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Mode coupling instability in friction-induced vibrations
and its dependency on system parameters including
damping
Jean-Jacques Sinou, Louis Jezequel
To cite this version:
Jean-Jacques Sinou, Louis Jezequel. Mode coupling instability in friction-induced vibrations and its
dependency on system parameters including damping. European Journal of Mechanics - A/Solids,
Elsevier, 2007, 26 (1), pp.106-122. �10.1016/j.euromechsol.2006.03.002�. �hal-00207759�
1
Journal home page : http://www.sciencedirect.com/science/journal/09977538
Mode coupling instability in friction-induced vibrations and its dependency on system parameters
including damping
European Journal of Mechanics - A/Solids, Volume 26, Issue 1, January-February 2007, Pages 106-122
Jean-Jacques Sinou and Louis Jézéquel
MODE COUPLING INSTABILITY IN FRICTION-INDUCED VIBRATIONS AND ITS
DEPENDENCY ON SYSTEM PARAMETERS INCLUDING DAMPING
Jean-Jacques SINOU*, Louis JEZEQUEL
Laboratoire de Tribologie et Dynamique des Systèmes UMR CNRS 5513
Ecole Centrale de Lyon, 36 avenue Guy de Collongues, 69134 Ecully, France.
ABSTRACT
Friction-induced vibrations due to coupling modes can cause severe damage and are recognized as one of
the most serious problems in industry. In order to avoid these problems, engineers must find a design to
reduce or to eliminate mode coupling instabilities in braking systems. Though many researchers have
studied the problem of friction-induced vibrations with experimental, analytical and numerical
approaches, the effects of system parameters, and more particularly damping, on changes in stable-
unstable regions and limit cycle amplitudes are not yet fully understood.
The goal of this study is to propose a simple non-linear two-degree-of-freedom system with friction in
order to examine the effects of damping on mode coupling instability. By determining eigenvalues of the
linearized system and by obtaining the analytical expressions of the Routh-Hurwitz criterion, we will
study the stability of the mechanical system’s static solution and the evolution of the Hopf bifurcation
point as functions of the structural damping and system parameters. It will be demonstrated that the
effects of damping on mode coupling instability must be taken into account to avoid design errors. The
results indicate that there exists, in some cases, an optimal structural damping ratio between the stable and
unstable modes which decreases the unstable region. We also compare the evolution of the limit cycle
amplitudes with structural damping and demonstrate that the stable or unstable dynamic behaviour of the
coupled modes are completely dependent on structural damping.
1. INTRODUCTION
Though friction-induced vibration has received considerable attention from a number of researchers [1-
23], there are still no methods which completely eliminate or reduce instabilities. Solving potential
friction-induced vibration problems requires a complete understanding and appropriate analysis to
identify the effects of all physical parameters on system stability.
For example, it is well known that the stability of the static solution can be affected by the friction
coefficient and the associated friction mechanisms. In order to find the most suitable mechanism to
describe friction-induced vibration in brake systems, these various mechanisms must be examined. They
fall into four categories [1-2]: stick-slip, variable dynamic friction coefficient, sprag-slip [7] and
2
geometric coupling of degrees of freedom [8,10,12-16]. The first two approaches rely on changes in the
friction coefficient with relative sliding speed affecting system stability. The last two approaches use
kinematic constraints and modal coupling in order to develop the instability; in these cases, instability can
occur with a constant brake friction coefficient.
Stick-slip is a low sliding speed phenomenon caused when the static friction coefficient is higher than the
dynamic coefficient. In 1938, a study by Jarvis and Mills [10] led to an initial understanding that brake
squeal was associated with a decrease in friction coefficient with rubbing speed. Due to this negative
slope, the steady state sliding becomes unstable and causes friction-induced vibrations. Although this
mechanism is still recognised as explaining some low frequency brake vibration problems, it was soon
realised that a decrease in friction coefficient was insufficient to explain some friction-induced vibrations.
It was later realised that this tribological property was not the only cause of brake squeal, and that
vibration could occur when the friction coefficient remained largely constant with speed. Spurr [7]
proposed the sprag-slip action which does not depend on a friction coefficient varying with the relative
rotation speed of the brake disc. Later researchers (Earles [8, 11-12], North [17], Miller [9], D’Souza and
Dweib [18]) developed a more generalised theory describing the mechanism as a geometrically induced
or kinematic constraint instability. For example, Jarvis and Mills [16] demonstrated that the variation of
the friction coefficient with sliding speed was insufficient to cause friction-induced vibrations and so the
instability was due to coupling even if the friction coefficient was constant. It is now accepted that there is
no uniform theory for the characterisation of the problem and that stick-slip phenomena [19], negative
friction velocity slope [20], sprag-slip phenomena and geometric coupling of the structure involving
sliding parts [7,8,10,12-16] contribute to the description of mechanisms causing dynamic instability of
brake systems.
The analysis of friction-induced instability is still a vast problem in spite of numerous recent studies on
the subject. The stability of the static solution of the mechanical system can be affected by the design and
layout of brake components and the effects of all system parameters are not yet clearly understood. This is
especially the case for structural damping: in a broad variety of engineering systems, incorporating
additional damping into one part of the brake system is undertaken in order to reduce or eliminate
significantly friction-induced vibrations. However, it has been observed that the addition of damping to
one part of the mechanical system may have a worse effect. Moreover, many studies on mode coupling
instability due to friction have relied on undamped model systems. Then, when structural damping is
taken into account, its role and effects in the analysis of mode coupling instability is not fully
investigated. For example, Earles and Chambers [8] discussed the concept of geometrically induced
instability on damped systems and concluded that the effects of damping are too complex to make
predictions intuitively and could not be readily anticipated. Recently, Shin et al. [21-22] investigated the
effect of damping on a two-degree-of-freedom model and clearly indicated that the amount of damping is
a key factor in order to avoid unstable vibrations and stick-slip phenomena. They showed that damped
model systems connected through a sliding friction interface can become unstable if damping is added
only on one side of the sliding interface. Finally, Hoffmann and Gaul [23] studied the effects of damping
on mode-coupling instability in friction-induced oscillations. They also conclude that increasing damping
may destabilize friction-induced vibrations w and that the role of structural damping is not a side effect
which can be easily ignored. They show that the effects of damping on mode coupling instability in
friction-induced vibrations is a surprising and complex phenomenon.
Given that the influences of structural damping and the associated mode coupling phenomena are not yet
fully understood, the goal of this study is to clarify the effects and roles of damping for friction-induced
mode coupling instability. For the sake of simplicity a two-degree-of-freedom model will be developed
and analysed. One of the biggest advantages of this model is that the stability analysis via the Routh-
Hurwitz criterion can be undertaken and the analytical expressions of the stable/unstable boundary
regions can be calculated. The results not only illustrate the effects of damping on the determination of
the stable and unstable regions, but also indicate that, in some cases, the most efficient damping ratio
between the stable and unstable coupling modes needs to be taken into account to avoid design errors.
3
Moreover, mode coupling instability and flutter phenomena will be studied in detail by showing the
change in stable and unstable modes due to the variations of structural damping. We will thus
demonstrate that stable or unstable dynamic behaviour of the coupled modes are completely dependant on
structural damping and that the role of the damping ratio of the coupled modes is essential to the unstable
or stable motion of these coupled modes. Then, various parametric studies will be made in order to show
the influence of system parameters such as the friction coefficient and the natural frequency ratio between
the coupling modes. Finally, the changes in the limit cycle amplitudes for various system parameters will
be shown in order to better understand the effects of structural damping on unstable amplitudes.
2. DESCRIPTION OF THE MECHANICAL MODEL
Because flutter instability is a mode-coupling phenomenon, a simple self-excited mechanism proposed by
Hulten [24-25] will be investigated and developed. This is shown in Figure 1. This model was introduced
by Hulten in order to study squeal vibration in drum brakes. For this study, Hulten’s model will be
sufficient to investigate friction-induced vibration, and to develop analytical expressions in order to better
understand the roles and effects of damping. This model is composed of a mass
m
held against a moving
band; the contact between the mass and the band is modelled by two plates supported by two different
springs. For the sake of simplicity, we assume that the mass and band surfaces are always in contact. This
assumption may be due to a preload applied to the system. The contact can be expressed by two cubic
stiffnesses. Damping is integrated as shown in Figure 1. The friction coefficient at contact is assumed to
be constant and the band moves at a constant velocity. Then it is assumed that the direction of friction
force does not change because the relative velocity between the band speed and
1
X
or
2
X
is assumed to
be positive. All these assumptions are taken into account in order to study a simple non-linear theoretical
two-degree-of-freedom system with friction such that the effects of damping on mode coupling instability
and the associated analytical developments may be easily investigated.
The tangential force
T
F
due to friction contact is assumed to be proportional to the normal force
N
F
as
given by Coulomb’s law:
TN
FF
=
µ
. Assuming the normal force
N
F
is linearly related to the
displacement of the mass normal to the contact surface, the resulting equations of motion can be
expressed as
(
)
++MX+CXKXFNLX=0
(1)
with
()
12
T
XX=X
.
X
,
X
and
X
are, respectively, the acceleration, velocity, and displacement
response 2-dimensional vectors of the degrees-of-freedom. The mass matrix
M
, the damping matrix
C
,
the stiffness matrix
K
and the non-linear vector
(
)
FNL X
are given by
0
0
m
m
⎡
⎤
=
⎢
⎥
⎣
⎦
M
(2)
1
2
0
0
c
c
⎡
⎤
=
⎢
⎥
⎣
⎦
C
(3)
12
12
kk
kk
µ
µ
−
⎡
⎤
=
⎢
⎥
⎣
⎦
K
(4)
()
33
11 22
33
11 22
NL NL
NL NL
kX kX
kX kX
µ
µ
⎛⎞
−
=
⎜⎟
+
⎝⎠
FNL X
(5)
K
is asymmetric as a result of the friction force. Therefore this system may become unstable.
By dividing these equations by m and considering the relative damping coefficients
ii ii
cmk
η
=
(i=1,2) and natural pulsations
0,
iii
km
ω
= (i=1,2) , the following relations can be established
4
where
22
33
10,1
1
0,1 0,2
11 22
11
22
33
20,2
2
0,1 0,2
11 22
22
0
10
0
01
NL NL
NL NL
X
XXXX
X
XXXX
ηω
ωµω
ϕµϕ
ηω
µω ω
µϕ ϕ
⎡⎤
−⎛⎞ ⎛⎞
⎛⎞
⎡⎤
−+
⎛⎞
⎡⎤
++ =
⎢⎥
⎜⎟ ⎜⎟ ⎜ ⎟
⎢⎥
⎜⎟
⎢⎥
−−
⎣⎦
⎝⎠
⎣⎦
⎝⎠
⎝⎠ ⎝⎠
⎣⎦
(6)
and where
11
NL NL
km
ϕ
=
and
22
NL NL
km
ϕ
=
.
The base parameters are given by
1
0,1
2 1000 .rad s
−
=×
ωπ
;
1
0,2
2 800 .rad s
−
=×
ωπ
;
12
0.02
η
η
==
;
2
10,1
NL
ϕ
ω
=
and
2
20,2
NL
ϕω
=
.
3. STABILITY ANALYSIS
The stability of the static solution is investigated by calculating the eigenvalues
λ
of the linearized
system [14-15]. For the non-linear system being studied, the non-linear static solution corresponds to the
origin of the system. So the eigenvalues of the linear system can be found by solving the characteristic
equation [14-15]
(
)
2
det 0
λλ
+
+=MCK (7)
As long as the real part of all the eigenvalues remains negative, the static solution of the system is stable.
When at least one of the eigenvalues has a positive real part, the static solution is unstable. The imaginary
part of this eigenvalue represents the frequency of the unstable mode. By considering the friction
coefficient
µ
as a control parameter, the Hopf bifurcation point is defined by the following conditions:
(
)
(
)
()
()
()
()
()
0
0
0
,
,
,
Re 0
Re 0
Re 0
center
non center
d
d
µµ
µµ
µµ
λµ
λµ
λµ
µ
=
−
=
=
=
≠
≠
X=0
X=0
X=0
(8)
The first condition implies that the system has a pair of purely imaginary eigenvalues
center
λ
, while all of
the other eigenvalues
non center
λ
−
have nonzero real parts at
(
)
0
,
µ
µ
=
X=0
. The second condition of
equation (8), called a transversal condition, implies a transversal or nonzero speed crossing of the
imaginary axis.
First, the change in the real and imaginary parts of eigenvalues against the friction coefficient
µ
is
studied for various damping ratios. Figures 2-3 show the effects of these. As illustrated in Figure 3, the
well known behaviour for friction-induced mode coupling is obtained: there are two stable modes at
different pulsations when
0
µ
µ
<
. As the control parameter
µ
increases, these two modes move closer
until they reach the bifurcation zone at the Hopf bifurcation point
0
µ
. For
0
µ
µ
= , there is one pair of
purely imaginary eigenvalues and all other eigenvalues have negative real parts (as indicated in Figures 2
and 4). After the Hopf bifurcation point (
0
µ
µ
> ), the two stable and unstable modes couple. As indicated
in Figure 2, the real part of the stable and unstable modes (indicated by the stable and unstable branches)
are negative and positive, respectively. These effects of damping appear to be very complex. For
example, it may be observed that the unstable mode varies due to the damping ratio (see Figure 3): for
()
12
0.02, 0.02
ηη
==, the unstable mode corresponds to the smaller imaginary part of the two coupling
modes; for
()
12
0.02, 0.005
ηη
==, the unstable mode corresponds to the larger imaginary part of the two
coupling modes.
To further our understanding of the effects of damping, Figure 6 shows the stable and unstable regions of
the static solution for various damping ratios
12
η
η
(with
1
0.02
η
=
) and various friction coefficients
µ
.
It clearly appears that increasing damping in only one part of the system may induce mode coupling
