The in-plane wave motion is analytically examined to address the stationary deflection, natural frequency splitting, and mode contamination of the rotationally ring-shaped periodic structures (RRPS). The governing equation is developed by the Hamilton's principle where the structure is modeled as a thin ring with equally-spaced particles, and the centrifugal effect is included. The free responses are captured by the perturbation method and determined as closed-form expressions. The results imply that the response of stationary RRPS is characterized as standing wave, and the natural frequencies can split when the wave number n and particle number N satisfying 2n/N = int. Also the splitting behavior is determined by the relative angle between the particle and wave antinode. The coefficients of the mode contamination are also obtained. For rotating RRPS, the invariant deflections due to the centrifugal force are estimated at different rotating speeds. It is found that, for certain waves satisfying 2n/N = int, the natural frequency exceeds that of the corresponding smooth ring at the critical speed, and furthermore, the critical speed of the backward traveling wave is lower than that of the forward one. The contamination coefficients of the two kinds of waves are also obtained and they have different magnitudes. All results verify that the splitting and contamination can be determined by the relationship among the mode order, wave number, particle number, and relative position between the particle and antinode. Numerical examples and comparisons with the existing results in the literature are presented.

References

References
1.
Schwartz
,
D.
,
Kim
,
D. J.
, and
M'Closkey
,
R. T.
,
2009
, “
Frequency Tuning of a Disk Resonator Gyro Via Mass Matrix Perturbation
,”
ASME J. Dyn. Syst. Meas. Control
,
131
(
6
), p.
061004
.10.1115/1.3155016
2.
Zhao
,
C. S.
,
2007
,
Ultrasonic Motors Technologies and Applications
,
Science Press
,
Beijing
.
3.
Canchi
,
S. V.
, and
Parker
,
R. G.
,
2006
, “
Parametric Instability of a Circular Ring Subjected to Moving Springs
,”
J. Sound Vib.
,
293
(
1
), pp.
360
379
.10.1016/j.jsv.2005.10.007
4.
Tseng
,
J. G.
, and
Wickert
,
J. A.
,
1994
, “
On the Vibration of Bolted Plate and Flange Assemblies
,”
ASME J. Vib. Acoust.
,
116
(
4
), pp.
468
473
.10.1115/1.2930450
5.
Kim
,
M.
,
Moon
,
J.
, and
Wickert
,
J. A.
,
2000
, “
Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures
,”
ASME J. Vib. Acoust.
,
122
(
1
), pp.
62
68
.10.1115/1.568443
6.
Chang
,
J. Y.
, and
Wickert
,
J. A.
,
2001
, “
Response of Modulated Doublet Modes to Travelling Wave Excitation
,”
J. Sound Vib.
,
242
(
1
), pp.
69
83
.10.1006/jsvi.2000.3363
7.
Chang
,
J. Y.
, and
Wickert
,
J. A.
,
2002
, “
Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks
,”
J. Sound Vib.
,
250
(
3
), pp.
379
400
.10.1006/jsvi.2001.3942
8.
Wang
,
S. Y.
,
Xiu
,
J.
,
Cao
,
S. Q.
, and
Liu
,
J. P.
,
2014
, “
Analytical Treatment With Rigid-Elastic Vibration of Permanent Magnet Motors With Expanding Application to Cyclically Symmetric Power-Transmission Systems
,”
ASME J. Vib. Acoust.
,
136
(
2
), p.
021014
.10.1115/1.4025993
9.
Fox
,
C. H. J.
,
1990
, “
A Simple Theory for the Analysis and Correction of Frequency Splitting in Slight Imperfect Rings
,”
J. Sound Vib.
,
142
(
2
), pp.
227
243
.10.1016/0022-460X(90)90554-D
10.
McWilliam
,
S.
,
Ong
,
J.
, and
Fox
,
C. H. J.
,
2005
, “
On the Statistics of Natural Frequency Splitting for Rings With Random Mass Imperfections
,”
J. Sound Vib.
,
279
(
1
), pp.
453
470
.10.1016/j.jsv.2003.11.034
11.
Chang
,
C. O.
,
Chou
,
C. S.
, and
Lai
,
W. F.
,
2004
, “
Vibration Analysis of a Three-Dimensional Ring Gyroscope
,”
Bull. Coll. Eng. NTU
,
91
, pp.
65
73
.
12.
Gallacher
,
B. J.
,
Hedley
,
J.
,
Burdess
,
J. S.
,
Harris
,
A. J.
,
Rickard
,
A.
, and
King
,
D. O.
,
2005
, “
Electrostatic Correction of Structural Imperfections Present in a Microring Gyroscope
,”
J. Microelectromech.
,
14
(
2
), pp.
221
234
.10.1109/JMEMS.2004.839325
13.
Bisegna
,
P.
, and
Caruso
,
G.
,
2007
, “
Frequency Split and Vibration Localization in Imperfect Rings
,”
J. Sound Vib.
,
306
(
3
), pp.
691
711
.10.1016/j.jsv.2007.06.027
14.
Xi
,
X.
,
Wu
,
Y. L.
,
Wu
,
X. M.
,
Tao
,
Y.
, and
Wu
,
X. Z.
,
2012
, “
Investigation on Standing Wave Vibration of the Imperfect Resonant Shell for Cylindrical Gyro
,”
Sens. Actuators A: Phys.
,
179
, pp.
70
77
.10.1016/j.sna.2012.03.031
15.
Wu
,
X. H.
, and
Parker
,
R. G.
,
2006
, “
Vibration of Rings on a General Elastic Foundation
,”
J. Sound Vib.
,
295
(
1–2
), pp.
194
213
.10.1016/j.jsv.2006.01.007
16.
Wang
,
S. Y.
,
Xiu
,
J. Y
,
Gu
,
J. P.
,
Xu
,
J.
, and
Shen
,
Z. G.
,
2010
, “
Prediction and Suppression of Inconsistent Natural Frequency and Mode Coupling of a Cylindrical Ultrasonic Stator
,”
Proc. Inst. Mech. Eng., Part J
,
224
(
9
), pp.
1853
1862
.10.1243/09544062JMES1993
17.
Wang
,
S. Y.
,
Huo
,
M. N.
,
Zhang
,
C.
,
Liu
,
J. P.
,
Song
,
Y. M.
,
Cao
,
S. Q.
, and
Yang
,
Y. H.
,
2011
, “
Effect of Mesh Phase on Wave Vibration of Spur Planetary Ring Gear
,”
Eur. J. Mech. A/Solid
,
30
(
6
), pp.
820
827
.10.1016/j.euromechsol.2011.06.004
18.
Wang
,
S. Y.
,
Xu
,
J. Y.
,
Xiu
,
J.
,
Liu
,
J. P.
,
Zhang
,
C.
, and
Yang
,
Y. H.
,
2011
, “
Elastic Wave Suppression of Permanent Magnet Motors by Pole/Slot Combination
,”
ASME J. Vib. Acoust.
,
133
(
2
), p.
024501
.10.1115/1.4002954
19.
Fox
,
C. H. J.
,
Hwang
,
R. S.
, and
McWilliam
,
S.
,
1999
, “
The In-Plane Vibration of Thin Rings With In-Plane Profile Vibrations Part II: Application to Nominally Circular Rings
,”
J. Sound Vib.
,
220
(
3
), pp.
517
539
.10.1006/jsvi.1998.1962
20.
Choi
,
S. Y.
, and
Kim
,
J. H.
,
2011
, “
Natural Frequency Split Estimation for Inextensional Vibration of Imperfect Hemispherical Shell
,”
J. Sound Vib.
,
330
(
9
), pp.
2094
2106
.10.1016/j.jsv.2010.11.014
21.
Yu
,
R. C.
, and
Mote
, Jr.,
C. D.
,
1987
, “
Vibration of Circular Saws Containing Slots
,”
Holz. Roh- Werkst.
,
45
(
4
), pp.
155
160
.10.1007/BF02627571
22.
Singh
,
R.
,
1988
, “
Case History: The Effect of Radial Slots on the Noise of Idling Circular Saws
,”
Noise Control Eng. J
.,
31
(
3
), pp.
167
172
.10.3397/1.2827720
23.
Nishio
,
S.
, and
Marui
,
E.
,
1996
, “
Effects of Slots on the Lateral Vibration of a Circular Saw Blade
,”
Int. J. Mach. Tool Manuf.
,
36
(
7
), pp.
771
787
.10.1016/0890-6955(95)00088-7
24.
Raman
,
A.
, and
Mote
, Jr.,
C. D.
,
2001
, “
Effects of Imperfection on the Non-Linear Oscillations of Circular Plates Spinning Near Critical Speed
,”
Int. J. Nonlinear Mech.
,
36
(
2
), pp.
261
289
.10.1016/S0020-7462(00)00014-7
25.
Kim
,
H.
, and
Shen
, I
. Y.
,
2009
, “
Ground-Based Vibration Response of a Spinning, Cyclic, Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects
,”
ASME J. Vib. Acoust.
,
131
(
2
), p.
021007
.10.1115/1.3025847
26.
Huang
,
S. C.
, and
Soedel
,
W.
,
1987
, “
Effects of Coriolis Acceleration on the Free and Forced In-Plane Vibration of Rotating Rings on Elastic Foundation
,”
J. Sound Vib.
,
115
(
2
), pp.
253
274
.10.1016/0022-460X(87)90471-8
27.
Asokanthan
,
S. F.
, and
Cho
,
J.
,
2006
, “
Dynamic Stability of Ring-Based Angular Rate Sensors
,”
J. Sound Vib.
,
295
(
3
), pp.
571
583
.10.1016/j.jsv.2006.01.028
28.
Esmaeili
,
M.
,
Durali
,
M.
, and
Jalili
,
N.
,
2006
, “
Ring Microgyroscope Modeling and Performance Evaluation
,”
J. Vib. Control
,
12
(
5
), pp.
537
553
.10.1177/1077546306064445
29.
Canchi
,
S. V.
, and
Parker
,
R. G.
,
2006
, “
Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs
,”
ASME J. Vib. Acoust.
,
128
(
2
), pp.
231
243
.10.1115/1.2159040
30.
Lesaffre
,
N.
,
Sinou
,
J. J.
, and
Thouverez
,
F.
,
2007
, “
Stability Analysis of Rotating Beams Rubbing on an Elastic Circular Structure
,”
J. Sound Vib.
,
299
(
4
), pp.
1005
1032
.10.1016/j.jsv.2006.08.027
31.
Ghasemloonia
,
A.
,
Rideout
,
D. G.
, and
Butt
,
S. D.
,
2013
, “
Vibration Analysis of a Drillstring in Vibration-Assisted Rotary Drilling: Finite Element Modeling With Analytical Validation
,”
ASME J. Energy Resour. Technol.
,
135
(
3
), p.
032902
.10.1115/1.4023333
32.
Natsiavas
,
S
.,
1995
, “
On the Dynamics of Rings Rotating With Variable Spin Speed
,”
Nonlinear Dyn.
,
7
(
3
), pp.
345
363
.10.1007/BF00046308
33.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
John Wiley & Sons
,
New York
, Chap. II.
34.
Lacarbonara
,
W.
, and
Yabuno
,
H.
,
2006
, “
Refined Models of Elastic Beams Undergoing Large In-Plane Motions: Theory and Experiment
,”
Int. J. Solids Struct.
,
43
(
17
), pp.
5066
5084
.10.1016/j.ijsolstr.2005.07.018
You do not currently have access to this content.