t) = fP (t), ws(t) = KP (t)(

z(t− Tz)− z(t))

. (3.104)

The self-excited part corresponds to the influence of the regenerative effect, so that, as previously

defined in (3.54), ws(t) = wR(t).

Considering the dynamics of the coupled system formed by the process and the plant structure

and using the superposition principle, the resulting state vector x(t) can be divided into two

corresponding parts, i.e. a forced vibration part xf and a self-excited vibration part xs such that

x(t) = xf (t) + xs(t). (3.105)


In steady state conditions and using the periodicity property of fP , for a tool without runout,

one can write

xf (t) = xf (t+ Tz). (3.106)

If no controller action is considered, i.e. u = 0, relations (3.78) and (3.79) lead to the definition

of the following two dissociated dynamic systems.

xf (t) = A xf (t) +Bw fP (t), (3.107)

xs(t) = A xs(t) +Bw KP (t)Cz

(

xs(t− Tz)− xs(t))

. (3.108)

The controller input can thus also be divided into two parts,

y(t) = α yf (t) + ys(t) (3.109)

where α is a scalar equal to 1 for a full feedback controller or equal to 0 for a perturbation feedback,

i.e. when the forced vibration part of the signal is not considered.

Two corresponding independent LTI controllers without delay term are considered, such that

u(t) = uf (t) + us(t) (3.110)

where

uf (t) = KC xCf (t) + αKC,0 yf (t), us(t) = KC xCs(t) +KC,0 ys(t) (3.111)

with

xCf (t) = AC xCf (t) + αBC,0 yf (t) (3.112)

and

xCs(t) = AC xCs(t) +BC,0 ys(t). (3.113)

The resulting controller inputs are

yf (t) = Cy xf (t) +Dyw fP (t) +Dyu uf (t), (3.114)

ys(t) = Cy xs(t) +Dyw KP (t)Cz

(

xs(t− Tz)− xs(t))

+Dyu us(t). (3.115)

Using these relations, the global dynamics is described by the superposition of the two following

independent dynamic systems, namely, the forced vibration dynamics and the self-excited vibra-

tion dynamics. The forced vibration dynamics is given by the following set of ordinary differential

equations.

xgf (t) = Agf xgf (t) +Bgwf fP (t) (3.116)

with

xgf (t) =

[

xf (t)

xCf (t)

]

,

3.3 Stability Analysis 67

Agf =

[

A+ αBu KC,0 Sgα Cy Bu (I+ αKC,0 Sgα Dyu)KC

αBC,0 Sgα Cy AC + αBC,0 Sgα Dyu KC

]

,

Bgwf =

[

Bw + αBu KC,0 Sgα Dyw

αBC,0 Sgα Dyw

]

(3.117)

where

Sgα = (I− αDyu KC,0)−1 . (3.118)

The self-excited vibration dynamic is given by

xgs(t) = Ag,0(t) xgs(t) +Ag,T (t) xgs(t− Tz) (3.119)

where

xgs(t) =

[

xs(t)

xCs(t)

]

(3.120)

and the matrices Ag,0, Ag,T are given by (3.99).

The process stability is fully described by (3.119). One notes that this system does not depend

on the forced vibration feedback, i.e. no α term is involved. This means that a perturbation

feedbacks is sufficient to stabilize the process. The difference between the full and perturbation

feedback lies in the forced vibration dynamics. The perturbation feedback does not modify the

open-loop dynamics of the forced vibration system where the full feedback, when α = 1, changes

its eigenvalues.

Considering a stable and stabilizing controller and a perturbation feedback (α = 0), xf and thus

yf as well as ys tend to zero when time goes towards infinity. This makes u also tending to zero.

This is not the case when α = 1, where u tends to uf , which is different from zero.

It can thus be concluded that using a perturbation feedback does not modify the influence of

a stabilizing controller on the process stability and allows to reduce the effort provided by the

actuating system.

3.3 Stability Analysis

The main issue for the simulation tools concerns the prediction of the process stability under the

influence of the selected controller. The objective is to determine if regenerative chatter occurs

or not for the given range of considered machining conditions and if the process can be stabilized

using the proposed control strategy. Some complementary aspects must also be investigated such

as, for instance, the amount of effort (force and power) required by the actuating system to

stabilize the process or the internal reaction forces induced by the process and the controller.

Even if the dynamics of the machine tool structure and the controller can be satisfactorily approx-

imated by linear systems, the cut interaction makes the overall system fundamentally nonlinear.


This is due to the regenerative phenomenon but also to the presence of some effects such as, for

example, the tool jumps out of the cut or the nonlinear dependency of the cutting forces with

the chip thickness. The time varying coefficients, induced by the rotational kinematics of the

cutting tool, make also the global system susceptible to present several types of bifurcation. In

nonlinear dynamic systems theory, a bifurcation is defined as a qualitative change of the phase

space, i.e. a change in the number of the limit sets, e.g. periodic solutions or equilibria, related

to the variation of a system parameter.

Four different stability analysis methods, namely: time-domain simulation (TDS), semi-discretization

(SD), zeroth order approximation (ZOA) and linear time-invariant approximation (LTI), are used

in this work. They differ by their computational costs and the level of approximation of the con-

sidered model.

3.3.1 Time-Domain Simulation

In this method, the time-domain is discretized in intervals to numerically integrate the differential

equations given by (3.72). TDS allows to consider more exact process models than the one

presented in section 3.1.2. Using topological analysis, the exact undeformed chip geometry can

be computed, considering the trochoidal tool path and the influence of the TCP deviations in

all three space dimensions coupled to the exact macroscopic tool geometry as well as the jumps

out of cut. It can take into account several other second order phenomena, like the nonlinear

dependency of the cutting forces with the chip thickness. Transient phenomena, such as the tool

engagement, may be considered and the resulting cutting forces and the surface finish are also

predictable.

The stability analysis using TDS method requires the definition of criteria for the detection of

chatter occurrence and a sufficient amount of simulation time is necessary to identify if the process

is stable or not. The stability criteria detect the presence of self-excited vibrations based on the

fact that they occur at frequencies distinct from the forced vibrations generated by the process

kinematics. The most straightforward technique consists in computing the Fourier transform

of the TCP deviations z and seeing if any relevant frequency components lies outside of the

tooth passing frequency harmonics, or in case of runout, outside of the spindle rotation frequency

harmonics. This technique implies stationary process conditions, which means that the simulation

must run until the transient effects disappear or reach stable limit set. Furthermore, in order to

get enough resolution in the frequency-domain, the time period in stationary conditions must be

sufficiently long. This leads to time-consuming simulation runs over long time periods. This is why

another criterion is preferred here based on the Poincaré map, representing the intersection of a

periodic orbit in the state-space of a continuous dynamic system with a certain lower dimensional

subspace, called the Poincaré section. The TCP deviations are resampled at the fundamental

frequency of the forced vibrations, i.e. either the tooth passing or the spindle rotation frequency.

If only forced vibrations occur, the obtained data remain at the same location in the state-space


along the time. If self-excited vibrations are superimposed, the points are distributed at different

locations. So, looking at the variability along the time of the resampled data allows to detect the

presence of chatter. As a reduced state-space is considered, the resampled TCP deviations can

be seen as the reduced Poincaré map. The following chatter detection condition can be defined.

Given a predefined threshold value σlim, it is admitted that chatter occurs if and only if

σP > σlim (3.121)

where

σP =

√

√

√

√

1

NT

NT∑

k=1

‖z(kT )− zkT ‖22 (3.122)

and

zkT =1

NT

NT∑

k=1

z(kT ). (3.123)

The number of considered tooth passing periods or spindle rotations is NT and ‖.‖2 represents

the Euclidean norm. The threshold value σlim must be calibrated. This criterion may also be

used for the chatter detection during experimental tests. In this case, as the TCP deviations can

usually not be measured, the criterion is not based on z but replaced by the information delivered

by a monitoring system.

A key feature of the chatter phenomenon is its chatter frequency. In milling, due to the pro-

cess kinematics, multiple self-excited vibration frequencies simultaneously coexist. The chatter

frequency is defined as the self-excited vibration component presenting the highest magnitude.

Two main types of bifurcation are susceptible to occur in milling operation around the peri-

odic solution, namely: the quasiperiodic or Hopf bifurcation and the period-doubling or flip

bifurcation.

In case of Hopf bifurcation, the reduced Poincaré map shows a distributed pattern. This

means that the chatter frequency does not correspond to a harmonic or a subharmonic of the

tooth passing frequency and lies somewhere in between two harmonics of forced vibrations.

In case of flip bifurcation, the chatter frequency of the self-excited vibrations exactly lies in

the middle of two harmonics of forced vibrations. In other words, the tool tip is making some

integer and a half periods of oscillation relatively to the workpiece between two subsequent

cuts. Thus, the evolution of the reduced Poincaré map presents two distinct levels where the

resampled data are aligned.

More information about these types of bifurcation can be found in [125].


3.3.2 Semi-Discretization Method

The semi-discretization method, developed by Insperger and Stépán [35], is based on the extended

Floquet theory.

Considering the solution of the motion equations composed by a periodic orbit and the move-

ment resulting from a perturbation around this orbit, the stability is given by the equations

governing the dynamics of the perturbation. Due to the transcendental nature of its charac-

teristic equation induced by the time delay, the system presents a infinite dimensional phase

space and the monodromy operator has no closed-form solution. The monodromy operator

must thus be approximated by a finite dimensional matrix, called Floquet transition matrix

or monodromy matrix.

The method consists in the construction of the finite dimensional approximate monodromy op-

erator to determine the stability around the periodic solution associated with the trivial fixed

point. To do so, the period of the time delay is discretized into equal intervals. The stability of

the resulting approximated system is given by the eigenvalues of its monodromy matrix, named

Floquet multipliers. If the largest Floquet multiplier, corresponding to the spectral radius of the

monodromy matrix, is strictly smaller than one, the process is locally asymptotically stable. If

greater than one, the location of this largest eigenvalue gives the type of bifurcation encountered

and its imaginary part, the corresponding chatter frequencies.

Hopf bifurcation is characterized by complex Floquet multipliers with magnitude greater than

one. Flip bifurcation corresponds to purely real Floquet multipliers with magnitude equal to −1.

As demonstrated in [126], another type of bifurcation can occur when a tool with an even number

of teeth and presenting runout is considered. This third bifurcation is called period one or fold

bifurcation and corresponds to a Floquet multiplier equal to 1.

The computation of the monodromy matrix using the SD method, from the dynamic system

expressed by (3.119), is detailed in [35]. The model (3.119) considers a LTI controller but can be

extended to the general form of the controller expressed by (3.56) and (3.57).

The monodromy matrix explicitly depends on the spindle speed and the depths of cut. This means

that, in order to build the SLD, the considered ranges for the spindle speed and the depths of

cut must be discretized and the monodromy matrix must be computed for each combination.

3.3.3 Zeroth Order Approximation Method

By approximating the time-periodic coefficients with zeroth order terms, such as proposed in

section 3.2, an autonomous system is derived. The stability of the resulting autonomous delayed

system can be studied in the frequency-domain. The following procedure is drawn from [12].


In order to derive an analytical form for the zeroth order approximation of KP , one considers a

zero helix angle and a straight tool, which means κ equal to 90 deg. In these conditions, only the

perturbation dynamics in the radial plane needs to be considered for the stability analysis. This

allows to write the perturbation force wR,r in the radial plane as

wR,r(t) = KP,r(t)(

zr(t− Tz)− zr(t))

(3.124)

where

zr =

[

zX

zY

]

, KP,r(t) = −1

2ap Kt Aφ(t) (3.125)

with

Aφ(t) =

[

axx(t) axy(t)

ayx(t) ayy(t)

]

(3.126)

and

axx(t) =

Nt∑

j=1

−(

sin 2φj +Kr

Kt

(

1− cos 2φj)

)

gj ,

axy(t) =

Nt∑

j=1

−(

(

1 + cos 2φj)

+Kr

Ktsin 2φj

)

gj ,

ayx(t) =

Nt∑

j=1

(

(

1− cos 2φj)

−Kr

Ktsin 2φj

)

gj ,

ayy(t) =

Nt∑

j=1

(

sin 2φj −Kr

Kt

(

1 + cos 2φj)

)

gj . (3.127)

For convenience, the subscript (.),r referring to the radial plane is omitted in the rest of this

section.

Considering a tool without runout, the zeroth order approximation of KP gives

KP =1

Tz

∫ t0+Tz

t0

KP (t) dt

= −1

2ap Kt A0 (3.128)

where

A0 =Nt

2π

[

αxx αxy

αyx αyy

]

(3.129)


with

αxx =1

2

(

cos 2φ− 2Kr

Ktφ+

Kr

Ktsin 2φ

)

∣

∣

∣

∣

φout

φin

,

αxy =1

2

(

− sin 2φ− 2φ+Kr

Ktcos 2φ

)

∣

∣

∣

∣

φout

φin

,

αyx =1

2

(

− sin 2φ+ 2φ+Kr

Ktcos 2φ

)

∣

∣

∣

∣

φout

φin

,

αyy =1

2

(

− cos 2φ− 2Kr

Ktφ−

Kr

Ktsin 2φ

)

∣

∣

∣

∣

φout

φin

. (3.130)

Considering a LTI controller, the transfer function of the radial TCP dynamics in closed-loop is

given by

z(s) = Fzw(s)w(s) (3.131)

where

Fzw =

[

Fzw,XX Fzw,XY

Fzw,Y X Fzw,Y Y

]

. (3.132)

The closed-loop transfer function Fzw is obtained from the open-loop transfer functions by

Fzw(s) = Gzw(s) +Gzu(s)C(s) S(s)Gyw(s) (3.133)

and

S =(

I−Gyu(s)C(s))−1

. (3.134)

Coupling the TCP dynamics with the approximated cutting forces using zeroth order approxi-

mation and expressing the result in the Fourier domain, i.e. s = iωc, one obtains

wR∼=(

e−iωcTz − 1)

KP,lim Fzw(ωc)wR (3.135)

where i2 = −1, Fzw(ωc) = Fzw(s)|s=iωcand KP,lim corresponds to the cutting stiffnesses at the

stability limit. It is to note that this relation uses the fact that the forced vibrations of the TCP

are not influencing the regenerative phenomenon, as previously mentioned in section 3.2.3.

The limit of stability is given by the eigenvalue problem,

λwR = A0 Fzw(ωc)wR. (3.136)

One defines a complex scalar Λ such that

Λ = ΛR + iΛI = −1/λ. (3.137)

From relation (3.135), one gets

Λ =1

2ap,lim Kt(e

−iωcTz − 1). (3.138)


The nontrivial solution of the eigenvalue problem gives the characteristic equation

det(

I+ ΛA0 Fzw(ωc))

= 0. (3.139)

The eigenvalues of the product A0 Fzw(ωc) leads to ΛR and ΛI .

It is thereby possible to derive the critical axial depth of cut, given by

ap,lim = −ΛR

Kt(1 + χ2) (3.140)

where

χ =ΛI

ΛR. (3.141)

As the characteristic equation is a second order algebraic equation, two values are obtained.

The stability limit is given by the smallest positive value.

Similarly as previously demonstrated for the case of turning in section 2.1, the corresponding

spindle speed is

n =60 ωc

Nt

(

2πq + π − 2ψ) (3.142)

where q ∈ N and ψ = tanχ.

The stability lobes can thus be drawn by computing the critical axial depth of cut and the spindle

speed for a given range of corresponding chatter frequencies ωc. The major advantage of this

method is its low computational cost as the limit of stability is explicitly computed by opposition

to the previously described time-domain methods, where the stability limit is iteratively searched.

Furthermore, no complete dynamic model of the machine structure is required but only the FRFs

at the tool tip. This means that also experimental FRFs can directly be used to compute the

SLD.

In spite of this, the different approximations implied by this method limit its application to

operations with simple tool geometries and large radial immersions of the cutter in the workpiece.

3.3.4 Linear Time-Invariant Approximation Method

By approximating the time-periodic coefficients with constants and the time delay terms using

Padé approximation, as proposed in 3.2, a standard LTI autonomous system is obtained.

The resulting global system is described by (3.101). Its poles, given by the eigenvalues of its

system matrix A′g expressed by (3.102), determine its stability. The global system is locally

asymptotically stable if and only if

Re(

λj(

A′g

) )

< 0, ∀j ∈ [1, . . . , (nd + 1)n+ nC ] (3.143)


where λj(.) corresponds to the jth eigenvalue.

As for time-domain methods, the construction of the SLD is made by discretizing and scanning

the spindle speed and axial depth of cut ranges.

For a sufficiently high order of the Padé approximation, the results provided by the LTI approxi-

mation method tend to those obtained using the ZOA method. The zeroth order approximation

of the time-periodic coefficients implies that this method is also limited to the study of machining

operations with important radial immersion ratio. However, by opposition to the ZOA method,

more complex tool geometries can be considered.

3.4 Validation

The different simulation tools described above must be validated before making any analysis.

First, the model of the mechatronic passive structure of the prototype spindle, presented in

section 3.1.1, is verified. To do so, a simple rotating FE model is used. The results obtained with

the proposed model are compared with the results provided by the commercial software ANSYS.

More specifically, the Campbell diagrams — representing the natural frequencies evolution of the

backward and forward whirl modes with the rotational speed — derived with both methods are

compared.

The milling process model, described in section 3.1.2, is verified using cutting forces measured

during machining tests realized in stable conditions.

Finally, the stability analysis methods are validated by comparing the obtained stability lobes

diagrams with those provided from the experimental results presented by Bayly et al. in [127].

75

Chapter 4

Active Structural Control for Process

Stability Improvement

In this chapter, the development of the control concept used to mitigate the occurrence of milling

process instability is described. The concept is based on the collaboration between an active

system integrated into the machine tool spindle and a strategy implemented in the controller.

First, the selected control strategies are presented. Finally, the design of the active system is

discussed.

4.1 Control Strategies

As previously seen, different approaches have been developed to minimize the apparition of chatter

with the help of an active structural control system. Several classifications exist to group all the

possible strategies. One considers here a classification based on the formulation of the control

objective. Two main categories can be defined, namely: disturbance rejection and stabilization

schemes.

4.1.1 Disturbance Rejection Scheme

Figure 4.1 represents the principle of disturbance rejection. In this case, the milling process model

is not explicitly taken into account and the disturbance coming from the process is considered

generically. The global system corresponds to the flowchart of figure 3.6.

Based on the defined specifications of the process disturbance w, the controller tries to reduce

the influence of w on the TCP deviations z, which corresponds to a minimization of the tool

tip receptance function or an increasing of the damping of the dominant resonance frequencies.

76 4. Active Structural Control for Process Stability Improvement

Through the reduction of the TCP deviations, the variation of the chip thickness resulting

from the regenerative effect is also reduced. As explained in section 2.1, this produces a direct

positive influence on the resulting process stability. More specifically, it tends to increase the

unconditionally stable DOC.

The disturbance rejection schemes have the advantage of being independent of the process pa-

rameters and are thus designed for a wide range of machining conditions. However, in some

conditions, the controller may be suboptimal or even worse, can destabilize the process. This is

why such approach is recommended to be used at low spindle speeds, where the stability limit is

almost independent of the spindle speed and the stability limit approximately corresponds to the

unconditionally stable depth of cut.

Finishing surface

from previous cut

with AVC

Finishing surface

from current cut

with AVC

Finishing surface

from current cut

without AVC

Cutter

Workpiece

Nominal

chip thickness

Effective

chip thickness

Finishing surface

from previous cut

without AVC

vc

Figure 4.1: Disturbance rejection principle. AVC: active vibration control; vc: cutting speed.

4.1.2 Stabilization Scheme

Stabilization schemes explicitly consider the interaction between the milling process and the plant

and thus try to generate a controller such that, for some given machining conditions, the process

stability is guaranteed. The considered global system corresponds to the flowchart in figure 3.1.

The principle of the stabilization schemes is described in figure 4.2.

The regenerative effect corresponds to an interaction between two periodic phenomena,

namely the structural vibrations between the tool and the workpiece, and the spindle ro-

tation. The relation between both determines the phase between the current oscillation of

the cutter in workpiece material and the undulation left by the previous cut. This phase

is the key parameter on which the controller tries to act. More specifically, the controller

influences the plant dynamics such that this phase-shift becomes equal to zero.

4.2 Observability and Controllability Aspects 77

Finishing surface

from previous cutFinishing surface

from current cut

with AVC

Finishing surface

from current cut

without AVC

Cutter

Workpiece

Nominal

chip thickness

Effective

chip thickness

vc

Figure 4.2: Stabilization principle.

In this work, feedback optimal control strategies are studied. The control objective is formally

defined and model-based procedures are used to synthesize a controller reaching optimal closed-

loop conditions.

First, some aspects related to the observability and controllability characteristics of the plant are

addressed. The control objectives for disturbance rejection and stabilization schemes are then

formulated. Finally, the design choices, which determined the construction of the active system,

are described.

4.2 Observability and Controllability Aspects

The control efficiency, intuitively defined by the ratio of the performance over the control effort,

depends on the placement of the sensors and actuators. More specifically, the location of the

sensing and actuating points relatively to the modal shape of the natural modes of interest

determines the ability of the active system to influence them. This can be quantitatively evaluated

using observability and controllability concepts which are intrinsic properties of the plant. They

are thus independent of the control strategy and may help the positioning of the actuators and

sensors in the design step of the active system.

The observability and controllability concepts are usually defined as binary criteria using the

observability and controllability matrices, but in the control of structures with multiple DOFs,

it is more interesting to define measures able to quantify the level of observability, respectively

controllability, for a given location of the sensing and actuating points regarding the mode shapes.

In control theory, the notions of state and pole are used. Pole refers to the frequency-domain

as state refers to the time-domain. To one structural natural mode corresponds two complex

conjugate poles and two corresponding states in the state-space. Considering the first order

system defined by (3.25) and (3.26), the poles are the solutions of the characteristic equation of


the system

det (sI−A) = 0 (4.1)

and are therefore common to all transfer functions between all inputs and outputs. As the system

corresponds to an underdamped second order structural system, the poles are complex conjugate

pairs. Each pair pr, p∗r corresponds to the rth eigenmode characterized by a natural circular

frequency ωn,r and modal damping ratios ζr according to the following relations.

pr = σr + i ωd,r, σr = −ζr ωn,r, ωd,r = ωn,r

√

1− ζ2r (4.2)

where i2 = −1 and r = 1, . . . , n/2. ωd,r corresponds to the damped circular frequency. Figure

4.3 shows the geometrical relation between the pole location and the modal parameters in the

complex plane where cosφr = ζr.

Im

Re

Figure 4.3: Geometrical relation between pole location and modal parameters.

In order to verify that all the important states, and thus the corresponding eigenmodes, are

observable and controllable, the output, respectively, the input pole vectors can be used. The

pole vectors have the advantage over the observability and controllability matrices to provide

information related to a specific pole of the system and are susceptible to quantify the relative

degree of observability and controllability.

The output pole entry yp,jr of the rth pole from the jth output is defined, in [128], as

yp,jr := cj tr (4.3)

where cj is the jth row of the C output matrix and tr, the right eigenvector of the A matrix

associated with the rth pole. The rth pole is observable if and only if all the entries of the output

pole vector related to the output matrix Cy are non-zero for all right eigenvectors tr. The system

is state observable if and only if all its poles are state observable.

Similarly, the input pole entry up,kr of the rth pole from the kth input is defined as

up,kr := bHk qr (4.4)

where bk is the kth column of the B input matrix, the operator (.)H corresponds to the conjugate

transpose and qr is the left eigenvector of the A matrix associated with the rth pole. The

4.2 Observability and Controllability Aspects 79

corresponding pole is said to be state controllable if and only if all the entries of the input pole

vector related to the input matrix Bu are non-zero for all left eigenvectors qr. The system is

state controllable if and only if all its poles are state controllable.

For an underdamped second order structural system, the right eigenvectors represent the modal

shapes. In the general case, where the left and right eigenvectors are distinct, for instance due to

gyroscopic effects, the left eigenvectors are called the normal excitation shapes, as suggested in

[13]. Intuitively, the output pole vector can be interpreted as the image of the rth mode shape

seen by the jth output and the input pole vector, as the influence induced by the kth input on

the mode shape. The combination of both input-output pole vectors can be interpreted as the

generalization of the notion of residue, sometimes also called modal constant in modal analysis.

This is more explicit by looking at the dyadic expansion of the transfer function of the system.

The transfer function (3.31) can be written as

G(s) = C (sI−A)−1B+D

=

n/2∑

r=1

(

yp,r uHp,r

s− pr+

y∗p,r u

Tp,r

s− p∗r

)

+D (4.5)

where (.)∗ is the complex conjugate and (.)T, the transpose. It is important to note that the right

and left eigenvectors are normalized such that

QH T = I (4.6)

where T and Q are the matrices containing the right, respectively, the left eigenvectors. In other

words, T and Q are orthonormal to each other and this can be interpreted as the generalization

of the mass normalization used in modal analysis.

It is now interesting to quantify the degree of observability and controllability of each eigen-

mode. The input-output pole vectors can be used to do so. Their product gives an image of the

magnitude of the frequency response function around the natural frequency of a specific mode,

as

lims→iωn,r

Gjk(s) ∼=yp,jr u

∗p,kr

ζr ωn,r. (4.7)

A representative measure of the observability of the rth eigenmode can thus be defined as

ym,r :=yp,r

√

ζr ωn,r

. (4.8)

This value is called here output modal vector. Similarly, an input modal vector is defined as

um,r :=up,r

√

ζr ωn,r

. (4.9)

Thereby, their product

Rr = ym,r uHm,r


gives an image of the resonance amplitudes relative to the rth mode between all inputs and outputs

of the system. Rr is named here modal gain. The modal gain is, up to a factor, equivalent to the

Hankel singular values derived from the observability-controllability Gramians.

4.3 Optimal Control

In optimal control, the structure of the controller is a priori known and a cost functional formally

defines the control objective. A model-based procedure attempts to synthesize a the controller

minimizing this cost functional. The achievement of this minimization automatically guarantees

the stability of the closed-loop system. This can be used to stabilize the cutting process by

explicitly integrating a model of the process into the control design.

Different structures for the controller and different cost functionals may be used. Here, linear

time-invariant controllers with finite dimension, as defined by (3.96) and (3.97), and two types of

cost functionals are considered. The state-space realization over xC ∈ RnC of the LTI controller

is given by

Cs=

[

AC BC,0

KC KC,0

]

. (4.10)

The developments described in this section are mainly based on the theoretical aspects presented

in the book of Skogestad and Postlethwaite [128].

4.3.1 System Norms Definition

A general class of optimal controllers based on frequency-domain cost functionals became very

popular in the last decades, where the synthesis tends to minimize a certain norm of the closed-

loop transfer function of a generalized plant. Mainly two norms have been investigated, namely

the H2 and the H∞ norms. The H stands for the Hardy space of transfer functions with the

corresponding bounded norm. In the case of H2, it corresponds to the set of stable and strictly

proper transfer functions and H∞ corresponds to the set of stable and only proper transfer

functions.

Given a matrix strictly proper transfer function F between an input w and an output z, with

F(sj) ∈ Cm×n for a specific sj ∈ C, the H2 norm is defined as

‖F(s)‖2 :=

√

1

2π

∫ ∞

−∞

tr(

F(ω)H F(ω))

dω (4.11)

where F(ω) = F(s)|s=iω and tr(.) is the matrix trace. Using the relation between the Frobenius

4.3 Optimal Control 81

norm and the singular values, one gets

‖F(s)‖2 =

√

√

√

√

√

1

2π

∫ ∞

−∞

min(m,n)∑

j

σ2j(

F(ω))

dω (4.12)

where σj(

F(ω))

is the jth singular value of F(ω). One notes that if the transfer function is not

strictly proper, the H2 norm is infinite. From Parseval’s theorem, it is possible to express the

previous relation in time-domain and to obtain

‖F(s)‖2 =

√

√

√

√

m∑

j=1

n∑

k=1

∫ ∞

0|Fjk(τ)|

2 dτ (4.13)

where Fjk(t) is the jkth element of the impulse response of F(s). The minimization of the H2

norm is equivalent to minimize the square root of the energy of the system response signal z to

an impulse excitation w or the root mean square of z in response to a white noise excitation.

The H∞ norm is defined as

‖F(s)‖∞ := supω

σ(

F(ω))

(4.14)

where σ(

F(ω))

is the maximum singular value of F(ω). The H∞ norm represents the peak value

of the transfer function F or the highest possible gain between the input and output energies.

Thus, the system only need to be proper to guaranty a finite value.

4.3.2 Generalized Plant Definition

Given a generalized plant GP represented in figure 4.4, the H2 and H∞ controllers attempt

to minimize the H2, respectively the H∞ norm of the closed-loop transfer function FP given

by

FP (s) = GPzw(s) +GP

zu(s)C(s) SP (s)GPyw(s) (4.15)

where SP (s) =(

I−GPyu(s)C(s)

)−1.

The generalized output vector zP is extended by integrating all variables to be minimized and

similarly, the extended exogenous input vector wP integrates all input variables susceptible to

disturb the system. Moreover, the generalized plant may integrate some frequency dependent

weighting functions in order to adjust the performance and stability robustness of the resulting

closed-loop system.


C

y

wP

u

zP

GP

FP

Figure 4.4: Closed-loop system with generalized plant.

The generalized plant is therefore defined as

GP (s) =

[

GPzw(s) GP

zu(s)

GPyw(s) GP

yu(s)

]

, (4.16)

so that[

zP (s)

y(s)

]

=

[

GPzw(s) GP

zu(s)

GPyw(s) GP

yu(s)

][

wP (s)

u(s)

]

. (4.17)

The state-space realization of GP over xP ∈ RnP is

GP s=

AP BPw BP

u

CPz DP

zw DPzu

CPy DP

yw DPyu

. (4.18)

4.3.3 Control Objectives

The great interest of such controller is that they synthesize an optimal or suboptimal controller

Copt which minimizes a given norm of the transfer function FP and implicitly guarantees the

stability of the resulting closed-loop system.

The H2 control objective is to find the optimal stabilizing controller minimizing

∥

∥FP (s)∥

∥

2.

Similarly, given a scalar γ > γmin, the H∞ control objective is to find all the suboptimal

stabilizing controllers leading to∥

∥FP (s)∥

∥

∞< γ

where γmin corresponds to the minimum value of∥

∥FP (s)∥

∥

∞over all stabilizing controllers.


4.3.4 Optimal Controller Resolution

According to the separation principle, the resulting controller can be seen as a state estimator E

coupled to a state feedback regulator K, as represented in figure 4.5.

EK

C

yu

wP zP

GP

Figure 4.5: Feedback control using state estimator.

A state observer is required to estimate the current states of the system from the available values

of y. The full-state observer delivers an estimate of the state vector xP from the differential

equations

˙xP (t) = AP xP (t) +BPu u(t) + L

(

y(t)− y(t))

(4.19)

y(t) = CPy xP (t) +DP

yu u(t), (4.20)

trying to capture the dynamics of the generalized plant GP . The matrix L is derived by solving

the filter algebraic Riccati equation (FARE). The state regulator is given by

u(t) = −Kc xP (t). (4.21)

The matrix Kc is obtained by solving the continuous-time algebraic Riccati equation (CARE).

According to (4.10), this leads to the following state-space matrices

xC = xP , AC = AP −BPu Kc − L

(

CPy −DP

yu Kc

)

, BC,0 = L,

KC = −Kc, KC,0 = 0.

One notes that the size nC of the synthesized controller is equal to the state dimensions nP of

the generalized plant GP . If the problem is well-posed, the solution of the H2 optimal problem is

unique and is obtained by solving the two aforementioned algebraic Riccati equations. In the case

of the H∞ suboptimal problem, a bisection method on γ iteratively solves the Riccati equations

to approach γmin within a certain tolerance. Several conditions must be verified before starting

the synthesis procedure in order to guarantee that the problem is well-posed. These conditions are:

-(

AP ,BPu ,C

Py

)

is stabilizable and detectable (C1). This implies that a stabilizing controller

exists.


- DPzu has full column rank and DP

yw has full row rank (C2). These conditions mean that

the controller is proper and thus realizable.

-

[

AP − iωI BPu

CPz DP

zu

]

has full column rank for all ω and

[

AP − iωI BPw

CPy DP

yw

]

has full row

rank for all ω (C3). These conditions are required to prevent the cancellation of poles or

zeros on the imaginary axis leading to closed-loop instability.

- DPzw = 0 (C4). This is required by the H2 problem where a finite value of the norm only

exists for strictly proper transfer function FP . However, this condition is not required for

H∞ problems.

The H2 and H∞ optimal controllers are synthesized using the Matlab functions h2syn and

hinfsyn respectively, available in the Robust Control toolbox.

More information can be found in [129, 130, 128].

4.3.5 Disturbance Rejection Scheme

Figure 4.6 represents the chosen structure of the generalized plant for the application of distur-

bance rejection scheme. The output vector is extended by considering also the variable u in

order to be able to influence the control effort. The resulting both output variables z and u are

weighted using the weighting functions Wz and Wu, respectively. Also, the exogenous input

variables to consider are extended with a new input variable v representing the measurement

noise influencing the control input y, so that

y(t) = Cy x(t) +Dyw w(t) +Dyu u(t) + v(t). (4.22)

Two weighting functions, Ww and Wv, are used to influence the extended exogenous input

variables w and v, respectively. As represented in figure 4.6, the resulting extended input and

output vectors of the generalized plant are

wP =

[

w1

w2

]

, zP =

[

z1

z2

]

. (4.23)

Considering proper transfer functions for the weighting function, their corresponding state-space

realizations over, respectively, xw1 ∈ Rnw1 , xw2 ∈ R

nw2 , xz1 ∈ Rnz1 and xz2 ∈ R

nz2 , are

Wws=

[

Aw1 Bw1

Cw1 Dw1

]

, Wvs=

[

Aw2 Bw2

Cw2 Dw2

]

,

Wzs=

[

Az1 Bz1

Cz1 Dz1

]

, Wus=

[

Az2 Bz2

Cz2 Dz2

]

. (4.24)


C

Gzw

Gyw

Gzu

yu

+

+

+

+

GP

Wzz1

Wuz2

Gyu

w1

+

+

w2

Ww

Wv

G

w

FP

v

z

Figure 4.6: Signal-based design of norm controller for disturbance rejection scheme.

Considering the state-space realization of the plant G over x ∈ Rn,

Gs=

A Bw Bu

Cz Dzw Dzu

Cy Dyw Dyu

, (4.25)

and the previous state-space realizations for the weighting functions, the state-space matrices of

the generalized plant GP become

AP =

A Bw Cw1 0 0 0

0 Aw1 0 0 0

0 0 Aw2 0 0

Bz1 Cz Bz1 Dzw Cw1 0 Az1 0

0 0 0 0 Az2

,

BPw =

Bw Dw1 0

Bw1 0

0 Bw2

Bz1 Dzw Dw1 0

0 0

, BPu =

Bu

0

0

Dzu

Bz2

,

CPz =

[

Dz1Cz Dz1DzwCw1 0 Cz1 0

0 0 0 0 Cz2

]

, CPy =

[

Cy DywCw1 Cw2 0 0

]

,

DPzw =

[

Dz1 Dzw Dw1 0

0 0

]

, DPzu =

[

Dz1 Dzu

Dz2

]

,


DPyw =

[

Dyw Dw1 Dw2

]

, DPyu = Dyu. (4.26)

The state vector of the resulting generalized plant is equal to

xP =

x

xw1

xw2

xz1

xz2

(4.27)

and its size nP , equal to

nP = n+ nw1 + nw2 + nz1 + nz2. (4.28)

The corresponding optimal controller is then obtained using the aforementioned Matlab func-

tions with the system matrices of the so defined generalized plant.

4.3.6 Stabilization Scheme

In order to be able to apply optimal control techniques susceptible to stabilize the plant, a

representative LTI model of the coupled system formed by the plant and the process is used. The

approximated model described by equations (3.92), (3.93) and (3.94) is used here.

In this case, one only cares about the process stability. As demonstrated in section 3.2.3, the

forcing term fP does not play any role on the stability of the process so it can be replaced by

another more general exogenous variable w′ that is used to formulate a control objective focused

on the TCP dynamics. This resulting system is represented by figure 4.7. It corresponds to

the case where an exogenous input w′ is added to the cutting forces wR resulting from the

regenerative effect and where w′ represents the deterministic forcing term fP and some eventual

stochastic additional loads occurring at the TCP. This system is represented by the following

state-space realization over x′ ∈ R(nd+1)n, where nd is the order of the Padé approximation,

GGRs=

A′GP B′

GPw B′GPu

C′GPz 0 0

C′GPy Dyw Dyu

(4.29)

and where the state-space matrices are given in section 3.2.1.

Similarly as for the disturbance rejection case, one builds a generalized plant by extending the

exogenous input and output vectors and adding weighting functions as represented in figure 4.8.

The corresponding state-space realization over xP ∈ RnP is given by

GP s=

AP BPw BP

u

CPz DP

zw DPzu

CPy DP

yw DPyu

(4.30)


C

Gzw

Gyw

Gzu

yu

+

+

+

+Gyu

G

w z

R

+

+

GGR

'

Figure 4.7: Considered system for stabilization schemes.

where

xP =

x′

xw1

xw2

xz1

xz2

, AP =

A′GP B′

GPw Cw1 0 0 0

0 Aw1 0 0 0

0 0 Aw2 0 0

Bz1 C′GPz 0 0 Az1 0

0 0 0 0 Az2

,

BPw =

B′GPw Dw1 0

Bw1 0

0 Bw2

0 0

0 0

, BPu =

B′GPu

0

0

0

Bz2

,

CPz =

[

Dz1 C′GPz 0 0 Cz1 0

0 0 0 0 Cz2

]

, DPzw =

[

0 0

0 0

]

, DPzu =

[

0

Dz2

]

,

CPy =

[

C′GPy DywCw1 Cw2 0 0

]

, DPyw =

[

DywDw1 Dw2

]

, DPyu = Dyu. (4.31)

The dimension nP of the resulting generalized plant is equal to

nP = (nd + 1)n+ nw1 + nw2 + nz1 + nz2. (4.32)


C

Gzw

Gyw

Gzu

yu

+

+

+

+

G

Wzz1

Wuz2

Gyu

w1

+

+

w2

Ww

Wv

G

w

FP

v

z+

+

G

'

Figure 4.8: Signal-based design of norm controller for stabilization scheme.

4.3.7 Computation of Control Effort

The control effort required by the actuating system can be derived in time-domain by computing

u in stable conditions with fP as exogenous input. Considering a stable cut and assuming a

finite-dimension controller, defined by (3.96) and (3.97), the resulting global system is given by

[

x(t)

xC(t)

]

= AGC,0

[

x(t)

xC(t)

]

+BGCw,0 fP (t) (4.33)

u(t) = CCG,u

[

x(t)

xC(t)

]

+DCG,u fP (t) (4.34)

where

AGC,0 =

[

A+Bu SGC KC,0 Cy Bu SGC KC

BC,0 Cy +BC,0 Dyu SGC KC,0 Cy AC +BC,0 Dyu SGC KC

]

,

BGCw,0 =

[

Bw +Bu SGC KC,0 Dyw

BC,0 Dyw +BC,0 Dyu SGC KC,0 Dyw

]

,

CGC,u =[

SGC KC,0 Cy SGC KC

]

,

DGC,u = SGC KC,0 Dyw, SGC =(

I−KC,0 Dyu

)−1. (4.35)


In frequency-domain, the control effort can be evaluated via the transfer function Fuw given by

Fuw(s) = C(s)(

I−Gyu(s)C(s))−1

Gyw(s). (4.36)

The H2 or H∞ norm of this transfer function can be used as a measure.

4.3.8 Choice of Weighting Functions

The choice of the weighting functions is determinant for the performance of the resulting con-

troller. As the synthesized controller has the dimension of the generalized plant GP , i.e. the

order of the considered plant model plus the order of the weighting functions, these latter ones

must be chosen as simple as possible. In practice, the controller design starts generally using

constant weights. If the resulting performances do not fulfill the requirements, some frequency

dependent penalties may be applied.

By choosing constant values for the weighting functions, so that

Ww = Dw1, Wv = Dw2, Wz = Dz1, Wu = Dz2, (4.37)

the H2 problem becomes equivalent to a LQG control problem, corresponding to the association

of a linear-quadratic regulator (LQR) coupled to a Kalman filter. In this case, the following

stochastic system is considered.

x(t) = A x(t) +Bu u(t) +Bw w(t) (4.38)

y(t) = Cy x(t) +Dyu u(t) +Dyw w(t) + v(t) (4.39)

where w and v are the process, respectively, the measurement noises. As presented in [131], the

Kalman estimator assumes that they are uncorrelated zero-mean Gaussian stochastic processes

with constant power spectral densities given by the matrices Qe, Re and Ne, so that

E

[

w(t)

v(t)

]

[

w(τ)T v(τ)T]

=

[

Qe Ne

NTe Re

]

δ (t− τ) (4.40)

where E . is the expectation operator and δ(.) is the delta function defined as

δ(t) =

1 if t = 0,

0 otherwise.(4.41)

The minimization of the steady state error covariance leads to a problem dual to the LQR.

In our case, a priori no correlation exists between process and measurement noises, so that Ne =

0. The stochastic nature of the measurement noise is assumed to be true and Re is a diagonal

matrix with entries equal to the square of the variance σv of the sensor signal noise. However, for

some given machining parameters, the nature of the process noise is more deterministic than


stochastic. In spite of this, considering the machining process disturbances for a variety of

machining parameters, one can assume a quasi-stochastic phenomenon having a spectral density

Qe. In this sense, the Kalman filter will only provide a suboptimal observer.

The LQG control problem is to find the optimal control u that minimizes the cost functional

J = E

limT→∞

1

T

∫ T

0zP (t)T zP (t) dt

(4.42)

where

zP (t) =

[

Dz1 0

0 Dz2

][

z(t)

u(t)

]

(4.43)

with

z(t) = Cz x(t) +Dzu u(t). (4.44)

It is to note, that the transfer function between w and z is considered as strictly proper. After

replacing zP by the expression (4.43), the cost functional can be rewritten as

J = E

limT→∞

1

T

∫ T

0

(

z(t)T Qz z(t) + u(t)T Rz u(t))

dt

(4.45)

or as

J = E

limT→∞

1

T

∫ T

0

(

x(t)T Q x(t) + 2 x(t)T Nu(t) + u(t)T Ru(t))

dt

where

Qz = DTz1 Dz1, Rz = DT

z2 Dz2

and

Q = CTz Qz Cz, N = CT

z Qz Dzu, R = Rz +DTzu Qz Dzu. (4.46)

Considering an exogenous process noise wP with white noise characteristics and unit intensity,

defined as

wP (t) =

[

Dw1 0

0 Dw2

][

w(t)

v(t)

]

, (4.47)

the cost functional J is equivalent to the square of the H2 norm of the transfer function FP

between wP and zP ,

J =∥

∥FP (s)∥

∥

2

2. (4.48)

The LQG problem then becomes equivalent to the H2 optimal problem with the generalized plant

GP presenting the following state-space realization

GP s=

A BwDw1 0 Bu

Dz1Cz 0 0 Dz1Dzu

0 0 0 Dz2

Cy DywDw1 Dw2 Dyu

. (4.49)

4.4 Design of Active System 91

The existence and uniqueness of the solution is guaranteed if (A,Bu) is a stabilizable pair and(

Q−NR−1NT,A−BuR−1NT

)

has no unobservable mode on the imaginary axis.

R must be a strictly positive definite weighting matrix and Q − NR−1NT must be a positive

semidefinite weighting matrix. However, this choice for N leads to a negative definite matrix

Q−NR−1NT. By setting Dzu to zero, N becomes equal to zero. The resulting influence of this

consideration is a reduction of the stability gain margin.

Assuming diagonal covariance and weighting matrices, this allows to draw the following equiva-

lencies for the static weighting functions,

Dw1 = Q1/2e , Dw2 = R1/2

e Dz1 = Q1/2z , Dz2 = R1/2

z . (4.50)

Still, it is to note that the LQR controller loses its guaranty for stability when combined with a

state observer. Its stability robustness property must thus be verified.

4.4 Design of Active System

As explained in section 1.3, the active system is integrated into an existing high performance

milling motor spindle. The specifications of this spindle are listed in table 1.1. The design of

the active system is mainly dictated by industrial constraints. To ensure its interchangeability

with a standard spindle, the basic functionalities of the prototype active spindle, such as the tool

clamping system, power and torque specifications, maximal spindle speed as well as the external

geometry of its housing, must be maintained. This implies that the dimensions of the shaft must

remain unchanged.

The implementation of the active system in the force path of the spindle bearings inevitably

induces a decrease of the structural stiffness that may lead to some accuracy problems during

machining operations. The final design must thus be able to guaranty a sufficient static structural

stiffness at the tool tip.

The different features of the spindle, like the cool core option, vibration and axial movement

monitoring, etc., must also be maintained. The access to the automatic tool changing system

and the freedom of movement required for the five-axes machining impose the external geometry

at the spindle nose.

The last industrial constraint concerns the production costs of the active system which must lead

to an affordable sale price.

The design procedure starts with the validation by simulation of the proposed concept using a

simple model of principle. This model also helps the dimensioning of the actuating and sensing

systems. Based on this, the integration design of the selected active system is made and a more

accurate physical model can be built to perform representative simulations.

In the following sections, the choices leading to the design of the actuating and sensing systems


are presented. The proposed concept is represented in figure 3.2 and has been patented. More

information on this patent can be found in [132]. Figure 4.9 shows the resulting prototype spindle.

Figure 4.9: Prototype spindle with integrated active system. Courtesy of Step-Tec.

4.4.1 Actuating System

As discussed in chapter 1, chatter problems mainly arise due to vibrations of the TCP in the

radial plane of the spindle rotational axis, coming from the first bending modes of the spindle

shaft assembly. The objective of the actuating system is to be able to compensate such vibrations

by applying counteracting forces on the spindle shaft or imposing movements.

Basically, three types of actuating systems may be used to act on the spindle shaft. Actuators

can drive the supports of the shaft bearings. This solution is referred here to as active bearing

support and employ actuators transmitting force through mechanical contact, like solid state

actuators.

A second possibility is to use active bearing. In this case, as the actuator makes the link between

the non-rotating spindle housing and the rotating spindle shaft, non-contact actuators are used,

such as electromagnetic bearings.

The third type uses an additional location to the existing bearing supports required for its main

functionality to act on the spindle shaft structure. Such solution is called here active ancillary

bearing. Even if the ability of this latter solution to positively influence the TCP vibration seems

to be important (see subsection 2.5.2.3), its major inconvenience comes from the required space

for the integration of the ancillary bearing. The previously mentioned constraints on the motor

size and shaft length make this third possibility unsuitable.


Finally, it is decided to opt for an active bearing support solution using low voltage PZT (ceramic

perovskite material) piezoelectric stack actuators driven by a charge amplifier. This choice, in

comparison with the active bearing possibility, is motivated by several reasons. Its high cost

constitutes the main drawback of active magnetic bearings even if it becomes less obvious for

an active spindle than for a conventional spindle without active system. Also, their maximal

loading rate is quite limited compared to rolling bearings. So, even if they are particularly

well suited for high speed applications, this makes them inappropriate for high performance

machining applications involving high cutting loads. Finally, the use of active bearing support

allows to keep the same architecture of the spindle shaft and thus to take advantage of the know-

how of the spindle manufacturer. Another advantage of using solid state actuators is that the

resulting stiffness of the shaft in passive conditions, i.e. without controller action, is given by

the structural stiffness of the piezoelectric stacks. Therefore, if no specific action is necessary in

quasistatic domain, no steering action but only regulation actions are required by the controller

and vibration sensors, such as accelerometers, are sufficient to provide the necessary information.

This may play a significant role for the costs of the final solution.

Because of its proximity with the process and after the analysis of the spindle housing architecture

as well as the controllability of the first bending modes of the shaft, the front bearing seems

to be the most suitable location to implement the actuating system. Furthermore, a push-pull

configuration of the actuators is used. This means that in each radial direction, a pair of actuators

located in front of each other is working together with opposite sign signals. This allows to

guarantee sufficient actuator stroke together with a compact distribution of the actuating system

around the front of the spindle housing.

The piezoelectric stack actuators are working longitudinally, which means that their applied elec-

trical field is parallel to their polarity direction. Several aspects need to be considered when using

piezoelectric stacks. The first one is their nonlinear behavior leading to creep and hysteresis ef-

fects (see [133] for more information) between applied electrical field and resulting deformation.

This can be in a large extend compensated using position feedback and charge amplifiers.

Also, piezoelectric stack actuators only support compressive stresses and require mechanical

preloading which is determinant for the dynamics of the resulting actuating system. The ac-

tuators must thus be sufficiently prestressed so that no tensile stresses occur between the stacked

layers. The amplifiers driving the actuators only deliver positive voltage and, in push-pull config-

uration, the pair of actuators works around a nominal setting point given by the middle of their

input voltage range. The resulting preloading is determined by the mounting geometry and the

nominal working condition. In this work, the adjective passive refers to the condition when the

actuating system is set to this nominal working condition but when no control action is imposed.

In order to prevent the ceramic stack from shear stresses, only the axial DOF of the actuator,

corresponding to the radial directions of the spindle, is constrained using a dedicated connecting

system.

The overheating effect of the ceramic is also an issue. This is why a special cooling system is used


to maximize the dissipation of the heat generated by the actuators outside of the spindle and to

minimize thermal problems. The use of push-pull configuration allows a good compensation of

thermal drift.

Furthermore, the radial DOFs of the support of the front bearing are controlled by the precon-

strained actuators. The radial structural stiffness of the shaft is therefore determined by the

structural stiffness of the piezoelectric ceramic stacks in series with the connecting system and

the ball bearings. To prevent any shear stresses in the ceramic, the axial and rotational DOFs

of the ring supporting the front bearing must be constrained by a guiding system. This guiding

system must thus guarantee high axial stiffness to the support ring — as this stiffness determines

the axial stiffness of the whole shaft — but leave the radial DOFs as free as possible. A flexible

guiding system is designed to this end.

Finally, each piezoelectric actuator is driven by a special hybrid amplifier. This amplifier uses

two different working modes over different frequency ranges. In quasistatic range, the voltage

at the actuator terminal is controlled by a feedback loop. In dynamic range, the amplifier is

controlling the charge of the piezoelectric actuator. This allows to compensate some hysteresis

effects but, above all, this theoretically leads to an increase of the passive structural stiffness,

which is determinant in our application.

4.4.2 Sensing System

The sensing system must provide useful information to the controller on the radial deviations of

the shaft induced by the process loading. Two types of sensors are a priori foreseen. Accelerom-

eters are able to measure the absolute acceleration at a specific location. Also, eddy current

non-contact displacement transducers (NCDTs) can give the relative displacement between two

points of the structure and can cope with the presence of oil or some other liquids in between.

NCDTs have the advantage of being able to measure directly on the rotating shaft and can be

placed close from the spindle nose, therefore from the process. The fact that they capture the

relative displacement between the shaft and the housing implies that their signals are less influ-

enced by the axes movements of the machine than the accelerometers.

Accelerometers must be placed on a non-rotating part. It is decided to mount two single axis

piezoelectric accelerometers orthogonally positioned and nearly collocated with the actuators, i.e.

on the support of the external rings of the front bearing. This arrangement allows to minimize the

phase lag between the sensing and actuating points susceptible to be detrimental to the control

performance. In order to be able to place them in the same radial plane, they are shifted around

the rotational axis of the spindle (see figure 3.2). NCDTs can be placed along the shaft to provide

complementary information to the controller.

In the prototype spindle presented here, it is decided to use only accelerometers as sensing system.

This choice is mainly motivated by the fact that active damping action are foreseen to be applied.


Active damping controls basically use the product of a velocity signal and an actuating force. In

the case of accelerometer, the signal is time integrated by the controller before generating a force

reference signal. In the case of displacement transducer, the time derivative of the signal must

be computed. The −20 dB/decade roll-off induced by the time integration tends to attenuate

the high frequency components of the signal, where the time derivative amplifies them. This

makes the use of acceleration signals much more convenient in the presence of high frequency

measurement noise.

96 5. Experiments using Prototype Spindle

Chapter 5

Experiments using Prototype Spindle

This chapter presents the different experimental investigations realized with the prototype spindle

and starts with a brief description of the preliminary experiments performed during the design step

and the commissioning. The machining tests with the prototype spindle mounted in a machine

tool are then discussed. The main objectives of these tests are the validation of the concept in

representative machining conditions and the identification of the limitations of the system in order

to guide an eventual subsequent design optimization. Some appropriate and relevant machining

conditions are selected. To analyze and document the tests, a monitoring system needs to be

installed on the machine tool. This system is described in the next section. The specific criterion

used for the detection of chatter occurrence during the milling tests is described. Before starting

the machining experiments, the dynamics at the TCP is analyzed by measuring the frequency

responses for every selected tool. The procedure used for the identification of the cutting force

coefficients for the chosen machining conditions is presented. The next section details the design

of the different controllers up to their implementation in a real-time digital controller for the

realization of the milling tests. The TCP dynamics is measured once again but this time under

the influence of the control action. Finally, the influence of the active system on the process

stability is investigated and a comparison between experiments and predictions is presented.

5.1 Preliminary Experiments

Before the final design of the prototype, the proposed concept is first validated on a test bench

where the active system is implemented on a dummy spindle. These first tests allow to verify

the functioning and the size of the actuating and sensing systems as well as to make the model

validation. They also make possible the test of different control strategies and identify the most

promising ones.

The active system is then integrated into the prototype spindle. After its commissioning and

before being mounted into the machine tool, the prototype spindle is investigated on a test bench

5.2 Selected Machining Parameters 97

Tool Free Number Helix Entering

assembly Type Diameter length of teeth angle angle

(mm) (mm) (deg) (deg)

T1 Insert end mill 40 175 5 +10 +90



T4 (1) Carbide end mill 20 158 2 +30 +90


T5 Carbide end mill 16 155 2 +40 +90






T10 Face mill 63 90 5 +20 +45






T16 Round insert end mill 40 116 5 +4 –

Table 5.1: Specifications of selected tool assemblies.

where the objective is to test the active system when the spindle shaft is rotating. A magnet

bearing is used to generate an excitation emulating the cutting process loads.

These tests pointed out that beyond a spindle speed approximately equal to 15’000 rpm, the

rolling elements of the bearings excite the natural frequencies of the piezoelectric accelerometers

leading to signal overloading problems and restricting the use of the active system to lower spindle

speed range.

5.2 Selected Machining Parameters

The process stability as well as the characteristics of the active system strongly depend on the

selected tool assembly and machining parameters. In order to get a representative overview of

the active system capabilities, a broad range of machining conditions must be investigated.

The machining tests are realized with the prototype spindle integrated into a Mikron Agie

Charmilles five-axes machining center type HPM of the line G20 (see figure 5.1). The axes

configuration is AC − workpiece − tool − ZXY . Different tool assemblies (T) and workpiece

materials (M) are selected. The chosen tool assemblies are presented in figure 5.2 and their

specifications are listed in table 5.1. Two workpiece materials are selected. The material M1


Figure 5.1: Mikron Agie Charmilles HPM five-axes machining center. Courtesy of Mikron GF Agie

Charmilles.

corresponds to hot-rolled carbon steel Ck45 and M2, to aluminium alloy EN AW-6082 T6.

The tool, workpiece and machining parameter combinations are chosen in order to get relevant but

also predictable conditions. The selection of the tool assembly is mainly guided by its structural

characteristics, namely that the dynamic compliance at tool tip is dominated by the first bending

modes of the spindle shaft.

The first investigations show that several tool-workpiece combinations are not properly suited

for the test of the prototype spindle. For instance, the dynamic stiffness of the tool assemblies

T2 and T3 is so important that no chatter phenomenon is susceptible to occur for the selected

cutting conditions up to the maximal axial DOC allowed by the inserts.

In steel workpiece material, the occurrence of chatter during the tests almost systematically leads

to a tool breakage caused by the important forces due to a higher material strength compared

to aluminium. Therefore, due to economical reasons, the stability tests using solid carbide end

mills are limited to aluminium workpiece material.

Another limiting factor is the process damping occurring when the ratio between chatter frequency

and spindle speed is very large. This corresponds to the cases where the critical mode generating

chatter presents a high natural frequency. Such problem is encountered with the tools T7 and T8

where the chatter frequencies are identified in the range of 5 kHz and higher, corresponding to

5.2 Selected Machining Parameters 99

T1

T2

T3

T4

T5

T6

T7

T8

T12

T13

T14

T15

T9

T10

T11

T16

Figure 5.2: Selected tool assemblies.

bending deformations of the tool. Process damping results from frictional forces induced by an

effective negative clearance angle. It is very sensitive to the tool wear and can greatly influence

the process stability. This is why under process damping, it becomes very difficult to guaranty a

sufficient repeatability of the stability measurements.

Finally, for some tool assemblies, cutting tests point out that the natural mode of the spindle

shaft assembly responsible for chatter occurrence lies outside of the controller bandwidth.

Table 5.2 shows the different selected tool-workpiece material combinations. From the above-

mentioned reasons and in order to limit the size of this document, only the conditions P11 and

P12 are further discussed in the following sections. The investigation of these conditions show

different representative aspects and problematic which can be transposed to the other machining


Tool Workpiece material

assembly M1 M2

T1 P11 P12

T2 P21 –T3 P31 P32

T4 (1) P41 –T4 (2) – P42

T5 – P52T6 – P62

T7 (1) P71 –T7 (2) – P72

T8 P81 –T9 – P92T10 P101 –T11 P111 –T12 P121 –T13 – P132T14 – P142T15 – P152T16 P161 –

Table 5.2: Tool-workpiece material combinations.

Combinations

Machining parameters P11 P12

Up-milling Up-millingFeed direction Y+ Y+

Feed rate, fz (mm/tooth) 0.08 0.08Radial DOC, ae (mm) 25 25Spindle speed, n (rpm) 2’000–3’100 4’000–11’000

Cutting speed, vc (m/min) 251–390 503–1382

Table 5.3: Machining parameters of presented tool-workpiece combinations.

conditions. The corresponding machining parameters are listed in table 5.3. The feed directions

in this table and in the following of the document correspond to the directions of the machine

coordinate system. The ranges of cutting speed are selected based on the tool manufacturer

recommendations.

5.3 Monitoring System 101

5.3 Monitoring System

During the machining tests, the chatter occurrence must be detected and characterized. The most

simple method for chatter detection is to look at the surface finish (see figure 5.7). However, in

order to define a quantitative detection criterion and to identify the corresponding chatter fre-

quency, a sensing system is employed. Using the prototype spindle, chatter phenomena coming

from the tool side can be identified using the information delivered by the integrated accelerom-

eters of the active system.

Using a conventional spindle, additional sensors must be used. This is why a 3D-piezoelectric

accelerometer is mounted on the spindle housing. Another one is located close to the workpiece

to detect from which side of the structural loop come the critical vibrations. The cables must be

protected against the chip and lubrication fluid projections. For chatter detection, microphone

may also be used. One advantage of accelerometer over microphone is to provide directional

information relative to the vibrations.

To identify the coefficients used by the process model, the cutting forces must be monitored. A

3D-dynamometer table is used to this end. The frequency bandwidth of the dynamometer table

directly depends on its size. A compromise must be found between the measurement bandwidth

and the size of the workpiece.

Figure 5.3 shows the monitoring system installed on the machine.

5.4 Chatter Detection

The detection and identification of chatter phenomenon is necessary in this study. Several meth-

ods exist to detect the apparition of unstable self-excited vibrations in milling process. The key

aspect that allows the differentiation between forced and self-excited vibrations is the frequency

content. Forced vibrations occur at the frequency of the periodic excitation and self-excited vi-

brations occur at frequencies distinct from the excitation. A similar criterion as the one defined

in section 3.3.1, based on the variance of the Poincaré map, is used. However, here the TCP

deviations z are a priori not known. Instead of z, the signals provided by the monitoring system

ym are used to define the detection criterion.

Chatter occurs if the standard deviation

σP,y > σy,lim, (5.1)

where

σP,y =

√

√

√

√

1

NT

NT∑

k=1

‖ym(k T )− ykT ‖22 (5.2)


Triaxial

accelerometers

3D-dynamometer

table

Tool assembly

Workpiece

Figure 5.3: Monitoring system for machining tests.

with

ykT =1

NT

NT∑

k=1

ym(k T ) (5.3)

and σy,lim being a threshold value that must be experimentally calibrated.

NT is the number of considered spindle revolutions and ‖.‖2 is the Euclidean norm.

In practice, the tool always presents a certain runout that can influence the frequency content

of the excitation. In this case, the fundamental frequency of the process excitation becomes the

spindle rotation. The image of the reduced Poincaré map must thus be based on the sampling

5.4 Chatter Detection 103

period Tn, so that, in the above relations, T = Tn. The measured samples are resampled at a

multiple of the spindle rotation frequency. The choice of starting sample is made in order to

maximize the standard deviation σP,y. In order to be able to compare the different values, the

considered time period must correspond to a window where the process is in steady state, i.e.

where the exponential unstable increase of vibration amplitudes is already limited by second order

nonlinearities.

5.4.1 Illustrative Examples

The following figures present examples of the signals obtained by the integrated accelerometers

during machining tests with the combination P11 in three different process stability conditions.

As for this combination, the tool possesses an odd number of teeth, only secondary Hopf and flip

bifurcations are susceptible to occur. Figure 5.4 corresponds to an unstable machining condition

where a Hopf bifurcation is originated. In this case, the prototype spindle is passive — which

means that no action from the control system is generated — and the resulting σP,y is equal to

60.6 m/s2. Figure 5.5 corresponds to the same machining condition but with a counteraction of

the active system stabilizing the process. In this case, the standard deviation σP,y is equal to

5.4 m/s2. The case of a flip bifurcation is presented in figure 5.6 where σP,y = 120.5 m/s2.

k fTP

k fSR

fSR

Normal

Feed

Mag

.se

nso

r(m/s

2)

Frequency (Hz)

Linear spectrumSensor normal (m/s2)

Radial vibrations

fSR samples

All samples

Sen

sor

feed

(m/s

2)

Time (s)

Time signal

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−100 0 1000 0.5 1 1.5

5

10

15

20

25

30

−100

−50

0

50

100

−100

−50

0

50

100

Figure 5.4: Hopf bifurcation: condition P11, n = 2’300 rpm, ap = 1.7 mm, passive spindle. fSR: Spindle

revolution frequency, k fSR: Spindle revolution frequency harmonics, k fTP : Tooth passing frequency

harmonics.


k fTP

k fSR

fSR

Normal

Feed

Mag

.se

nso

r(m/s

2)

Frequency (Hz)


Radial vibrations

fSR samples

All samples

Sen

sor

feed

(m/s

2)

Time (s)

Time signal

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−100 0 1000 0.5 1 1.5

5

10

15

20

25

30

−100

−50

0

50

100

−100

−50

0

50

100

Figure 5.5: Stable machining: condition P11, n = 2’300 rpm, ap = 1.7 mm, active spindle.

The resulting surface finish is also a good indicator of the chatter presence. Figure 5.7 shows the

resulting peripheral surface for the machining corresponding to the conditions of figures 5.4 and

5.5. In case of Hopf bifurcation, chatter marks are visible. In the lower plot, the corresponding

waviness for both conditions are represented. One notes that the undulations of the surface finish

left during chatter is several times larger than in stable condition.

5.4 Chatter Detection 105

k fTP

k fSR

fSR

Normal

Feed

Mag

.se

nso

r(m/s

2)

Frequency (Hz)


Radial vibrations

fSR samples

All samples

Sen

sor

feed

(m/s

2)

Time (s)

Time signal

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−200 0 2000 0.5 1 1.5

10

20

30

40

−200

−100

0

100

200

−200

−100

0

100

200

Figure 5.6: Flip bifurcation: condition P11, n = 2’400 rpm, ap = 1.9 mm, passive spindle.

Figure 5.7: Measured waviness of peripheral surface finish from machining tests where stable cut and

Hopf bifurcation occur.


5.5 Frequency Responses at Tool Tip

In order to predict the process stability, the relative receptance functions between the tool and

the workpiece must be determined. For this end, FRF measurements are led at the tool tip and

on the workpiece side. The axes of the machine are positioned at a location representative of the

coordinates used during the machining tests.

The natural modes of the whole machine identified by experimental modal analysis are almost not

influenced by the change of tool assembly. However, the resulting receptance at TCP is greatly

dependent on the tool assembly. This is why FRF measurements at TCP must be performed for

each selected tool assembly.

The measurements at the tool tip are made on the non-rotating tool. The FRFs are obtained

using an instrumented impulse hammer to excite the structure at the TCP or on the workpiece

and an lightweight uniaxial accelerometer to measure the resulting deviations (see figure 5.8).

The receptance functions are thus obtained by double time integration of the accelerance transfer

functions, defined in [13] as the output acceleration over the input force.

For the considered workpieces, the influence of the machine axis displacements as well as the

variation of the workpiece geometry during the machining on the tool-workpiece dynamics is

negligible. The influence of the shaft rotation and some other factors is discussed in section 5.9.1.

Uniaxial

accelerometer

Instrumented

impulse hammer

Tool assembly

T1

Figure 5.8: Measurement of frequency response functions at tool tip using impulse hammer and accelerom-

eter.

Figures 5.9 and 5.10 represent the direct receptance functions measured in three directions on the

tool assembly T1 and on the workpiece M1, respectively. The measurements on the workpiece

M2 are very similar to those obtained with the workpiece M1 and are thus not shown here. The

magnitude of the cross FRFs in the radial directions X and Y are also represented.

The tool T1 shows the largest compliance around 700 Hz. This mode corresponds to the first

bending mode of the shaft assembly coupled to the front bearing. This mode is identified using

5.5 Frequency Responses at Tool Tip 107

finite element modal analysis and its modal shape is represented in figure 5.11. Both radial direc-

tions show a good symmetry. The cross terms at the tool tip of the critical modes remain about

one order smaller than the direct terms. The first axial mode of the spindle shaft lies around

550 Hz and its flexibility is much smaller than the radial modes.

On the workpiece side, the main critical mode in the axial direction is located at 160 Hz corre-

sponding to a bending mode of the A-axis cradle in Z-direction, as identified by experimental

modal analysis. In X and Y -directions, the first critical modes are located around 380 Hz, which

corresponds to the modes of the C-axis coupled with the rotation of the A-pendulum around the

X-direction. It can be seen that these modes are strongly coupled by looking at the cross FRFs.

Two critical modes are also visible around 1200 Hz. They probably correspond to local modes of

the workpiece and dynamometer table assembly. It can be noticed that the compliances on the

workpiece side are more than one order smaller than those observed at the tool tip.

The resulting relative receptance between the tool and the workpiece, as both compliances are in

series, is obtained by the addition of both absolute receptance functions at the tool tip and on

the workpiece.

More generally, based on the analysis of all considered tool assemblies (see figure 5.2), it can

be said that long tool assemblies exacerbate the bending modes of the spindle shaft increasing

the propensity of chatter to occur. The use of long and massive tool assemblies, like T1, leads

to critical modes essentially induced by the compliance of the front bearing coupled to the first

bending mode of the shaft. Figure 5.12 shows the receptance functions measured at the tool tip for

the tool assemblies presenting critical modes coming from the first bending modes of the spindle

shaft assembly coupled to the deformation of the front bearing. In the figure, the maximum

singular values (MSVs) of the receptance matrix is used to represent the maximal amplitude of

the receptance functions over all direct and cross receptance functions.

Tool assemblies with slender tools, such as T15, show critical modes local to the tool. The modal

shape of the critical mode relative to the tool assembly T15, corresponding to a local bending

of the tool, is shown in figure 5.13. The measured receptance function of the tool assemblies

indicating the presence of a critical mode induced by a local bending deformation of the tool is

presented in figure 5.14.


Frequency (Hz)

Phas

e(d

eg)

GTzw,Y X

GTzw,XY

GTzw,ZZ

GTzw,Y Y

GTzw,XX

Mag

nit

ude

(µm/N

)Receptance at tool tip

200 400 600 800 1000 1200 1400 1600 1800 2000−200

0

20010−4

10−2

100

Figure 5.9: Frequency response functions measured at tool tip with tool T1. GTzw,XY corresponds to the

receptance at tool tip between an excitation DOF in Y -direction and a measurement DOF in X-direction.

Frequency (Hz)

Phas

e(d

eg)

GMzw,Y X

GMzw,XY

GMzw,ZZ

GMzw,Y Y

GMzw,XX

Mag

nit

ude

(µm/N

)

Receptance on workpiece

200 400 600 800 1000 1200 1400 1600 1800 2000−200

0

20010−4

10−2

100

Figure 5.10: Frequency response functions measured on workpiece M1. GMzw,XY corresponds to the recep-

tance on workpiece between an excitation DOF in Y -direction and a measurement DOF in X-direction.

5.5 Frequency Responses at Tool Tip 109

Figure 5.11: Modal shape of the critical mode of the spindle shaft with tool assembly T1 at 704 Hz

obtained using finite element modal analysis.

T16

T13

T10

T9

T4

T3

T2

T1

Frequency (Hz)

MSV

(µm/N

)

Receptance at tool tip

500 1000 1500 2000 2500 3000 3500 4000 4500 500010−3

10−2

10−1

100

Figure 5.12: Receptance functions of tool assemblies presenting critical mode corresponding to the first

bending mode of the spindle shaft assembly. MSV: maximum singular value.


Figure 5.13: Modal shape of the critical mode of the spindle shaft with tool assembly T15 at 1416 Hz

obtained using finite element modal analysis.

T15

T14

T12

T11

T8

T7

T6

T5

Frequency (Hz)

MSV

(µm/N

)

Receptance at tool tip

500 1000 1500 2000 2500 3000 3500 4000 4500 500010−3

10−2

10−1

100

Figure 5.14: Receptance functions of tool assemblies presenting critical mode local to the tool.

5.6 Identification of Average Cutting Force Model Coefficients 111

5.6 Identification of Average Cutting Force Model Co-

efficients

The shear and edge cutting force coefficients, Kc and Ke respectively, of the average cutting

force law, expressed by (3.34), must be experimentally identified in order to be able to model

the cutting interaction. A mechanistic approach is used, which means that these coefficients

are valid for a given tool geometry and workpiece material. They are determined from the

cutting forces measured with a dynamometer table during machining tests in stable conditions,

i.e. without chatter occurrence. The tool wear plays an important role on the cutting forces and

must eventually be taken into account. They may also depend to a smaller extent on the cutting

speed but are theoretically independent of the selected DOCs and feed rate. However, due to the

tool geometry, this statement may not be true and some secondary order effects can generate a

dependency between these parameters and the average cutting force coefficients.

The average cutting force model linearizes the dependency between the cutting forces and the

uncut chip thickness given by the feed rate, the number of rotations per minute, the number of

teeth and the radial depth of cut. The identification of the cutting force coefficients is based on

experiments realized at given cutting speed and depths of cut but with three different feed rates.

This allows the determination of the cutting forces coefficients and the verification of the linear

dependency between the cutting forces and the chip thickness. The measured cutting forces are

averaged over the time period of one or several integer tool revolutions. From these values, the

shear and edge cutting force coefficients are extracted by linear regression. The procedure is

presented in [12].

The variation of the averaged cutting forces with the feed rate for the case of the tool T1 with

only one insert cutting the workpiece material M1 is presented in figure 5.15. It can be noticed

that the linear dependency of the cutting forces with the chip thickness is verified in all three

directions.

Figure 5.16 compares the cutting forces in feed, normal and axial directions predicted by the pro-

cess model with the forces measured using the dynamometer table. The measurement bandwidth

of the dynamometer table is limited by the first natural frequency of the table coupled to the

workpiece. According to the manufacturer data, the bandwidth goes up to 1 kHz. The output

signals are thus lowpass filtered at 1 kHz. In spite of this filtering, some remaining oscillations

on the delivered signals, induced by the dynamics of the dynamometer table, are visible between

two cuts. Except these oscillations, one notes that the forces predicted by the milling process

model are in good agreement with the measured forces.

It is generally assumed that the cutting force coefficients do not significantly vary with the spindle

speed and the DOCs such that a single value of the cutting force coefficients, identified in nominal

conditions, is sufficient to compute reliable stability predictions over the whole range of considered

spindle speeds and DOCs. In order to investigate the validity of this nominal value of cutting


replacements

Axial

Normal

Feed

Ave

rage

dcu

ttin

gfo

rce

(N)

Feed rate (mm/tooth)0 0.02 0.04 0.06 0.08 0.1 0.12

−150

−100

−50

0

50

100

Figure 5.15: Dependency of the averaged cutting forces with the feed rate using the tool T1 with only

one insert in workpiece M1, n = 2’500 rpm and ap = 3 mm. The dots represent the values obtained from

the measurements and the lines represent the linear regression leading to the identification of the average

cutting force coefficients.

force coefficients and its variation for the selected conditions, the spindle speed and axial DOC

ranges of the combinations P11 and P12 are discretized in three values, leading to nine different

machining combinations. The influence of the tool wear is also studied. For each condition, the

validity of the obtained values is verified by comparing the measured signals with the simulated

forces using the empirical coefficients.

The results indicate non negligible variations of the coefficient values. However, as discussed in

section 5.9.1, their influence on the stability predictions remains very weak in comparison with

the structural parameters. This allows to state that a single value for the cutting force coefficients

is sufficient to compute the stability lobes diagrams.

5.7 Control Design

To design the controller, the transfer functions between the different inputs and outputs of the

passive mechatronic structure must be determined. To do so, the corresponding FRFs are mea-

sured and, using curve fitting procedures, the transfer functions can be extracted.

Figure 3.2 shows a schematic representation of the plant with its inputs and outputs. Figure 5.17

5.7 Control Design 113

Experiment, axial

Experiment, normal

Experiment, feed

Model, axial

Model, normal

Model, feed

Time (s)

Cutt

ing

forc

e(N

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−600

−400

−200

0

200

400

600

Figure 5.16: Measured and predicted cutting forces using the tool T1 with only one insert in workpiece

M1, n = 2’500 rpm and ap = 3 mm.

represents the corresponding flowchart between the input and output variables and the transfer

functions.

The FRFs between the process input w and both outputs z and y are measured through im-

pact testing and the FRFs from the actuator inputs are determined using a sweep sine signal

u delivered to the actuator amplifiers. As the sensing system of the prototype spindle only

use accelerometers, y corresponds to the signals delivered by the conditioners connected to the

integrated accelerometers.

Neglecting the influence of the workpiece and considering only both radial directions, X and

Y , of the spindle at the tool tip, the global model of the plant consists in a multiple inputs,

multiple outputs (MIMO) system with the four inputs wX , wY , uX , uY and the four outputs

zX , zY , yX , yY . If the cross terms between both radial directions are considered, sixteen FRFs

must be taken into account for the modeling.

In figure 5.18, the FRFs matrix in both radial directions between the inputs and outputs of the

plant with the tool assembly T1 is represented. The dynamics seen by the process corresponds

to the FRFs Gzw (upper left corner). As previously mentioned, the most critical natural mode

is visible around 700 Hz. The FRFs Gyw and Gzu give a qualitative representation of the ability

of the controller to detect and influence this natural mode. For the control design, the FRFs

Gyu are determining. Due to the nearly collocated configuration between the sensors and the


Gzw

Gyw

Gzu

Gyu y

w

u

+

+

+

+

z

G

Figure 5.17: Flowchart of the plant.

actuators of the active system, the phase almost remains between the boundaries of 180 and 0 deg

over the considered frequency range. This means that the poles and zeros distribution forms an

alternating pattern along the imaginary axis. This also signifies that the plant between y and u

is almost minimum phase, i.e. the system ans its inverse are causal and stable. This property

makes possible the design of collocated controllers, such as direct velocity feedback or second

order filter. More information on collocated controls can be found in [134, 135].

In order to investigate the influence of the controller on the poles of the plant, a representative

model must be built. This model may also be used by the observer to estimate the current states

of the plant for a model-based controller. The physical model, based on finite element method

and presented in section 3.1.1, is not sufficiently reliable to be used to this end. A black box

model must thus be elaborated from the measured FRFs using curve fitting techniques. A method

able to synthesize a LTI MIMO system in state-space form based on frequency-domain data is

employed: the subspace method N4SID (numerical algorithm for subspace state-space system

identification) [136]. This method allows a construction of the model by considering all FRFs of

the inputs-outputs matrix simultaneously. To identify a MIMO system, state-space techniques

generally provide better results than polynomial-based techniques, such as ARMA (autoregressive

moving average) for instance.

This model can also be used to perform time-domain simulations and process stability analysis.

It must thus be representative over a sufficiently wide frequency range. Also, in order to be able

to correctly predict the stability, frequencies above the critical natural mode must be captured

in order to prevent any spillover effect coming from unmodeled high frequency dynamics.

In the present case, a 24th order model considering simultaneously the direct and cross FRFs in

both radial directions is fitted over the frequency range from 200 to 1800 Hz. The model order is

determined by looking at the agreement between the measured and the resulting fitted FRFs over

the considered frequency range. Before leading the identification, a scaling of the different FRFs

is required. The identification results are very sensitive to the values used to scale the inputs and


Y X

XY

Y Y

XX

Frequency (Hz)

(V/V

)

Gyu

(µm/V

)

Gzu

Phase

(deg

)

Frequency (Hz)

Magnitude

(V/N

)

Gyw

Phase

(deg

)M

agnitude

(µm/N

)Gzw

500 1000 1500 2000500 1000 1500 2000−200

0200

10−3

10−2

10−1

−2000

200

10−2

10−1

100

−2000

200

10−4

10−3

10−2

−2000

200

10−3

10−2

10−1

100

Figure 5.18: Matrix of FRFs of plant with tool assembly T1.

the outputs of the plant. Different types of scaling can be used. One chooses here to scale the

inputs and outputs of the plant such that the maximal value of their corresponding time signals

is approximately equal to one.

The identified state-space model is a square — the number of inputs equals those of the outputs

— stable, linear continuous time-invariant system of the following form,

x(t) = A x(t) +[

Bw Bu

]

[

w(t)

u(t)

]

(5.4)

[

z(t)

y(t)

]

=

[

Cz

Cy

]

x(t) +

[

Dzw Dzu

Dyw Dyu

][

w(t)

u(t)

]

(5.5)

with x ∈ Rnid and

w =

[

wX

wY

]

, u =

[

uX

uY

]

, z =

[

zX

zY

]

, y =

[

yX

yY

]

. (5.6)

As previously mentioned, in this case, nid = 24. As a reminder, w correspond to the radial

disturbing forces generated by the milling process, u are the reference signals at the input of the


actuators, z represent the tool center point deviations from the nominal tool path and y are the

signals delivered by the integrated sensing system to the controller.

In figure 5.19, both measured and fitted data of the FRFs matrix with the tool assembly T1 are

represented. For convenience, only the FRFs corresponding to the excitation in X-direction are

shown. The FRFs generated by the identified model show a satisfactory agreement over the whole

frequency range of interest in both directions.

Frequency (Hz)

Gyu,jX

(V/V

)

Gzu,jX

(µm/V

)

Phase

(deg

)

Frequency (Hz)

Magnitude

(V/N

)

Gyw,jX

Phase

(deg

)

Exp.

Model

Gzw,jX

Magnitude

(µm

/N

)

500 1000 1500 2000500 1000 1500 2000−200

0200

10−3

10−2

10−1

−2000

200

10−2

10−1

100

−2000

200

10−4

10−3

10−2

−2000

200

10−3

10−2

10−1

100

Figure 5.19: Comparison of FRFs matrix with excitation in X-direction between experimental data from

tool assembly T1 and identified model (24th order based on frequency range from 200 to 1800 Hz in both

directions). j = X,Y .

5.7.1 Plant Characteristics

Before leading the control design, the intrinsic properties of the plant should be more deeply

investigated in order to determine the eventual performance limitations that may be faced.

By looking at figure 5.18, it is visible that the cross FRFs are less influential in comparison with

the direct terms over the frequency range of the critical modes. This allows to study each radial

direction independently. Two state-space models are thus identified from each four direct FRFs


in radial directions and the cross FRFs are not considered. In this case, for each radial direction,

a 12th order model is fitted over the frequency range from 200 to 1800 Hz. In figure 5.20, both

measured and fitted data of the direct FRFs matrix in the X-direction are represented. The

following steps are conducted on the model obtained in the X-direction but similar results are

found in the other radial direction.

Frequency (Hz)

Gyu,XX

(V/V

)

Gzu,XX

(µm/V

)

Phase

(deg

)

Frequency (Hz)

Gyw,XX

Magnitude

(V/N

)P

hase

(deg

)

Exp.

Model

Gzw,XX

Magnitude

(µm/N

)

500 1000 1500 2000500 1000 1500 2000−200

0200

10−3

10−2

10−1

−2000

200

10−1

100

−2000

200

10−4

10−3

10−2

−2000

200

10−2

10−1

100

Figure 5.20: Comparison of direct FRFs matrix in X-direction between experimental data from tool

assembly T1 and identified model (12th order based on frequency range from 200 to 1800 Hz in X-

direction).

The first relevant property of the plant is the location of its poles. Table 5.4 lists the poles and

the corresponding natural frequencies and modal damping ratios of the model in X-direction.

As expected, the identified system is an underdamped second order asymptotically stable system,

i.e. all its poles are distinct complex conjugate pairs located in the closed left-half plane (LHP).

Each complex conjugate pair corresponds to a natural mode of the plant. A lower damping ratio

on the second and the fifth eigenmodes is visible.

In a second step, the critical eigenmodes for the process stability must be identified and it must

be found out how well can the controller detect and influence these critical modes. This is related

to the state observability and state controllability and their combination. This can be done using


Mode Pole Natural frequency Damping ratio

r pr, p∗r (103 rad/s) fn,r (Hz) ζr (%)

1 −0.483± i 3.55 570.8 12.52 −0.271± i 4.31 688.1 6.33 −0.529± i 5.38 861.0 9.84 −0.589± i 6.53 1043.1 9.05 −0.303± i 8.51 1354.8 3.66 −0.665± i 9.42 1502.9 7.1

Table 5.4: List of the poles of the identified system in X-direction with tool assembly T1.

the notions of modal gain, input and output modal vector previously defined in section 4.2.

In this case, all output and input poles entries between the input uX and the output yX are

non-zero. This means that all poles, and thus all eigenmodes, are observable and controllable by

the controller. The same condition is verified for the three transfer functions between the other

input and output of the plant. This means that the identified system is in minimal realization

and no order reduction is required.

Figure 5.21 represents the values of the modal gains Rr for all four transfer functions. The modal

gains of the transfer function between wX and zX (upper left diagram) provide an image of the

criticity of each natural mode seen by the process. It confirms here that the second eigenmode

located at 688 Hz is the most critical. It is now interesting to look at the degree of observability

and controllability of this eigenmode but on the transfer function between yX and uX (lower

right diagram), where it can be noticed that the corresponding value is low in comparison with

the other eigenmodes. This indicates that the control system must provide higher efforts to be

able to influence this second eigenmode than the first one, for instance. The values of the modal

gains of Gyu depend on the location of the sensing and actuating points of the control system

regarding the modal shapes. It indicates that in this case, the positioning of the actuator and

the sensor is not ideal to influence this second mode shape.

It is now interesting to determine if the lack of information transmitted from the plant input to

its output relative to the critical mode comes more from a poor observability or controllability.

In other words, to improve the influence of the controller on the critical mode, is it preferable to

modify the location of the sensors or the actuators? By computing the input-output modal vectors

for the input uX , respectively, the output yX , one can evaluate the relative degree of the modal

controllability, respectively observability, for all the identified modes. Figure 5.22 represents the

magnitude of the modal gains and the components of the input-output modal vectors for the

transfer function Gyu in X-direction. The second mode shows a modal observability much lower

than for all the other modes. On the contrary, its modal controllability is high. Thus, the fact

that the controller possesses only a weak potential of influencing this second mode comes from

a poor observability due to the use of sensors not ideally located. It is worth to remind here


Mode r

|Rr,yu|

Mode r

|Rr,yw|

|Rr,zu|

|Rr,zw|

1 2 3 4 5 61 2 3 4 5 6

×10−6×10−7

0

0.05

0.1

0.15

0.2

0

0.005

0.01

0.015

0

0.5

1

1.5

2

0

2

4

6

Figure 5.21: Magnitude of the modal gains between the inputs and outputs of the plant in X-direction.

that the choice of the sensing point was mainly guided in order to get a nearly collocated control

system and so, is a priori not ideally positioned in the sense of observability. On the other hand,

the high relative controllability confirms the judicious choice for the actuating system.

The next important characteristic of the plant is the location of its zeros and more precisely the

configuration between the poles and zeros. The poles of the model are common to all transfer

functions and the zeros are given by the input-output configuration and are thus different for each

transfer function. They depend on the sensing and actuating points related to the modal shapes

and are thus closely related to the input-output pole vectors seen previously. A zero corresponds

to a value of s where the transfer function loses rank. Physically, they can be interpreted as the

frequency where a non-zero input excitation produce no output, i.e. where the superposition of

all mode shapes cancels the resulting vibrations, corresponding to a singularity.

The location of the zeros can drastically limit the performances of the controller. More specifically,

right-half plane (RHP) zeros of the transfer function between the sensors and the actuators, Gyu,

restrict the achievable bandwidth and gain of the controller (see [128, 134, 137]). Also, zeros

closely located to poles logically reduce the potential of influencing them.

As previously seen, the obtained identified model is in minimal realization so that no unobservable

or uncontrollable states must be eliminated before analyzing its zeros in order to get only the

transmission zeros (no invariant zero).

Figure 5.23 represents the location of the poles and zeros of the transfer function Gyu,XX . It can

be seen that the plant possesses five complex conjugate pairs (dynamic) of zeros and two purely


Mode r

|um

,ur|

Modal controllability

|ym

,yr|

Modal observability

|Rr,yu|

Modal gain

1 2 3 4 5 60

0.005

0.01

0.015

0

5

10

15

0

0.05

0.1

0.15

0.2

Figure 5.22: Magnitude of the modal gains and the components of the input-output modal vectors between

the input uX and the output yX .

real (static) zeros: one in the LHP and one in the RHP. The RHP zero is close to the origin

and is due to the nature of the transfer function — more precisely due to the sensing principle

(accelerometers). As the input u from the actuators can be interpreted as a force, the transfer

function Gyu represents an accelerance function which has one zero located at the origin and a

feedthrough term, i.e. the frequency response function tends to a non-zero value as the frequency

goes towards infinity. However, the presence of this RHP zero is not critical as it is far enough

from the control bandwidth and is thus not excited.

It can be seen that the pole corresponding to the second mode is located very close to a zero.

This explains the corresponding poor modal gain previously observed. Moreover, the distribution

of the poles and zeros is not alternating along the imaginary axis. This means that a pole-zero

flipping effect is involved. The plant is thus non-minimum phase due to the fact that actuators

and sensors are not strictly collocated and their dynamics is not ideal. This pole-zero flipping

explains the shifting of the phase of the FRF below 0 deg observed in figure 5.20. The presence

of the zero is also visible on the magnitude function of the FRF, corresponding to the drop of

magnitude around 900 Hz.


Zeros of Gyu,XX

Poles of Gyu,XX

Imag

inar

ypar

t

Real part

0.10.20.3

0.6

0.7

0.8

0.9 2.5e+003

5e+003

7.5e+003

−6000 −5000 −4000 −3000 −2000 −1000 0 1000−1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Figure 5.23: Location of the poles and zeros of the transfer function Gyu,XX .

To conclude on the characteristics of the plant, the major limitation that the controller must face

comes from the presence of zero close to the critical pole that seems to be due to a suboptimal

location of the sensing point.

These observations remain valid for the other tool assemblies presenting critical modes corre-

sponding to the first bending modes of the shaft (see section 5.5) but also for the tool assemblies

demonstrating critical modes with deflections mainly restricted to the tool.

5.7.2 Optimal Controllers Design

The design of the optimal controllers starts with the construction of the plant model. In the case

of disturbance rejection schemes, the plant consists in the dynamics of the mechatronic system


between the TCP and its inputs and outputs. A black box model of G is identified based on

FRF measurements using the N4SID method, as previously explained in this chapter. In the case

of stabilization schemes, a milling process model must be included. Such semi-empirical process

model is based on cutting force coefficients that need to be determined with machining tests,

as described in section 5.6. Using the approximations described in section 3.2.1, the LTI model

GGR of the machine coupled to the process and defined by (4.29) can be constructed.

The next step consists in defining the weighting functions Wj as described in section 4.3.8.

Considering the identified plant model, expressed by (5.4) and (5.5), used for the model-based

control design, the weighting functions are of the form

Wj(s) = diag(

wj,X(s), wj,Y (s))

, j = z, u, w, v. (5.7)

In a first step, frequency-independent weighting functions are selected. If the performance re-

quirements cannot be reached using these constants, then higher order functions are employed.

The weighting functions allow to influence the distribution and the focus of the control effort and

performance over the frequency-domain. The dimensioning of the controller consists in choosing

satisfactory values for the coefficients involved in the weighting functions.

Considering these weighting functions, the generalized plant GP , expressed by (4.18), can be

designed. From the state-space realization of GP , the optimal H2 or H∞ controllers C can be

synthesized using the Matlab functions h2syn or hinfsyn, respectively.

To perform the synthesis procedure, the generalized plant must satisfy the conditions C1 to C4

listed in section 4.3.4. By considering constant weighting functions and negligible feedthrough

terms of the plant between both inputs w and u and the output z, so that Dzw = Dzu = 0,

these conditions are fulfilled for both, the disturbance rejection and stabilization schemes. The

optimal problem is thus well-posed and can be solved. The conditions on the feedthrough terms

are required for the stabilization scheme, as the approximated model used for the control design

lies on these assumptions. For the disturbance rejection, these conditions are not necessary for

the synthesis of a H∞ controller and for the H2 case, only the condition on Dzw is required.

In order to be implementable, the closed-loop of the synthesized controller with the mechatronic

structure must be stable. This latter condition is requested to guarantee the stability also when

the cutter is not in interaction with the workpiece. In the case of stabilization schemes, the

process stability must also be guaranteed for depths of cut smaller than the condition used for

the control design. This can be verified through robust stability analysis. Furthermore, even if

unstable controllers are theoretically usable, the observations made during the tests show that

only stable controllers can be successfully implemented in the RT system. This implies that the

considered plant must not only be stabilizable but strongly stabilizable, i.e. stabilizable using a

stable controller. The maximal achievable control performance is mainly limited by these stability

matters.

The order of the resulting synthesized controller is equal to the order of the generalized plant

GP , given by (4.32), which, in the case of stabilization schemes, is typically larger than 200. Due

to numerical problems coming from rounding errors, such high order transfer functions cannot be


implemented in a digital RT controller. They first need to be reduced. The full order controller

is first transferred in its Gramian-based input/output balanced realization, i.e. in a form where

the observability and controllability Gramians are equal and diagonal. The states corresponding

to the lowest Hankel singular values are then truncated with a method guarantying the same

static gain of the reduced transfer function. In this case, a maximal 12th order transfer function

is supported by the RT system ensuring a correct representation.

Before implementing the resulting reduced and discrete controller, its stability and the stability

of its predicted closed-loop transfer function with the plant must be verified. In the case of

stabilization scheme, also the stability of the predicted closed-loop system including the milling

process must be checked.

After its implementation, FRF measurements at the tool tip are performed. Based on these

FRFs, process stability predictions are made and compared with the design expectations before

leading any machining test.

To illustrate the design procedure, an example is presented corresponding to the synthesis of a H2

controller with constant weighting functions designed for the improvement of the process stability

involving the tool assembly T1 in interaction with the workpiece material M1 (combination P11).

Both, disturbance rejection and stabilization schemes are applied. The design is based on an

identified model of 14th order on the frequency range between 200 and 1200 Hz.

5.7.2.1 Disturbance Rejecting Controller

The results obtained using the disturbance rejection scheme are presented in figures 5.24 and

5.25. The upper chart in figure 5.24 compares the FRFs at the TCP when no control is applied

(passive) with the closed-loop FRFs Fzw (see equation (3.133)), obtained applying the disturbance

rejecting H2 controller (active). These FRFs are obtained from the identified model used for the

control design. A magnitude reduction of 31.7% of the critical resonance peak in closed-loop is

visible. In this case, the stability of the synthesized controller limits the control performances.

The lower chart represents the FRF of the controller.

Figure 5.25 shows the stability charts corresponding to the FRFs of figure 5.24 predicted by the

ZOA method. The disturbance rejecting controller achieves an increase of the unconditionally

stable axial DOC, i.e. the lowest value of the stability boundary, equal to 45.3%. One notes that

the influence of the controller globally tends to increase the process stability limit. However, it

does not guarantee an improvement of the stability for all spindle speeds, as it is noticeable for

the spindle speed range around 2’800 rpm.

5.7.2.2 Stabilizing Controller

For the stabilization scheme, the control design considers two spindle speeds where three differ-

ent axial DOCs are investigated. A 15th order Padé approximation is necessary to accurately


ActivePassive

Frequency (Hz)

MSV

(V/V

)

FRF of controller

MSV

(µm/N

)FRF at TCP

200 400 600 800 1000 1200 1400 1600

×108

0

1

2

30

0.1

0.2

0.3

0.4

0.5

Figure 5.24: Predicted FRFs of closed-loop system at TCP and of controller by applying disturbance

rejecting H2 controller.

approximate the time delay. As described in the subsection 3.2.1, this leads to a 224th order

controller. The controller is then reduced to a 10th order transfer function. The obtained results

are presented in figures 5.26 to 5.29.

The upper chart in figure 5.26 compares the FRFs at the TCP when no control is applied with

the closed-loop FRFs Fzw obtained applying the stabilizing controllers designed for the spindle

speed: n = 2’300 rpm and three different axial DOCs: ap = 1.5, 2.0, 2.5 mm. One notes that,

in order to improve the process stability at 2’300 rpm, the controller attempts to generate an

additional resonance close to a harmonic of the tooth passing frequency. In the lower chart of

figure 5.26, it can be seen that, to create such additional resonance at the TCP, the required

controller is quite complex as it must consider the whole dynamics of the active system between

the actuating and sensing points and the TCP. One also notes that the peak magnitude of the

controller transfer function is one order smaller than for the case of the disturbance rejecting

controller.

The stability charts corresponding to the transfer functions of figure 5.26 are represented in figure

5.27. The round, square and diamond dots correspond to the machining conditions considered for

the controller design. All three synthesized controllers succeed to stabilize the selected machining

conditions. The chosen spindle speed corresponds to a lower bound of the stability limit. At


ActivePassive

Axia

lD

OC

(mm

)

Spindle rotational speed (rpm)

Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 34001

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Figure 5.25: Stability lobes diagrams predicted by ZOA method based on predicted FRF at tool tip by

applying a disturbance rejecting H2 controller. DOC: depth of cut.

this spindle speed, more than a doubling of the stability limit is predicted using the synthesized

controllers.

Similarly, stabilizing controllers are synthesized for the spindle speed: n = 2’700 rpm and three

axial DOCs: ap = 3.0, 3.5, 4.0 mm. The resulting closed-loop transfer functions at the TCP are

shown in figure 5.28 and the corresponding stability charts, in figure 5.29. In this case again, the

stabilizing controllers attempt to generate additional resonances at the TCP close to the tooth

passing frequency harmonics. The predicted stability charts indicate a successful stabilization of

the selected machining conditions.

It is here worth reminding that, unlike for the disturbance rejecting controller, the stabilizing

controller is always parametrized and thus designed for one machining condition. A change of

rotational speed or DOC requires a redesign.


k fTP

Active for 2’300 rpm, 2.5 mm



Passive

Frequency (Hz)

MSV

(V/V

)

FRF of controller

MSV

(µm/N

)FRF at TCP

200 400 600 800 1000 1200 1400 1600

×107

0

1

2

30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.26: Predicted FRFs of closed-loop system at TCP and of controller by applying a stabilizing H2

controller designed for machining conditions: n = 2’300 rpm and ap = 1.5, 2.0, 2.5 mm.




Passive

Axia

lD

OC

(mm

)


Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 34000.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 5.27: Stability lobes diagrams predicted by ZOA method based on predicted FRF at tool tip

by applying a stabilizing H2 controller designed for the machining conditions: n = 2’300 rpm and ap =

1.5, 2.0, 2.5 mm.


k fTP




Passive

Frequency (Hz)

MSV

(V/V

)

FRF of controller

MSV

(µm/N

)FRF at TCP

200 400 600 800 1000 1200 1400 1600

×107

0

1

2

30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.28: Predicted FRFs of closed-loop system at TCP and of controller by applying a stabilizing H2

controller designed for the machining conditions: n = 2’700 rpm and ap = 3.0, 3.5, 4.0 mm.




Passive

Axia

lD

OC

(mm

)


Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 34000.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 5.29: Stability lobes diagrams predicted by ZOA method based on predicted FRF at tool tip

by applying a stabilizing H2 controller designed for the machining conditions: n = 2’700 rpm and ap =

3.0, 3.5, 4.0 mm.


5.7.2.3 Controllers Comparison

By comparing the results obtained with both types of controllers, one notes that their work prin-

ciple is completely different. Where disturbance rejection scheme attempts to minimize critical

resonance peaks, the stabilization scheme generates additional resonances at specific frequencies.

This makes disturbance rejecting controllers more suitable when the active system is used to damp

some forced vibrations at the TCP. However, in terms of process stabilization, in this example,

the stabilizing controller achieves greater performance than the disturbance rejecting controller.

Also, even if an increase of the stability limit for every considered spindle speed can not be

guaranteed, stabilization schemes present the advantage, at least theoretically, of preventing any

decrease of the process stability, which is not the case for the disturbance rejection schemes.

From the control effort point of view, the look at the controller FRFs indicates that, in the worst

case, i.e. for a machining process presenting a tooth passing frequency harmonic corresponding

to the frequency of the peak magnitude of the controller frequency response, the disturbance

rejecting controller would require approximately ten times more effort from the actuating sys-

tem. Figure 5.30 compares the control outputs for both, disturbance rejection and stabilization

schemes, for a stabilized machining condition corresponding to a spindle speed of 2’300 rpm and

an axial depth of cut equal to 1.5 mm. At this condition, the performance of the correspond-

ing stabilizing controller can be considered as equivalent to the performance of the disturbance

rejecting controller. In this case, the tooth passing frequency harmonics do not correspond to

the frequency of the peak magnitude of the disturbance rejecting controller transfer function.

However, even in this more favorable case for the disturbance rejecting controller, the amplitude

of the control outputs and their time-derivative of the stabilizing controller in Y -direction are

almost the half of the amplitudes required by the disturbance rejecting controller, as visible in

the two lower right-hand side plots of the figure. These plots correspond to the results of time-

domain simulations considering the machining condition represented by the gray dot in the upper

right-hand side stability chart.

It is still interesting to investigate the influence of the design parameters on the control effort of

stabilizing controllers. To do so, one looks at the effort required by two different stabilizing con-

trollers designed for the same spindle speed but for two different axial depths of cut. Considering

an axial DOC stabilized by both controllers, the controller designed for this specific axial DOC

requires less effort than the controller designed for a higher axial DOC. This is due to the working

principle of the stabilizing controller tending to generate additional resonances. The larger the

axial DOC to stabilize, the greater are these resonance peaks. This also implies higher required

effort by the actuating system.

This is illustrated by figure 5.31, where the corresponding FRFs at TCP, stability charts and

control efforts are represented for two stabilizing controllers designed for the machining condition

P11, a spindle speed of 2’300 rpm and two different axial DOCs, 1.5 and 2.5 mm. For an axial

DOC equal to 1.5 mm, the efforts required by the controller designed for an axial DOC of 2.5 mm

are much more important than for the other one, even if this former controller is susceptible to


uY

(V/s)

uX (V/s)

uY

(V)

uX (V)

Cond. for effort simulation

Axia

lD

OC

(mm

)Spindle rotational speed (rpm)

Stability chart

k fTP

Dist. reject.

Stab.

Passive

Frequency (Hz)

MSV

(µm/N

)FRF at TCP

×104−1 0 1−2 0 2

2000 2500 3000200 400 600 800 1000 1200 1400 1600

×104

−1

0

1

−2

0

2

1

1.5

2

2.5

3

0

0.1

0.2

0.3

0.4

0.5

Figure 5.30: Comparison of control effort between disturbance rejection and stabilization schemes.

stabilize higher axial DOCs.

From a general observation made during the tests, the use of the H∞ norm instead of the H2

does not lead to fundamentally different results. In some particular cases, depending on the type

of encountered critical mode and the considered machining parameters, one norm could achieve

slightly more performance than the other but no general design criterion can be established. So

every case must be carefully evaluated.

Even if the choice of higher order weighting functions might improve the controller performances,

frequency-independent functions are sufficient in our case because no specific influence outside of

the control bandwidth is required. This is due to the fact that no steering but only compensation

actions are necessary and the characteristics of the input and output signals provided by the

processing devices are adequate.

Generally, the determination of stability lobes diagrams only considers the direct FRFs at the

TCP due to the fact that the cross terms usually do not play an important role. However, in

the case of the stabilization scheme, the synthesized controller uses the cross terms to tailor the

process stability and the cross FRFs must thus be taken into account to compute the stability

prediction.


uY

(V/s)

uX (V/s)

Act. rangeuY

(V)

uX (V)


Axia

lD

OC

(mm


Stability chart

k fTP

Stab. for 2’300 rpm, 2.5 mm

Stab. for 2’300 rpm, 1.5 mm

Passive

Frequency (Hz)

MSV

(µm/N

)FRF at TCP

×104−2 0 2−5 0 5

2000 2500 3000200 400 600 800 1000 1200 1400 1600

×104

−2

0

2

−5

0

5

0.5

1

1.5

2

2.5

3

3.5

0

0.2

0.4

0.6

0.8

1

1.2

Figure 5.31: Comparison of control effort between two stabilizing controllers designed for different axial

depths of cut.

5.7.3 Controller Implementation

The designed controllers are implemented into a PXI system using NI LabVIEWTM. The control

design is made with Matlab in continuous-time domain. The resulting continuous transfer

function of the controller C needs to be discretized to be implemented into the digital RT system.

To do so, the bilinear method is used. This method is a conformal mapping that maintains an

equivalent location of the poles and thus the stability properties of the resulting system. The

accuracy of the discretized controller Cz depends on the sampling rate of the RT system. A rule

of thumb recommends to use a sampling rate at least equal to ten times the upper bound of the

control bandwidth. In our case, a sampling rate of 20 kS/s is selected.

Before the filtering of the sensing signals by the transfer function Cz, the controller must first

reorient the sampled signals delivered by the accelerometers in the coordinate system of the

machine. The signals are then bandpass-filtered to increase the roll-off of the controller. Finally,

range and rate limiters prevent the controller to deliver damageable outputs to the actuator

amplifiers. Also, a security routine sets the control output to zero if a maximal number of

overload samples is reached.

5.8 Closed-Loop Frequency Responses at Tool Tip 131

5.8 Closed-Loop Frequency Responses at Tool Tip

Before testing the performance of the implemented control strategy in machining conditions, the

modification of the structural dynamics at TCP induced by the control action is investigated.

As the control objective is based on the transfer functions seen by the process, it is essential to

accurately predict the influence of the control strategy on the TCP. From the flow chart of figure

3.1, the resulting closed-loop frequency response functions Fzw at the tool tip is determined from

the open-loop FRFs matrix of the plant G and from the controller C according to

Fzw(ω) = Gzw(ω) +Gzu(ω)C(ω) S(ω)Gyw(ω) (5.8)

where S is the output sensitivity defined by

S(ω) =(

I−Gyu(ω)C(ω))−1

(5.9)

and G(ω) corresponds to the frequency response function G(iω) with i2 = −1.

The transfer function of the controller is derived from the control design. The experimental open-

loop FRFs of the plant, measured between the TCP and inputs-outputs of the control system

and used for the model identification, can be employed. The identified model may also be used

to derive the FRFs matrix G.

The resulting predictions are then compared with the FRFs measured at the tool tip after the

implementation of the control strategy.

5.8.1 Comparison between Predictions and Experiments

The tool assembly T1 is investigated along each radial direction independently using two uncou-

pled (decentralized) single input single output (SISO) disturbance rejecting H2 controllers with

constant weighting functions. The use of SISO controller presents the advantage of being able

to apply simple design criteria to check the robustness of the resulting closed-loop system. The

control design is based on two 12th order models identified over the frequency range from 200 to

1800 Hz.

The upper chart of figure 5.32 represents the predicted closed-loop FRF Fzw, where a decrease

of the resonance peak of approximately 30% is achieved. This figure shows the predictions based

on the experimental open-loop FRFs and the identified model. The predictions provided by both

methods are very similar, especially in the range of the main resonance peak.

The implementation of this controller leads to the experimental results presented in the lower

chart of figure 5.32. The decrease of 30% is verified and the resulting influence of the controller

on the closed-loop behavior is similar to the predictions in spite of a difference of the resonance

peak amplitude at 700 Hz between the measured open-loop FRF Gzw used for the prediction and

the FRFs after the implementation. This allows to claim that the predictions of the closed-loop

behavior are in good agreement with the experiments.


ActivePassive

Frequency (Hz)

MSV

(µm/N

)

Measured FRF at TCP

Active, based on open-loop exp.Passive, experimentsActive, based on modelPassive, based on model

MSV

(µm/N

)Predicted FRF at TCP

200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Figure 5.32: Predicted and measured FRFs at TCP obtained using decentralized disturbance rejecting

H2 controller designed for the tool assembly T1.

In this case, the use of a MIMO controller, designed from a coupled model of the plant dynamics,

does not achieve better disturbance rejection performances.

5.8.2 Influence of Sensing and Actuating Point on Control Per-

formance

In the design of the above presented controller, the performance is restricted by stability robust-

ness criteria. In order to investigate the best achievable performance, it is interesting to consider

three extreme cases. The first one is when the controller input corresponds to the TCP deviations

z. The controller gets the best available information on the deviations resulting from the process

disturbances that must be minimized. In the second case, the controller gets the information

provided by the integrated sensors but acts at the TCP, so that u is added to w. This can

be considered as the case where the controllability of the controller on the critical modes is the

best. The third case combines the two precedent to get the ideal controller where the sensed

inputs corresponds to the TCP deviations and the actuators act at the TCP. For each of these

5.8 Closed-Loop Frequency Responses at Tool Tip 133

three cases, decentralized SISO H2 controllers are designed in each radial direction to achieve the

highest performance according to the selected robustness criteria.

The results are shown in figure 5.33, where the predictions of the closed-loop point FRFs Fzw

are plotted for the real case and the three extreme cases. The ideal control system (upper left

plot), corresponding to the case where the sensed inputs are the TCP deviations and the actu-

ators act at the same location than the process disturbance, reveals a high level of performance

as the resonance peak of the critical mode is decreased by 66%. In the extreme case where the

sensed signals are the TCP deviations (upper right plot), the controller achieves the same level

of performance as the peak decrease reaches 68%. In the last extreme case, where the sensed

inputs are the signals from the integrated accelerometers and the actuators act directly at the

TCP, the level of performance is limited by the excitation of a higher mode around 900 Hz and

the achievable peak reduction is equal to 19%.

These observations confirm the affirmations made in the previous section 5.7.1 about the poor

observability of the control system and that the main limiting factor of the actual system comes

from the sensing system.

Frequency (Hz)

Control input: y, output: u

Control input: z, output: u

Frequency (Hz)

MSV

(µm/N

)

Control input: y, output: w

ActivePassive

MSV

(µm/N

)

Control input: z, output: w

500 1000 1500 2000500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Figure 5.33: Predicted closed-loop FRFs at TCP based on measured open-loop FRFs for tool assembly

T1 and decentralized H2 controller from different control input and output configurations.


5.9 Stability Charts

In order to evaluate the influence of the active system on the process, the stability charts, repre-

senting the stability of the global system composed by the active machine structure in interaction

with the process in the phase plane of the machining parameters n, ap, are investigated.

First, for a given tool-workpiece combination, based on the measured FRFs at the tool tip and

the values of the machining parameters recommended by the tool manufacturer, the SLDs are

predicted using the proposed methods described in section 3.3. Due to its convenience, the ZOA

method is preferred to the other methods. The tool geometry as well as the radial depth of cut are

selected in order to guarantee good agreement with the conditions assumed by the ZOA method.

In this first step, the values from available data base of the average cutting force coefficients are

used. Based on these predictions, a range of spindle speeds and axial DOCs is selected. Machin-

ing tests are then performed to identify more accurate values of the cutting force coefficients (see

section 5.6) and get refined predictions. Some complementary machining tests are led to validate

the predictions.

The stability charts are experimentally determined by scanning a range of selected spindle speeds.

For each selected spindle speed, an initial stable axial DOC is incrementally increased until reach-

ing the chatter occurrence. At the stability limit, the presence of chatter can be ambiguous. Very

light self-excited vibrations can be present in spite of the fact that the process seems to be stable

(no chatter marks on the surface). For each tool-workpiece combination, the threshold value

σy,lim, used by the chatter detection criterion (see section 5.4), needs to be defined. This value

may differ from one combination to another and must be calibrated. This is also the reason why,

for each spindle speed and axial DOC, a machining test is performed with constant cutting pa-

rameters over a sufficiently long time period so that the overall coupled system can reach a stable

or unstable steady state, that can be accurately determined by post-processing of the acquired

data.

The size of the spindle speed and axial DOC increments are chosen so that a trade-off between

experimental effort and accuracy of results can be found. The range of machining parameters is

selected around the nominal values recommended by the tool manufacturers.

First, the validity of the stability predictions using the proposed methods is investigated by look-

ing at the passive, or open-loop, prototype spindle. The influence on the process stability of the

different implemented control strategies is then predicted, based on the closed-loop FRFs mea-

sured at the tool tip. Finally, these closed-loop stability predictions are experimentally verified.

5.9.1 Process Stability in Passive Configuration

The validity of the stability predictions is a major issue especially for the design of stabilization

schemes. The results provided by the different prediction methods are first compared. Secondly,

in order to validate the stability predictions, the uncertainties related to the predicted SLDs and

5.9 Stability Charts 135

the corresponding experiments are evaluated.

5.9.1.1 Comparison between Stability Prediction Methods

To predict the stability of the overall system formed by the machine structure, the process inter-

action and the active system, four methods are proposed, namely, the zeroth order approximation

(ZOA), semi-discretization (SD), linear time-invariant (LTI) and time-domain simulation (TDS)

methods (see section 3.3). The results provided by these methods are first compared considering

both combinations P11 and P12 and the active system in open-loop configuration. A 14th order

model identified from 200 to 1200 Hz is used to make the computation. This model is the same

as the one used for the validation of the model-based control principle in section 5.7.2. Figure

5.34 shows the obtained predictions using the ZOA, LTI and SD methods, for the case of the

combination P11 (see table 5.3). The upper chart represents the chatter stability limit in terms of

axial depth of cut vs spindle speed and, in the lower chart, the corresponding chatter frequencies

are indicated. The results show a perfect correspondence between ZOA and LTI methods. The

SD method shows very similar results. The same observations can be drawn from the investiga-

tion of the machining condition P12, shown by figure 5.35. The presented results obtained from

the SD method, especially the occurrence of the flip bifurcation around 2’480 rpm for P11 and

6’800 rpm for P12, is verified using time-domain simulation.

As previously mentioned, compared to the other prediction methods, the ZOA method is the

most convenient due to its low computational effort and the fact that it may directly consider the

use of experimental FRFs to predict the SLD. From the above observations, it can be stated that

the ZOA method is representative of the results provided by more accurate methods, such as SD

or TDS, and can thus be used for the stability predictions. So in the following, if not explicitly

specified, the ZOA method is used to predict the process stability.

5.9.1.2 Stability Chart Uncertainty

The agreement between the stability predictions using the ZOA method and the experiment are

now investigated. In order to be able to make a representative comparison, the uncertainty related

to the predictions as well as to the experiments needs to be evaluated.

The procedure used here to determine the uncertainty related to the stability predictions has been

proposed by Duncan [138]. Using Monte Carlo simulation techniques, it evaluates the uncertainty

propagation on the stability chart from several input uncertainty contributors. These contributors

are either related to the TCP dynamics or to the cutting force coefficients.

In a first step, the uncertainty related to some parameters difficult to evaluate is assumed to be

negligible, even if it is probably not the case. For instance, the influence of the spindle rotation

and the cut interaction on the tool tip dynamics as well as the machine dynamics variation due

to the axes displacement during the machining tests are neglected. The influence of the machine


k fTP

Chat

ter

freq

uen

cy(H

z)


SD

LTI

ZOA

Axia

lD

OC

(mm

)Predicted stability charts

1800 2000 2200 2400 2600 2800 3000 3200500

600

700

800

900

10000.5

1

1.5

2

2.5

3

3.5

4

Figure 5.34: Comparison of stability charts predicted by different methods and based on plant model for

combination P11.

temperature is also neglected. The uncertainty related to some other contributors as the effective

depths of cut and the values delivered by the monitoring system can be considered as negligible.

The uncertainty on the TCP dynamics is evaluated from five impact tests made after removing

and putting the tool assembly again in the spindle. The FRF measurements are all realized in

the identical machine axes position and spindle shaft orientation. The figure 5.36 represents the

frequency dependent uncertainties considered on the average frequency responses at the tool tip

in both radial directions for the tool T1.

The uncertainty related to the cutting force coefficients considers all the values identified in the

nine different combinations of spindle speeds and axial depths of cut (see section 5.6). For exam-

ple, in the case of the combination P11, the average values with their corresponding uncertainties

of the radial and tangential shear cutting force coefficients are equal to Krc = 579± 162 N/mm2

and Ktc = 1561± 111 N/mm2, respectively.

The considered uncertainty contributors are thus the frequency response functions at the TCP

Gzw and the radial and tangential shear cutting force coefficients. Before leading the Monte

Carlo simulation process, the correlation between these parameters must be determined. Krc

and Ktc are assumed to be 100% correlated — some tests show a value higher than 95% for

some identical cutting conditions and around 65% over the whole range of selected machining

conditions — as they are obtained from the same cutting tests. So only one independent random


k fTP

Chat

ter

freq

uen

cy(H

z)


SD

LTI

ZOA

Axia

lD

OC

(mm

)Predicted stability charts

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000500

600

700

800

900

10002

4

6

8

10

12

14

Figure 5.35: Comparison of stability charts predicted by different methods and based on plant model for

combination P12.

variable is used for both parameters. The real and imaginary parts of the point FRFs are mainly

uncorrelated even if, at some frequency regions, a correlation close to 1 is observed. Moreover,

the correlation of the FRFs in function of the frequency is close to one. Two independent random

variables are thus considered for each FRF. This represents a conservative assumption. The FRFs

from different radial DOFs can be considered as uncorrelated. So, by taking the workpiece into

account, eight FRFs must be considered to represent the TCP dynamics. This leads to a total of

sixteen independent random variables for the representation of the uncertainty on the machine

structure dynamics. For all the seventeen considered independent random variables a Gaussian

distribution is assumed.

The contribution on the stability prediction uncertainty of the structural parameters — the fre-

quency responses — are compared with the contribution of the cutting force coefficients. The

results (not shown here) point out a greater and spindle speed dependent influence of the struc-

tural parameters.

Figures 5.37 and 5.38 show the superposition between the uncertainty from the stability pre-

dictions after thousand Monte Carlo simulation runs and the experiments for both machining

conditions P11 and P12, respectively. The black dots represent the average values from experi-

ments with their corresponding 2σ uncertainty bar, where σ represents the standard deviation.

The predicted average values are represented by the continuous red line and the 2σ uncertainty


Uncertainty on FRF at TCP in Y direction

Mag

nit

ude

(µm/N

)

Frequency (Hz)

Average ±σ

Average values

Uncertainty on FRF at TCP in X directionM

agnit

ude

(µm/N

)

200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Figure 5.36: Considered uncertainties related to the direct frequency responses at the tool tip in both

radial directions for the tool T1. σ: standard deviation.

range by the interrupted red lines. Moreover, in figure 5.38, the experimental stability limit for

the spindle speed equal to 6’500 rpm is not represented due to the fact that the chatter limit

could not be reached using the maximal axial DOC for the insert corresponding to 11 mm.

In the case of P11, predictions and experiments are in good agreement, as an overlapping between

their respective uncertainty is visible for the axial DOCs as well as for the chatter frequencies.

However, this is not the case for P12. Indeed, in figure 5.38, a gap is obvious between the pre-

dictions and the experiments around the spindle speed range of 9’000 rpm. At 6’500 rpm, the

experimental stability limit is larger than 11 mm, which is above the predicted uncertainty range.

This is the case at 4’000 rpm as well. Generally speaking, for the case of P12, the zones of lowest

stability are in good agreement with the experiment but, by looking at the global pattern of the

predictions, the experimental stability lobes seem to be slightly shifted on the left-hand side of

the predictions. This may typically result from a shifting in lower frequencies of the most critical

mode at the TCP located around 700 Hz. Physically, a decrease of the stiffness or an increase of

the mass of the corresponding mode could explain this gap.

From a more global point of view, based on the results provided by the other tool assemblies

but also with different spindle types, some gaps in the stability predictions are visible in the

majority of the investigated machining conditions. Moreover, a priori no systematic tendency


kfTP

kfSR

Experiments

Average values ±2σ

Average values from all runs

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 3400

600

800

1000

0.5

1

1.5

2

2.5

3

3.5

4

Figure 5.37: Predicted and experimental stability charts with corresponding uncertainty for combination

P11.

can be identified in order to give some indications on the potential error sources.

In order to identify and try to compensate these prediction gaps, the modeling procedure must

be deeper investigated. The predictions are derived from the stability analysis of the machine

structure dynamics coupled with the milling process. Both corresponding models are based on

certain assumptions and on parameters that must be experimentally identified. The machine

structure model is obtained through frequency response measurement via impulse testing and

the cutting force coefficients are derived from the cutting forces measured using dynamometer

table during machining tests.

In our case, the main modeling assumptions related to the structure dynamics correspond to

a linear, asymptotically stable and time-invariant behavior. The model of the milling process

assumes cutting coefficients independent of spindle speed and depths of cut. The validity of these

assumptions has been verified which allows to state that the most probable source of error comes

from a difference between the experimental conditions used for the identification of the model

coefficients and the real machining conditions. For example, the TCP dynamics is identified on

the non-rotating tool which of course does not correspond to the reality of the process. This is

equivalent to assume that the TCP dynamics is independent of the spindle speed and the cutting

depths. Also, the excitation force generated by the impulse hammer might not be representative

of the forces generated by the process.

In the considered combinations P11 and P12, the influence of the rotation or the cutting force


k fTP

k fSR

Experiments

Average values ±2σ

Average values from all runs

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)Stability chart

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000

500

1000

15000

5

10

15

20

Figure 5.38: Predicted and experimental stability charts with corresponding uncertainty for combination

P12.

is not able to explain the gap with the effective stability limit. The most plausible source seems

to come from the influence on the TCP dynamics of the additional mass induced by the internal

cooling lubricant required for the machining of the workpiece material M2, but not for M1, and

flowing through the cutting head.

The improvement of stability predictions would require some further efforts. Unfortunately, a

deeper investigation of this aspect goes beyond the scope of this thesis.

5.9.2 Process Stability in Active Configuration

The next step consists in experimentally investigating the influence of the implemented control

strategies in different machining conditions. Undoubtedly this constitutes the key part of this

work. First, the disturbance rejection and then the stabilization schemes are presented.

5.9.2.1 Disturbance Rejection Scheme

Figure 5.39 represents the experimental and predicted stability charts corresponding to the com-

bination P11 in passive and active configurations using the H2 controller described in section 5.8

and corresponding to the FRFs shown in figure 5.32.


The dots represent the investigated experimental machining conditions. In the upper plot, the

green dots correspond to stable process with and without the influence of the active vibration

control (AVC) action. The red squares are unstable conditions with and without control. The

blue diamonds represent conditions stabilized by the control action and the black triangles coin-

cide with conditions destabilized by the control. The continuous black line traces the resulting

experimental stability limit in passive state and the interrupted line, the stability limit in active

state. The interrupted red line is the stability limit predicted by the ZOA method based on the

measured FRFs at TCP with control and the identified average cutting force coefficients. This

line should coincide with the black interrupted line. The blue continuous line corresponds to the

predicted stability limit in passive state, which can be compared with the continuous black line.

The experimental dots surrounded by a gray dot represent the conditions where the controller

gain has to be reduced due to an overload of the actuator input range from the controller output.

In the lower plot, the blue diamonds are the chatter frequencies measured at the stability limit

during the machining tests using the prototype spindle in passive state. They should correspond

to the continuous blue line provided by the predictions. The interrupted red line represents the

chatter frequencies predicted with the influence of the control action.

k fTP

k fSR

Exp. passive

Experimental activeExperimental passiveActuator overloadChat. w/ and w/o AVCDestabilized by AVCStabilized by AVCStable w/ and w/o AVCZOA activeZOA passive

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)

Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 3400500

600

700

800

900

10000.5

1

1.5

2

2.5

3

3.5

4

Figure 5.39: Predicted and experimental stability charts for combination P11 obtained using disturbance

rejecting controller.

The stability predictions in both open and closed-loop configurations are globally in accordance

with the experimental results, especially for the spindle speeds corresponding to the zones of


lowest stability.

The active system is susceptible to increase the unconditional experimental stability limit by 55%

and a maximal stability increase equal to 73% is locally achieved at 2’300 rpm.

It is still to note that for most of the spindle speeds the controller reference output u reaches the

limits of the actuating system capabilities before attaining the limit of process stability.

The control objective of disturbance rejecting controllers is not only to stabilize the process but

to reduce the amplitudes of vibrations. It is thus interesting to look at the influence of the

control action on the amplitudes of the vibrations measured by the integrated sensing system

along the stability limit. Figure 5.40 represents this influence. The blue bars correspond to the

variations due to the active system in case of self-excited vibrations coinciding with the upper

dots represented in the stability chart (upper plot) and the green bars, to the lower dots. The

horizontal interrupted lines correspond to the average value for all investigated spindle speeds.

The results show a general reduction of the amplitudes of 17% in case of forced vibrations and

22% in case of self-excited vibrations. Except at 2’600 rpm, the controller is able to reduce the

amplitude of the forced vibrations for all tested spindle speeds. In the cases where the controller

is able to stabilize the process, an important decrease of vibration amplitudes is visible. On

the contrary, when the control action destabilizes the process, a logical increase of the vibration

amplitudes is noticeable at 2’100 rpm. However, at 2’800 rpm, a reduction of the amplitudes is

visible. This is explained by the fact that the amplitudes of the self-excited vibrations induced

by the controller are lower than the forced vibrations without control action.

5.9.2.2 Stabilization Scheme

The stabilizing controllers for the machining conditions previously presented are investigated

here. The design step explicitly considers the process and tries to guarantee the process stability

for a specific spindle speed and axial depth of cut. The design is based on the LTI model of the

coupled system consisting of the plant dynamics and the process. In the case of the machining

condition P11, the model used for the design of the stabilizing controllers lies on a 24th order

model of the plant identified over the frequency range from 200 to 1800 Hz (see figure 5.19) and

a 20th order Padé approximation of the time delay. Based on this approximated model, a H2

controller with constant weighting functions is synthesized. The resulting controller corresponds

to a 504th order filter reduced to a 10th order transfer function.

To validate the stabilization scheme, several controllers are designed to improve the process sta-

bility limit at specific spindle speeds. Figure 5.41 shows the comparison between the SLD in

passive and active configurations predicted by the model used for the control design and the

experimental results performed at the corresponding spindle speeds. The signification of the dif-

ferent represented lines and dots is similar to those previously explained and related to figure

5.39. The magenta circles indicate the machining parameters selected for the control design.

One notes a general tendency to get in practice slightly lower stabilizing performance than pre-


Average SEVAverage FVSelf-excited. vib.Forced vib.

Chat. w/ and w/o AVCDestabilized by AVCStabilized by AVCStable w/ and w/o AVCZOA activeZOA passive

Var

iati

on(%

)


Variation of vibration amplitudes (RMS) due to AVC

Axia

lD

OC

(mm

)Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 3400

−50

0

50

1

2

3

4

Figure 5.40: Influence of disturbance rejecting controller on RMS values of vibrations measured by inte-

grated sensors for combination P11. FV: forced vibration; SEV: self-excited vibration.

dicted and, for the spindle speeds of 2’100 and 2’900 rpm, the controller does not succeed to

stabilize the process up to the axial DOC selected for the control design. This trend towards

a reduction of the performance is most probably due to some modeling errors. However, for

all investigated spindle speeds, an effective increase of the stability limit, representative of the

tendency predicted by the control design, is noticeable.

After the implementation of the controller into the RT system, the FRF at the tool tip in open

and closed-loop are measured. Based on these measurements, the stability lobes diagrams are

predicted using the ZOA method. These stability diagrams must correspond to the SLDs pre-

dicted by the control design presented in figure 5.41.

Figure 5.42 shows the SLD predictions based on the measured FRFs for the stabilizing controller

designed at n = 2’300 rpm and ap = 2.0 mm. A good agreement is visible between these SLDs

and those derived from the identified model shown in figure 5.41 (middle left chart).

Figure 5.42 also compares the predictions with the experimental results obtained from machining

tests, represented by the dots. In this case, not only the spindle speed selected for the control

design is tested but the whole spindle speed range is experimentally investigated. Therefore, the

predicted influence of the active system on the process stability can be more broadly verified.

Based on the experiments without AVC, the experimental stability limit in passive condition is

represented by the continuous black line. The comparison of this line with the corresponding


Selected cond. for contr. des.Chatter w/ and w/o AVCStabilized by AVCStable w/ and w/o AVCZOA activeZOA passive


n = 3’100 rpm, ap = 2.0 mm

n = 2’900 rpm, ap = 2.5 mm

n = 2’700 rpm, ap = 3.0 mmA

xia

lD

OC

(mm

)


n = 2’500 rpm, ap = 2.0 mm

Axia

lD

OC

(mm

)

n = 2’300 rpm, ap = 2.0 mm

Axia

lD

OC

(mm

)n = 2’100 rpm, ap = 2.5 mm

1800 2000 2200 2400 2600 2800 3000 3200 34001800 2000 2200 2400 2600 2800 3000 3200 3400

1

2

3

4

1

2

3

4

1

2

3

4

Figure 5.41: Predicted stability charts based on plant model obtained using stabilizing controllers designed

for different machining conditions.

predictions, given by the blue line, indicates a quite good agreement between both. This is also

verified by looking at the chatter frequency chart where the values identified during the machining

tests, represented by the blue diamonds, are located close to the predictions.

Due to the limitations imposed by the actuating system, the predictions in active configuration can

not be fully verified. However, at the spindle speeds where a stabilizing effect of the AVC system

is predicted, an increase of the stability limit is observed. Also, for the spindle speeds predicting

a detrimental effect from the AVC system, effective destabilizing influences are detected. This

allows to state that, in this case, the correspondence between the SLD predictions and experiments

is satisfactory.

Furthermore, the machining condition selected for the control design is successfully stabilized by

the active system. At 2’300 rpm, an increase equal to 91% of the maximal stable material removal

rate, which is directly proportional to the axial DOC, is achieved.


k fTP

k fSR

Exp. passive

Cond. for control designExperimental passiveActuator overloadChatter w/ and w/o AVCDestabilized by AVCStabilized by AVCStable w/ and w/o AVCZOA activeZOA passive

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)Stability chart

1800 2000 2200 2400 2600 2800 3000 3200 3400500

1000

15000.5

1

1.5

2

2.5

3

3.5

4

Figure 5.42: Predicted and experimental stability charts obtained using stabilizing controller designed for

combination P11 with n = 2’300 rpm and ap = 2.0 mm.

All these observations allow to state that the concept of stabilization scheme is valid and can

be successfully implemented. However, several interrogations remain. The first one concerns the

performance robustness of the concept regarding modeling errors. To investigate this aspect, the

machining condition P12 is used, as in this case some gaps between the stability predictions and

the reality have been pointed out (see subsection 5.9.1).

Four different spindle speeds are investigated. Figures 5.43 and 5.44 show the obtained results

for a selected spindle speed equal to 5’000 rpm and 11’000 rpm, respectively. At these spindle

speeds, the stability predictions without control action, i.e. in passive configuration, are in good

agreement with the experiments, as noticeable on figure 5.38. In these cases, even if like for the

combination P11 the effective performance of the stabilizing controller is less than the predictions,

the stabilizing influence corresponds to the tendency predicted by the design.

At 9’000 rpm, the prediction in passive configuration does not correspond to the reality. The

resulting control design at this spindle speed is shown in figure 5.45. In this case, even if a gap

between the open-loop predictions and the reality is obvious, the stabilizing influence predicted

by the control design is in good agreement with the experiments. This means that a gap in

the open-loop predictions does not necessarily induce a reduction of the model-based control

performances.

The fourth investigated spindle speed corresponds to 7’000 rpm. From the SLD without control

action, shown in figure 5.38, it can be seen that, at this spindle speed, chatter problem does


k fTP

Exp. passive

Cond. for control designChatter w/ and w/o AVCStabilized by AVCStable w/ and w/o AVCZOA activeZOA passive

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)Stability chart

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000500

1000

15002

4

6

8

10

12

14

Figure 5.43: Predicted stability charts based on plant model obtained using stabilizing controller designed

for combination P12 with n = 5’000 rpm and ap = 6.0 mm.

not come from the most critical natural mode at 700 Hz but is mainly due to the mode close

to 900 Hz and may also be generated by the modes located around 1’500 Hz. In this case, the

predictions in passive configuration are in accordance with the experiments. Nevertheless, the

comparison between the control predictions and the results from the machining tests, represented

in figure 5.46, indicates two completely different behaviors. Indeed, where the control design

predicts a stabilization of the axial depth of cut equal to 10 mm, the experiments show a desta-

bilizing influence from the control action that decreases the stability limit from 9, in open-loop,

to 8 mm, with control. By looking at the SLDs predicted from the FRFs measured at the tool

tip, represented in figure 5.47, an additional lobe at the spindle speed of 7’000 rpm is visible.

This lobe decreases the stability limit predicted by the control design in figure 5.46 and explains

the destabilizing influence observed in practice. Moreover, by looking at the corresponding pre-

dicted chatter frequency, this lobe is induced by the natural mode located close from 1’500 Hz.

Figure 5.48 shows the predicted and measured FRFs at the TCP in both, passive and active

configurations. The predictions and the experiments show globally similar trends. At 1’500 Hz,

an obvious difference is visible where an increase of the resonance peak under the control action

is noticeable in practice. This increase is due to modeling errors coming from the identification

of the plant and explains the resulting detrimental effect observed on the stability limit. Indeed,

the closed-loop behavior at the tool tip is determined by all the transfer functions — sixteen if

the coupled system in the radial plane is considered — between the TCP and the inputs-outputs

of the control system. The combination of a priori acceptable modeling errors in each of these


k fTP

Exp. passive


Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)Stability chart

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000500

1000

15002

4

6

8

10

12

14



transfer functions can generate unacceptable errors in the resulting closed-loop behavior at the

TCP. This is especially susceptible to occur when looking at less critical natural modes which

can play a dominant role for some spindle speed ranges of higher stability, as it is the case here.

To summarize, one can conclude that some prediction errors in the open-loop behavior do not

necessary induce a decrease of the control performance. Conversely, a good agreement of the

open-loop predictions does not guarantee a satisfactory performance due to the fact that the

modeling errors of the plant dynamics, not only at the TCP but in the whole internal structure

and not only around the most critical modes but also around less dominant dynamics, play a

determinant role in the stability prediction in closed-loop.


k fTP

Exp. passive


Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)Stability chart

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000500

1000

15002

4

6

8

10

12

14



k fTP

Exp. passive

Cond. for control designChatter w/ and w/o AVCDestabilized by AVCStable w/ and w/o AVCZOA activeZOA passive

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)

Stability chart

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000500

1000

15002

4

6

8

10

12

14




k fTP

k fSR

Exp. passive

Cond. for control designChatter w/ and w/o AVCDestabilized by AVCStabilized by AVCStable w/ and w/o AVCZOA activeZOA passive

Chat

ter

freq

uen

cy(H

z)


Axia

lD

OC

(mm

)

Stability chart

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000500

1000

15002

4

6

8

10

12

14

Figure 5.47: Predicted stability charts based on measured FRFs obtained using stabilizing controller

designed for combination P12 with n = 7’000 rpm and ap = 10.0 mm.


Frequency (Hz)

MSV

(µm/N

)

Measured FRF at TCP

k fTP

Active for 7’000 rpm, 10 mm

Passive

MSV

(µm/N

)

Predicted FRF at TCP

200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5.48: Predicted and measured FRF at TCP obtained using stabilizing controller designed for

combination P12 with n = 7’000 rpm and ap = 10.0 mm.


5.9.3 Intermediate Conclusions

From the above observations, one can state that the proposed active system using optimal control

strategies is able to improve the process stability using both types of control approaches: distur-

bance rejection and stabilization schemes. In conformance with the predictions, the stabilization

scheme shows greater performance and efficiency for the mitigation of chatter. But disturbance

rejecting controllers present the advantage of being susceptible to damp forced vibrations as well.

The comparison between the stability predictions and the experiments shows a good agreement

at the lower boundaries of the stability charts, i.e. close from the unconditional stability limit.

In these zones, the potential for stability improvement is also greater than for the spindle speeds

corresponding to high stability pockets. The reason is that in the areas of high stability, less

dominant dynamics of the machine structure is involved. The potential of action from the control

system on this dynamics is usually much lower than on the resonance peaks. This less dominant

dynamics is also more difficult to be accurately modeled which makes its control more risky.

152 6. Main Performance Limitations of Active System

Chapter 6

Main Performance Limitations of

Active System

From the previously described experiments, several limiting factors for the performances of the

active system are identified. The main limiting factors result from its observability properties and

the control effort exceeding the actuating system capabilities. These limitations are discussed in

the following sections and some ways to deal with are proposed.

6.1 Sensing Point

Some previous observations pointed out the possibility to improve the performance of the system

by relocating the sensing point (see subsection 5.8.2). To investigate the potential of performance

and efficiency improvement, a complementary sensing point y′ at the spindle nose is considered.

Using an accelerometer located on the tool assembly at the root of the HSK clamping system, the

FRFs matrix G is measured. A 14th order model is identified from these measurements. The per-

formance and the efficiency of the controllers synthesized with this complementary model are then

compared with the model considering the original sensing point y. First, the maximal achievable

performance of disturbance rejecting controllers based on these two models is compared. The

results are shown in figure 6.1, representing the case of the machining condition P11. On this

figure, a larger reduction (50% instead of 32%) of the critical resonance peak is achieved by the

controller considering the sensing point y′. A corresponding higher unconditional stability limit

is also noticeable. However, considering a machining condition at 2’300 rpm and an axial DOC

equal to 1.5 mm, the control effort required by this complementary controller is greater than for

the original one. This is visible in the lower right-hand side charts representing the controller

outputs u in both radial directions X and Y and their time derivatives during the stabilized

machining process corresponding to the gray dot in the chart above.

To compare the efficiency, two stabilizing controllers, based on both identified models, are de-

6.1 Sensing Point 153

signed for the same machining conditions, namely 2’300 rpm and an axial DOC equal to 2.5 mm.

The obtained results are presented in figure 6.2. Both controllers achieve the same performance

as they both are able to stabilize the selected machining condition. Nevertheless, for this level of

performance, one notes that the controller based on the complementary sensing point y′ requires

much less effort than the original one. This might be intuitively understood by the fact that a

sensing point closer from the TCP provides more relevant information making the controller able

to focus its effort on the most important disturbances and to act more efficiently.

These observations allow to state that the relocation of the sensing point close to the spindle nose,

using eddy current sensors for instance, could improve the performances of the active system. The

use of non-contact displacement sensor instead of piezoelectric accelerometers would also solve

the problem of spindle speed limitation, discussed in section 5.1.

uY

(V/s)

uX (V/s)

Act. range

uY

(V)

uX (V)


Axia

lD

OC

(mm

)


Stability chart

k fTP

Dist. reject. with y′

Dist. reject. with y

Passive

Frequency (Hz)

MSV

(µm/N

)

FRF at TCP

×104−2 0 2−10 0 10

2000 2500 3000200 400 600 800 1000 1200 1400 1600

×104

−2

0

2

−10

0

10

1

1.5

2

2.5

3

0

0.1

0.2

0.3

0.4

0.5

Figure 6.1: Comparison of control effort between disturbance rejecting controllers with two different

sensing points, y and y′.


uY

(V/s)

uX (V/s)

Act. rangeuY

(V)

uX (V)


Axia

lD

OC

(mm


Stability chart

k fTP

Stabilization with y′

Stabilization with y

Passive

Frequency (Hz)

MSV

(µm/N

)FRF at TCP

×104−4 −2 0 2 4−10 0 10

2000 2500 3000200 400 600 800 1000 1200 1400 1600

×104

−4

−2

0

2

4

−10

0

10

0.5

1

1.5

2

2.5

3

3.5

0

0.2

0.4

0.6

0.8

1

1.2

Figure 6.2: Comparison of control effort between stabilizing controllers with two different sensing points,

y and y′.

6.2 Filtering of Forced Vibration Component

As noticed during the experimental tests, the output capabilities of the actuating system often

constitute a limiting factor for the performances of the active system. A size increase of the actu-

ating system can hardly be imagined due to the design constraints imposed by the functionality

of the spindle into the machine tool. As demonstrated in the previous section, the use of another

sensing point could improve the control efficiency and thus, for a given performance level, reduces

the required control effort. Another way to improve the active system efficiency consists in fo-

cusing the control effort on the stabilization of the process. Indeed, the current control system

receives information delivered by the integrated sensors on the structural deviations of the plant.

A part of these deviations corresponds to the forced vibrations induced by the kinematics of the

process and another part, in case of chatter occurrence, comes from self-excited vibrations. So, in

the current configuration, the control system receives both information and generates an action

in response to the forced and self-excited vibrations. As demonstrated in subsection 3.2.3, if the

stabilization of the process is the only control objective, the action relative to the self-excited

vibration is sufficient. Assuming that the input signal y coming from the sensors can be divided

6.2 Filtering of Forced Vibration Component 155

into a forced (synchronous) and a self-excited vibrations (asynchronous) part and feeding the con-

troller only with the self-excited component, the control effort can be reduced. This is especially

true in high stability pockets between two lobes, where the forced vibration part is important.

The decomposition of the signal can be done based on the frequency properties of the forced

and self-excited vibrations, as by definition both should occur at different frequencies. Figure

6.3 represents the decomposition of the signal obtained from the integrated sensors in the case of

chatter apparition. This decomposition is obtained using the recursive Kalman filter described

in the following text.

Frequency (Hz)Frequency (Hz)

k fSR

FFT

mag

nit

ude

(V)

Frequency (Hz)

Asynchronous signal, ya

Time (s)

Synchronous signal, ys

Time (s)

Full signal, y = ys + ya

Am

plitu

de

(V)

Time (s)

600 800 1000600 800 1000600 800 1000

0 0.50 0.50 0.5

0

0.1

0.2

0.3

0.4

0.5

−2

−1

0

1

2

Figure 6.3: Decomposition in synchronous and asynchronous components of the signal delivered by the

sensor during chatter.

As the spindle speed and the number of teeth are a priori known, the frequencies corresponding

to the forced vibrations components can be determined. However, their amplitude and phase

must be estimated. The idea is to build an estimate ys of the synchronous part of the signal

delivered by the sensing system and to subtract this component from the full signal y to get the


self-excited part ya. The estimate of the synchronous component has the following form

ys(t) =N∑

j=N0

aj sin jωt+ bj cos jωt (6.1)

where ω is the tooth passing angular frequency or the pulsation of the spindle rotation in case of

runout. N0 and N are the initial, respectively the final number of the considered harmonics of

ω. The above relation can be written in the compact form as

ys(t) = h(t) x (6.2)

with

h(t) =[

uTN0

(t) uTN0+1(t) · · · uT

N (t)]

, uj(t) =

[

sin jωt

cos jωt

]

, j = N0, . . . , N,

x =[

aN0bN0

aN0+1 bN0+1 · · · aN bN

]T. (6.3)

The estimate of the self-excited part of the signal is given by

ya(t) = y(t)− ys(t). (6.4)

Recursive procedure, using recursive Kalman filter (RKF) or NLMS adaptive filter, are used here

to estimate the coefficient vector x. An estimation error

e(t) = y(t)− ys(t) (6.5)

is defined. This error tends to ya as the filter coefficients aj , bj converge.

Figure 6.4 shows the convergence of the coefficients estimated in open-loop, i.e. without any con-

trol action, using the RKF in the case of an unstable operation. The upper left chart represents

the superimposition of the recorded time signal y with the corresponding estimate ys of the syn-

chronous part and the resulting error signal e. The upper right chart presents the corresponding

linear spectra after the convergence of the filter coefficients where it is visible that the resulting

estimate of the asynchronous part of the signal contains all frequency components of y without

the harmonics of the spindle rotational speed. The lower plot shows the evolution with the time

of the coefficients of the filter for the first considered harmonic. In this case, a good estimation

of the synchronous part is visible with a convergence time of approximately 0.15 s. Figure 6.5

represents the case where no chatter occurs. In this case, the asynchronous part is close to zero

and the convergence is faster. In some cases, the estimation of the synchronous part becomes

difficult. For instance, when the process is not in steady state or when the chatter frequencies are

very close to the forced vibration harmonics. The latter case is represented in figure 6.6, where

problem of convergence and estimation are noticeable.

Figure 6.7 shows the results obtained by the NLMS filter based on the same measurements used

in figure 6.4 with the RKF. Even if the coefficients converge towards different values than for the

RKF, the estimation seems to be satisfactory.


k fSR

Signals, frequency-domain

FFT

mag

nit

ude

(V)

Frequency (Hz)

b N0,k

(V)

Time (s)

Coefficients

aN

0,k

(V)

e

ys

y

Signals, time-domainA

mplitu

de

(V)

500 600 700 800 900 1000

0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

−0.02

0

0.02

−0.02

0

0.02

−2

−1

0

1

2

Figure 6.4: Estimation of synchronous part of signal using recursive Kalman filter during chatter occur-

rence. y: full signal; ys: estimate of synchronous part; e: error signal; aN0,k, bN0,k: filter coefficients of

the N0th spindle rotational speed harmonic estimated at the kth iteration.

Several difficulties make a correct signal decomposition in real-time, before that the signals are

delivered to the controller, not an easy task. First, the computational time required at each

iteration by the filter must be sufficiently shorter than the sampling period to guarantee real-

time properties for the controller. The sampling frequency required by the RT controller to

guarantee an adequate bandwidth is 20 kS/s. The implementation of the RKF shows a required

computational time too important to be implemented in real-time. NLMS filter is used instead,

as it satisfies the real-time constraints.

Figure 6.8 presents the experimental results using the NLMS. On the left-hand side of the figure

is represented the case of control without filtering of the signal delivered by the sensors in X-

direction during the machining condition P11 at 2’900 rpm with an axial DOC equal to 1.5 mm.

In the upper plot, one notes that the control output reaches the limit of the actuator input range

and after approximately 2.4 s, a security switches off the controller. From this time, without

control action, chatter appears, as noticeable on the spectrogram of the acceleration signal by

the presence of chatter frequencies in-between the spindle rotational speed harmonics. Using

the NLMS filtering, a reduction of the control output is achieved and no actuator overloading


k fSR


FFT

mag

nit

ude

(V)

Frequency (Hz)

b N0,k

(V)

Time (s)

Coefficients

aN

0,k

(V)

e

ys

y

Signals, time-domainA

mplitu

de

(V)

500 600 700 800 900 1000

0 0.05 0.1 0.15 0.2

×10−3

×10−3

0

0.02

0.04

0.06

0.08

0.1

0.12

−5

0

5

−5

0

5

−1

−0.5

0

0.5

1

Figure 6.5: Estimation of synchronous part of signal using recursive Kalman filter during stable operation.

appends anymore. This allows to stabilize the process over the whole time period.

A fundamental issue of this technique concerns the convergence time required by the algorithm.

During this convergence time, the filter tends to make an identification of the answer delivered

by the closed-loop system formed by the plant, the process and the controller. During this

convergence time, the estimate of the synchronous part is wrong and thus, the corresponding

wrong estimate of the self-excited component can lead to severe problems of convergence or even

instability if transmitted to the controller. The easiest way to deal with this problem of system

identification in closed-loop is to wait until the convergence of the recursive filter is reached before

transmitting the filtered information to the controller. This means that, at the beginning of the

cut interaction, the controller must work without filtering of its input signal. In parallel, the

recursive filter must start the estimation and as soon as a convergence criterion is reached, the

controller uses the filtered signal. The treatment of this aspect would require further investigation

before being implemented.


k fSR


FFT

mag

nit

ude

(V)

Frequency (Hz)

b N0,k

(V)

Time (s)

Coefficients

aN

0,k

(V)

e

ys

y

Signals, time-domain

Am

plitu

de

(V)

700 750 800 850 900

0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

−0.05

0

0.05

−0.05

0

0.05

−4

−2

0

2

4

Figure 6.6: Estimation of synchronous part of signal using recursive Kalman filter during chatter operation

with chatter frequencies close to the forced vibration components.


k fSR


FFT

mag

nit

ude

(V)

Frequency (Hz)

b N0,k

(V)

Time (s)

Coefficients

aN

0,k

(V)

e

ys

y

Signals, time-domain

Am

plitu

de

(V)

500 600 700 800 900 1000

0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

−0.1

0

0.1

−0.05

0

0.05

−2

−1

0

1

2

Figure 6.7: Estimation of synchronous part of signal using NLMS adaptive filter during chatter occurrence.


(dB

)

Time (s)

Spectrogram of acceleration

k fSR

Fre

quen

cy(H

z)

Time (s)

Spectrogram of acceleration

Time (s)

Acc

eler

ation

(m/s2

)

Time (s)

With NLMS filtering

Act. range

Without filtering

Act

uato

rsi

gnal(V

)

0 2 4 6 80 2 4 6 8

−10

0

10

20

30

40

50

60

70

80

90

600

650

700

750

800

−100

0

100−4

−2

0

2

4

Figure 6.8: Machining condition P11 at 2’900 rpm with an axial DOC of 1.5 mm, with (right-hand side)

and without (left-hand side) the use of normalized least mean square adaptive filter for the conditioning of

the controller input signals. Without filtering, actuator overloading occurs switching off the active system

after approximately 2.4 s. From this time, chatter appears. With filtering, the reference signal remains

within the actuator output limits and the process is stabilized over the whole machining period.

162 7. Conclusion and Outlook

Chapter 7

Conclusion and Outlook

This work demonstrates the possibility to improve the productivity of milling operations in stable

conditions by mitigating the occurrence of regenerative chatter based on an active structural

control concept using a mechatronic system integrated in the main machine tool spindle. The

system is composed of accelerometers measuring the radial deviations of the front bearing support

of the shaft driven by piezoelectric stack actuators.

The innovative design of the presented active system respects the constraints imposed for its even-

tual industrialization and has been patented. Several recommendations susceptible to improve

its performances are proposed. A model describing the physics of the overall system, formed

by the interaction between the machine structure, the active system and the milling process, is

defined and validated. The elaborated control strategy, based on optimal control, demonstrates

the ability to mitigate chatter. A general formulation of the control objective is presented. This

formulation makes possible the consideration of two different approaches, namely disturbance re-

jection and stabilization schemes, as well as different norms for the minimization problem. This

allows to opt for the most appropriate control objective according to the situation. Disturbance

rejecting controllers are specially well suited in low spindle speed range for the increase of the

unconditionally stable depth of cut as well as for the damping of forced vibrations. For the miti-

gation of chatter at moderate and high spindle speeds, stabilization scheme presents the highest

performance and efficiency. Both types of control strategy are experimentally validated in repre-

sentative machining conditions. However, their industrial application remains challenging due to

the lack of reliability of process stability predictions. This is especially true for tools presenting

critical modes with high natural frequency, complex tool geometries and operations with high

variability of the process conditions. The complexity of the actual controller setup procedure is

a second aspect that makes the use of predictive optimal control strategies hardly implementable

in an industrial framework. For example, in order to design a stabilizing controller, the frequency

responses at the tool tip as well as between the inputs and outputs of the controller must be

measured. The milling forces also need to be monitored during some preliminary cutting tests.

These procedures requiring time and expensive measurement devices are usually not economically

163

justifiable in production.

Nevertheless, on the basis of these previous remarks, it can be concluded that the main objec-

tives of this thesis have been attained but the requirements for the industrial implementation

need further investigations.

This thesis constitutes a first step in the study of active structural control of regenerative chatter

at inspire AG, respectively at the Institute for Machine Tools and Manufacturing (IWF) of the

ETH Zurich, and is attempting to facilitate the transfer of such mechatronic solutions from the

academic field to the machine tool industry. The applicability of the proposed concept is limited

by several aspects. As demonstrated in chapter 6, a modification in the design of the mechatronic

system, such as the location and the dimension of the actuating system, is susceptible to improve

its performances but the fundamental factor limiting its implementation comes from the lack of

reliability of the process stability predictions. In section 5.9, the comparison between the pre-

dicted stability charts and the experiments indicates a general good agreement especially along

the stability limits at low depths of cut. In these regions, the influence of the active system on

the process stability predicted by the control design corresponds to the reality. However, the in-

vestigations in higher stability regions show a greater sensitivity to modeling errors which makes

their active control more complicated. As a result, it is at the moment not possible to reliably

predict if the active system is able to provide satisfactory performances or not for a given ma-

chining condition. So, before investigating the proposed concept on broader range of machining

conditions, further efforts must be brought in order to define more accurate model of the system

representing the plant dynamics coupled with the milling process. As seen in this work, this is not

an easy task and it will require the definition of novel experimental procedures able to capture

the behavior of the tool tip during its interaction with the workpiece material in representative

machining conditions.

A predictive approach is used here for the control design, which means that the controller corre-

sponds to a static filter whose parameters are determined during the control design step and are

not influenced by the data collected during the cut. However, it is conceivable to use this infor-

mation to adapt or eventually correct the initial model and improve the controller performances.

If the stability predictions cannot be sufficiently improved, at least a reliable uncertainty model

must be determined so that a work-around strategy based on robust stabilization control could

be developed.

Another issue comes from the high diversity of machining conditions encountered in the shop floor

susceptible to face some chatter problems and making the stability predictions very challenging.

In the present study, milling operations with only rectilinear tool paths are considered. The

structure of the workpiece has been selected such that its structural compliance can be neglected

compared to the tool tip compliance. Furthermore, simple tool geometries have been chosen so

that the modeling assumptions can be considered as valid. If the previously mentioned limita-

tions related to the stability prediction can be solved, a great potential of development remains

by applying the proposed concept to the cases of simultaneous multiple axes machining, flexible

164 7. Conclusion and Outlook

workpieces and complex tool geometries.

A single type of methods is not able to provide the best benefit for the customer over the wide

spectrum of machining conditions found in practice. This is why the ultimate objective is to

combine different approaches, such as process, regeneration disturbing and structural passive

methods, in order to reach a more general control scheme. Also, several active structural systems

with different time constants might be associated in order to cover a wider frequency range. For

instance, using high dynamic drives, the main axes of the machine could be employed to influence

the first natural modes of the machine structure.

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