This paper investigates the vibration of an annular disk that is subjected to rotation and in-plane frictional traction distributed over a sector of the disk’s two faces. Technical applications include noise, vibration, and harshness in automotive and aircraft disk brakes, clutches, transmissions, and other rotating machine components. To the degree that the rotor-to-stator friction in such cases is directed along the disk’s deformable surface, it is treated here as a nonconservative follower-type load. The vibration model incorporates membrane stiffness which derives both from rotation, and from the stresses established as a result of friction. The plane stress state is determined in closed form as a Fourier series, and that solution is compared with the companion, but computationally intensive, results from finite element analysis. For the cases of sector-shaped and full annular loading, the vibration model predicts the critical mode, which is defined as the one that becomes dynamically unstable at the lowest friction level. Vibration modes and propagating waves that fall into opposite symmetry classes are shown to have opposite stability characteristics in the presence of frictional loading.

1.
Beck
M.
,
1953
, “
Die Knicklast des Einseitig Eingespannten, Tangential Gedru¨ckten Stabes
,”
Zeitschrift fu¨r Angewendte Mathematik und Physik
, Vol.
3
, pp.
225
228
.
2.
Bolotin, V. v., 1963, “Nonconservative Problems of the Theory of Elastic Stability,” Journal of Sound and Vibration, Pergammon Press, New York.
3.
Celep
Z.
,
1979
, “
Axially Symmetric Stability of a Completely Free Circular Plate Subjected to a Non-conservative Edge Load
,”
Journal of Sound and Vibration
, Vol.
65
, pp.
549
556
.
4.
Chan
S. N.
,
Mottershead
J. E.
, and
Cartmell
M. P.
,
1995
, “
Instabilities at Subcritical Speeds in Discs with Rotating Frictional Follower Loads
,”
ASME JOURNAL OF VIBRATION AND ACOUSTICS
, Vol.
117
, pp.
240
242
.
5.
Chen
J.-S.
, and
Bogy
D. M.
,
1993
, “
The Effects of a Space-Fixed Friction Force on the In-Plane Stress and Stability of Transverse Vibrations of a Spinning Disk
,”
ASME Journal of Applied Mechanics
, Vol.
60
, pp.
646
648
.
6.
Chen
L.-W.
, and
Ku
D.-M.
,
1994
, “
Stability of Nonconservatively Elastic Systems Using Eigenvalue Sensitivity
,”
ASME JOURNAL OF VIBRATION AND ACOUSTICS
, Vol.
116
, pp.
168
172
.
7.
Chonan
S.
,
1982
, “
Vibration and Stability of Annular Plates Under Conservative and Nonconservative Loads
,”
Journal of Sound and Vibration
, Vol.
80
, pp.
413
420
.
8.
Dym, C. L., 1974, Stability Theory and Its Applications to Structural Mechanics, Leyden Noordhoff.
9.
Herrmann
G.
, and
Jong
I.-C
,
1965
, “
On the Destabilizing Effect of Damping in Non-conservative Elastic Systems
,”
ASME Journal of Applied Mechanics
, Vol.
32
, pp.
592
597
.
10.
Herrmann
G.
,
1967
, “
Stability of Equilibrium of Elastic Systems Subjected to Nonconservative Forces
,”
Applied Mechanics Reviews
, Vol.
20
, pp.
103
108
.
11.
Higuchi
K.
, and
Dowell
B. H.
,
1989
, “
Dynamic Stability of a Completely Free Plate Subjected to a Controlled Nonconservative Follower Force
,”
Journal of Sound and Vibration
, Vol.
132
, pp.
115
128
.
12.
Higuchi
K.
, and
Dowell
B. H.
,
1990
, “
Dynamic Stability of a Rectangular Plate with Four Free Edges Subjected to a Follower Force
,”
AIAA Journal
, Vol.
28
, pp.
1300
1305
.
13.
Leipholz
H. H. E.
,
1975
, “
On the Buckling of Thin Plates Subjected to Non-conservative Follower Forces
,”
Transactions of the CSME
, Vol.
3
, pp.
25
34
.
14.
Leipholz
H. H. E.
, and
Pfendt
F.
,
1981
, “
On the Stability of Rectangular, Completely Supported Plates with Uncoupling Boundary Conditions Subjected to Uniformly Distributed Follower Forces
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
30
, pp.
19
52
.
15.
Mottershead
J. E.
, and
Chan
S. N.
,
1995
, “
Flutter Instability of Circular Discs with Frictional Follower Loads
,”
ASME JOURNAL OF VIBRATION AND ACOUSTICS
, Vol.
117
, pp.
161
163
.
16.
Ono
K.
,
Chen
J.-S.
, and
Bogy
D. B.
,
1991
, “
Stability Analysis for the Head-Disk Interface in a Flexible Disk Drive
,”
ASME Journal of Applied Mechanics
, Vol.
58
, pp.
1005
1014
.
17.
Reissner
E.
, and
Wan
F. Y. M.
,
1992
, “
A Follower Load Buckling Problem for Rectangular Plates
,”
ASME Journal of Applied Mechanics
, Vol.
59
, pp.
674
676
.
18.
Sasaki
M.
, and
Chonan
S.
,
1985
, “
Vibration and Stability of Elastically Supported Circular Plates under Conservative and Non-conservative Loads
,”
Journal of Sound and Vibration
, Vol.
103
, pp.
99
108
.
19.
Timoshenko, S. P., and Goodier, J. N., 1970, Theory of Elasticity, McGraw-Hill.
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