A mass–spring–damper slider excited into vibration in a plane by a moving rigid belt through friction is a major paradigm of friction-induced vibration. This paradigm has two aspects that can be improved: (1) the contact stiffness at the slider–belt interface is often assumed to be linear and (2) this contact is usually assumed to be maintained during vibration (even when the vibration becomes unbounded at certain conditions). In this paper, a cubic contact spring is included; loss of contact (separation) at the slider–belt interface is allowed and importantly reattachment of the slider to the belt after separation is also considered. These two features make a more realistic model of friction-induced vibration and are shown to lead to very rich dynamic behavior even though a simple Coulomb friction law is used. Both complex eigenvalue analyses of the linearized system and transient analysis of the full nonlinear system are conducted. Eigenvalue analysis indicates that the nonlinear system can become unstable at increasing levels of the preload and the nonlinear stiffness, even if the corresponding linear part of the system is stable. However, they at a high enough level become stabilizing factors. Transient analysis shows that separation and reattachment could happen. Vibration can grow with the preload and vertical nonlinear stiffness when separation is considered, while this trend is different when separation is ignored. Finally, it is found that the vibration magnitudes of the model with separation are greater than the corresponding model without considering separation in certain conditions. Thus, ignoring the separation is unsafe.

References

References
1.
Akay
,
A.
,
2002
, “
Acoustics of Friction
,”
J. Acoust. Soc. Am.
,
111
(
4
), pp.
1525
1548
.
2.
Ibrahim
,
R. A.
,
1994
, “
Friction-Induced Vibration, Chatter, Squeal, and Chaos—Part I: Mechanics of Contact and Friction
,”
ASME Appl. Mech. Rev.
,
47
(
7
), pp.
209
226
.
3.
Ibrahim
,
R. A.
,
1994
, “
Friction-Induced Vibration, Chatter, Squeal, and Chaos—Part II: Dynamics and Modeling
,”
ASME Appl. Mech. Rev.
,
47
(
7
), pp.
227
253
.
4.
Berger
,
E. J.
,
2002
, “
Friction Modeling for Dynamic System Simulation
,”
ASME Appl. Mech. Rev.
,
55
(
6
), pp.
535
577
.
5.
Kinkaid
,
N. M.
,
O'Reilly
,
O. M.
, and
Papadopoulos
,
P.
,
2003
, “
Automotive Disc Brake Squeal
,”
J. Sound Vib.
,
267
(
1
), pp.
105
166
.
6.
Ouyang
,
H.
,
Nack
,
W.
,
Yuan
,
Y.
, and
Chen
,
F.
,
2005
, “
Numerical Analysis of Automotive Disc Brake Squeal: A Review
,”
IJVNV
,
1
(
3–4
), pp.
207
231
.
7.
Elmaian
,
A.
,
Gautier
,
F.
,
Pezerat
,
C.
, and
Duffal
,
J. M.
,
2014
, “
How Can Automotive Friction-Induced Noises be Related to Physical Mechanisms?
,”
Appl. Acoust.
,
76
, pp.
391
401
.
8.
Mills
,
H. R.
,
1938
, “
Brake Squeak
,” Institution of Automobile Engineers, London, Technical Report No. 9000 B.
9.
Spurr
,
R. T.
,
1961
, “
A Theory of Brake Squeal
,”
Proc. Automob. Div. Inst. Mech. Eng.
,
15
(
1
), pp.
33
52
.
10.
Hoffmann
,
N.
, and
Gaul
,
L.
,
2004
, “
A Sufficient Criterion for the Onset of Sprag-Slip Oscillations
,”
Arch. Appl. Mech.
,
73
(
9–10
), pp.
650
660
.
11.
Sinou
,
J. J.
,
Thouverez
,
F.
, and
Jezequel
,
L.
,
2003
, “
Analysis of Friction and Instability by the Centre Manifold Theory for a Non-Linear Sprag-Slip Model
,”
J. Sound Vib.
,
265
(
3
), pp.
527
559
.
12.
Popp
,
K.
, and
Stelter
,
P.
,
1990
, “
Stick-Slip Vibrations and Chaos
,”
Philos. Trans. R. Soc. London, Ser. A
,
332
(
1624
), pp.
89
105
.
13.
Luo
,
A. C. J.
, and
Gegg
,
B. C.
,
2006
, “
On the Mechanism of Stick and Nonstick, Periodic Motions in a Periodically Forced, Linear Oscillator With Dry Friction
,”
ASME J. Vib. Acoust.
,
128
(
1
), pp.
97
105
.
14.
North
,
N. R.
,
1976
, “
Disc Brake Squeal
,”
Proc. Inst. Mech. Eng.
,
C38/76
, pp.
169
176
.
15.
Hoffmann
,
N.
,
Fischer
,
M.
,
Allgaier
,
R.
, and
Gaul
,
L.
,
2002
, “
A Minimal Model for Studying Properties of the Mode-Coupling Type Instability in Friction Induced Oscillations
,”
Mech. Res. Commun.
,
29
(
4
), pp.
197
205
.
16.
Sinou
,
J.-J.
, and
Jézéquel
,
L.
,
2007
, “
Mode Coupling Instability in Friction-Induced Vibrations and Its Dependency on System Parameters Including Damping
,”
Eur. J. Mech. A/Solid.
,
26
(
1
), pp.
106
122
.
17.
Duffour
,
P.
, and
Woodhouse
,
J.
,
2004
, “
Instability of Systems With a Frictional Point Contact. Part 1: Basic Modelling
,”
J. Sound Vib.
,
271
, pp.
365
390
.
18.
Chan
,
S. N.
,
Mottershead
,
J. E.
, and
Cartmell
,
M. P.
,
1994
, “
Parametric Resonances at Subcritical Speeds in Discs With Rotating Frictional Loads
,”
Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci.
,
208
(
6
), pp.
417
425
.
19.
Ouyang
,
H.
,
Mottershead
,
J. E.
, and
Li
,
W.
,
2003
, “
A Moving-Load Model for Disc-Brake Stability Analysis
,”
ASME J. Vib. Acoust.
,
125
(
1
), pp.
53
58
.
20.
Jarvis
,
R. P.
, and
Mills
,
B.
,
1963
, “
Vibrations Induced by Dry Friction
,”
Proc. Inst. Mech. Eng.
,
178
(
1
), pp.
847
857
.
21.
Flint
,
J.
, and
HultÉN
,
J.
,
2002
, “
Lining-Deformation-Induced Modal Coupling as Squeal Generator in a Distributed Parameter Disc Brake Model
,”
J. Sound Vib.
,
254
(
1
), pp.
1
21
.
22.
Ouyang
,
H.
, and
Mottershead
,
J. E.
,
2005
, “
Dynamic Instability of an Elastic Disk Under the Action of a Rotating Friction Couple
,”
ASME J. Appl. Mech.
,
71
(
6
), pp.
753
758
.
23.
Kinkaid
,
N. M.
,
O'Reilly
,
O. M.
, and
Papadopoulos
,
P.
,
2005
, “
On the Transient Dynamics of a Multi-Degree-of-Freedom Friction Oscillator: A New Mechanism for Disc Brake Noise
,”
J. Sound Vib.
,
287
(
4–5
), pp.
901
917
.
24.
Hoffmann
,
N.
, and
Gaul
,
L.
,
2003
, “
Effects of Damping on Mode-Coupling Instability in Friction Induced Oscillations
,”
ZAMM Z. Angew. Math. Mech.
,
83
(
8
), pp.
524
534
.
25.
Li
,
Y.
, and
Feng
,
Z. C.
,
2004
, “
Bifurcation and Chaos in Friction-Induced Vibration
,”
Commun. Nonlinear Sci. Numer. Simul.
,
9
(
6
), pp.
633
647
.
26.
Hetzler
,
H.
,
Schwarzer
,
D.
, and
Seemann
,
W.
,
2007
, “
Analytical Investigation of Steady-State Stability and Hopf-Bifurcations Occurring in Sliding Friction Oscillators With Application to Low-Frequency Disc Brake Noise
,”
Commun. Nonlinear Sci. Numer. Simul.
,
12
(
1
), pp.
83
99
.
27.
Luo
,
A. C. J.
, and
Thapa
,
S.
,
2009
, “
Periodic Motions in a Simplified Brake System With a Periodic Excitation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
5
), pp.
2389
2414
.
28.
Shin
,
K.
,
Brennan
,
M. J.
,
Oh
,
J. E.
, and
Harris
,
C. J.
,
2002
, “
Analysis of Disc Brake Noise Using a Two-Degree-of-Freedom Model
,”
J. Sound Vib.
,
254
(
5
), pp.
837
848
.
29.
Andreaus
,
U.
, and
Casini
,
P.
,
2001
, “
Dynamics of Friction Oscillators Excited by a Moving Base and/or Driving Force
,”
J. Sound Vib.
,
245
(
4
), pp.
685
699
.
30.
Andreaus
,
U.
, and
Casini
,
P.
,
2002
, “
Friction Oscillator Excited by Moving Base and Colliding With a Rigid or Deformable Obstacle
,”
Int. J. Non-Linear Mech.
,
37
(
1
), pp.
117
133
.
31.
von Wagner
,
U.
,
Hochlenert
,
D.
, and
Hagedorn
,
P.
,
2007
, “
Minimal Models for Disk Brake Squeal
,”
J. Sound Vib.
,
302
(
3
), pp.
527
539
.
32.
Butlin
,
T.
, and
Woodhouse
,
J.
,
2009
, “
Friction-Induced Vibration: Should Low-Order Models be Believed?
,”
J. Sound Vib.
,
328
(
1–2
), pp.
92
108
.
33.
D'Souza
,
A. F.
, and
Dweib
,
A. H.
,
1990
, “
Self-Excited Vibrations Induced by Dry Friction, Part 2: Stability and Limit-Cycle Analysis
,”
J. Sound Vib.
,
137
(
2
), pp.
177
190
.
34.
Ouyang
,
H.
,
Baeza
,
L.
, and
Hu
,
S.
,
2009
, “
A Receptance-Based Method for Predicting Latent Roots and Critical Points in Friction-Induced Vibration Problems of Asymmetric Systems
,”
J. Sound Vib.
,
321
(
3–5
), pp.
1058
1068
.
35.
Sinou
,
J. J.
,
Thouverez
,
F.
, and
Jézéquel
,
L.
,
2004
, “
Methods to Reduce Non-Linear Mechanical Systems for Instability Computation
,”
Arch. Comput. Method Eng.
,
11
(
3
), pp.
257
344
.
36.
Soobbarayen
,
K.
,
Sinou
,
J. J.
, and
Besset
,
S.
,
2014
, “
Numerical Study of Friction-Induced Instability and Acoustic Radiation–Effect of Ramp Loading on the Squeal Propensity for a Simplified Brake Model
,”
J. Sound Vib.
,
333
(
21
), pp.
5475
5493
.
37.
Brunetti
,
J.
,
Massi
,
F.
,
D'Ambrogio
,
W.
, and
Berthier
,
Y.
,
2015
, “
Dynamic and Energy Analysis of Frictional Contact Instabilities on a Lumped System
,”
Meccanica
,
50
(
3
), pp.
633
647
.
38.
Leine
,
R. I.
,
Brogliato
,
B.
, and
Nijmeijer
,
H.
,
2002
, “
Periodic Motion and Bifurcations Induced by the Painlevé Paradox
,”
Eur. J. Mech. A/Solid
,
21
(
5
), pp.
869
896
.
39.
Saha
,
A.
,
Wiercigroch
,
M.
,
Jankowski
,
K.
,
Wahi
,
P.
, and
Stefański
,
A.
,
2015
, “
Investigation of Two Different Friction Models From the Perspective of Friction-Induced Vibrations
,”
Tribol. Int.
,
90
, pp.
185
197
.
40.
Lee
,
U.
,
1996
, “
Revisiting the Moving Mass Problem: Onset of Separation Between the Mass and Beam
,”
ASME J. Vib. Acoust.
,
118
(
3
), pp.
516
521
.
41.
Medio
,
A.
, and
Lines
,
M.
,
2001
,
Nonlinear Dynamics: A Primer
,
Cambridge University Press
,
Cambridge, UK
.
42.
Meirovitch
,
L.
,
1990
,
Dynamics and Control of Structures
,
Wiley
,
New York
.
43.
Pollard
,
H.
, and
Tenenbaum
,
M.
,
1964
,
Ordinary Differential Equations
,
Harper and Row
,
New York
.
44.
Andreaus
,
U.
, and
Nisticò
,
N.
,
1998
, “
An Analytical-Numerical Method for Contact-Impact Problems. Theory and Implementation in a Two-Dimensional Distinct Element Algorithm
,”
Comput. Model. Simul. Eng.
,
3
(
2
), pp.
98
110
.
45.
Baeza
,
L.
, and
Ouyang
,
H.
,
2008
, “
Dynamics of a Truss Structure and Its Moving-Oscillator Exciter With Separation and Impact-Reattachment
,”
Proc. R. Soc. London, Ser. A
,
464
(
2098
), pp.
2517
2533
.
You do not currently have access to this content.