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PROCESS DAMPING ANALYTICAL STABILITY ANALYSIS ANDVALIDATION

by

Christopher Tyler

A thesis submitted to the faculty ofThe University of North Carolina at Charlotte

in partial fulfillment of the requirementsfor the degree of Master of Science in

Mechanical Engineering

Charlotte

2012

Approved by:

Dr. Tony Schmitz

Dr. John Ziegert

Dr. Brigid Mullany

ii

c2012Christopher Tyler

ALL RIGHTS RESERVED

iii

ABSTRACT

CHRISTOPHER TYLER. Process damping analytical stability analysis andvalidation. (Under the direction of DR. TONY SCHMITZ)

A substantial fraction of modern machining research is directed towards increasing

productivity. One limitation this research addresses is unstable cutting, or chatter

vibration. Stability lobe diagrams, which relate the allowable chip width to spindle

speed, may be used to select stable spindle speeds based on the system dynamics.

However, at lower spindle speeds the stability limit asymptotically approaches a nearly

constant chip width (using standard stability analyses) and variations in spindle speed

do not have a significant effect. Thermo-mechanical properties normally limit hard-

to-machine metals, such as titanium, and nickel alloys to machining at lower spindle

speeds to avoid prohibitive tool wear. However, at these low speeds, the process

damping effect enables increased chip widths to be realized.

This thesis extends analytical modeling of machining processes to include process

damping effects in turning and milling processes. The velocity-dependent process

damping model applied in this analysis relies on a single process damping coefficient.

Comparisons between the analytical model, time-domain simulation, and experiment

are provided. The process damping coefficient is identified experimentally using a

flexure-based machining setup for a selected tool-workpiece pair (carbide insert-AISI

1018 steel). The effects of tool wear and cutting edge relief angle are also evaluated.

It is shown that a smaller relief angle and higher wear results in increased process

damping and improved stability at low spindle speeds.

iv

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Tony Schmitz, for his patience, his ideas, and

for providing the opportunity to participate in this experience. I have grown both

professionally and personally under his guidance. I would also like to thank the other

members of the supervisory committee, Dr. John Ziegert and Dr. Brigid Mullany,

for their participation.

I would like to thank all the students and faculty of the Machine Tool Research

Center at the University of Florida for making the research possible. Furthermore,

I would like to specifically recognize Jaydeep Karandikar and Mark Pope for their

assistance and guidance throughout this effort.

Lastly, I thank my family for their continued support and encouragement.

vTABLE OF CONTENTS

CHAPTER 1: INTRODUCTION 1

CHAPTER 2: LITERATURE REVIEW 6

CHAPTER 3: STABILITY ALGORITHM 10

3.1 Linear Stability Analysis: An Overview 10

3.2 Stability Analysis Including Process Damping 13

3.2.1 Single Degree of Freedom Turning 14

3.2.2 Two Degree of Freedom Turning 17

3.2.3 Milling 19

3.2.3.1 Up Milling 20

3.2.3.2 Down Milling 21

3.3 Time-Domain Simulation 22

3.3.1 Single Degree of Freedom Turning Simulation 22

3.3.2 Two Degree of Freedom Turning Simulation 24

3.3.3 Up Milling Simulation 25

3.3.4 Down Milling Simulation 26

CHAPTER 4: EXPERIMENTAL IDENTIFICATION OF PROCESS DAMP-

ING COEFFICIENT 28

4.1 Experimental Single Degree of Freedom Setup 28

4.2 Experimental Stability Identification 31

4.3 Process Damping Coefficient Identification 34

vi

CHAPTER 5: RESULTS AND DISCUSSION 38

5.1 Experimental Identification of the Process Damping Coefficient 38

5.1.1 Relief Angle Effects on Process Damping Coefficient 40

5.1.2 Tool Wear Effects on Process Damping Coefficient 42

5.1.3 Repeatability 43

5.1.4 Sensitivity Analysis for Analytical Solution 43

CHAPTER 6: CONCLUSIONS AND FUTURE WORK 47

6.1 Completed Work 47

6.2 Future Work 49

REFERENCES 51

vii

LIST OF FIGURES

FIGURE 1: Chip thickness variation resulting from cutting tool vibrations. 1

FIGURE 2: Stability lobe diagram for a single degree of freedom system. 2

FIGURE 3: Physical description of process damping. The clearance angle

varies with the instantaneous surface tangent as the cutter re-

moves material on the sinusoidal surface. 4

FIGURE 4: Stability lobe boundary including process damping effects. 5

FIGURE 5: Demonstration of instantaneous chip thickness calculation. 11

FIGURE 6: Single degree of freedom turning model. 12

FIGURE 7: Example stability lobe diagram. The stability boundary sepa-

rates stable(chip width, spindle speed) pairs (denoted by circle,

o) and unstable pairs (denoted by x). 13

FIGURE 8: Stability diagram for single degree of freedom model from Fig-

ure 6 with no process damping. 16

FIGURE 9: Convergence demonstration (N = 20, 10 iterations) for single

degree of freedom model from Figure 6 with C = 6.11105 N/m. 17

FIGURE 10: Stability diagram for single degree of freedom model from Fig-

ure 6 with C = 6.11 105 N/m. 18

FIGURE 11: Two degree of freedom turning model. 19

FIGURE 12: Geometry for up milling using average tooth angle stability

analysis. 21

viii

FIGURE 13: Geometry for down milling using average tooth angle stability

analysis. 22

FIGURE 14: Example time-domain force and displacement for = 1625 rpm

and b = 1 mm for 50 revolutions. 24

FIGURE 15: Example time-domain force and displacement for = 1500 rpm

and b = 1 mm for 50 revolutions. 24

FIGURE 16: Single degree of freedom turning model with = 0. The

time domain simulation results are identified as: (circle) stable;

(cross) unstable; and (square) marginally stable. 25

FIGURE 17: The analytical solution and time-domain results for the two

degree of freedom turning model. 26

FIGURE 18: The analytical solution and time-domain results for the 50%

radial immersion up milling model. 26

FIGURE 19: The analytical solution and time-domain results for the 50%

radial immersion down milling model. 27

FIGURE 20: Full view of test setup showing cutting tool, accelerometer, test

coupon, and flexure. 29

FIGURE 21: Frequency response function comparison of flexure and cutting

tool (x and y directions were similar). 30

FIGURE 22: Time domain signal of a stable cut. 32

FIGURE 23: Frequency domain signal of a stable cut performed at a spindle

speed of 300 rpm (5 rps). 32

ix

FIGURE 24: Time domain signal of an unstable cutting performance. 33

FIGURE 25: Frequency domain signal of an unstable cutting performance.

Strong frequency content is observed at the flexures 228 Hz

natural frequency for the test cut completed at 700 rpm (11.7

rps). 33

FIGURE 26: Surface finish of workpiece after a stable cut. 34

FIGURE 27: Surface finish of workpiece after an unstable cut. 34

FIGURE 28: Stability lobe validation for the 228 Hz setup. Stable cuts (o)

and unstable cuts (x) are identified. 35

FIGURE 29: Stability lobe validation for the 156 Hz setup. 36

FIGURE 30: Description of variables for RSS estimate of process damping

coefficient. 36

FIGURE 31: Sweep of process damping coefficients used to select the final

stability boundary. 37

FIGURE 32: Up milling stability boundary for 50% radial immersion, 15 deg

relief angle, low wear milling tests using the 228 Hz flexure

setup (C = 2.5 105 N/m). 39

FIGURE 33: Tool wear measurement of: a) new 15 deg relief angle insert;

and b) moderately worn insert with approximately 200 m of

FWW . 39

FIGURE 34: Tool wear monitoring set-up using 60 digtal microscope. 40

xFIGURE 35: Up milling stability boundary for 50% radial immersion, 15 deg


setup (C = 2.6 105 N/m). 41



setup (C = 3.3 105 N/m). 41



setup (C = 3.3 105 N/m). 42

FIGURE 38: Up milling stability confidence region for 50% radial immersion,

11 deg relief angle milling tests using the 228 Hz flexure setup

with an unworn cutting edge (C = (3.2 0.15) 105 N/m). 44

FIGURE 39: Three axial depth of cut (points 1, 2, and 3 corresponding to

4 mm, 3 mm, and 2 mm) used in the sensitivity analysis. 45

FIGURE 40: Relative-sensitivities for a 5% increase in parameter value. 46

CHAPTER 1: INTRODUCTION

In modern manufacturing, subtractive processes such as milling and turning play

a signicant role. The need to produce parts accurately and eciently has become

increasingly important as part complexity has increased. In some cases, productivity

in machining processes is limited by self-excited vibrations, referred to as chatter.

These self-excited vibrations arise due to the regeneration of surface waviness that

occurs as the tool removes the chip from the surface that was created during the

previous pass; see Figure 1. In turning operations, this is developed after successive

rotations of the workpiece. In milling, the surface is generated from one tooth to the

next.

Figure 1: Chip thickness variation resulting from cutting tool vibrations.

Prior to research in the 1950s and 60s, the primary mechanisms that cause re-

generative chatter, or unstable cutting conditions remained a mystery. Pioneering

research, led by Tobias, Tlusty, and Merrit [13], established that the process stabil-

2ity depends on the stiness of the tool and its damping. Their work helped to develop

a mechanistic model to estimate stability based on the natural vibratory modes of

the cutting tool with respect to the direction of the cutting force. This, in turn, led

to several eective methods of predicting chatter, including the stability lobe dia-

gram. The analytical stability lobe diagram oers an eective predictive capability

for selecting stable chip width-spindle speed combinations in machining operations.

These diagrams require knowledge of the system dynamics (in the form of the sys-

tems frequency response function, or FRF) and the cutting conditions (cutting force

coecients, radial immersion, and geometric tool specications). These diagrams de-

pict regions of stable cutting with respect to spindle speed and chip width segregated

by a stability boundary; see Figure 2.

Figure 2: Stability lobe diagram for a single degree of freedom system.

At high spindle speeds, more ecient cutting is possible because the lobes are more

widely spaced. Consequently, the popularity of aluminum alloys in the aerospace

industry has shifted much of the machining dynamics research in the high speed

3machining direction, where high surface speeds can be achieved. More recently, the

focus on stability prediction has shifted to lower spindle speed regions where hard-

to-machine materials cannot take advantage of the higher speed stability zones due

to prohibitive tool wear at excessive speeds.

Materials such as titanium, Inconel, and steel alloys are now commonly used in

the medical, defense, and energy manufacturing sectors. However, achieving high

material removal rates with acceptable tool life is challenging due to, for example,

low thermal conductivity, signicant work hardening, and nite dynamic stiness. To

address these challenges, companies invest in expensive machining centers that are

extremely sti and equipped with elaborate cooling delivery systems in order to meet

production requirements.

In order to maintain productivity and prevent excessive tool wear, these materials

must be machined at low spindle (surface) speeds in the process damping regime.

Process damping occurs when there is interference between the cutting tools ank

face and the undulations left behind on the cut surface. This serves as an energy

dissipation mechanism and increases stability. The result is the ability to cut at

much higher chip widths at low spindle speeds and, therefore, avoid prohibitive tool

wear.

To describe the physical mechanism for process damping, consider a tool moving

along a wavy surface while shearing away the chip; see Figure 3. Four locations are

identied: 1) the clearance angle, , between the ank face and the work surface

tangent is equal to the nominal relief angle for the tool; 2) there is a greater potential

for tool/surface interference; 3) the relief angle returns to the original nominal value;

4and 4) is signicantly larger than the nominal value.

Figure 3: Physical description of process damping. The clearance angle varies withthe instantaneous surface tangent as the cutter removes material on the sinusoidalsurface.

The damping eect is larger for shorter vibration wavelengths because the slope

of the sinusoidal surface increases and, subsequently, the variation in clearance angle

increases. Therefore, lower cutting speeds or higher vibrating frequencies gives shorter

wavelengths and, subsequently, increased process damping. The interference of the

tool described at point 2 (Figure 3) creates an increase in stability at these lower

cutting speeds known as the process damping regime; see Figure 4.

Identifying and verifying a model that can predict these process damping condi-

tions is becoming more important in todays manufacturing industry. This research

provides an analytical solution for machining stability that includes process damping

eects. A velocity-dependent process damping force model, which relies on a single

coecient, the cutting depth, the cutter velocity, and cutting speed, is applied. The

process damping coecient is identied experimentally for a selected tool-workpiece

pair and the eects of tool wear and relief angle are evaluated.

5Figure 4: Stability lobe boundary including process damping eects.

CHAPTER 2: LITERATURE REVIEW

Process damping can be described as the energy dissipation due to relative velocity

and interference between the relief angle of a cutting tool and the existing vibrations

on the machined workpiece surface. Modeling the process damping mechanism in

metal cutting has been the subject of several research eorts. This chapter summa-

rizes prior process damping research.

Many researchers have investigated process damping in turning and milling oper-

ations. Underestimations of stability limits at low cutting velocities motivated early

studies by Sisson and Kegg [4], Wallace and Andrew [5] , and Tlusty and Ismail [6].

In their studies, they established that contact between the cutter ank and the vibra-

tions imprinted on the machined surface inuenced the dynamic cutting forces and

lead to process damping. Tlusty and Ismail [6] concluded that the stability increased

with a reduction in cutting speed is largely due to process damping. In their 1978

CIRP keynote, Tlusty et al. [7] discussed the direct inuence of tool geometry, work

material, and cutting speed on the dynamic cutting force coecients at process damp-

ing cutting speeds. The important factors described in these investigations provided

the basis for subsequent process damping models.

Process damping is particularly important for machining modern hard-to-machine

materials, such as titanium and nickel alloys. For this reason, more recent eorts have

7been made to analytically predict process damping behavior. In his 1989 publication,

Wu [8] developed a model in which plowing forces present during the tool-workpiece

contact are assumed to be proportional to the volume of interference between the ank

of the cutter and the undulations existing on the workpiece. Several others later

implemented this method of estimating process damping forces by calculating the

volume of material displaced by the cutter. Tarng et al. [9] expanded the indentation

model to calculate process damping forces using feedforward neural networks. In the

proposed model, they analyzed the stability in turning processes using dierent relief

and rake angles to identify the stability boundary. Elbestawi et al. [10] modied

Wus indentation model to include milling. An important aspect of this study was

the ability to simulate the increase in stability limit due to tool wear eects at process

damping speeds.

In 1994, the tool wear stabilizing eect in turning processes was observed by Chiou

and Liang [11, 12]. Here, a rst-order Fourier transform representation of the inter-

ference between the tool and workpiece was developed to model the nonlinear process

as a linear one. Analytical stability lobe diagrams were generated using the linear

approximation and qualitative agreement was observed in turning experiments at

several levels of ank wear.

Altintas et al. [13] also provided a method for identifying the dynamic cutting force

coecients and stability at low cutting velocities in turning. In a series of cutting

tests, a fast tool servo was used to modulate the cutting tool at the desired frequencies

and amplitudes. This removed the regenerative aspect of orthogonal cutting and

isolated the process damping eect. This study also considered the eects of tool

8wear on the process coecients and illustrated the inuence on stability at low cutting

speeds.

Continuing in their study of process damping in machining, Altintas et al. [14]

extended their prior research to include milling at low cutting velocities. By modeling

process damping as a linear function of the velocity and modeling both rotating and

xed structures present in the cutting process, they produced the stability boundary

for the process damping regime. Using Wus indentation method to obtain the process

damping coecients, they were able to compare simulations against a series of slot

milling experiments. They observed stability dependence both on cutting speed and

symmetry in the rotating systems structural dynamics.

Huang and Wang [15] showed that the plowing force (more than the shear force)

was the dominant contributor in process damping for peripheral milling operations.

In this study, the eects of cutting conditions (cutting speed, feed, axial and radial

depths of cut) on process damping were examined. Separating the process damping

force into shearing and plowing components enabled them to identify that plowing

was the dominant element in process damping.

Budak and Tunc [16] used an energy-based method to identify the process damping

force coecient and reinforced the concept that decreasing the relief angle increases

the process damping eect for milling and turning processes. They found the hone

radius of a particular cutter could increase the process damping eect at higher cutting

speeds in orthogonal cutting. The proposed method was veried by time domain

simulation and experiments.

In a follow-up study, Tunc and Budak [17] extended their model to include addi-

9tional eects of cutting conditions and tool geometry on process damping. In this

paper, they observed a slight increase in process damping when using a cylindrical

ank face as compared to a planar ank geometry.

Sims and Turner [18] developed a time domain model in milling operations that

included non-linear process damping eects. Their method calculated: 1) the current

tooth position, 2) the instantaneous chip thickness based on the tooth position, and

3) the forces that arise due to the ank-surface interference. Their data revealed

a strong connection between the feed rate of the cutter and the process damping

wavelength. Experimental comparison with the time domain simulation suggested

further calibration of the model parameters was required either through empirical

experimentation or detailed chip formation modeling.

Turner et al. [19] experimentally assessed the role of the geometric cutter parame-

ters on process damping. Cutting tests performed on a exure were used to evaluate

the inuence of edge geometry, relief angle, rake angle, and variable helix/pitch angle

on process damping in milling operations. The exible workpiece study revealed a

signicant performance increase using variable helix/pitch cutters and tools with in-

creased edge radius. To a lesser extent, low relief angle/low rake angle combinations

also increased the process damping performance.

CHAPTER 3: STABILITY ALGORITHM

In this chapter, an iterative, analytical stability analysis is described that incorpo-

rates the eects of process damping. An overview of the traditional stability analysis

is presented followed by a detailed description of the new analysis that includes pro-

cess damping. The dynamic model is used to describe both up and down milling

operations, as well as single and multi-degree of freedom turning. The analytical

results are then compared to time-domain simulations.

3.1 Linear Stability Analysis: An Overview

Frequency-domain solutions for machining stability are well established and have

been applied for many years. Eorts initiated by Tobias, Tlusty, and Merrit [1

3] identied regenerative chatter as a mechanism inuenced largely by the cutting

systems dynamic response. The frequency response function (FRF) of the cutting

system is mapped to create stability lobe diagrams that display chip width and spindle

speed as variables to identify stable and unstable cutting regions.

In descriptions of regenerative chatter in machining, the instantaneous cutting

force, F , is typically written as:

F = Ksbh(t) (1)

where Ks is the specic cutting force (which depends on the tool-workpiece combi-

11

nation and, to a lesser extent, the cutting parameters), b is the chip width, and h(t)

is the time-dependent instantaneous chip thickness:

h(t) = hm + y(t ) y(t). (2)

The constant portion of the force containing the mean chip thickness, hm, does

not inuence stability. The feedback between the current vibration, y(t), and the

time-delayed vibration from the previous cut, y(t ), is the mechanism that leads

to self-excited vibration; see Figure 5, where is the time delay between cuts.

Figure 5: Demonstration of instantaneous chip thickness calculation.

To describe the stability algorithm, consider the single degree of freedom turning

model displayed in Figure 6. Tlusty [20] denes the limiting stable chip width, blim,

for regenerative chatter using:

blim =1

2KsRe(Gor), (3)

where Gor is the oriented frequency response function, Gor = cos()cos()Gu for

the model displayed in Figure 6. In this expression, is the force angle relative to

the surface normal, is the angle between the u direction and the surface normal,

12

and Gu is the frequency response function in the u direction, which can be described

using the modal mass, m, modal stiness, k, and modal (viscous) damping, c. The

spindle speed, , relationship is:

fc

= N +

2, (4)

where N = 0, 1, 2... is the integer number of vibration waves left on the workpiece

surface per revolution, fc is the valid chatter frequencies (identied by the negative

real portion of Gor), and is the relative phase between the current and time-delayed

vibration; see Eq. 5, where Re denotes the real part and Im the imaginary part of

the oriented frequency response function.

= 2 2tan1(Re(Gor)

Im(Gor)

)(rad) (5)

Figure 7 shows an example stability lobe diagram where the curves generated from

the governing equations separate the region into stable and unstable chip width-

spindle speed combinations.

Figure 6: Single degree of freedom turning model.

13

Figure 7: Example stability lobe diagram. The stability boundary separates sta-ble(chip width, spindle speed) pairs (denoted by circle, o) and unstable pairs (denotedby x).

3.2 Stability Analysis Including Process Damping

From Figure 7, the chatter-free depths of cut are observed to diminish substantially

at low spindle speeds, where the stability lobes become more closely spaced. Fortu-

nately, the process damping eect can serve to increase the allowable chip width for

low spindle speeds. The process damping force, Fd, is characterized as a 90 deg phase

shift relative to the displacement and opposite in sign from the velocity. Given the

preceding description, the process damping force is modeled as the viscous damping

force shown in Eq. 6.

Fd = C bVy (6)

Here, the process damping force in the y direction (perpendicular to the cut surface)

is expressed as a function of the cutter velocity, y, chip width, b, cutting speed, V ,

14

and a process damping coecient, C. The following sections demonstrate how an

analytical stability solution is derived for both milling and turning applications with

the inclusion of the process damping force.

3.2.1 Single Degree of Freedom Turning

Consider the single degree of freedom system presented in Figure 6. To incorporate

the process damping force into the stability analysis, it is rst projected into the

u direction:

Fu = Fdcos() = C bVycos() =

(C

b

Vcos()

)y. (7)

The nal form of the u projection of the process damping force is eectively a

viscous damping term. Therefore, the force can be incorporated in the traditional

regenerative chatter stability analysis by modifying the structural damping in Gu. As

shown in Figure 6, the single degree of freedom, lumped parameter dynamic model

can be described using the modal mass, m, modal viscous damping coecient, c, and

modal spring stiness, k. In the absence of process damping, the equation of motion

in the u direction is:

mu+ cu+ ku = Fcos( ). (8)

The corresponding frequency response function in the u direction is:

Gu =U

Fcos( ) =1

m2 + ic + k , (9)

where is the excitation frequency (rad/s). When process damping is included,

15

however, the equation of motion becomes:

mu+ cu+ ku = Fcos( )(C

b

Vcos()

)y. (10)

Replacing y in Eq. 10 with cos()u gives:

mu+ cu+ ku = Fcos( )(C

b

Vcos2()

)u. (11)

Rewriting Eq. 11 to combine the velocity terms yields:

mu+

(c+ C

b

Vcos2()

)u+ ku = Fcos( ), (12)

where the new viscous damping coecient is cnew = c + CbVcos2(). Replacing

the original damping coecient, c, (from the structural dynamics only) with cnew

enables process damping to be incorporated in the analytical stability model. The

new frequency response function is:

Gu =U

Fcos( ) =1

m2 + icnew + k . (13)

The modied frequency response function is a function of the spindle speed-dependent

limiting chip width and the cutting speed (which, in turn, depends on the spindle

speed). Therefore, the chip width and spindle speed values must be known beforehand

to incorporate process damping eects. This leads to an iterative, converging analysis

in which the new damping coecient is updated after each iteration. The stability

boundary is established for each lobe number, N , using the following steps [21]:

16

1. the analytical stability boundary is calculated with no process damping to iden-

tify initial b and vectors

2. these vectors are used to determine the corresponding cnew vector

3. the stability analysis is repeated with the new damping value to determine

updated b and vectors

4. the process is repeated until the stability boundary converges.

To demonstrate the approach, consider the model in Figure 6 with = 0, k =

6.48 106 N/m, m = 0.561 kg, c = 145 Ns/m, Ks = 2927106 N/m2, = 61.8 deg,

and d = 35 mm. The stability boundary with no process damping (C = 0) is shown

in Figure 8 for N = 0 to 100. It is observed that the limiting chip width approaches

the asymptotic stability limit of 0.37 mm for spindle speeds below 1000 rpm.

Figure 8: Stability diagram for single degree of freedom model from Figure 6 with noprocess damping.

17

Results of the converging procedure with process damping for the N=20 stability

boundary are provided in Figure 9. Converging behavior is observed for the 10 it-

erations as the lobes move up and slightly to the right. A practical selection of 20

iterations was applied for the diagrams in this study to ensure convergence. Figure 10

displays the new stability diagram for N = 0 to 100 with C = 6.11 105 N/m.

Figure 9: Convergence demonstration (N = 20, 10 iterations) for single degree offreedom model from Figure 6 with C = 6.11 105 N/m.

3.2.2 Two Degree of Freedom Turning

In some cases it is not sucient to consider the system to be exible in one direc-

tion only. The process damping model can be extended to include vibration in two

directions as shown in Figure 11. The procedure for establishing stability lobes is

similar, although now the new damping term is calculated in both directions. The

new damping values are:

18

Figure 10: Stability diagram for single degree of freedom model from Figure 6with C = 6.11 105 N/m.

cnew,1 = c1 + Cb

Vcos2(1) (14)

for the u1 direction and

cnew,2 = c2 + Cb

Vcos2(2) (15)

for the u2 direction. These two new damping terms are applied to the frequency re-

sponse functions Gu1 and Gu2 in the u1 and u2 directions. The directional orientation

factors, 1 = cos( 1)cos(1) and 2 = cos( 2)cos(2), are used to calculate

the oriented frequency response function as a linear combination of the two modes:

Gor = 1Gu1 + 2Gu2 . (16)

19

Figure 11: Two degree of freedom turning model.

3.2.3 Milling

Tlusty [20] modied the previously described turning analysis to accommodate the

milling process. A primary obstacle to dening an analytical solution for milling

stability (aside from the inherent time delay) is the time dependence of the cutting

force direction. Tlusty solved this problem by assuming an average angle of the

tooth in the cut, ave, and, therefore, an average force direction. This produced an

autonomous, or time-invariant, system. He then made use of directional orientation

factors, x and y, to rst project this force into the x and y mode directions and,

second, project these results onto the surface normal (in the direction of ave). The

new blim and expressions for milling are provided in Eqs. 17 and 18, where Nt is

the number of teeth on the cutter and Nt is the average number of teeth in the cut;

see Eq. 19, where s and e (deg) are the start and exit angles dened by the radial

depth of cut and milling direction (up, or conventional, and down, or climb milling).

The equation remains the same as Eq. 5.

20

blim =1

2KsRe(Gor)Nt, (17)

fcNt

= N +

2, (18)

Nt =e s360/Nt

(19)

3.2.3.1 Up Milling

The process damping force model dened in Eq. 6 may be extended to up milling

operations. The geometry is shown in Figure 12, where n is the surface normal

direction, which is dened by ave. The projection of the process damping force from

the n direction onto the x direction is:

Fx = Fdcos(90 ave) = (C

b

Vcos(90 ave)

)n. (20)

Note that the velocity term is now n. Substituting n = cos(90 ave)x in Eq. 20

gives:

Fx = (C

b

Vcos2(90 ave)

)x (21)

The new damping in the converging stability calculation for the x direction frequency

response function, Gx, is therefore:

cnew,x = cx + Cb

Vcos2(90 ave) (22)

21

Figure 12: Geometry for up milling using average tooth angle stability analysis.

Similarly, the new y direction damping is:

cnew,y = cy + Cb

Vcos2(180 ave) (23)

The oriented frequency response function for this case is Gor = xGx + yGy, where

x = cos( (90ave))cos(90ave) and y = cos(180ave)cos(180ave).

3.2.3.2 Down Milling

The geometry for the down milling case is shown in Figure 13. Using the same

approach as described in the up milling case, the x and y direction damping values

are provided in Eqs. 24 and 25.

cnew,x = cx + Cb

Vcos2(ave 90) (24)

cnew,y = cy + Cb

Vcos2(180 ave) (25)

The oriented frequency response function for this case is Gor = xGx + yGy, where,

22

x = cos(+(ave90))cos(ave90) and y = cos((180ave))cos(180ave).

Figure 13: Geometry for down milling using average tooth angle stability analysis.

3.3 Time-Domain Simulation

Both analytical analyses and time-domain simulations were completed to deter-

mine the process damping eects on turning and milling stability. The time-domain

milling simulations were based on the Regenerative Force, Dynamic Deection Model

described by Smith and Tlusty [22]. Comparisons are then drawn between simulation

and analytical models.

3.3.1 Single Degree of Freedom Turning Simulation

The model shown in Figure 6 was considered. In this case = 0, k = 6.48

106 N/m, m = 0.561 kg, c = 145 Ns/m, Ks = 2927 106 N/m2, and = 61.8 deg.

The workpiece diameter, d, was 35 mm. The damping force was included directly

in the numerical integration of the system equations of motion and the time-domain

forces and tool displacements were calculated using the integration method detailed

in [23]. The simulation was carried out in small time steps, dt. The equation of

23

motion derived in Eq. 12 was rewritten to solve for the current acceleration in the

mode direction, u, as

u =Fu cnewu ku

m(26)

where the velocity, u, and position, u, are each from the previous time step (they are

initally zero). Using a numerical (Euler) integration scheme the current velocity and

position are calculated using:

u = u+ u dt and u = u+ u dt, (27)

where the right sides of Eq. 27 are retained from the previous time step. The

displacement is then projected into the y direction by y = ucos() in order to nd

the tool displacement with respect to the surface normal. Finally, the current chip

thickness, h(t), and cutting force are updated using Eqs. 2 and 1 and the process is

repeated for all time steps in the simulation.

Figure 14 depicts simulation results for = 1625 rpm and b = 1 mm for the

single degree of freedom turning model. The cut is clearly unstable, as the force

and displacement are growing signicantly over time. Alternatively, Figure 15 shows

the response approaching a steady state force and tool displacement during a stable

cut. A comparison between the analytical stability lobes and time-domain simulation

results is provided in Figure 16. In all instances, the time-domain results agree with

the analytical stability limit.

24

Figure 14: Example time-domain force and displacement for = 1625 rpm andb = 1 mm for 50 revolutions.

Figure 15: Example time-domain force and displacement for = 1500 rpm andb = 1 mm for 50 revolutions.

3.3.2 Two Degree of Freedom Turning Simulation

The model shown in Figure 11 was considered. A comparison between the analyt-

ical stability lobes and time-domain simulation results is provided in Figure 17. The

25

Figure 16: Single degree of freedom turning model with = 0. The time domainsimulation results are identied as: (circle) stable; (cross) unstable; and (square)marginally stable.

model parameters were: 1 = 30 deg, 2 = 60 deg, the dynamics in both directions

were described by k1,2 = 6.48 106 N/m, m1,2 = 0.561 kg, and c1,2 = 145 Ns/m,

and the force constants were Ks = 2927 106 N/m2, and = 61.8 deg, with

C = 6.11 105N/m. Good agreement between the analytical solution and simu-

lation was obtained obtained.

3.3.3 Up Milling Simulation

The model parameters for the three-tooth cutter were: s = 0 deg and e = 90 deg

(50% radial immersion), the dynamics in both the x and y directions were described

by k = 9 106 N/m, a natural frequency of fn = 900 Hz, and a viscous damping

ratio of = 0.03, and the force constants were Ks = 2000 106 N/m2, = 70 deg,

and C = 6.11 105 N/m. The comparison between the analytical stability lobes and

time-domain simulation is provided in Figure 18. The results match well.

26

Figure 17: The analytical solution and time-domain results for the two degree offreedom turning model.

Figure 18: The analytical solution and time-domain results for the 50% radial im-mersion up milling model.

3.3.4 Down Milling Simulation

Figure 13 displays the model for the down milling conditions considered. The

starting and exit angles for a 50% radial immersion cut using the three-tooth cutter

27

were s = 90 deg and e = 180 deg. All other parameters were the same as for

the up milling example. The comparison between the analytical stability lobes and

time-domain simulation is given in Figure 19. Good agreement between the analytical

lobes and simulation is observed for the down milling example.

Figure 19: The analytical solution and time-domain results for the 50% radial im-mersion down milling model.

CHAPTER 4: EXPERIMENTAL IDENTIFICATION OF PROCESS

DAMPING COEFFICIENT

The experimental verication of the analytical solution and procedure for obtaining

the process damping coecient is provided in this chapter. A series of cutting tests

for selected tool-workpiece combinations were performed on a single degree of freedom

exure. This setup provided controllable and repeatable dynamics. The stability limit

was identied over a grid of axial depth of cut and spindle speed pairs. Using the

experimental stability boundary, the process damping coecient was then identied.

The eects of insert relief angle and tool wear on the process damping coecient were

examined. The exure dynamics were also adjusted to determine the sensitivity of

the coecient to changes in the system dynamics.

4.1 Experimental Single Degree of Freedom Setup

The experimental setup used a parallelogram leaf-type exure in order to control

the dynamics of the system and approximate single degree of freedom behavior. The

exure was constructed to provide a exible foundation for individual AISI 1018 steel

workpieces/coupons. Figure 20 shows the full experimental setup for the cutting tests.

The dimensions of the exure were chosen so that the exibility of the platform was

much greater than that of the cutting tool. The exure leafs were selected to be

3.13 mm thick AISI 6061 aluminum sheets. An AISI 1080 steel sheet was used as the

29

top platform to increase the mass of the exible system. The entire system was xed

to the machine table using T-slot nuts to ensure a sti connection. An accelerometer

was also mounted to the exure platform to provide in-process data for stability

evaluation.

Figure 20: Full view of test setup showing cutting tool, accelerometer, test coupon,and exure.

To study the inuence of relief angle on the process damping behavior, two single-

tooth indexable square endmills of similar diameter were used. The rst was an

18.54 mm diameter inserted endmill (Kennametal model KICR-0.73-SD3-033.3C)

with a 15 deg relief angle (SDCW322 KC725M). The second was a 19.05 mm di-

ameter endmill (Cutting Tool Technologies model DRM-03) with an 11 deg relief

angle insert (SPEB322 KC725M). Both cutting tools had a zero deg rake angle. The

inserts had no edge preparation and similar TiAlN coatings. The tools were placed

in a Schunk Tribos R holder with an overhang length of 55.2 mm. Individual AISI

1018 steel workpieces, or test coupons, were bolted on the exure so that test cuts

30

could be completed at any specied radial immersion.

A preliminary frequency response function for the single degree of freedom setup

was obtained by impact testing the exure setup with an uncut test part mounted

to the top. FRFs for the sti direction of the exure and the tool (both x and y

directions) were also obtained in order to conrm that they were much stier than

the intended exible direction for the exure. Measurements showed the peak imag-

inary amplitudes for the tool response were approximately 10 times smaller than the

intended exure direction and were therefore considered negligible when modeling the

dynamic system; see Figure 21.

Figure 21: Frequency response function comparison of exure and cutting tool (x andy directions were similar).

The sensitivity of the process damping coecient to changes in the system dynam-

ics was also evaluated experimentally. To reduce the natural frequency, mass was

added to the platform; the added mass decreased the natural frequency by approxi-

31

mately 32%. The modal parameters for both cases are provided in Table 1. The x and

y directions correspond to the exible and sti directions of the exure, respectively,

where x is the feed direction for the milling tests.

Table 1: Modal parameters for exure with and without mass.

Direction Viscous damping ratio Modal stiness (MN/m) Natural frequency (Hz)

No massx 0.063 2.77 228y 0.037 174 1482

Added massx 0.018 4.37 156y 0.028 276 1137

The cutting force coecients were identied under stable cutting conditions using a

cutting force dynamometer (Kistler model 9257B). The cutting tests were performed,

as described in reference [23], using a spindle speed of 500 rpm, an axial depth of cut of

3 mm, and a feed per tooth range of 0.10-0.15 mm/tooth. For the 18.54 mm diameter

cutter, the specic cutting force, Ks, and cutting force direction, , were determined

to be 2359.1 N/mm2 and 63.5 deg, respectively. For the 19.05 mm diameter cutter,

the values were 2531.0 N/mm2 and 62.0 deg. A linear regression to the mean cutting

force over a series of tests at the selected feed per tooth values was used to identify

the cutting force model values.

4.2 Experimental Stability Identication

An accelerometer (PCB Piezotronics model 352B10) was used to measure the vi-

bration during cutting. The frequency content of the accelerometer signal was used

in combination with the machined surface nish to establish stable/unstable perfor-

mance. Cuts that exhibited uniform vibrations in the time-domain and exhibited

content in the frequency-domain at the only the tooth passing frequency and its har-

32

monics were considered to be stable. Figure 22 and Figure 23 represent an example

stable cut in the time and frequency domain, respectively.

Figure 22: Time domain signal of a stable cut.

Figure 23: Frequency domain signal of a stable cut performed at a spindle speed of300 rpm (5 rps).

Alternatively, cuts that had vibrations which increased signicantly over the du-

ration of the cut and exhibited signicant frequency content near the exures x di-

33

rection natural frequency were considered to be unstable. Figure 24 and Figure 25

represent an unstable cutting condition in the time and frequency domain, respec-

tively.

Figure 24: Time domain signal of an unstable cutting performance.

Figure 25: Frequency domain signal of an unstable cutting performance. Strongfrequency content is observed at the exures 228 Hz natural frequency for the testcut completed at 700 rpm (11.7 rps).

34

In addition, the surface of each workpiece was analyzed visually after each test cut.

The surface texture of a stable cut exhibited very little surface aws, as shown in

Figure 26. Inspection of the workpiece surface for an unstable cut revealed irregular

aws in the surface nish due to the cutter vibration; see Figure 27.

Figure 26: Surface nish of workpiece after a stable cut.

Figure 27: Surface nish of workpiece after an unstable cut.

4.3 Process Damping Coecient Identication

Conventional linear stability analysis (i.e., C = 0 N/m) was rst used to validate

the stability behavior at higher speeds for the exure setup. Using the experimental

exure modal parameters and cutting force coecients, analytical stability lobes were

generated without including the eects of process damping. Several test cuts were

then chosen to conrm that key features predicted by the analytical lobes existed

35

at higher spindle speeds. As seen in Figure 28, the predicted behavior was observed

experimentally. Additionally, the critical limiting chip width, blim,cr, was identied to

be approximately 1 mm for the 228 Hz exure setup; this result also agreed with the

analytical prediction. A similar approach was used to validate the stability boundary

for the 156 Hz setup. The critical stability limit was approximately 0.4 mm for this

case; see Figure 29.

Figure 28: Stability lobe validation for the 228 Hz setup. Stable cuts (o) and unstablecuts (x) are identied.

A grid of test points at low spindle speeds was next selected to investigate the

process damping behavior. Based on the stable/unstable cutting test results, a single

variable residual sum of squares (RSS) estimation was applied to identify the process

damping coecient that best represented the experimental stability boundary; see

Figure 30. The spindle speed-dependent experimental stability limit, bi, was selected

to be the midpoint between the stable and unstable points at the selected spindle

speed. The sum of squares of residuals is given by Eq. 29, where f(i) is the analytical

36

Figure 29: Stability lobe validation for the 156 Hz setup.

stability boundary and n is the number of test points.

RSS =n

i=1

(bi f(i))2 (28)

Figure 30: Description of variables for RSS estimate of process damping coecient.

A range of process damping coecients was selected and the RSS value was calcu-

37

Figure 31: Sweep of process damping coecients used to select the nal stabilityboundary.

lated for each corresponding stability limit; see Figure 31. The C value that corre-

sponded to the minimum RSS value was selected to identify the best-t nal stability

boundary for all test conditions.

CHAPTER 5: RESULTS AND DISCUSSION

Milling tests were conducted to observe the process damping behavior and sub-

sequent stability increase at low cutting speeds. Quantitative eects of clearance

angle, ank wear width (FWW ), and the natural frequency of the dynamic system

were evaluated using the process damping coecient identication method described

in Chapter 4. Stability boundaries were generated for each case using the best-t

process damping coecient value. The condence interval for one test condition was

established to evaluate the repeatability of the method for determining the process

damping coecient. A sensitivity analysis for the analytical solution was also com-

pleted.

5.1 Experimental Identication of the Process Damping Coecient

A preselected grid of axial depths was chosen at low spindle speeds to identify

the process damping regime. Stability testing was rst performed for the 18.54 mm

diameter, 15 deg relief angle end mill. The axial depths ranged from below the

conrmed blim,cr value of 1 mm to a maximum depth of 3 mm. All cuts were performed

using 50% radial immersion up-milling conditions. The feed per tooth was also held

constant at 0.05 mm/tooth. Using the minimum RSS method described in Chapter 4,

a process damping coecient of C = 2.5 105 N/m was obtained to best-t the data

for the 228 Hz system. The corresponding stability boundary is provided in Figure 32.

39

Figure 32: Up milling stability boundary for 50% radial immersion, 15 deg reliefangle, low wear milling tests using the 228 Hz exure setup (C = 2.5 105 N/m).

Because ank wear can aect the process damping behavior due to changes in

the tool/surface interference, the ank wear width (FWW ) was limited to less than

100 m for these stability tests and approximately 200 m for moderately worn tests;

see Figure 33. The tool wear was monitored using a 60 digital microscope (Dino-

Lite: model Pro-AM413T); see Figure 34.

Figure 33: Tool wear measurement of: a) new 15 deg relief angle insert; and b)moderately worn insert with approximately 200 m of FWW .

The procedure was repeated for the 156 Hz setup and a process damping coecient

40

Figure 34: Tool wear monitoring set-up using 60 digtal microscope.

of C = 2.6 105N/m was identied. These results are displayed in Figure 35.

The lower natural frequency of the system causes a shift in the process damping

regime to lower spindle speeds and limiting chip widths compared to the 228 Hz

system. However, a comparison of the two process damping coecients indicates less

than a 4% dierence. Therefore, the process damping coecient is determined to be

insensitive to moderate changes in the system dynamics.

5.1.1 Relief Angle Eects on Process Damping Coecient

Tests were then performed using the 19.05 mm diameter, 11 deg relief angle end

mill. The same procedure was following and the FWW was again limited to be less

than 100 m for all cuts. The process damping coecient for both the 228 Hz and

156 Hz setups was 3.3 105 N/m; see Figures 36 and 37.

The low wear stability test results are summarized in Table 2. The process damping

coecient for the 228 Hz setup increased by 32% for the 11 deg relief angle tool relative

to the 15 deg relief angle tool. A 27% increase was observed for the 156 Hz setup.

41



These trends make sense given that additional interference would be encouraged by

the smaller relief angle.

42


Table 2: Comparison of process damping coecients for low wear tests.

Relief angle (deg) C (N/m) for the 228 Hz setup C (N/m) for the 156 Hz setup15 2.5 105 2.6 10511 3.3 105 3.3 105

5.1.2 Tool Wear Eects on Process Damping Coecient

In order to explore the eect of tool wear on the process damping performance,

tests were completed using worn tools where the FWW was maintained at a level of

approximately 200 m; see Figure 33. For the 15 deg relief angle tool, the specic

cutting force and cutting force direction were 2441.0 N/mm2 and 63.5 deg, respec-

tively; this represents a 3.5% increase in the specic cutting force relative to the

unworn tool tests. However, the process damping coecient was found to increase

from the unworn tool tests by 20% for the 228 Hz setup and 31% for the 156 Hz

setup. Similarly, for the 11 deg cutter, the cutting force parameters experienced only

a slight change (Ks = 2550.2 N/mm2 and = 62.0 deg). However, the process

43

damping coecient increased by 15.2% for both exure setups; see Table 3.

Table 3: Comparison of process damping coecients for moderate wear tests.

Relief angle (deg) C (N/m) for the 228 Hz setup C (N/m) for the 156 Hz setup15 3.0 105 3.4 10511 4.0 105 3.8 105

5.1.3 Repeatability

Repeat testing was performed using the 19.05 mm diameter, 11 deg relief angle

cutting tool in order to observe the variability in the process damping coecient

identication process. A series of three additional cutting tests were performed on

the 228 Hz system using an unworn insert. The three process damping coecients

were: 3.3 105 N/m, 3.3 105 N/m, and 2.9 105 N/m. Assuming a normal

distribution, a two-sided 90% condence level was computed for this small sample size.

The condence interval for the population mean was: C = (3.2 0.15) 105 N/m.

Figure 38 illustrates the corresponding condence region.

5.1.4 Sensitivity Analysis for Analytical Solution

A sensitivity analysis was completed to determine which system parameter has the

most signicant eect on the predicted stability boundary within the process damping

regime. For this activity the following parameters were considered: the stiness (kx)

and damping ratio (x) in the exible direction, the specic cutting force (Ks), the

force direction angle (), and the process damping coecient (C).

The sensitivity of the stability boundary to each parameter was identied by calcu-

lating the percent change in spindle speed at a prescribed chip width on the stability

44

Figure 38: Up milling stability condence region for 50% radial immersion, 11 degrelief angle milling tests using the 228 Hz exure setup with an unworn cutting edge(C = (3.2 0.15) 105 N/m).

boundary as the selected parameter was varied individually from a nominal value.

The relative sensitivity was then calculated using:

Sp =% change in

% change in p=

/

p/p, (29)

where is the spindle speed on the stability boundary and p is the selected parameter.

The points shown in Figure 39 were analyzed individually for each parameter to

evaluate whether the sensitivity varied along the process damping stability boundary.

Axial depths of 4 mm, 3 mm, and 2 mm were considered in the analysis.

Table 4: Nominal parameter values for sensitivity analysis.

Initial Parameters (fn = 228.9Hz)kx 2.77 106 N/mx 0.063Ks 2550.2 N/mm2

62 degC 3 106 N/m

45

Figure 39: Three axial depth of cut (points 1, 2, and 3 corresponding to 4 mm, 3 mm,and 2 mm) used in the sensitivity analysis.

The nominal values used for the calculation of the stability boundary are provided

in Table 4. These values were used in the calculation for percent change. The spindle

speeds at the stability limit obtained from the nominal parameter values were 1 =

418 rpm, 2 = 446 rpm, and 3 = 535 rpm for points 1, 2, and 3, respectively.

Using these nominal values, the sensitivities were calculated for 2%, 5%, and 10%

increases in each parameter value. Figure 40 illustrates the relative-sensitivity for

the 5% increase in the parameter values. It can be seen that, while the stiness,

damping, and process damping coecient exhibited a positive inuence (increase in

with increasing parameter values), increases in the cutting force parameters each

contributed strongly to a decrease in the process damping region.

Results for the 2% and 10% parameter increase follow the same trend as the 5%

increase. Based on Figure 40, slight changes in the stiness and damping ratio would

result in a small change to the stability boundary for higher axial depths (along the

46

Figure 40: Relative-sensitivities for a 5% increase in parameter value.

process damping stability boundary). However, at low axial depths, these system

parameters have a greater inuence on the stability limit. As expected, an increase in

the specic force, Ks, and force angle, , results in a negative inuence on the process

damping regime. A larger specic force value yields a lower stability limit. Again,

the strongest eect is observed at smaller axial depths. The sensitivity to the process

damping coecient is similar to the stiness and damping results, except that the

sensitivity is higher at higher axial depths. Finally, the specic cutting force, Ks, and

average cutting force angle, , emerge as the factors that play the strongest role in

modifying the stability limit in the process damping regime.

CHAPTER 6: CONCLUSIONS AND FUTURE WORK

6.1 Completed Work

In this research, an analytical stability solution was developed that considers pro-

cess damping in milling and turning applications. Derivation of the model, which

relied on a single process damping coecient, C, was presented and verication was

completed using time-domain simulation and experiments.

Accurate stability prediction at low machining speeds is sensitive to the work ma-

terial, tool geometry, and frequency of surface vibrations during cutting. Process

damping due to ank-workpiece interference serves to increase the stability at these

low cutting speeds. The approach used to model the process damping eect relied

on an equivalent viscous damping force that represents the contact force. This pro-

cess damping force depended proportionally on the chip width and cutter velocity

and inversely proportionally on the cutting speed. In addition, the damping force

was proportional to a process damping coecient, which is analogous to the specic

cutting force, Ks, approach used to model cutting force. Due to the chip width and

cutting speed dependence of the process damping force, the new stability analysis

followed an iterative, converging approach to produce the stability boundary. The

limiting chip width values were observed to increase for individual stability lobes as

spindle speeds decreased in the process damping regime.

48

Comparison with time-domain simulation demonstrated that the analytical stabil-

ity model accurately predicted the stability lobe boundary. Simulation of turning

and milling operations indicated accurate stability prediction in the process damp-

ing region, provided a correct process damping coecient was selected. The goal

of the second part of this research was to identify the process damping coecient

experimentally.

A single-tooth indexable end mill was used to mill AISI 1018 steel workpieces se-

cured to a single degree-of-freedom parallelogram exure. The stability limit was

identied over a grid of stable/unstable axial depths of cut and spindle speeds. Cut-

ting stability was identied in two ways: 1) the frequency content of an accelerometer

secured to the exure; and 2) qualitative analysis of the machined surface using a

digital microscope. Using the experimental stability boundary, the process damping

coecient was determined in a best-t manner. The eects of the cutting insert re-

lief angle and tool wear were examined. The exure dynamics were also adjusted to

determine the sensitivity of the process damping coecient to changes in the system

dynamics.

For a 50% radial immersion up milling cut, the stability boundary was discovered

to increase as the spindle speed decreased and a value for the process damping coef-

cient was obtained. Substituting the original 15 deg relief angle insert for an 11 deg

relief angle insert and repeating the cutting tests revealed that the process damping

coecient increased by approximately 30%. This indicated that the decreased relief

face angle increases process damping, as expected. Tests were also completed using

both new and worn inserts. Under worn conditions, the process damping coecient

49

increased by approximately 15% compared to new conditions. Finally, the dynamic

properties of the system were adjusted by increasing the mass xed to the exure.

No appreciable change in the process damping coecient was observed.

Repeat testing was performed in order to observe the variability in the process

damping coecient. A total of four stability grids were evaluated using the same ex-

perimental conditions and a condence interval for the stability boundary was estab-

lished. Results indicated acceptable repeatability in the process damping coecient

identied using the experimental approach.

Finally, a sensitivity analysis was performed to determine which system parame-

ter had the highest inuence on the stability boundary within the process damping

regime. The stiness and damping ratio in the exible direction for the workpiece

exure, the specic cutting force and force direction angle for the force model, and

the process damping coecient were considered. The process damping coecient

and average cutting force angle emerged as the factors that play the strongest role in

modifying the stability limit in the process damping regime.

6.2 Future Work

Process damping is particularly important for exotic metals, such as titanium,

nickel super alloys, and hardened steels. Therefore, in follow-up studies, the corre-

sponding process damping coecients will be identied using the established method.

The nal goal of this eort will be to populate a reference table so that process damp-

ing eects may be estimated for known tool/workpiece combinations.

Further studies will also be required to identify other relationships between cutting

50

parameters and the process damping coecient. Applying the analytical stability

method will provide quantitative characterization of process damping when factors

such as feed rate, cutting edge rake angle, and edge preparation are considered (in

addition to tool wear and relief angle eects). Finally, experimental identication

of the process damping coecient will be extended to include turning operations. A

exural-based setup will be designed and a similar process used for identifying process

damping in milling will be used in future turning experiments.

51

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ct_thesis_1List_of_Figuresct_thesis_2