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Dynamic modeling of ultra-precision fly cutting machine tool and theeffect of ambient vibration on its tool tip responseTo cite this article: Jianguo Ding et al 2020 Int. J. Extrem. Manuf. 2 025301

 

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Dynamic modeling of ultra-precision flycutting machine tool and the effect ofambient vibration on its tool tip response

Jianguo Ding1 , Yu Chang1, Peng Chen1, Hui Zhuang1, Yuanyuan Ding2,Hanjing Lu2,3 and Yiheng Chen2

1 School of Science, Nanjing University of Science and Technology, Nanjing, People’s Republic of China2 Institute of Launch Dynamics, Nanjing University of Science and Technology, Nanjing, People’sRepublic of China3Department of Engineering, Aarhus University, Aarhus, Denmark

E-mail: nustdjg@163.com

Received 25 December 2019, revised 14 February 2020Accepted for publication 1 March 2020Published 24 March 2020

AbstractThe dynamic performances of an ultra-precision fly cutting machine tool (UFCMT) has adramatic impact on the quality of ultra-precision machining. In this study, the dynamic model ofan UFCMT was established based on the transfer matrix method for multibody systems. Inparticular, the large-span scale flow field mesh model was created; and the variation in linear andangular stiffness of journal and thrust bearings with respect to film thickness was investigated byadopting the dynamic mesh technique. The dynamic model was proven to be valid by comparingthe dynamic characteristics of the machine tool obtained by numerical simulation with theexperimental results. In addition, the power spectrum density estimation method was adopted tosimulate the statistical ambient vibration excitation by processing the ambient vibration signalmeasured over a long period of time. Applying it to the dynamic model, the dynamic response ofthe tool tip under ambient vibration was investigated. The results elucidated that the tool tipresponse was significantly affected by ambient vibration, and the isolation foundation had a goodeffect on vibration isolation.

Keywords: ultra-precision fly cutting machine tool, transfer matrix method for multibodysystems, dynamic response of tool tip, power spectrum density estimation method, ambientvibration

1. Introduction

To meet the precision and close tolerance requirements ofindustries such as aerospace, precision instrumentation, andnational defense, the quality of manufacturing equipment iscontinuously being improved. An ultra-precision fly cuttingmachine tool (UFCMT) is one of the key manufacturingmachines used to produce ultraprecise components in thoseindustries. The surface morphology of UFCMT-machined

components can be influenced by a variety of factors,including the machining parameters and the structuraldynamics of the machine. However, the tool tip response ismainly responsible for the good or bad surface morphology ofa finished workpiece. Thus, in order to improve processingprecision, research on the dynamic performances of machinetools is becoming more and more important.

Scholars have been studying the dynamic modeling ofmachine tools for several decades. Gurney and Tobias [1]found that a machine tool can be equated to a dampedmass-spring system. Hijink and van der Wolf [2] transformeda milling machine tool into a beam model and calculated thenatural frequencies with corresponding mode shapes. Thefinite element method (FEM) has risen to become one of the

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most commonly used methods of analyzing the performancesof machine tools. Zatarain et al [3] conducted modal analysisof a milling machine by using FEM and showed that theaccuracy of the calculation increases as the finite elementnodes increase. Jiang et al [4] applied FEM and structuraltopology optimization to improve the connection of a simplemechanical structure. In recent years, with the development ofcomputer technology, commercial software is widely used inthe dynamic analysis of machine tools, which makes it pos-sible for the holistic modeling of complex and largeUFCMTs. Liang et al [5] combined FEM and MATLAB®

Simulink® to predict the processing surface of a workpiece,and harmonic response analysis was conducted on the wholemachine tool [6]. Similarly, Chen et al [7] determined therelationship between the tool tip displacement response andthe cutting time using FEM simulation. Based on the criterionof having the same dominant structural frequency as themachine tool, they also presented a novel simulation methodwhere the machine tool was replaced by an equivalentcylinder; and then the cutting process was simulated inAbaqus FEA software [8]. Thus, the surface profile of aworkpiece can be analyzed during the manufacturing process.Gao et al [9] established a dynamic model of an UFCMT byusing ANSYS FEA software, and the topography of amachined surface was simulated by combining with amathematic model. However, it is too time-consuming whenusing FEM to analyze the performances of machine tools; andill-conditioning may occur due to the high order of the systemmatrix and complex rigid-flexible coupling problems,although the accuracy is relatively high. Thus, the dampedmass-spring system is used by some researchers to builddynamic model of machine tools. In the research of Wanget al [10], the beams and columns of the milling machine toolwere considered as lumped mass; and the joints were modeledby spring elements. Sato et al [11] modeled a machine toolbased on a damped mass-spring dynamic model, and therelationship between machine tool behavior and cutting forcein time domain was analyzed. Nevertheless, when applyingthe damped mass-spring system, the degrees of freedom areoften not enough, which leads to low accuracy. In addition,the overall dynamic equations of the system are alwaysestablished, which complicates the modeling process.

In this study, the transfer matrix method for multibodysystems (MSTMM) was used in the dynamic modeling of anUFCMT. Rui [12–14] presented the transfer matrix methodfor multibody systems, also called ‘Rui method’ in academia,to overcome the shortcomings of the traditional modelingmethods described above. It is a highly efficient dynamiccalculation method that can decrease the order of a systemmatrix, avoid ill-conditioning, and significantly improvecalculation speed. Today, MSTMM has been widely appliedin various fields of scientific research and practical engi-neering [15–17].

In some of the literature mentioned above [5–9], aerostaticbearings are used because they meet the high precision require-ments of ultra-precision machine tools. The gas film was

regarded as a series of springs when analyzing the dynamicperformances of the spindle, which hints at the importance ofaccurate stiffness calculation of these springs. Moreover, it wasalready established by the previous research that the dynamicresponse of the machine tool is greatly influenced by aerostaticbearings. At present, the FEM [18, 19] and the finite differencemethod (FDM) [20, 21] are widely used to solve the Reynoldsequation to obtain the gas pressure distribution. However, theresults calculated by FEM or FDM largely depend on the dis-charge coefficient [22], which mainly depends on the orificeparameters and operational conditions which are difficult toaccurately determine. In order to overcome this problem andmake the results more accurate, computational fluid dynamics(CFD) was adopted to directly solve the Navier–Stokes (N–S)equations [23, 24]. Nevertheless, it takes substantial time whencalculating the gas film stiffness under different geometric para-meters due to the mesh generation. The dynamic mesh technique(DMT) provides an efficient approach to analyze the perfor-mances of aerostatic bearings through mesh deformation, whichgreatly reduces the time required for repeatedly meshing. Chenet al [25] and Zhuang et al [26] applied DMT to investigate thedynamic performances of gas thrust bearings. The grids wererepeatedly deformed in the direction of bearing gravity to obtainthe relationship between bearing displacement and gas pressure.In addition to linear stiffness, more and more attention is beingpaid to the tilt effect of bearings [19, 27]. However, DMT israrely used for the analysis of bearing angular stiffness, especiallyfor journal bearings. This is mainly because of the greater ratio ofthe diameter and length of a bearing to the gas film thickness,which results in increased difficulty in generating mesh. More-over, the negative grid is more likely to appear during meshdeformation due to the curvature of gas mesh in the circumfer-ential direction.

In this paper, the dynamic model of the UFCMT fitted witha gas bearing is established by applying MSTMM. The large-span scale mesh model of a radial-thrust aerostatic bearing wasbuilt; and the bearing stiffness was calculated by adopting DMTbased on CFD. The dynamic characteristics of the machine toolwere found to be in good accordance with the experimentalresults. Moreover, the field ambient vibration was tested, andthen the power spectrum of the ground vibration was obtainedaccording to the data measured. Based on the verified dynamicmodel, the dynamic response of a tool tip under measuredambient vibration was investigated. The results showed that theambient vibration cannot be ignored in order to obtain a moreaccurate dynamic response of any system, and the isolatedfoundation was proven to be effective.

2. Dynamic modeling of the UFCMT

2.1. Dynamic model of the UFCMT multibody system

As shown in figure 1, the UFCMT is one kind of two-axismachine tool with a hydrostatic guide rail and aerostaticspindle. During the machining process, the aerostatic spindle

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

rotates at a constant speed around the z axis; then the cut-terhead and tool holder are driven. The workpiece is fixed onthe sliding table and fed slowly along the hydrostatic guiderail in the y direction.

The components of UFCMT can be regarded as rigidbodies, elastic beams, lumped masses, and elastic hinges,according to the natural properties. The dynamic model ofUFCMT is composed of 17 bodies and 39 joints. Thetopology figure is shown in figure 2. As prescribed by theprinciple of numbering in the MSTMM:

• The ground boundary is numbered 0.• The foundation, lathe bed, sliding table, left and rightcolumns, and cross-member are numbered 1, 2, 3, 4, 5,and 10.

• The left and right guide rails are numbered 6, 7, 8, and 9.• In the spindle system, the torque motor, aerostaticspindle, cutterhead, and two tool holders are numbered11, 12, 14, 15, 16, and 17.

• The aerostatic spindle is considered to be two beams withsingle input and single output ends connected by amassless dummy body and is numbered 13.

All of the bodies are connected by spatial elastic hingeswhose numbers are clearly marked in figure 2.

For the topology figure of the closed-loop multibodysystem, after ‘cutting’ at the junction of elements 3/27, 3/28,10/30, 10/37, and 10/39, the original closed-loop systembecomes a tree system, as shown in figure 3.

The interactions among the body elements are reflectedin the vibration characteristics and their effect on the dynamic

response of the system. The solution procedures for vibrationanalysis of the UFCMT system are shown in figure 4.

2.2. State vector, transfer equation, and transfer matrix

The state vector is defined as a column matrix denoting themechanics state of a point. The state variables include the

Figure 1. Picture of the UFCMT.

Figure 2. Topology figure of the dynamic model.

Figure 3. Topology figure of the dynamic model of the treemultibody system.

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

displacements and the internal forces at that point. Forthe UFCMT system, the state vectors of each connection point can be expressed as

= Q Q QZ X Y Z M M M Q Q Q, , , , , , , , , , , 1i j x y z x y z x y z i jT

, ,[ ] ( )

in which the variables are the modal coordinates of the displacement, angular displacement, internal momentum, and internalforce along the x, y, and z directions, respectively.

After combining the main transfer equation and geometric equations of all relevant elements, the overall transfer equationof the UFCMT system can be obtained:

=U Z 0, 2all all ( )

where

=Z Z Z Z Z Z Z Z Z, , , , , , , 3all 18,0T

16,0T

17,0T

27,0T

28,0T

30,0T

37,0T

39,0T T[ ] ( )

=

- + + + + +

++

+ + ++ ++ ++ +

- - - - - - - - - - - -

- -

- -

- - - -

- - - - -

- - - - -

- - - - -

- -

- - - - - - - -

- - - - - - - - -

- - - - - - - -

- - - - - - - -

4U

I T T T T C T T C T T C T T C T T CO G O O O O O GO O G O O O O GO G G O O O G GO G G O O G C G GO G G O O O G G C GO G G O O O G G G CO O O G C G C O O OO G G O O G G C G G C G G CO G G G C G C G C G G C G G CO G G G O G C G G C G G CO G G O G G C G G C G G C

.

I I I I I

I

I

I

I I

I I I

I I I I I

I I I

I I I

all

16 18 17 18 27 18 3 18, 28 18 3 18, 30 18 10 18, 37 18 10 18, 39 18 10 18,

16 15 39 15

17 15 39 15

16 13 17 13 37 13 39 13

16 10 17 10 10 10, 37 10 39 10

16 10 17 10 37 10 10 10, 39 10

16 10 17 10 37 10 39 10 10 10,

3 3, 3 3,

16 2 17 2 30 2 10 2, 37 2 10 2, 39 2 10 2,

16 2 17 2 3 2, 3 2, 10 2, 37 2 10 2, 39 2 10 2,

16 2 17 2 27 2 10 2, 37 2 10 2, 39 2 10 2,

16 2 17 2 28 2 10 2, 37 2 10 2, 39 2 10 2,

1 2 2 3 4

2

3

4

1 2

2 3 4

1 2 2 3 4

2 3 4

2 3 4

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

( )

The equation describing the relations among the state vectors of each boundary end can be easily deduced, as shown inequation (5).

= + ++ + ++ + ++ + +

- - -

- - -

- - -

- - -

Z T Z T Z TT C Z T T C ZT T C Z T

T C Z T T C Z , 5

I I

I

I I

18,0 16 18 16,0 17 18 17,0 27 18

3 18, 27,0 28 18 3 18, 28,0

30 18 10 18, 30,0 37 18

10 18, 37,0 39 18 10 18, 39,0

1 2

2

3 4

() ( )

( ) () ( ) ( )

where

============

-

-

-

-

-

-

-

-

-

-

-

-

T U U U U U U U U U U U U U U U UT U U U U U U U U U U U U U U U UT U U U U U U U U UT U U U U U U U U UT U U U U U U UT U U U U U U U U U U U U U U UT U U U U U U U U U U U U U U UT U U U U U UT U U U U U UT U U U U U U U UT U U U U U U U UT U U U U U U U U

. 6

I I I I

I I I I

I

I

I

I I I

I I I I

I I I

I I I

I I I

I I I

I I I

16 18 18 1 19 2, 21 4 29 10, 38 13, 33 14 34 15, 35 16

17 18 18 1 19 2, 21 4 29 10, 38 13, 33 14 34 15, 36 17

27 18 18 1 19 2, 23 6 25 8 27

28 18 18 1 19 2, 24 7 26 9 28

30 18 18 1 19 2, 22 5 30

37 18 18 1 19 2, 21 4 29 10, 38 13, 32 12 31 11 37

39 18 18 1 19 2, 21 4 29 10, 38 13, 33 14 34 15, 39

3 18, 18 1 19 2, 20 3,

3 18, 18 1 19 2, 20 3,

10 18, 18 1 19 2, 21 4 29 10,

10 18, 18 1 19 2, 21 4 29 10,

10 18, 18 1 19 2, 21 4 29 10,

1 1 2 2

1 1 2 3

4

5

2

1 1 1

1 1 2 1

1 3 1

2 3 2

2 1 2

3 1 3

4 1 4

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪

( )

4

Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

Taking the example of body element 2, the geometric equations can be deduced

+ + + + + + + =+ + + + + + + + =+ + + + + + + =+ + + + + + + =

- - - - - - - -

- - - - - - - - -

- - - - - - - -

- - - - - - - -

7

G Z G Z G G C Z G G C Z G G C ZG Z G Z G CZ G CZ G CZ G G C Z G G C ZG Z G Z G Z G CZ G G C Z G G C ZG Z G Z G Z G CZ G G C Z G G C Z

00

00

,

I I I

I I I I I

I I I

I I I

16 2 16,0 17 2 17,0 30 2 10 2, 30,0 37 2 10 2, 37,0 39 2 10 2, 39,0

16 2 16,0 17 2 17,0 3 2, 27,0 3 2, 28,0 10 2, 30,0 37 2 10 2, 37,0 39 2 10 2, 39,0

16 2 16,0 17 2 17,0 27 2 27,0 10 2, 30,0 37 2 10 2, 37,0 39 2 10 2, 39,0

16 2 16,0 17 2 17,0 28 2 28,0 10 2, 30,0 37 2 10 2, 37,0 39 2 10 2, 39,0

2 3 4

1 2 2 3 4

2 3 4

2 3 4

⎧⎨⎪⎪

⎩⎪⎪

( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

where

= -= -= -= -= -= -= -

=====

-

-

-

-

-

-

-

-

-

-

-

-

G H U U U U U U U U U U U UG H U U U U U U U U U U U UG H U U U U U U U U U U UG H U U U U U U U U U U UG H U U U UG H U U U UG H U U U UG H U U U U UG H U U U U UG H U U UG H U UG H U U

. 8

I I I I

I I I I

I I I

I I I I

I I I

I I I

I I I

I

I

I

I I I

I I I

16 2 2, 21 4 29 10, 38 13, 33 14 34 15, 35 16

17 2 2, 21 4 29 10, 38 13, 33 14 34 15, 36 17

37 2 2, 21 4 29 10, 38 13, 32 12 31 11 37

39 2 2, 21 4 29 10, 38 13, 33 14 34 15, 39

10 2, 2, 21 4 29 10,

10 2, 2, 21 4 29 10,

10 2, 2, 21 4 29 10,

27 2 2, 23 6 25 8 27

28 2 2, 24 7 26 9 28

30 2 2, 22 5 30

3 2, 2, 20 3,

3 2, 2, 20 3,

1 1 2 2

1 1 2 3

1 1 1

1 1 2 1

2 1 2

3 1 3

4 1 4

4

5

2

1 3 1

2 3 2

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪

( )

The transfer matrices of rigid bodies, beams, and elastic hinges and the detailed process of deduction can be seen in [28].The state vectors are defined in the form of equation (1). Specifically, Z18,0 is a fixed boundary; and its displacement and angleare zero. The displacement and angle of boundary ends Z16,0 and Z17,0 are unknown, while the force and moment are zero.After cutting the closed-loops, new boundaries with equivalent state vectors are generated, i.e. Z27,0, Z28,0, Z30,0, Z37,0, andZ39,0

=

= Q Q Q =

= Q Q Q =

Z

Z

Z

M M M Q Q Q

X Y Z i

X Y Z M M M Q Q Q i

0, 0, 0, 0, 0, 0, , , , , ,

, , , , , , 0, 0, 0, 0, 0, 0 16, 17

, , , , , , , , , , , 27, 28, 30, 37, 39

. 9

x y z x y z

i x y z i

i x y z x y z x y z i

18,0 18,0T

,0 ,0T

,0 ,0T

{[ ]

[ ] ( )

[ ] ( )

( )

Substituting equation (9) into (2), the system’s characteristics equation can be obtained:

=U 0det . 10all¯ ( )

Solving equation (10), natural frequencies w =k 1, 2,k ( ···) can be obtained.

2.3. Bending of the revolving cutterhead

Due to the height difference between the centroids of the cutterhead and tool holder in the direction of the z axis, as theUFCMT starts rotating, the centrifugal force causes the tool holder to produce a moment on the cutterhead. Thus, thecutterhead bends upward during the machining process; and the higher the spindle speed, the more the bending. Although Dinget al [29] and Lu et al [30] analyzed the dynamic performances of UFCMT using MSTMM, they did not consider the influenceof cutterhead bending.

The FDM is applied to calculate the elastic deformation of the cutterhead. The cutterhead is regarded as a multistep beamcomposed of n sections with uniform cross-section and equal length, as shown in figure 5.

For the section ij( j=0, 1, 2, 3), the displacement of the output end in the z axis is described as z .ijThe displacements of

the adjacent sections satisfy equation (11)

- + =+ -z z z dM

EI2 , 11i i i

j

j

2j j j1 1 ( )

where Mj and EIj are moment and flexural rigidity on the jth section, and d is the length of each section.

5

Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

Assuming the slope of the deflection at the section i0equals zero, then

»-

=¢dz

dy

z z

d20. 12i i

0

1 1⎛⎝⎜

⎞⎠⎟ ( )

By iterative computation, the displacements of eachsection can be obtained.

2.4. System response

The UFCMT multibody system is composed of body ele-ments, such as rigid bodies, beams, and others, connected byvarious hinges. Meanwhile, as the damping forces are inclu-ded in the external forces, the dynamic equation of any body

Figure 4. Vibration characteristics and dynamic response procedure.

Figure 5. Partition of the cutterhead.

6

Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

element i can be written in the matrix form

+ =M v K v f , 13i i tt i i i, ( )

where Mi and Ki are parameter matrices of mass and stiffness,vi is a column matrix composed of displacements and angulardisplacements indicating the motion state of the body ele-ment, and fi is the column matrix of external forces andexternal torques acting on the body element. The subscript tdenotes differentiation with respect to time.

For a rigid body i vibrating in space with N input and Loutput ends, the motion can be described by linear displace-ments ri and angular displacements θi of the first input end I1.The motion equation of the rigid body is

å åq- - + == =

r l q q Fm m . 14i i i I C i in

N

I il

L

O i i,1

,1

,n l1 ˜ ( )

The rotation equation of the rigid body rotating about I1is

å å å

å

q+ + - -

+ = +

= = =

=

m l r J m m l q

l q m l F ,

15

l

l

i I C i i I i in

N

I i

L

O in

N

I I i I i

L

I O i O i i I D i i

, ,1

,1

,2

, ,

1, , ,

l

l l

n n n1 1 1

1 1

˜ ˜

˜ ˜

( )

where mi and Fi are external principal moment and forceacting on the rigid body. The internal force and moment of theinput end and output end are q ,I i,n

q ,O i,lmI i,n

and m .O i,l Thecoordinate matrices of the center of mass, output point, inputpoints, expect the first input point, and the force acting pointrelative to the first input point I1 are l ,I C i,1

˜ l ,I O i,l1˜ l ,I I i,n1

˜ and l ,I D i,1˜

respectively. The moment of inertia matrix of the rigid bodywith respect to point I1 is J .I i,1

The simplified center ofexternal force is D.

For a beam undergoing transversal, longitudinal, andtorsional vibrations, the dynamic equations are

rq q

¶¶

+¶¶

= - ¶¶

¶¶

+ ¶¶

= - ¶¶

¶¶

- ¶¶

=

¶¶

-¶¶

=

16

EIy

zm

y

tf z t

zm z t z l

EIx

zm

x

tf z t

zm z t z l

mz

tEA

z

zf z t z l

Jt

GJz

m z t z l

, , 0

, , 0

, 0

, 0

,

x i i y x i

y i i x y i

i i z i

p iz

p iz

z i

,

4

14

2

2 11

1 1

,

4

14

2

2 11

1 1

2

2

2

12 1 1

2

2

2

12 1 1

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪( )

( ) ¯ ( ) ( ) ( )

( ) ¯ ( ) ( ) ( )

¯ ( ) ( ) ( )

( ) ( ) ( ) ( )

where x (z1, t), y (z1, t), and z (z1, t) denote the transversal andlongitudinal displacements of any point on the beam relativeto its equilibrium position, respectively. The torsion angle ofan arbitrary cross-section relative to its equilibrium position is

Figure 6. Configuration of aerostatic spindle system.Figure 7. Mesh model and boundary conditions.

Figure 8. Pressure contour of aerostatic bearings system.

7

Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

θz (z1, t). The external distributional force and moment in thex, y, and z directions are denoted by fx (z1, t), fy (z1, t), fz (z1, t),mx (z1, t), my (z1, t), and mz (z1, t), respectively.

Arranging the body dynamic equations of each element,the body dynamic equation of the system can be constituted

+ =Mv Kv f , 17tt ( )

where

=

= = =

M

MM

M

K

KK

K

v

vv

v

f

f

f

f

,

, , . 18

B

B

B

B

B

B

B

B

B

B

B

B

n

nn

n

1

2

1

2

1

2

1

2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥( )

The subscripts B1, B2, K and Bn are the sequencenumbers of the body elements.

Applying modal technology, assume that

å=v V q t , 19k k ( ) ( )

where = =V V V V k, , , 1, 2, .kBkT

BkT

BkT T

n1 2[ ··· ] ( )

Figure 9. The variation of linear stiffness of the journal and thrust bearings with respect to film thickness: (a) journal bearing and (b) thrustbearing.

Figure 10. The variation of angular stiffness of the journal and thrust bearings with respect to film thickness: (a) journal bearing and (b) thrustbearing.

Figure 11. Modal test.

8

Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

Table 1. Natural frequencies and mode shapes.

Order Modal test (Hz) MSTMM (Hz) Relative error (%) Mode shape

1 7.45 7.4 −0.67 The rotation of the UFCMT and foundation around the z axis in the same direction2 9.99 10.00 0.10 The movement of the UFCMT and foundation along the z axis in the same direction3 42.72 42.83 0.26 The wiggle of the UFCMT and foundation around the x axis in the opposite direction4 53.56 51.60 −3.66 The wiggle of the UFCMT and foundation around the y axis in the opposite direction5 70.98 71.98 1.41 The movement of the UFCMT and foundation along the z axis in the opposite direction6 100.09 101.07 0.98 The rotation of the cutterhead and lathe bed in the same direction7 106.75 110.03 3.07 The rotation of the UFCMT around the x axis8 180.66 176.36 −0.03 The torsion of the cutterhead and cross-member around the x axis in the same direction9 222.48 224.1 0.73 The torsion of the cross-member to lathe bed around the z axis in the opposite direction10 261.92 253.36 −1.46 The movement of the cutterhead along the z axis11 366.74 370.12 0.92 The vibration of the cutterhead and machine tool along the z axis in the opposite direction12 442.49 449.81 1.65 The vibration of the cutterhead and cross-member along the z axis in the opposite direction13 582.37 582.66 0.05 The transformation of the cutterhead14 690.48 687.79 −0.39 The transformation of the sliding table, the torsion of lathe bed15 808.58 808.09 −0.06 The vibration of the tool holder and sliding table along the z axis in the same direction16 961.27 946.69 −1.52 The vibration of the tool holder along the z axis

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Int.J.

Extrem

.Manuf.

2(2020)

025301JDing

etal

The augmented eigenvectors V k include the modalcoordinates of displacements and angular displacements ofeach discrete element and the mode shape of each continuouselement corresponding to the kth eigenfrequency ωk of themachine tool system.

Substituting equation (19) into (17) yields

å å+ ==

¥

=

¥

MV KV fq t q t . 20k

k k

k

k k

1 1

( ) ( ) ( )

Taking the inner product for both sides of equation (20)using augmented eigenvectors V p (p=1,2, Kn) and theorthogonality of the augmented eigenvectors,

d dá ñ = á ñ =MV V KV VM K, , , . 21k pk p p

k pk p p, , ( )

The generalized coordinate equations of the system canbe obtained:

w+ =á ñ

=f V

q t q tM

p n,

1, 2, , . 22pp

pp

p

2 ( ) ( ) ( ··· ) ( )

Finally, according to the initial condition v (x, 0) and vt(x, 0), the solution’s generalized coordinates qp (t) are easilydetermined, and the dynamics response v can be computed.

3. Stiffness analysis of aerostatic bearings

3.1. Geometric model of aerostatic bearings

Figure 6 illustrates the configuration of a vertical aerostaticbearings system in the UFCMT, which consists of a journalbearing and a thrust bearing. Externally pressurized gassupplied at gas inlets, flows through the gas channel andcavity and then forms the bearings’ gas film of journal andthrust, which maintains the rotational and translational motionof bearings.

3.2. Governing equations of fluid field

For compressible Newtonian fluid, the pressure distribution inthe fluid domain is governed by N–S equations, assuming thatthe fluid domain is isothermal during the calculation process.The mass and momentum conversation equations in the x, y,and z orientation are depicted as follows:

rr

¶

¶+ =u

tdiv 0 23

gg( ) ( )

Figure 12. Power spectrum density functions.

Figure 13. Welch method procedure.

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

rr t t t

rr

t t t

rr

t t t

¶

¶+ = -

¶¶

+¶¶

+¶

¶+

¶¶

+

¶

¶+ = -

¶¶

+¶

¶+

¶

¶+

¶

¶+

¶

¶+ = -

¶¶

+¶¶

+¶

¶+

¶¶

+

24

u

u

u

u

tu

p

x x y zF

v

tv

p

y x y zF

w

tw

p

z x y zF

div

div

div

,

gg

xx yx zxx

gg

xy yy zyy

gg

xz yz zzz

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

( )

( )( )

( )( )

( )( )

where

t h l t t h

t h l t t h

t h l t t h

=¶¶

+ = =¶¶

+¶¶

=¶¶

+ = =¶¶

+¶¶

=¶¶

+ = =¶¶

+¶¶

u

u

u

u

x

u

y x

v

x

u

z

w

x

w

x z

w

y

2 div ;v

2 div ;

2 div ;v

, 25

xx xy yx

yy xz zx

zz yz zy

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( )

where t is time; u is the velocity vector; u, v, and w are thevelocity components in the three coordinate orientations; p ispressure; Fx, Fy, and Fz are the body force in the threecoordinate orientations; τ is viscous stress; and λ=−2/3.

3.3. Dynamic mesh technique

In this paper, the ratio of length to thickness of the gas film ismore than 1×104, which leads to grid distortion and eventhe generation of negative grids during the meshing and

calculating process. In order to improve the quality of themesh, the multiblock method has been applied to generate theflow field topology structure in the ANSYS® ICEM CFD™

software, hence sophisticated grid division of a large-spanscale flow field is achieved. The pressure distribution at theend of the inlet is complicated, the pressure drop is large, andO-block is an effective method of encrypting the mesh of acylindrical flow field. Therefore, grid encryption has beenperformed around the inlet hole, and the O-block has beencreated at the gas inlet. Figure 7 illustrates the mesh modeland boundary conditions of aerostatic bearings in the num-erical calculation process.

The DMT has been used to calculate the radial and axialstiffness of journal and thrust bearing in this paper. By uti-lizing the user define function microinstruction DEFINE_-GRID_MOTION in ANSYS® Fluent® to define thetranslational motion in x and z orientations of the rotationalwall, the load capacity and stiffness of bearings are obtained,and the calculation efficiency has also been improved. Thediffusion smooth method has been used in DMT to update themesh deformation, which is expressed as the followingequation:

g =u 0, 26m· ( ) ( )

where

g =al

1, 27( )

where um

is the mesh deformation velocity, g is the diffusioncoefficient which controls the influence of boundary meshmotion on internal mesh motion, and l is the orthogonaldistance from the boundary. In this paper, the value of α is setas zero so that the boundary mesh motion guarantees uniformdiffusion.

The position of mesh nodes is updated as follows:

= + D+x x u t, 28i im

1 ( )

where +x i 1is the position of updated mesh nodes, x i is the

position of mesh nodes before update, andDt is the time step.Newton’s law is solved, and the resultant force in the

radial and axial directions is obtained, as well as the momentof journal and thrust bearings. According to equations (29)and (30), the linear and angular stiffness of bearings can be

Figure 14. Ambient vibration acceleration signal: (a) experimental acceleration signal and (b) simulated acceleration signal.

Figure 15. Vibration test.

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

derived.

=DD

KW

h29h ( )

q=

DD

qKM

, 30( )

where ΔW and Δh are the variation of load capacity andthickness of the gas film, respectively; and ΔM and Δθ arethe variation of deflection moment and deflection angle,respectively.

3.4. Variation of bearings stiffness

In order to calculate the vibration response of the UFCMT,the stiffness of the journal and thrust bearings must beacquired first. Since the cutting force of vertical bearings isonly a few Newtons, the amount of radial eccentricity isextremely small, so the spindle and journal restrictor arealmost concentric during the calculation. When the gas filmthickness of the journal bearings is 15 μm, the sum of theupper and lower thrust film thickness is 30 μm, the inletpressure is 0.74 MPa, the rotational speed of the bearings (forthe purposes of this paper) is 280 rpm, and the total weight of

Figure 16. Acceleration response of the tool tip in the time domain: (a) experimental acceleration and (b) simulated acceleration.

Figure 17. Acceleration response of tool tip in the frequency domain: (a) experimental acceleration fast Fourier transform (FFT) and (b)simulated acceleration FFT.

Table 2. RMS and maximum amplitudes.

RMS (m s−2) Maximum (m s−2)

Experimental data 0.0062 0.0348Simulation result 0.0066 0.0350

Table 3. Comparison of frequencies (Hz).

Order 1 2 3

Experimental acceleration FFT 246.3 586.7 1000Simulated acceleration FFT 253.3 581.2 945.2Natural frequency 261.92 582.37 961.27

Figure 18. Ambient vibration monitoring points.

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

the bearings and cutterhead is 220 kg. The pressure contour ofthe aerostatic bearings system can be obtained, as shown infigure 8.

The maximum pressure is at the gas inlet, and the gasoutlet is approximately equal to atmospheric pressure. Due tothe eccentricity of the journal and thrust bearings, the thick-ness of the journal gas film is uneven, so the pressure is not aperiodic distribution. However, eccentricity of thrust onlyreduces the thickness of the lower gas film, so the pressure ofthe lower gas film is greater than that of the upper gas film.

The variations of linear and angular stiffness with respectto the film thickness of the journal and thrust bearings arecalculated as shown in figures 9 and 10, respectively.

It can be observed from figure 9 that the linear stiffnessof the journal and thrust bearings can be affected by the gasfilm thickness. The linear stiffness increases rapidly to peakand then decreases slowly with the growth of the filmthickness. The journal bearing radial stiffness is maintained ata relatively high level when the film thickness is between 9and 13 μm, and the maximum radial stiffness of 748.36N μm−1 can be obtained at a journal bearing film thickness of10 μm. For thrust bearing, a relatively high axial stiffnessexists at the sum of the upper and lower thrust film thicknessof 10–22 μm. The maximum axial stiffness of 2336.04N μm−1 can be obtained at the sum of the upper and lowerthrust film thickness of 14 μm.

Figure 10 presents the variation of angular stiffness of thejournal and thrust bearings with respect to gas film thickness.With the same results as shown in figure 9, the angularstiffness of the journal and thrust bearings also stronglydepends on the film thickness. The angular stiffness rises to apeak and then drops with the growth of film thickness. Theangular stiffness decreases sharply near the peak value. Thejournal and thrust bearings can obtain relatively high angularstiffness at a journal film thickness of 8–11 μm and a sum ofthe upper and lower thrust film thickness of 10–22 μm,

respectively. The maximum angular stiffness of 1.312×106 Nm rad−1 and 9.865×106 Nm rad−1 of the journal andthrust bearings can be acquired at a journal film thickness of 9μm and a thrust film thickness of 14 μm, respectively.

4. Dynamic characteristics analysis of the UFCMT

To validate the dynamic model of the UFCMT, a modal testwas conducted. By exerting the excitation at the test pointswith an impact hammer, as shown in figure 11, the vibrationresponse of each point was measured in sequence, which wasone column of the frequency response function matrix. Themodal parameters of the UFCMT were obtained by using thecurve fitting method. In this test, 364 test points were arran-ged to clearly describe the overall shape of the UFCMT, theanalyzing frequency fa was from 0 to 2048 Hz, the samplelength ΔT was 2 s, and the frequency resolution Δf was0.5 Hz.

Considering the lumped average curve of amplitude-frequency in three directions, the natural frequency and modeshape were selected and are shown in table 1. Meanwhile, thenatural frequencies w =k 1, 2,k ( ···) obtained by MSTMMwere compared with the experimental data. The maximumrelative error was 3.66%. Moreover, the frequencies andmode shapes were all in good agreement, which indicates thedynamic model of the UFCMT is appropriate.

5. Analysis and discussion

Due to their distinct traits, namely high precision, highresolution, and high sensitivity, precision machine tools, suchas the UFCMT, are able to machine components that havesurface roughness in nonmetric levels. Therefore, even theinterference of ambient vibration affects dynamic response ofthe machine tool. The vibration-sensitive frequency of ultra-precision machining equipment ranges from 1 to 150 Hz, andthe acceleration is generally below 2×10−2 m s−2. To pro-vide optimum performance, foundation isolation is critical,even in the toughest applications and the most demandingcircumstances. Considering the extreme requirements formachining precision, technical isolation foundation was

Figure 19. Acceleration response of tool tip: (a) without vibration isolation and (b) with vibration isolation.

Table 4. RMS and maximum amplitudes.

RMS (m s−2) Maximum (m s−2)

Without vibration isolation 0.0112 0.0515With vibration isolation 0.0066 0.0350

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

designed to avoid disturbance from the surroundingenvironment.

The fundamental requirement for designing a functioningfoundation isolation system is to establish the proper rela-tionship between the forcing frequency and the natural fre-quency of the isolation system, and the objective is to ensurethat the natural frequency is lower than the forcing frequency.The existing isolation foundation contains several proceduresfor achieving foundation isolation. An inertia block isolationsystem is the simplest, most practical approach with a naturalfrequency in the range of 8–19 Hz. It consists of a concreteinertia block sized to provide adequate support for theUFCMT being installed. It also supplies mass to give neededdamping to the UFCMT system being supported by a series offoundation isolation pad materials. Meanwhile, Regufoam®

isolation materials with a natural frequency range of 6–15 Hzwere applied as vibration isolating structural elements toensure maximum performance. These installations provide acost-effective, excellent approach to preventing the trans-mission of structure-born noise and vibration.

5.1. Power spectrum of ambient vibration

Due to the diversity of ambient vibration sources, thedynamic excitation characteristics of the actual system presentcomplex randomness; and it is difficult to study those usingaccurate analytical methods. Therefore, statistical analysisbecomes a more reasonable option, specifically the powerspectrum density estimation method. Common methods ofobtaining the power spectrum density function include theperiodogram method, autocorrelation function method, Burgmethod, and Welch periodic average method. The exper-imental ambient vibration signal is processed by these fourmethods and is shown in figure 12.

After comparing the power spectrum density functioncurves in figure 12 using the Welch method, the peak valuesare clearest, and the burr is minimal. The Welch method isapplicable to processing a smooth signal, and the ambientvibration signal is precisely the smooth one. Therefore, theWelch method was selected, and the solution procedure isshown in figure 13.

The Welch method divides the signal sequences x (n)with N samples into overlapping data sections, and eachsection has M samples. The section of data can be expressedas

= + - - u n x n iD n M i K, 0 1, 0 1,31

( ) ( )( )

where K is the number of sections, and iD is the origin of theith section. The overlap degree between data sections isdefined as D/M, e.g. as D=M/2, the overlap degree is 1/2.

Applying an appropriate window function w (n) to ensurethe non-negativity of the spectral estimation, the spectralestimation of each section is given by

å= w

=

--S w

MUu n w n e

1, 32xi

n

M

ij n

0

1 2

ˆ ( ) ( ) ( ) ( )

where å= =-

UM

w n1

n

M

0

1 2 ( ) is the normalization factor to

guarantee that the final power spectrum estimation isunbiased. Finally, Welch spectral estimation can be obtainedby averaging the spectral estimation of each section

å==

S wKU

S w1

. 33xxi

K

xi1

ˆ ( ) ˆ ( ) ( )

The original acceleration signal of ambient vibration isshown in figure 14(a). The simulated ambient vibrationacceleration signal can be obtained by the trigonometric seriesmethod, as in equation (34) and shown in figure 14(b).

å w f= +=

x t a tcos , 34k

N

k k k1

( ) ( ) ( )

where fk is the independent random phase angle evenlydistributed in [0, 2π), and ωk is the frequency obtained byequation (35)

w w w= + - D =k k N1

2, 1, , , 35k l ⎜ ⎟⎛

⎝⎞⎠ ··· ( )

where w w wD = - Nu l( )/ is the interval of frequency, and[ωl, ωu] is the frequency range in which the power spectraldensity function Sx(ω) has salient value. The amplitude ak canbe obtained by equation (36)

w w= D =a S k N2 , 1, , . 36k x k( ) ··· ( )

5.2. Vibration test of tool tip and analysis

The ambient vibration acceleration signal was applied to thefoundation, i.e. Body 1 in the dynamic model, as an externalexcitation. The dynamic response was calculated by solvingthe overall transfer equation and dynamics equation deducedin section 2. The vibration test to measure the vibrationacceleration response of the tool tip was set up as shown infigure 15. The acceleration signal was detected by accelera-tion sensor PCB 355B04, which was glued near the tool tip.The data was collected and stored by the micro dynamic dataacquisition system DH 5916, which was fixed at the bottomof the cutterhead.

The acceleration signal of ambient vibration, measuredunder start-up without rotating, was added into the dynamicmodel. Then the acceleration response of the tool tip could besimulated. The simulated results were compared with theexperimental vibration response of the tool tip both in thetime domain and frequency domain, as shown in figures 16and 17.

As shown in figure 16, the simulated acceleration of thetool tip is in good agreement with the experimental accel-eration. The root mean square (RMS) and maximum ampli-tudes are shown in table 2. The simulation values andexperimental results have negligible difference.

Fourier transform was applied to the vibration accelera-tion signal, and the corresponding frequency spectrums areshown in figure 17. The main frequencies of the experimental

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

acceleration and simulated acceleration were in good agree-ment. Furthermore, these frequencies also matched well withnatural frequencies from the modal test, as shown in table 3.

5.3. Effect of ambient vibration on the UFCMT

In figure 18, the ambient vibration monitoring point that liesoutside the isolation foundation is marked as P1, and themonitoring point on the foundation is P2. Using these twosensors, the ambient vibration signals with and without thevibration isolation were measured.

Using the power spectrum density estimation method, themeasured vibration acceleration signals of these two pointscan be simulated into statistical signals. As the externalexcitation, the vibration acceleration signals were added intothe dynamic model established in section 2. Then the accel-eration response of the tool tip was simulated (see figure 19).

From figure 19, it is apparent that the accelerationresponse of the tool tip decreased through the vibration iso-lation. The RMS and maximum amplitudes of acceleration areshown in table 4. The RMS decreased by 41.1%, and themaximum declined by 32.0%.

As the rotating speed of the spindle was 280 rpm and thefeed rate of the sliding table was 15 mmmin−1, the machiningprocess on the workpiece with a diameter of 50 mm was

simulated. Figure 20(a) shows the whole machining processafter obtaining the displacement response of the tool tiprelative to the workpiece, and the total machining time was200 s. The time for one revolution of the spindle was 0.214 s,and figure 20(b) shows the displacement in one revolution.Figure 20(c) represents the cutting process in which the tooltip came into contact with the workpiece.

With the vibration isolation foundation, the displacementresponse of the tool tip was smaller compared to the responsewithout vibration isolation (see figure 20). Selecting the cut-ting process, the maximum displacements were 140.47 and123.52 nm. Due to the role of isolation foundation, the dis-placement of the tool tip relative to the workpiece wasreduced by 12.1%.

After obtaining the displacement of the tool tip, thesimulated machined surface was acquired by using the surfacetopography model and considering the interference phenom-enon of the cutting profile and geometry of the tool tip, asshown in figure 21.

The corresponding peak-to-valley (PV) values of thesimulated machined surfaces were 302.68 and 289.51 nm,and the PV value decreased by 4.4%. Therefore, ambientvibration had great influence on the response of the tool tip,which would be the crucial factor in machining quality.

Figure 20. Displacement response of the tool tip relative to the workpiece: (a) whole machining process, (b) in one revolution, and (c) incutting process.

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Int. J. Extrem. Manuf. 2 (2020) 025301 J Ding et al

6. Conclusion

(1) The dynamic model of the UFCMT multi-rigid-flexiblesystem was established based on MSTMM. Thebending of the cutterhead due to rotation wasconsidered and calculated by FDM. The characteristicequation of the system was deduced theoretically, andthe simulated vibration characteristics were consistentwith the modal test results. The effective simulation ofthe dynamic response was realized by solving thedynamic equations, and the simulation results werevalidated by the vibration test.

(2) The mesh model of the large-span scale flow field wasestablished, and the DMT was proposed to calculate thelinear and angular stiffness of the journal and thrustbearings at different film thicknesses. Both linear andangular stiffness increased to the peak, and thendecreased with the growth of the film thickness. Themost suitable radial and angular stiffness of the journalbearing was obtained at a film thickness of 9–11 μm.When the sum of the upper and lower thrust filmthickness was 12–18 μm, the optimum axial andangular stiffness of the thrust bearing was also acquired.The angular stiffness of the thrust bearing was close toten times that of the angular stiffness of the spindlebearing, so the main anti-deflection capacity came fromthrust bearing.

(3) Due to the diversity of ambient vibration sources, theshort-time measured signal did not reflect the dynamicexcitation characteristics with complex randomness.Thus, the power spectrum density estimation methodwas adopted for statistical analysis. After comparing thefour methods of applying the power spectrum densityfunction calculation, the Welch method was chosen tosimulate the equivalent ambient vibration excitation byprocessing the long-time measured ambient vibrationsignal.

(4) Using the power spectrum density estimation method,the experimental ambient vibration acceleration signalswith complex randomness were transformed intostatistical signals, which were further applied to thedynamic model. Comparing the dynamic responses of

the tool tip with and without vibration isolation, thesimulated results indicated that the PV value declinedby 4.4% with the vibration isolation, which revealed thegreat influence of ambient vibration on the dynamicsresponse and machining quality.

Acknowledgments

The authors declare no potential conflicts of interest withrespect to the employment, consulting fees, research con-tracts, patent licenses, etc, of this article. This study is sup-ported by the Science Challenge Program (Grant No.TZ2016006-0104).

ORCID iDs

Jianguo Ding https://orcid.org/0000-0002-2569-4665

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