2006 cao spindle model

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International Journal of Machine Tools & Manufacture 47 (2007) 1342–1350

Modeling of spindle-bearing and machine tool systems for virtual

simulation of milling operations

Yuzhong Cao, Y. AltintasÃ

Department of Mechanical Engineering, University of British Columbia, 2054-6250, Applied Science Lane, Vancouver, BC, Canada V6T 1Z4

Received 27 June 2006; received in revised form 26 July 2006; accepted 3 August 2006

Available online 9 October 2006

Abstract

This paper presents a general, integrated model of the spindle bearing and machine tool system, consisting of a rotating shaft, tool

holder, angular contact ball bearings, housing, and the machine tool mounting. The model allows virtual cutting of a work material with

the numerical model of the spindle during the design stage. The proposed model predicts bearing stiffness, mode shapes, frequency

response function (FRF), static and dynamic deflections along the cutter and spindle shaft, as well as contact forces on the bearings with

simulated cutting forces before physically building and testing the spindles. The proposed models are verified experimentally by

conducting comprehensive tests on an instrumented-industrial spindle. The study shows that the accuracy of predicting the performance

of the spindles require integrated modeling of all spindle elements and mounting on the machine tool. The operating conditions of the

spindle, such as bearing preload, spindle speeds, cutting conditions and work material properties affect the frequency and amplitude of

vibrations during machining.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Spindle; Chatter vibration; Finite element method; Milling

1. Introduction

The successful application of high-speed machining

technology is highly dependent on spindles operating free

of chatter vibration without overloading the angular

contact ball bearings. Unless avoided, vibration instability

in the metal-cutting process leads to premature failure of

the spindle bearings [1]. The spindle, tool-holder, and tool

are the main sources of chatter vibrations on high-speed

machines. The objective of the design engineer is to predict

the cutting performance of the spindle during the designstage by relying on engineering model of the process and

system dynamics.

Early spindle research focused mainly on static and

quasi-static analysis, whereas current research is extended

to optimal design by using dynamic analysis. Ruhl et al. [2]

is one of the earliest researchers to use the finite element

(FE) method for modeling of rotor systems. His model

includes translational inertia and bending stiffness but

neglects rotational inertia, gyroscopic moments, shear

deformation, and axial load. Nelson [3] used the

Timoshenko beam theory to establish shape functions

and formulate system matrices, including the effects of

rotary inertia, gyroscopic moments, shear deformation,

and axial load.

In the past, little research has been conducted to model

the coupling of bearings and spindles. The effects of

preload and spindle speeds on bearing stiffness and thedynamics of the spindle system are seldom studied. Wardle

et al. [4] presented a very simplified model for describing

the dynamics of a spindle-bearing system with a constant

preload. The theoretical maximum operating speed of the

spindle system is increased by maintaining a constant

preload, but Wardle neglected the softening of bearing

stiffness due to rotational speeds. Chen et al. [5] built a

model for determining the response of a spindle-bearing

system at high speeds with an analytical method. His model

considers the spindle as a uniform Euler–Bernoulli beam

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doi:10.1016/j.ijmachtools.2006.08.006

ÃCorresponding author. Tel.: +1 604822 5622; fax: +1 604822 2403.

E-mail addresses: [email protected]

(Y. Cao), [email protected] (Y. Altintas).

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supported by a pair of angular contact bearings. Using

Jones’ [6] bearing model, Chen analyzed the dynamic

behavior of the spindle around the trivial equilibrium

configuration with zero end loads. Only the axial preload is

considered in this model. First, the Newton–Raphson

iteration method is used to calculate the bearing stiffness

matrix at a given spindle speed, then the dynamic behavioris computed at this speed using the obtained bearing

stiffness. Li and Shin [7] presented a coupled spindle-

bearing model that includes thermal effects to predict the

bearing stiffness and natural frequencies of the spindle

system, using DeMul’s bearing model. The bearing

configuration, however, is limited to several cases and the

gyroscopic effect is not included.

All of the above models predict the natural vibration and

frequency response for a specific spindle design, and

consider only the spindle shaft and bearings. The effects

of the machine tool on the spindle dynamics are neglected.

Neither centrifugal force nor gyroscopic effect is included

in modeling the spindle shaft. The contact forces on

bearing balls and the time response of the spindle-bearing

system under dynamic cutting forces have not been

reported in the literature.

In this paper, a general method is presented for modeling

the spindle machine tool system, which consists of the cutter,

tool-holder, spindle shaft, bearings, housing, and the machine

tool. A simplified model, representing the dynamics of the

whole machine tool without the spindle, is developed by

means of experimental modal analysis which needs to be done

only once for every machine tool. The model of the whole

machine tool system is then created by coupling the spindle

model developed by the authors [8,9] with the simplified

model of the machine tool without the spindle. The assembly

of the spindle unit and spindle head is modeled through

contact springs. The proposed method is validated by

performing frequency response and cutting tests.

2. FE model of spindle-bearing and machine tool system

An instrumented, experimental spindle is mounted on a

vertical machining center. The spindle moves vertically

with the spindle head, which travels on the guideway

attached to the machine column. The spindle head acts like

a cantilever beam elastically supported on the column due

to the contact with the guideway, therefore, the flexibility

of the spindle mounting has to be reflected in the model of

the spindle-machine system.

The spindle has five bearings in overall back-to-back

configuration as shown in Fig. 1. Three rear bearings are

placed in a floating housing, which can be moved by a

hydraulic preload unit. Through a hydraulic pump, the

preload is applied to the rear bearings by the floating

housing. At the same time, the force is transmitted to the

spindle shaft by the nut, and moves towards the rear. As a

result, the force is applied to the front bearings through

step A of the spindle shaft. The spindle housing prevents

the spindle shaft from moving further to the rear by step B

of the spindle housing. The whole spindle is self-balanced

in the axial direction under the preload.

A general FE model of the spindle-bearing and machine

tool system is presented (Fig. 1). The Timoshenko beam is

used to model the spindle shaft and housing. In the FE

model, the black dots represent nodes, and each node has

three translational displacements in the X -, Y-, and Z -axes,

ARTICLE IN PRESS

Nomenclature

A cross sectional area of the beam

E Young’s modulus of material

G shear modulus of material

I diametral moment of inertia for the beamJ polar moments of inertia for the beam

N number of bearing balls for each bearing

P axial force on the beam

k index of bearing balls

m mass of the bearing ball

O rotational speed of the shaft

r density of material

Dm pitch diameter of the bearing measured from the

ball center

J b mass moment of inertia for the bearing ball

K i; K o contact constants between bearing balls and

inner rings, outer rings, respectively, depending

on the geometry and the material of the bearingballs and bearing rings

k s transverse shear form factor, k s is 0.9 for the

circular cross-section beam

m y; mz distributed moments per unit length about axes

y and z, respectively

OE orbital speed of the bearing ball

OB angular speed of the bearing ball about its own

center

ak angle between the vector of the bearing ball’sangular velocity about its own center and shaft

axis

di; do normal contact deformations between the cen-

ters of the ball and inner ring, and the ball and

outer ring, respectively

dð1Þ; dð4Þ displacement vectors of the shaft and housing,

respectively

dð2Þ; dð3Þ displacement vectors of inner and outer ring,

respectively

dix; d

i y; d

iz; g

i y; g

iz displacements of the inner ring

dox; d

o y ; d

oz ; g

o y ; g

oz displacements of the outer ring

yi; yo inner and outer ring contact angles of the

bearingy y; yz rotations about axes y and z, respectively

Y. Cao, Y. Altintas / International Journal of Machine Tools & Manufacture 47 (2007) 1342–1350 1343



and two rotations about the Y - and Z -axes. The pulley is

modeled as a rigid disk, the bearing spacer as a bar

element, and the nut and sleeve as a lumped mass. Thespindle has two front bearings (nos.1 and 2) in tandem and

three bearings (nos. 3–5) in tandem at the rear. The preload

is applied on the outer ring defined as node A3, which can

move along the spindle housing with nodes A4 and A5.

The forces are transmitted to inner rings B3–B5 through

bearing balls, then to the spindle shaft through inner ring

B5, which is fixed to the spindle shaft. Finally, the forces

are transferred to the front bearings by inner ring B1,

which is also fixed to the spindle shaft, then to the housing

by outer ring A2, which is fixed to the housing. An initial

preload is applied during the assembly, and can be adjusted

later through the hydraulic unit. The inner ring and outer

ring of the bearing are related by nonlinear bearing

equations, from which bearing stiffness is obtained by

solving equations of the spindle machine tool system.

The tool is assumed to be rigidly connected to the tool

holder which is fixed to the spindle shaft rigidly or through

translational and rotational springs. An equivalent cylinder is

used to represent the spindle head. First, the modal

parameters are identified for the spindle head before the

spindle is installed. Then, two dominant modes from

experimental modal analysis are used to configure a simplified

model for the spindle head by using springs and the mass of

the spindle head. The spring constants are estimated through

the mass and natural frequencies of the spindle head. Springs

are also used between the spindle housing and spindle head,

whose stiffness is obtained experimentally. The modeling of

each part is described as follows.

2.1. Equations of motion for the spindle shaft with rotating

effects

The equations of motion for the spindle shaft with

centrifugal force and gyroscpic moment due to the rotating

are as follows [8]:

rAd2u

dt2À EA

q2u

qx2À qx ¼ 0,

rAd2v

dt2À

q

qxk sAG

qv

qxÀ yz À P

qv

qx ! À q y

ÀO2rAv ¼ 0,

rAd 2w

dt2À

q

qxk sAG

qw

qxþ y y

À P

qw

qx

!À qz

ÀO2rAw ¼ 0

rI d2y y

dt2þ OrJ

dyz

dtÀ EI

q2y y

qx2þ k sAG

qw

qxþ y y

Àm y ¼ 0,

rI d2yz

dt2À OrJ

dy y

dtÀ EI

q2yz

qx2À k sAG

qv

qxÀ yz

Àmz ¼ 0. ð1Þ

ARTICLE IN PRESS

ShaftHousing BearingHydraulic

fluid

ToolholderTool

Spindle nose

Pulley

Preload

Clamping

Displacement sensors

unit

Step A Step B NutBearing housing

Preload

Housing

Shaft

Node Bearing Rigidly connected Movable

Pulley

Spacer

Tool and tool-holder

Linear spring

Spindle head

Rotation spring

Inner ring

Outer ring A1A2

A3 A4A5

B1B2

B3 B4B5

Fig. 1. An experimental spindle and its Finite Element Model.

Y. Cao, Y. Altintas / International Journal of Machine Tools & Manufacture 47 (2007) 1342–13501344



Eq. (1) is also suitable for the spindle housing by setting

spindle speed to zero.

The following equations of the spindle shaft and housing

in matrix forms can be obtained by using the FE method:

bM bcf €qg À ObG bcf _qg þ bK bc þ bK bcPÀÀO

2

bM b

cCÁ

fqg ¼ fF b

g, ð2Þ

where bM bc is the mass matrix, bM bcC is the mass matrix

used for computing the centrifugal forces, bG bc is the

gyroscopic matrix which is skew-symmetric, bK bc is the

stiffness matrix, bK bcP is the stiffness matrix due to the

axial force, and {F b} is the force vector, including

distributed and concentrated forces. The superscript b

represents the spindle shaft and housing. The details of the

matrices are shown in our earlier publication [8]. The

damping matrix is not included here, and is estimated from

experimentally identified modal damping, which is mostly

constant for each spindle-bearing family developed by the

manufacturers.The bar element takes only the axial stiffness of the beam

element, and the rigid disk is treated as a short beam with a

large diameter by setting the Young’s modulus to zero.

2.2. Nonlinear bearing model

Jones’ bearing model, which considers the bearing balls

and rings as elastic parts, is used in this paper, see Fig. 2.

The Hertzian contact theory is used to calculate the contact

force and displacement. Contact forces between the bearing

ball and bearing rings:

Qi ¼ K id3=2i ; Qo ¼ K od

3=2o . (3)

Centrifugal force (F c) and gyroscopic moment (M g) are [6]:

F c ¼1

2mDmO

2 OE

O

2

, (4)

M g ¼ J bO2 OB

O

OE

O

sin a. (5)

The force acting on the bearing ring is

F ¼ XN

k ¼1

f di; do; dð2Þ; dð3Þ; yi; yo; Qi; Qo; F c; M gÀ Á. (6)

The derivative of force with respect to the displacement is

the bearing stiffness matrix as follows [8]:

KB½ � ¼KI ÀKI

ÀKo Ko

" #, (7)

where KI and Ko are 5 Â 5 matrices. The bearing stiffness

matrix depends on the displacements which are in turn

affected by the stiffness of the bearing.

2.3. Modeling of machine tool without the spindle

In order to avoid complex modeling of the wholemachine tool, a simplified model is used to simulate the

dominant vibration of the machine tool without the spindle

system. The spindle head is a casting which connects the

spindle to the machine tool (Fig. 3); therefore, it is used to

represent the dynamics of the whole machine tool for the

purpose of structural assembly of the spindle. The

dynamics is different in the X - and Y -directions because

of the asymmetry of the spindle head and machine column.

The experimental modal analysis in the X -direction is

presented here; however, the same method is applied in the

Y -direction.

An equivalent cylinder is used to represent the spindlehead. Both translational and rotational stiffness of the

springs supporting the spindle head is estimated by using

two dominant modes from the modal analysis. The same

equivalent cylinder is used for both X - and Y -directions,

but the stiffness of the springs is different. The simulated

and measured FRF at node 1 are illustrated in Fig. 3,

ARTICLE IN PRESS

inner ring

ball

δ = (δx, δy, δz, γ y, γ z)

δ(3)

δ(1)

δ(2)

δ(4)

outer ring

housing

shaft

θo

θi

Fc

Mg

Qi

Qo

outer ring

inner ring

δy

δz

δx

γ y

γ z

Fig. 2. Elastic model of the bearing.




In order to predict stability lobes in the frequency

domain, the FRF at the tool tip in both X - and Y -

directions needs to be evaluated. A CAT40 (i.e.,

CV40TT20M400) shrink-fit tool-holder and a four-fluted

carbide end mill with a diameter of 20 mm and a stick-out

of 50 mm were used in all experiments and simulations

presented in the paper. Both tool and holder are assumed

to be connected rigidly to the spindle shaft.

The magnitudes of FRF at the tool tip in both X - and

Y-directions for the rigid tool-holder–spindle connection

are shown in Fig. 6, where the bearing preload is 1200N.

The simulation matches measurements very well at lower

frequencies, but the errors increase at higher frequencies.

3.2. Prediction of stability lobes and chatter test

The predicted stability of the system was experimentally

evaluated by milling Aluminium 7050. The chatter stability

theory of Budak et al. [10] is used to predict the stability

lobes by considering both simulated and measured FRF at

the tool tip. The results are shown and verified by cutting

tests conducted under the cutting conditions marked with

boxes or circles in Fig. 7. It is shown that the simulated

FRF, which was obtained from the proposed FE model

of the spindle, can correctly predict the stability lobes.

It must be noted here that the inclusion of machine

tool–spindle connection dynamics is important to achieve

such accuracy.The measured cutting forces and the machined surfaces

for both stable and unstable depths of cut are shown

in Figs. 8 and 9. The process was stable in Fig. 8, where

the depth of cut was 2 mm. Although the depth of cut was

only doubled in Fig. 9, the process was unstable and the

cutting forces increased more than 500% due to chatter

vibrations. The dominant frequency of the cutting forces

at the depth of cut of 2 mm is the tooth passing frequency

of 400 Hz, while chatter occurs at the spindle mode of

1028Hz at 4mm.

The measured and simulated displacements at the tool-

holder for both depths of cut are shown in Figs. 10 and 11.

The simulated displacements are very close to the measured

displacements although the distribution is a little different.

However, the spectrum of displacements from both

simulation and measurement match very well.

The simulated radial stiffness of bearings nos. 1 and 5

for the two cutting tests is illustrated in Fig. 12.

The stiffness of the first bearing is affected more than

the fifth bearing. The bearing stiffness can even reach

zero during chatter. All forces are treated as dynamic

forces in simulation. The preload is applied to the

bearings first, and the cutting forces are applied to the

tool tip after the transient vibrations due to the preload

diminish.

ARTICLE IN PRESS

500 1000 1500 2000 2500 3000 35000

0.5

1

1.5

2

2.5

3

3.5

4x 10-8 FRF at spindle nose in the X direction

M a g n

i t u d e [

m / N ]

Frequency [Hz]

Measurement

Simulation (with the influence of the machine tool)

Simulation (without the influence of the machine tool)

Fig. 5. Comparison of the influence of the machine tool on spindle

dynamics.

500 1000 1500 2000 2500 3000 3500 4000 45000

2

4

6

x 10-7

FRF at tool tip in the X direction (preload =1200 N )

M a g n

i t u d e

[ m / N ]

500 1000 1500 2000 2500 3000 3500 4000 45000

2

4

6

x 10-7

FRF at tool tip in the Y direction (preload = 1200 N )

Frequency [Hz]

M a g n

i t u d e

[ m / N ]

Measurement

Simulation

Fig. 6. FRF at the tool tip for rigid connection of the tool-holder.

2000 4000 6000 8000 100000

2

4

6

8

10

Spindle speed [rpm]

D e p

t h o

f c u

t [ m m

]

From measured FRF

From simulated FRF

No chatter

Chatter

Fig. 7. Predicted stability lobes from measured and simulated FRF.




Contact forces on bearings nos. 1 and 5 are shown in

Fig. 13. Similar to the case of bearing stiffness, the first

bearing experiences higher contact forces than the fifth

bearing. Bearing stiffness and contact forces cannot be

measured directly. The correct prediction of the FRF and

displacement response, however, indirectly proves the

validity of their simulation since bearing stiffness is closely

related to the contact forces.

3.3. Effects of preload and speed on the dynamics of spindle

machine tool systems

In general, the natural frequencies of all modes

increase with preload due to increased bearing stiffness,

but decrease with spindle speed due to centrifugal

forces. In order to compare the simulati on and

measurement more clearly, the influence of the preload

and spindle speed on the second dominant natural

frequency is plotted separately in Fig. 14, by fixing

either the spindle speed or preload. The frequency

i ncreases from 1068 to 1142H z w hen preload i s

increased from 600 to 1800 N. However, the frequency

drops from 1140 to 1090 Hz when the speed is increased

from stationary to 10,000 rpm. It is shown that the

proposed model can correctly predict the effects of the

ARTICLE IN PRESS

-400

-200

0

200

0

500

C u t t

i n g f o r c e [ N ]

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

-400

-200

0

200

Time [s]

Z direction

Y direction

X direction

(a)

(b)

Fig. 9. The measured cutting forces and machined surface. Spindle speed:

6000rpm, depth of cut: 4 mm, feed rate: 0.1 mm/flute, chatter frequency

1028Hz.

0 0.01 0.02 0.03 0.04 0.05

-6

-4

-2

0

D i s p

l a c e m e n

t [ μ m

]

0 0.01 0.02 0.03 0.04 0.05-6

-4

-2

0

Time [s]

D i s p

l a c e m e n

t [ μ m

]

Measurement

Simulation

Fig. 10. The measured and simulated displacements in the X -direction at

the tool-holder. Spindle speed: 6000rpm, depth of cut: 2 mm, feed rate:

0.1 mm/flute, no chatter.

0 0.01 0.02 0.03 0.04 0.05-40

-20

0

20

40

D i s p

l a c e m e n

t [ μ m

]

0 0.01 0.02 0.03 0.04 0.05-40

-200

20

40

Time [s]

D i s p

l a c e m e n

t [ μ m

]

Measurement

Simulation

Fig. 11. The measured and simulated displacements in the X -direction at

the tool-holder. Spindle speed: 6000rpm, depth of cut: 4 mm, feed rate:

0.1 mm/flute, chatter.

-150

-100

-50

150

200

250

C u

t t i n g

f o r c e

[ N ]

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

-100

-50

0

Time [s]

X direction

Y direction

Z direction

(a)

(b

Fig. 8. The measured cutting forces and machined surface during a stable

cut. Spindle speed: 6000 rpm, depth of cut: 2 mm, feed rate: 0.1mm/flute.




preload and spindle speed on the dynamics of spindle

machine tool systems.

4. Conclusions

The numerical model of the spindle machine tool system

is developed to simulate the virtual cutting performance of

the machine–spindle system. It is shown that the reliability

of virtual cutting with the spindles require integrated

modeling of bearings, spindle shafts, tool and holders,

bearing preload, connection between the spindle and

machine tool housing, speed and machining process. The

study also demonstrates that the modeling of spindle alone

does not lead to correct prediction of its dynamics on the

machine tool, unless its mounting joints are included in the

mathematical models.

The experimentally verified mathematical model predicts

that the preload can increase the bearing stiffness, leading

to increased natural frequencies, which shifts the stability

lobes to the right towards higher speeds. However, higher

preload reduces the damping, which decreases the dynamic

stiffness at the tool tip hence reduces the chatter free, depth

of cuts. The preload cannot efficiently improve the stability

lobes, but it can enhance the static stiffness and reduce the

forced vibrations.

Acknowledgment

This research is jointly sponsored by NSERC, Pratt &

Whitney, Canada, Boeing Commercial Plane, and Weiss

Spindle Technology.

ARTICLE IN PRESS

0

2

4x 108

S t i f f n e s s

[ N / m ]

0

2

4x 108

S t i f f n e s s

[ N / m ]

Bearing No.1

Bearing No.5

Preload

period Cutting period

0

2

4x 108

S t i f f

n e s s

[ N / m ]

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

0

2

4x 108

Time [s]

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

Time [s]

S t i f f n e s s

[ N / m ]

Bearing No.1

Bearing No.5

Preloadperiod Cutting period

(a)

(b)

Fig. 12. Radial bearing stiffness under cutting forces (spindle speed:

6000 rpm, feed rate: 0.1 mm/flute).

0

100

200

C o n

t a c

t f o r c e

[ N ]

0

100

200

C o n

t a c

t f o r c e

[ N ]

Preloadperiod Cutting period

0

200

400

600

C o n

t a c

t f o r c e

[ N ]

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

0

100

200

Time [s]

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

Time [s]

C o n

t a c

t f o r c e

[ N ]

Preloadperiod

Cutting period

Bearing No.5

Bearing No.1

Bearing No.5

Bearing No.1

(a)

(b)

Fig. 13. Simulated bearing contact forces under cutting (spindle speed:

6000 rpm, feed rate: 0.1 mm/flute).

600 800 1000 1200 1400 1600 18001050

1100

1150The second dominant frequency (spindle speed = 0 rpm)

Preload [N]

F r e q u e n c y

[ H z

]

Measurement

Simulation

0 2000 4000 6000 8000 100001050

1100

1150The second dominant frequency (preload = 1200N)

Spindle speed [rpm]

F r e q u e n

c y

[ H z

] MeasurementSimulation

Fig. 14. The influence of the preload and spindle speed on the second

natural frequency.




References

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Annals of CIRP 53/2 (2004) 619–642.

[2] R.L. Ruhl, J.F. Booker, A finite element model for distributed

parameter turborotor systems, ASME Journal of Engineering for

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[3] H.D. Nelson, A finite rotating shaft element using Timoshenko beamtheory, ASME Journal of Mechanical Design 102 (4) (1980) 793–803.

[4] F.P. Wardle, S.J. Lacey, S.Y. Poon, Dynamic and static character-

istics of a wide spread range machine tool spindle, Precision

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Vibration and Acoustics 116 (4) (1994) 506–522.

[6] A.B. Jones, A general theory for elastically constrained ball and

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[7] H. Li, Y.C. Shin, Integrated dynamic thermo-mechanical modeling of

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Manufacturing Science and Engineering, Transactions of the ASME

126 (2004) 148–158.

[8] Y. Cao, Y. Altintas, A general method for the modeling of spindle-bearing systems, Journal of Mechanical Design, Transactions of the

ASME 126 (2004) 1089–1104.

[9] Y. Altintas, Y. Cao, Virtual design and optimization of machine tool

spindles, Annals of CIRP 54 (1) (2005) 379–382.

[10] E. Budak, Y. Altintas, Analytical prediction of chatter stability in

milling—Part I: general formulation, Journal of Dynamic Systems,

Measurement, and Control, Transactions of the ASME 120 (1998)

22–30.

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