This paper is concerned with the geometric nonlinear analysis of the lateral displacement of thin rotating disks when subjected to a space fixed stationary force. Of particular interest is the development of the stationary wave and the effect of this wave on the frequency response of the disk as a function of its rotational speed. The predictions of this analysis are compared with experimental data obtained in a companion paper (Khorasany and Hutton, “Vibration Characteristics of Rotating Thin Disks—Part I: Experimental Results,” ASME J. Appl. Mech., 79(4), p. 041006). The governing equations are based on Von Kármán plate theory. A Galerkin solution of the governing non linear equations is developed. The eigenfunctions derived from the linear analysis of a stationary disk are used as approximations to the spatial response of the disk, and the eigenfunctions of the biharmonic equation as approximations for the stress function. Using the developed solution, the equilibrium configuration of the disk under the application of a space fixed force is found. In order to facilitate the prediction of the frequency response, as a function of disk rotational speed, the governing nonlinear equations are linearized around the equilibrium solution. The linearized equations are then used to find the eigenvalues of the spinning disk under the application of a space fixed force. The effect of different levels of nonlinearity on the disk frequencies is studied and compared with experimental results. The analysis is shown to produce an accurate representation of the measured response. Of particular interest is the disk response at speeds close to and above the linear critical speed. In this region, both the analysis and the experimental results display frequency “lock-in” behavior in which the frequency of backward travelling waves becomes constant for supercritical speeds. No speed exists for which backward travelling waves have zero frequency. Thus, critical speeds do not exist in the presence of geometric nonlinearities.

References

References
1.
Lamb
,
H.
, and
Southwell
,
R. V.
, 1921, “
The Vibration of Spinning Disk
,”
Proc. R. Soc. London
,
99
, pp.
272
280
.
2.
Mote
,
C. D.
, 1965, “
Free Vibration of Initially Stressed Circular Plates
,”
J. Eng. Ind.
,
87
, pp.
258
264
.
3.
Hutton
,
S. G.
,
Chonan
,
S.
, and
Lehmann
,
B. F.
, 1987, “
Dynamic Response of a Guided Circular Saw
,”
J. Sound Vib.
,
112
, pp.
527
539
.
4.
Chen
,
J. S.
, and
Bogy
,
D. B.
, 1992, “
Effects of Load Parameters on the Natural Frequencies and Stability of a Flexible Spinning Disk With a Stationary Load System
,”
ASME J. Appl. Mech.
,
59
, pp.
S230
S235
.
5.
Tian
,
J. F.
, and
Hutton
,
S. G.
, 2001, “
Cutting Induced Vibration in Circular Saws
,”
J. Sound Vib.
,
242
(
5
), pp.
907
922
.
6.
Nowinski
,
J. L.
, 1964, “
Nonlinear Transverse Vibrations of a Spinning Disk
,”
ASME J. Appl. Mech.
,
31
, pp.
72
78
.
7.
Tobias
,
S. A.
, 1957, “
Free Undamped Non-Linear Vibrations of Imperfect Circular Disks
,”
Proc. Inst. Mech. Eng.
,
171
, pp.
691
701
.
8.
Jana
,
A.
, and
Raman
,
A.
, 2005, “
Nonlinear Dynamics of a Flexible Spinning Disc Coupled to a Precompressed Spring
,”
Nonlinear Dyn.
,
40
, pp.
1
20
.
9.
Nayfeh
,
A. H.
,
Jilani
,
A.
, and
Manzione
,
P.
, 2001, “
Transverse Vibrations of a Centrally Clamped Rotating Circular Disk
,”
Nonlinear Dyn.
,
26
, pp.
163
178
.
10.
Chen
,
J. S.
, 2001, “
On the Internal Resonance of a Spinning Disk Under Space-Fixed Pulsating Edge Loads
,”
ASME J. Appl. Mech.
,
68
, pp.
854
859
.
11.
Touze
,
C.
,
Thomas
,
O.
, and
Chaigne
,
A.
, 2002, “
Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory
,”
J. Sound Vib.
,
258
, pp.
649
676
.
12.
Yang
,
L.
, and
Hutton
,
S. G.
, 1998, “
Nonlinear Vibrations of Elastically-Constrained Rotating Discs
,”
ASME J. Vibr. Acoust.
,
120
, pp.
475
483
.
13.
Luo
,
A. C.
J., and
Mote
,
C.
D., 2000, “
Nonlinear Vibration of Rotating Thin Disks
,”
ASME J. Vibr. Acoust.
,
122
, pp.
376
383
.
14.
Khorasany
,
R. M. H.
, and
Hutton
,
S. G.
, “
Vibration Characteristics of Rotating Thin Disks—Part I: Experimental Results
,”
ASME J. Appl. Mech.
,
79
(
4
), p.
041006
. DOI:
15.
Tobias
,
S. A.
, and
Arnold
,
R. N.
, 1957, “
The Influence of Dynamical Imperfections on the Vibration of Rotating Disks
,”
Proc. Inst. Mech. Eng.
,
171
, pp.
669
690
.
16.
Khorasany
,
R. M. H.
, and
Hutton
,
S. G.
, 2010, “
The Effect of Axisymmetric Non-Flatness on the Oscillation Frequencies of a Rotating Disk
,”
ASME J. Vibr. Acoust.
,
132
, p.
051012
.
17.
D’Angelo
,
C.
, and
Mote
,
C. D.
, 1993, “
Aerodynamically Excited Vibration and Flutter of a Thin Disk Rotating at Supercritical Speed
,”
J. Sound Vib.
,
168
, pp.
15
30
.
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