This paper considers a moving beam in frictional contact with pads, making the system susceptible for self-excited vibrations. The equations of motion are derived and a stability analysis is performed using perturbation techniques yielding analytical approximations to the stability boundaries. Special attention is given to the interaction of the beam and the rod equations. The mechanism yielding self-excited vibrations does not only occur in moving beams, but also in other moving continua such as rotating plates, for example.

1.
Hochlenert
,
D.
,
Spelsberg-Korspeter
,
G.
, and
Hagedorn
,
P.
, “
Friction Induced Vibrations in Moving Continua and Their Application to Brake Squeal
,”
ASME J. Appl. Mech.
0021-8936, in press.
2.
Wickert
,
J. A.
, and
Mote
,
C. D.
, 1990, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME J. Appl. Mech.
0021-8936,
57
, pp.
738
744
.
3.
Wickert
,
J. A.
, and
Mote
,
C. D.
, 1991, “
Response and Discretization Methods for Axially Moving Materials
,”
Appl. Mech. Rev.
0003-6900,
44
, pp.
279
284
.
4.
Meirovitch
,
L.
, 1975, “
A Modal Analysis for the Response of Linear Gyroscopic Systems
,”
ASME J. Appl. Mech.
0021-8936,
42
(
2
), pp.
446
450
.
5.
D’Eleuterio
,
G. M. T.
, and
Hughes
,
P. C.
, 1984, “
Dynamics of Gyroelastic Continua
,”
ASME J. Appl. Mech.
0021-8936,
51
, pp.
415
422
.
6.
Parker
,
R. G.
, 1998, “
On the Eigenvalues and Critical Speed Stability of Gyroscopic Continua
,”
ASME J. Appl. Mech.
0021-8936,
65
, pp.
134
140
.
7.
Seyranian
,
A. P.
, and
Kliem
,
W.
, 2001, “
Bifurcations of Eigenvalues of Gyroscopic Systems With Parameters Near Stability Boundaries
,”
ASME J. Appl. Mech.
0021-8936,
68
, pp.
199
205
.
8.
Cheng
,
S. P.
, and
Perkins
,
N. C.
, 1991, “
The Vibration and Stability of a Friction-Guided, Translating String
,”
J. Sound Vib.
0022-460X,
144
(
2
), pp.
281
292
.
9.
Adams
,
G. G.
, 1996, “
Self-Excited Oscillations in Sliding With a Constant Friction Coefficient—A Simple Model
,”
ASME J. Tribol.
0742-4787,
118
, pp.
819
823
.
10.
Mottershead
,
J. E.
, and
Chan
,
S. N.
, 1995, “
Flutter Instability of Circular Discs With Frictional Follower Loads
,”
ASME J. Vibr. Acoust.
0739-3717,
117
, pp.
161
163
.
11.
Chen
,
J. S.
, and
Bogy
,
D. B.
, 1992, “
Mathematical Structure of Modal Interactions in a Spinning Disk Stationary Load System
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
390
397
.
12.
Bloch
,
A. M.
,
Krishnaprasad
,
P. S.
,
Marsden
,
J. E.
, and
Ratiu
,
T. S.
, 1994, “
Dissipation Induced Instabilities
,”
Ann. Inst. Henri Poincare, Anal. Non Lineaire
0294-1449,
11
(
1
), pp.
37
90
.
13.
Bolotin
,
V. V.
, 1963,
Non-conservative Problems of the Theory of Elastic Stability
,
Pergamon
, New York.
14.
Crandall
,
S. H.
, 1995, “
The Effect of Damping on the Stability of Gyroscopic Pendulums
,”
ZAMP
0044-2275,
46
, pp.
761
780
.
15.
Huseyin
,
K.
, 1978,
Vibrations and Stability of Multiple Parameter Systems
,
Noordhoff International Publishing
, Groningen, The Netherlands.
16.
Karapetjan
,
A. V.
, 1975, “
The Stability of Nonconservative Systems
,”
Vestn. Mosk. Univ., Ser. 1: Mat., Mekh.
0579-9368,
30
(
4
), pp.
109
113
.
17.
Kirillov
,
O. N.
, 2006, “
Gyroscopic Stabilization of Non-conservative Systems
,”
Phys. Lett. A
0375-9601,
359
(
3
), pp.
204
210
.
18.
Krechetnikov
,
R.
, and
Marsden
,
J. E.
, 2006, “
On Destabilizing Effects of Two Fundamental Non-conservative Forces
,”
Physica D
0167-2789,
214
, pp.
25
32
.
19.
Lakhadanov
,
V. M.
, 1975, “
On Stabilization of Potential Systems
,”
Prikl. Mat. Mekh.
0032-8235,
39
(
1
), pp.
53
58
.
20.
Lumijärvi
,
J.
, and
Pramila
,
A.
, 1995, “
Comment on Stability of Non-conservative Linear Discrete Gyroscopic Systems
,”
J. Sound Vib.
0022-460X,
185
(
5
), pp.
891
894
.
21.
Müller
,
P. C.
, 1981, “
Allgemeine lineare Theorie für Rotorsysteme ohne oder mit kleinen Unsymmetrien
,”
Ing.-Arch.
0020-1154,
51
, pp.
61
74
.
22.
Yang
,
S. M.
, and
Mote
,
C. D.
, 1991, “
Stability of Non-conservative Linear Discrete Gyroscopic Systems
,”
J. Sound Vib.
0022-460X,
174
(
3
), pp.
453
464
.
23.
Kirillov
,
O. N.
, and
Seyranian
,
A. O.
, 2005, “
The Effect of Small Internal and External Damping on the Stability of Distributed Non-conservative Systems
,”
J. Appl. Math. Mech.
0021-8928,
69
(
4
), pp.
529
552
.
24.
Kirillov
,
O. N.
, and
Seyranian
,
A. P.
, 2004, “
Collapse of the Keldysh Chains and Stability of Continuous Nonconservative Systems
,”
SIAM J. Appl. Math.
0036-1399,
64
(
4
), pp.
1383
1407
.
25.
Kirillov
,
O. N.
, and
Seyranian
,
A. P.
, 2005, “
Instability of Distributed Nonconservative Systems Caused by Weak Dissipation
,”
Dokl. Math.
1064-5624,
71
(
3
), pp.
470
475
.
26.
Mennicken
,
R.
, and
Möller
,
M.
, 2003,
Non-self-adjoint Boundary Eigenvalue Problems
,
Elsevier
, New York.
27.
Naimark
,
M. A.
, 1967,
Linear Differential Operators Part 1: Elementary Theory of Linear Differential Operators
,
G. G. Harrap
, London.
28.
Plaut
,
R. H.
, 1972, “
Determining the Nature of Instability in Nonconservative Problems
,”
AIAA J.
0001-1452,
10
, pp.
967
968
.
29.
Marsden
,
J. E.
, and
Ratiu
,
T. S.
, 2000,
Einführung in die Mechanik und Symmetrie
,
Springer
, Berlin, Germany.
30.
Arnold
,
V. I.
, 1985,
Geometrical Methods in the Theory of Ordinary Differential Equations
,
Springer
, New York.
31.
Vishik
,
M. I.
, and
Lyusternik
,
L. A.
, 1960, “
The Solution of Some Perturbation Problems for Matrices and Selfadjoint or Non-selfadjoint Differential Equations I
,
Russ. Math. Surveys
0036-0279,
15
, pp.
1
73
.
32.
O’Reilly
,
O. M.
,
Malhotra
,
N. K.
, and
Namachchivaya
,
N. S.
, 1996, “
Some Aspects of Destabilization in Reversible Dynamical Systems With Application to Follower Forces
,”
Nonlinear Dyn.
0924-090X,
10
, pp.
63
87
.
33.
Kirillov
,
O. N.
, 2005, “
A Theory of the Destabilization Paradox in Non-conservative Systems
,”
Acta Mech.
0001-5970,
174
, pp.
145
166
.
You do not currently have access to this content.