Abstract

This paper presents a new dynamic model of aerostatic spindle including the journal and thrust bearings. Reynolds equations are used to model the dynamics of a 4-degree-of-freedom (DOF) aerostatic journal bearing and a 3-DOF aerostatic thrust bearing. Finite element model of the spindle shaft is developed based on the Timoshenko beam theory considering the centrifugal and gyroscopic effects and is coupled with the bearing to construct the dynamic model of the whole aerostatic spindle. The effect of shaft tilt motion due to elastic deformation on the dynamic characteristics of the aerostatic bearing is considered for the first time. The finite difference method is used to determine the load capacity and moments provided by the bearings with changing air film thickness due to shaft vibration, and Newmark-β method is used to obtain the dynamic response of the spindle shaft. The simulated natural frequencies of the aerostatic spindle are verified through impact experiments under static and rotating conditions. Based on the developed model, the effects of tool overhang length, rotating speed, air film thickness, and supply air pressure on the frequency response function of the spindle are investigated comprehensively. The proposed dynamic model of the aerostatic spindle is able to provide useful guidance for structure design and process planning for micro-machining.

References

References
1.
Alting
,
L.
,
Kimura
,
F.
,
Hansen
,
H. N.
, and
Bissacco
,
G.
,
2003
, “
Micro Engineering
,”
CIRP Ann.
,
52
(
2
), pp.
635
657
. 10.1016/S0007-8506(07)60208-X
2.
Luo
,
X.
,
Cheng
,
K.
,
Webb
,
D.
, and
Wardle
,
F.
,
2005
, “
Design of Ultraprecision Machine Tools With Applications to Manufacture of Miniature and Micro Components
,”
J. Mater. Process. Technol.
,
167
(
2–3
), pp.
515
528
. 10.1016/j.jmatprotec.2005.05.050
3.
Powell
,
J. W.
,
1970
, “
Design of Aerostatic Bearings
,”
Machinery Publishing Co. Ltd, Brighton, UK.
4.
Hsia
,
Y. T.
, and
Domoto
,
G. A.
,
1983
, “
An Experimental Investigation of Molecular Rarefaction Effects in Gas Lubricated Bearings at Ultra-Low Clearances
,”
ASME J. Lubr. Tech.
,
105
(
1
), pp.
120
129
. 10.1115/1.3254526
5.
Cao
,
H.
,
Dörgeloh
,
T.
,
Riemer
,
O.
, and
Brinksmeier
,
E.
,
2017
, “
Adaptive Separation of Unbalance Vibration in Air Bearing Spindles
,”
Procedia CIRP
,
62
, pp.
351
356
. 10.1016/j.procir.2016.06.069
6.
Jin
,
X.
, and
Altintas
,
Y.
,
2013
, “
Chatter Stability Model of Micro-Milling With Process Damping
,”
ASME J. Manuf. Sci. Eng.
,
135
(
3
), p.
031011
. 10.1115/1.4024038
7.
Xi
,
S.
,
Cao
,
H.
,
Chen
,
X.
, and
Niu
,
L.
,
2018
, “
A Dynamic Modeling Approach for Spindle Bearing System Supported by Both Angular Contact Ball Bearing and Floating Displacement Bearing
,”
ASME J. Manuf. Sci. Eng.
,
140
(
2
), p.
021014
. 10.1115/1.4038687
8.
Lo
,
C.-Y.
,
Wang
,
C.-C.
, and
Lee
,
Y.-H.
,
2005
, “
Performance Analysis of High-Speed Spindle Aerostatic Bearings
,”
Tribol. Int.
,
38
(
1
), pp.
5
14
. 10.1016/j.triboint.2004.04.008
9.
Wang
,
X.
,
Xu
,
Q.
,
Wang
,
B.
,
Zhang
,
L.
,
Yang
,
H.
, and
Peng
,
Z.
,
2017
, “
Numerical Calculation of Rotation Effects on Hybrid Air Journal Bearings
,”
Tribol. Trans.
,
60
(
2
), pp.
195
207
. 10.1080/10402004.2016.1155786
10.
Chen
,
D.
,
Zhou
,
S.
,
Han
,
J.
,
Fan
,
J.
, and
Cheng
,
Q.
,
2017
, “
Characteristics Evaluation of Gas Film in the Aerostatic Thrust Bearing Within Rarefied Effect
,”
Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol.
,
231
(
2
), pp.
149
157
. 10.1177/1350650116649552
11.
San Andres
,
L.
,
1993
, “
The Effect of Journal Misalignment on the Operation of a Turbulent Flow Hydrostatic Bearing
,”
ASME J. Tribol.
,
115
(
3
), pp.
355
363
. 10.1115/1.2921643
12.
Sharma
,
S. C.
,
Phalle
,
V. M.
, and
Jain
,
S.
,
2012
, “
Combined Influence of Wear and Misalignment of Journal on the Performance Analysis of Three-Lobe Three-Pocket Hybrid Journal Bearing Compensated With Capillary Restrictor
,”
ASME J. Tribol.
,
134
(
1
), p.
011703
. 10.1115/1.4005644
13.
Rajput
,
A. K.
, and
Sharma
,
S. C.
,
2016
, “
Combined Influence of Geometric Imperfections and Misalignment of Journal on the Performance of Four Pocket Hybrid Journal Bearing
,”
Tribol. Int.
,
97
, pp.
59
70
. 10.1016/j.triboint.2015.12.049
14.
Jiang
,
S.
,
Yang
,
S.
,
Yin
,
Z.
, and
Xu
,
C.
,
2016
, “
Static and Dynamic Characteristics of Externally Pressurized Annular Porous Gas Thrust Bearings
,”
Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol.
,
230
(
10
), pp.
1221
1230
. 10.1177/1350650116631275
15.
Jiang
,
S.
,
Lin
,
S.
, and
Xu
,
C.
,
2018
, “
Static and Dynamic Characteristics of Externally Pressurized Porous Gas Journal Bearing With Four Degrees-of-Freedom
,”
ASME J. Tribol.
,
140
(
1
), p.
011702
. 10.1115/1.4037134
16.
Shi
,
J.
,
Cao
,
H.
, and
Chen
,
X.
,
2019
, “
Effect of Angular Misalignment on the Static Characteristics of Rotating Externally Pressurized Air Journal Bearing
,”
Sci. China Technol. Sci.
,
62
(
9
), pp.
1520
1533
. 10.1007/s11431-018-9429-4
17.
Shi
,
J.
,
Cao
,
H.
, and
Chen
,
X.
,
2019
, “
Effect of Angular Misalignment on the Dynamic Characteristics of Externally Pressurized Air Journal Bearing
,”
Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol.
, pp.
1
24
. 10.1007/s11431-018-9429-4
18.
Shi
,
J.
,
Cao
,
H.
, and
Jin
,
X.
,
2019
, “
Investigation on the Static and Dynamic Characteristics of 3-DOF Aerostatic Thrust Bearings With Orifice Restrictor
,”
Tribol. Int.
,
138
, pp.
435
449
. 10.1016/j.triboint.2019.06.026
19.
Pan
,
C. H.
, and
Kim
,
D.
,
2007
, “
Stability Characteristics of a Rigid Rotor Supported by a Gas-Lubricated Spiral-Groove Conical Bearing
,”
ASME J. Tribol.
,
129
(
2
), pp.
375
383
. 10.1115/1.2647443
20.
Zhang
,
S.
,
To
,
S.
,
Cheung
,
C.
, and
Wang
,
H.
,
2012
, “
Dynamic Characteristics of an Aerostatic Bearing Spindle and Its Influence on Surface Topography in Ultra-Precision Diamond Turning
,”
Int. J. Mach. Tools Manuf.
,
62
, pp.
1
12
. 10.1016/j.ijmachtools.2012.04.007
21.
Larsen
,
J. S.
,
Santos
,
I. F.
, and
von Osmanski
,
S.
,
2016
, “
Stability of Rigid Rotors Supported by Air Foil Bearings: Comparison of Two Fundamental Approaches
,”
J. Sound Vib.
,
381
, pp.
179
191
. 10.1016/j.jsv.2016.06.022
22.
Chen
,
D.
,
Huo
,
C.
,
Cui
,
X.
,
Pan
,
R.
,
Fan
,
J.
, and
An
,
C.
,
2018
, “
Investigation the Gas Film in Micro Scale Induced Error on the Performance of the Aerostatic Spindle in Ultra-Precision Machining
,”
Mech. Syst. Signal Process.
,
105
, pp.
488
501
. 10.1016/j.ymssp.2017.10.041
23.
Wang
,
C.-C.
,
2006
, “
Nonlinear Dynamic Behavior and Bifurcation Analysis of a Rigid Rotor Supported by a Relatively Short Externally Pressurized Porous Gas Journal Bearing System
,”
Acta mechanica
,
183
(
1–2
), pp.
41
60
. 10.1007/s00707-006-0323-x
24.
Xu
,
C.
, and
Jiang
,
S.
,
2015
, “
Dynamic Analysis of a Motorized Spindle With Externally Pressurized Air Bearings
,”
ASME J. Vib. Acoust.
,
137
(
4
), p.
041001
. 10.1115/1.4029675
25.
Dupont
,
R.
,
2015
, “
Robust Rotor Dynamics for High-Speed Air Bearing Spindles
,”
Precis. Eng.
,
40
, pp.
7
13
. 10.1016/j.precisioneng.2014.12.008
26.
Gao
,
S.
,
Cheng
,
K.
,
Ding
,
H.
, and
Fu
,
H.
,
2016
, “
Multiphysics-Based Design and Analysis of the High-Speed Aerostatic Spindle with Application to Micro-Milling
,”
Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol.
,
230
(
7
), pp.
852
871
. 10.1177/1350650115619609
27.
Müller
,
C.
,
Kirsch
,
B.
, and
Aurich
,
J. C.
,
2017
, “Compact air Bearing Spindles for Desktop Sized Machine Tools,”
Small Machine Tools for Small Workpieces
, pp.
21
34
.
Springer
.
28.
Cao
,
H.
,
Riemer
,
O.
, and
Brinksmeier
,
E.
,
2017
, “
Dynamic Modelling and Vibration Simulation of Air Bearing Spindle Systems Due to Unbalance
,”
Int. J. Mechatronics Manuf. Syst.
,
10
(
3
), pp.
260
276
. 10.1504/IJMMS.2017.087549
29.
Guo
,
Z.
,
Feng
,
K.
,
Liu
,
T.
,
Lyu
,
P.
, and
Zhang
,
T.
,
2018
, “
Nonlinear Dynamic Analysis of Rigid Rotor Supported by Gas Foil Bearings: Effects of Gas Film and Foil Structure on Subsynchronous Vibrations
,”
Mech. Syst. Signal Process.
,
107
, pp.
549
566
. 10.1016/j.ymssp.2018.02.005
30.
Miyatake
,
M.
, and
Yoshimoto
,
S.
,
2010
, “
Numerical Investigation of Static and Dynamic Characteristics of Aerostatic Thrust Bearings with Small Feed Holes
,”
Tribol. Int.
,
43
(
8
), pp.
1353
1359
. 10.1016/j.triboint.2010.01.002
31.
Nishio
,
U.
,
Somaya
,
K.
, and
Yoshimoto
,
S.
,
2011
, “
Numerical Calculation and Experimental Verification of Static and Dynamic Characteristics of Aerostatic Thrust Bearings with Small Feedholes
,”
Tribol. Int.
,
44
(
12
), pp.
1790
1795
. 10.1016/j.triboint.2011.07.004
32.
Jang
,
G.
, and
Kim
,
Y.
,
1998
, “
Calculation of Dynamic Coefficients in a Hydrodynamic Bearing Considering Five Degrees of Freedom for a General Rotor-Bearing System
,”
Am. Soc. Mech. Eng. (Paper)(98-TRIB-1-61)
,
121
(
3
), pp.
499
505
. 10.1115/1.2834095
33.
Cao
,
Y.
, and
Altintas
,
Y.
,
2004
, “
A General Method for the Modeling of Spindle-Bearing Systems
,”
ASME J. Mech. Des.
,
126
(
6
), pp.
1089
1104
. 10.1115/1.1802311
34.
Cao
,
Y.
, and
Altintas
,
Y.
,
2007
, “
Modeling of Spindle-Bearing and Machine Tool Systems for Virtual Simulation of Milling Operations
,”
Int. J. Mach. Tools Manuf.
,
47
(
9
), pp.
1342
1350
. 10.1016/j.ijmachtools.2006.08.006
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