Abstract

The damped mass-spring model is often used for the dynamic modeling and vibration analysis of aerostatic bearing systems by taking the air film as equivalent springs, especially when the bearing is used as a key component in mechanical equipment. However, the stiffness and damping of the air film are frequency-dependent, making the commonly used approach of taking static stiffness or fixed value as the spring coefficient no longer applicable for a bearing subject to a complex external force containing different frequencies. To address this issue, this paper develops the damped mass-spring model for the aerostatic thrust bearing considering the frequency-varying stiffness and damping by means of the linear superposition method. It indicates that the air bearing is still a linear system under a micro disturbance despite the frequency-dependent character of dynamic coefficients because the bearing vibration satisfies the superposition principle. The improved dynamic modeling approach is able to accurately and efficiently predict the overall dynamic response of the thrust plate with both axial and tilting motion when the plate is subjected to a multi-frequency vibration. In solving the overall dynamic response, the stiffness and damping associated with the responses of the transient part and steady part correspond to the natural vibration frequency and external disturbance frequencies, respectively. The feasibility and accuracy of the improved modeling approach are partly or completely verified by the direct trajectory calculation method, the computational fluid dynamics, dynamic mesh simulation, and a modal test. The proposed modeling method provides an effective way for the vibration analysis of air bearings, and in the meantime avoids the possible numerical errors caused by the traditional modeling approach.

References

1.
Fourka
,
M.
,
Tian
,
Y.
, and
Bonis
,
M.
,
1996
, “
Prediction of the Stability of Air Thrust Bearings by Numerical, Analytical and Experimental Methods
,”
Wear
,
198
(
1–2
), pp.
1
6
.
2.
Belforte
,
G.
,
Colombo
,
F.
,
Raparelli
,
T.
,
Trivella
,
A.
, and
Viktorov
,
V.
,
2010
, “
Performance of Externally Pressurized Grooved Thrust Bearings
,”
Tribol. Lett.
,
37
(
3
), pp.
553
562
.
3.
Charki
,
A.
,
Diop
,
K.
,
Champmartin
,
S.
, and
Ambari
,
A.
,
2013
, “
Numerical Simulation and Experimental Study of Thrust Air Bearings With Multiple Orifices
,”
Int. J. Mech. Sci.
,
72
, pp.
28
38
.
4.
Ma
,
W.
,
Cui
,
J.
,
Liu
,
Y.
, and
Tan
,
J.
,
2016
, “
Improving the Pneumatic Hammer Stability of Aerostatic Thrust Bearing With Recess Using Damping Orifices
,”
Tribol. Int.
,
103
, pp.
281
288
.
5.
Li
,
J.
, and
Liu
,
P.
,
2019
, “
Dynamic Analysis of 5DOFs Aerostatic Spindles Considering Tilting Motion With Varying Stiffness and Damping of Thrust Bearings
,”
J. Mech. Sci. Technol.
,
33
(
11
), pp.
5199
5207
.
6.
Chen
,
D.
,
Li
,
N.
,
Pan
,
R.
, and
Han
,
J.
,
2019
, “
Analysis of Aerostatic Spindle Radial Vibration Error Based on Microscale Nonlinear Dynamic Characteristics
,”
J. Vib. Control
,
25
(
14
), pp.
2043
2052
.
7.
An
,
C.
,
Zhang
,
Y.
,
Xu
,
Q.
,
Zhang
,
F.
,
Zhang
,
J.
,
Zhang
,
L.
, and
Wang
,
J.
,
2010
, “
Modeling of Dynamic Characteristic of the Aerostatic Bearing Spindle in an Ultra-Precision Fly Cutting Machine
,”
Int. J. Mach. Tools Manuf.
,
50
(
4
), pp.
374
385
.
8.
Huo
,
D.
,
Cheng
,
K.
, and
Wardle
,
F.
,
2010
, “
A Holistic Integrated Dynamic Design and Modelling Approach Applied to the Development of Ultraprecision Micro-Milling Machines
,”
Int. J. Mach. Tools Manuf.
,
50
(
4
), pp.
335
343
.
9.
Zhang
,
S.
,
To
,
S.
, and
Wang
,
H.
,
2013
, “
A Theoretical and Experimental Investigation Into Five-DOF Dynamic Characteristics of an Aerostatic Bearing Spindle in Ultra-Precision Diamond Turning
,”
Int. J. Mach. Tools Manuf.
,
71
, pp.
1
10
.
10.
Huang
,
P.
,
Lee
,
W.
, and
Chan
,
C.
,
2015
, “
Investigation of the Effects of Spindle Unbalance Induced Error Motion on Machining Accuracy in Ultra-Precision Diamond Turning
,”
Int. J. Mach. Tools Manuf.
,
94
, pp.
48
56
.
11.
Gao
,
Q.
,
Zhao
,
H.
,
Lu
,
L.
,
Chen
,
W.
, and
Zhang
,
F.
,
2020
, “
Investigation on the Formation Mechanism and Controlling Method of Machined Surface Topography of Ultra-Precision Flycutting Machining
,”
Int. J. Adv. Manuf. Technol.
,
106
(
7–8
), pp.
3311
3320
.
12.
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
.
13.
Wu
,
Q.
,
Sun
,
Y.
,
Chen
,
W.
,
Liu
,
H.
, and
Luo
,
X.
,
2019
, “
An Mechatronics Coupling Design Approach for Aerostatic Bearing Spindles
,”
Int. J. Precis. Eng. Manuf.
,
20
(
7
), pp.
1185
1196
.
14.
Zhang
,
G.
, and
Ehmann
,
K. F.
,
2015
, “
Dynamic Design Methodology of High Speed Micro-Spindles for Micro/Meso-Scale Machine Tools
,”
Int. J. Adv. Manuf. Technol.
,
76
(
1–4
), pp.
229
246
.
15.
Shi
,
J.
,
Cao
,
H.
,
Maroju
,
N. K.
, and
Jin
,
X.
,
2019
, “
Dynamic Modeling of Aerostatic Spindle With Shaft Tilt Deformation
,”
ASME J. Manuf. Sci. Eng.
,
142
(
2
), p.
021006
.
16.
Chen
,
W.
,
Liang
,
Y.
,
Sun
,
Y.
,
Bai
,
Q.
, and
An
,
C.
,
2015
, “
A Novel Dynamic Modeling Method for Aerostatic Spindle Based on Pressure Distribution
,”
J. Vib. Control
,
21
(
16
), pp.
3339
3347
.
17.
Lund
,
J.
,
1968
, “
Calculation of Stiffness and Damping Properties of Gas Bearings
,”
ASME J. Lubr. Tech.
,
90
(
4
), pp.
793
803
.
18.
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
.
19.
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
.
20.
Bhat
,
N.
,
Kumar
,
S.
,
Tan
,
W.
,
Narasimhan
,
R.
, and
Low
,
T. C.
,
2012
, “
Performance of Inherently Compensated Flat Pad Aerostatic Bearings Subject to Dynamic Perturbation Forces
,”
Precis. Eng.
,
36
(
3
), pp.
399
407
.
21.
Yang
,
G.
,
Du
,
J.
,
Ge
,
W.
,
Liu
,
T.
, and
Yang
,
X.
,
2017
, “
Dynamic Characteristics of Spiral-Grooved Opposed-Hemisphere Gas Bearings
,”
ASME J. Tribol.
,
139
(
3
), p.
031704
.
22.
Yu
,
Y.
,
Pu
,
G.
, and
Jiang
,
K.
,
2017
, “
Numerical Modelling and Analysis of Hydrostatic Thrust Air Bearings for High Loading Capacities and Low Air Consumption
,”
IOP Conf. Ser.: Mater. Sci. Eng.
,
280
, p.
012005
.
23.
Chen
,
X.
,
Zhu
,
J.
, and
Chen
,
H.
,
2013
, “
Dynamic Characteristics of Ultra-Precision Aerostatic Bearings
,”
Adv. Manuf.
,
1
(
1
), pp.
82
86
.
24.
Ishibashi
,
K.
,
Kondo
,
A.
,
Kawada
,
S.
,
Miyatake
,
M.
,
Yoshimoto
,
S.
, and
Stolarski
,
T.
,
2019
, “
Static and Dynamic Characteristics of a Downsized Aerostatic Circular Thrust Bearing With a Single Feed Hole
,”
Precis. Eng.
,
60
, pp.
448
457
.
25.
Ruan
,
B.
,
2002
, “
A Semi-Analytical Solution to the Dynamic Tracking of Non-Contacting Gas Face Seals
,”
ASME J. Tribol.
,
124
(
1
), pp.
196
202
.
26.
Liu
,
J.
, and
Du
,
X.
,
2005
,
Dynamics of Structures (in Chinese)
,
China Machine Press
,
Beijing
.
27.
Belforte
,
G.
,
Raparelli
,
T.
,
Trivella
,
A.
,
Viktorov
,
V.
, and
Visconte
,
C.
,
2015
, “
CFD Analysis of a Simple Orifice-Type Feeding System for Aerostatic Bearings
,”
Tribol. Lett.
,
58
(
2
), p.
25
.
28.
Chang
,
Y.
,
Ding
,
J.
,
He
,
Z.
,
Shehzad
,
A.
,
Ding
,
Y.
,
Lu
,
H.
,
Zhuang
,
H.
, et al
,
2020
, “
Effect of Joint Interfacial Contact Stiffness on Structural Dynamics of Ultra-Precision Machine Tool
,”
Int. J. Mach. Tools Manuf.
,
158
, p.
103609
.
29.
Ding
,
J.
,
Chang
,
Y.
,
Chen
,
P.
,
Zhuang
,
H.
,
Ding
,
Y.
,
Lu
,
H.
, and
Chen
,
Y.
,
2020
, “
Dynamic Modeling of Ultra-Precision Fly Cutting Machine Tool and the Effect of Ambient Vibration on Its Tool Tip Response
,”
Int. J. Extrem. Manuf.
,
2
(
2
), p.
025301
.
30.
Ding
,
Y.
,
Rui
,
X.
,
Lu
,
H.
,
Chang
,
Y.
, and
Chen
,
Y.
,
2020
, “
Research on the Dynamic Characteristics of the Ultra-Precision Fly Cutting Machine Tool and Its Influence on the Mid-Frequency Waviness of the Surface
,”
Int. J. Adv. Manuf. Technol.
,
106
(
1–2
), pp.
441
454
.
31.
LaTray
,
N.
, and
Kim
,
D.
,
2021
, “
Novel Thrust Foil Bearing With Pocket Grooves for Enhanced Static Performance
,”
ASME J. Tribol.
,
143
(
11
), p.
111803
.
32.
Jang
,
G. H.
, and
Kim
,
Y. J.
,
1999
, “
Calculation of Dynamic Coefficients in a Hydrodynamic Bearing Considering Five Degrees of Freedom for a General Rotor-Bearing System
,”
ASME J. Tribol.
,
121
(
3
), pp.
499
505
.
33.
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
.
34.
Zhuang
,
H.
,
Ding
,
J.
,
Chen
,
P.
,
Chang
,
Y.
,
Zeng
,
X.
,
Yang
,
H.
,
Liu
,
X.
, and
Wei
,
W.
,
2021
, “
Effect of Surface Waviness on the Performances of an Aerostatic Thrust Bearing With Orifice-Type Restrictor
,”
Int. J. Precis. Eng. Manuf.
,
22
(
10
), pp.
1735
1759
.
35.
Zhuang
,
H.
,
Ding
,
J.
,
Chen
,
P.
,
Chang
,
Y.
,
Zeng
,
X.
,
Yang
,
H.
,
Liu
,
X.
, and
Wei
,
W.
,
2019
, “
Numerical Study on Static and Dynamic Performances of a Double-Pad Annular Inherently Compensated Aerostatic Thrust Bearing
,”
ASME J. Tribol.
,
145
(
5
), p.
051701
.
You do not currently have access to this content.