Acta Polytechnica
DOI:10.14311/AP.2020.60.0081
Acta Polytechnica 60(1):81–87, 2020 © Czech Technical University in Prague, 2020
available online at https://ojs.cvut.cz/ojs/index.php/ap
EIGENVALUES EVALUATION OF GENERALLY DAMPED
ELASTIC DISC BRAKE MODEL LOADED WITH
NON-CONSERVATIVE FRICTION FORCES
Juraj Úradníčeka, ∗, Miloš Musila, Michal Bachratyb
a Slovak University of Technology, Faculty of Mechanical Engineering, Institute of Applied Mechanics and
Mechatronics, Námestie Slobody 17, 82131 Bratislava, Slovak republic.
b Slovak University of Technology, Faculty of Mechanical Engineering, Institute of Manufacturing Systems,
Environmental Technology and Quality Management, Námestie Slobody 17, 82131 Bratislava, Slovak republic.
∗ corresponding author: juraj.uradnicek@stuba.sk
Abstract. This paper deals with the evaluation of eigenvalues of a linear damped elastic two-
degrees-of-freedom system under a non-conservative loading. As a physical interpretation of a proposed
mathematical model, a simplified disk brake model is considered. A spectral analysis is performed to
predict an eigenvalues bifurcation, known as the Krein collision, leading to double eigenvalues, one
of them having a positive real part causing a vibration instability of the mechanical systems. This
defective behaviour of eigenvalues is studied with respect to a magnitude of non-conservative Coulomb
friction force, through the variation of the friction coefficient. The influence of a proportional versus
general damping on the system stability is further analysed. The generalized non-symmetric eigenvalue
problem calculation is employed for spectral analyses, while a modal decomposition is performed to
obtain a time-domain response of the system. The analyses are compared with an experiment.
Keywords: Eigenvalues bifurcation, Krein collision, non-conservative force, modal decomposition,
brake squeal.
1. Introduction
Brake systems represent one of the most important
safety and performance components of vehicles, such
as cars, trains, airplanes or industrial machines. Re-
liability, braking power and a fluent operation are
important properties of brake systems. However, dur-
ing the braking, the components of a disc brake tend
to vibrate and a noise and harshness such as a brake
squeal can occur. It is generally accepted that the
brake squeal is determined by a friction-induced vi-
bration via a rotating disc [1].
In 1952, Ziegler [2], studying a double mathemati-
cal pendulum exposed to the non-conservative force,
found that the limits of the system stability were
reduced by an application of minor damping to the
system. Damping reduces the stability of the system,
which contradicts the classical theory of structural
stability, in which the damping plays a purely stabiliz-
ing role. Thus, an unstable behaviour of mechanical
systems can also be caused by a dissipation induced
instability, which was, in the case of brake system,
pointed out in the work of Hoffman and Gaul in
2003 [3]. In a braking system, the non-conservative
friction force plays a major role. For this reason, the
disc brake system is considered as the system with a
potential occurrence of the damping destabilization
paradox.
Several studies focus on analyses of the behaviour
of such systems using simple analytical minimal mod-
els [4, 5], complex numerical Finite Element (FE) mod-
els [1] and experiments on real physical systems [6].
Analytical and FE studies are mainly based on the
complex Eigenvalue Analysis [7] to predict the mode
coupling. These analyses subsequently allow an opti-
mization of brake systems to avoid a potential brake
squeal occurrence.
In the process of disc brake development, the com-
plex eigenvalue analysis of finite element model is
usually used to perform the modal optimization of
the system to reduce any potential system instabil-
ity [8]. In this process, no damping or simple propor-
tional damping is usually considered. According to
the above-mentioned destabilization paradox, a damp-
ing can possibly cause a reduction of stability limits.
Significance of this effect should be closely examined
for disc brake applications.
In this paper, the damping in the system and its
influence on the system stability is studied on a sim-
ple two-degree-of-freedom (DOF) analytical model of
a disc brake. Observations are experimentally veri-
fied by measurements on an experimental pad-on-disc
system.
2. Physical problem description
In the following study, a simple two-degree-of-freedom
elastic system is considered (Figure 1). The system
consists of spring/damping elements and a mass mov-
ing in the horizontal and vertical direction. The spring
and damping elements in a skew direction couple both
degrees of freedom so that the mode local bifurcation
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https://doi.org/10.14311/AP.2020.60.0081
https://ojs.cvut.cz/ojs/index.php/ap
J. Úradníček, M. Musil, M. Bachraty Acta Polytechnica
Figure 1. Two-degrees-of-freedom mechanical model
including non-conservative friction force in the system
equilibrium position.
of complex conjugate eigenvalues (mode coupling) can
occur [9]. The mass element represents the mass of
the friction material (a part of the braking pad which
is in contact with the disc), while spring/damping
elements represent the stiffness and damping of the
friction material in particular directions. The mov-
ing belt acts as a generator of the non-conservative
friction force whose magnitude depends linearly on
state variables and a coefficient of friction, consid-
ering Coulomb friction law. To avoid the nonlinear
behaviour of the friction force related to changing its
sign, the following assumption is taken into account.
The solution is evaluated around the equilibrium point.
This case corresponds to the situation when the pre-
stress effect is applied on the mass in a negative x2
direction and after the solution is obtained around
this system’s equilibrium point. Under this assump-
tion, the overall friction force can exhibit only positive
values. In this paper, the evaluation of the eigenval-
ues with respect to the friction coefficient is studied
to investigate the limits of stability of the generally
damped simple minimal brake model.
3. Mathematical problem
description
The mechanical system in Figure 1 is described by
a set of second order differential equations in matrix
form
Mẍ + Cẋ + Kx = F (1)
M =
[
m 0
0 m
]
, C =
[
c2 + 12c3
1
2c3
1
2c3 c1 +
1
2c3
]
,
K =
[
k2 + 12k3
1
2k3
1
2k3 k1 +
1
2k3
]
, F =
[
Ft
0
]
, x =
[
x1
x2
]
,
where coordinates x1,x2 represent the displacement
of the mass around the equilibrium position, M, B, K
are coefficients’ matrices and Ft represents the dynam-
ical part of the overall friction force Ftc = Fo + Ft.
Constant friction force Fo = µFp results from the
pre-stress normal force Fp and inequality Fp > |Ft|/µ
has to be fulfilled so that the slider cannot detach
the moving belt. If the Coulomb friction model is
considered and the slider in the model cannot detach
the moving belt, the friction force Ft linearly depends
on the normal force acting between the slider and the
moving belt:
Ft = µ(k1x2 + c1ẋ2) (2)
where the µ represents the Coulomb friction coefficient.
Substituting (2) into (1), the system is reorganized to
the form {
Mẍ + C′(µ)ẋ + K′(µ)x = 0
Mẋ − Mẋ = 0
(3)
K′ =
[
k2 + 12k3
1
2k3 −µk1
1
2k3 k1 +
1
2k3
]
,
C′ =
[
c2 + 12c3
1
2c3 −µc1
1
2c3 c1 +
1
2c3
]
.
Adding the identical equation in (3), the system is
transformed into the state-space representation with
state variables vector y,
A(µ)ẏ + B(µ)y = 0 (4)
A =
[
C′ M
M 0
]
, B =
[
K′ 0
0 −M
]
, y =
[
x
ẋ
]
.
It can be seen that matrices A and B are non-
symmetric and smoothly depend on the scalar param-
eter µ.
4. System eigenvalues evaluation
The equation (4) can be rewritten in the form of a
generalized non-symmetric eigenvalue problem,
(B + λiA)vri = 0 (5)
where λi is the i−th generalized eigenvalue and vri is
the corresponding i−th right complex eigenvector. An
imaginary part of the eigenvalue is the damped natural
frequency ωd = ω0
√
1 − ζ2, where ω0 represents an
undamped natural frequency and the real part is ζω0.
A damping ratio ζ is calculated from a geometrical
relation cos θ = ζ, where θ is an angle to a pole from
a negative real axis.
In the undamped case, the local bifurcation of the
complex conjugate eigenvalues occurs in the point
µ = 0.5 (Figure 2) and the damped natural frequencies
are being coupled (Figure 3). At this point, the system
loses its stability. In the case of the proportional
damping, the system loses its stability slightly over
the point µ = 0.5 and damped natural frequencies
cross each other but don’t couple. In the case of the
general non-proportional damping, the bifurcation
point is unclear and the system loses its stability
below µ = 0.5.
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vol. 60 no. 1/2020 Eigenvalues evaluation of generally damped elastic disc brake model. . .
Figure 2. 1. and 2. modes damping ratios’ evolution with the friction coefficient, considering the proportionally
damped, non-proportionally damped and undamped system.
Figure 3. Damped natural frequencies’ evolution of the 1. and 2. modes with the friction coefficient considering
proportionally damped, non-proportionally damped and non-damped system.
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J. Úradníček, M. Musil, M. Bachraty Acta Polytechnica
(a). (b).
(c). (d).
(e). (f).
Figure 4. The transient response of the non-proportionally damped system to the initial conditions (IC) y(0) =
[1, 0, 0, 0] for a) µ = 0 (λ1 = −15.6 ± 5000i; λ2 = −21.9 ± 7071i), b) µ = 0.47 (λ1 = −37.5 ± 5886i; λ2 = 0 ± 6352i),
c) µ = 0.48 (λ1 = −44 ± 5943i; λ2 = 6.6 ± 6300i).
5. Modal decomposition
Homogeneous response of the linear state determined
system can be expressed in terms of state-transition
matrix Φ(t) as follows
yh(t) = Φ(t)y(0), (6)
where the state-transition matrix Φ(t) may be ex-
pressed in terms of the eigenvalues/eigenvectors as
Φ(t) = VreλtV−1r , (7)
defined by the modal matrix, consisting of right eigen-
vectors in columns, where n is a number of columns
of the modal matrix
Vr =
[
vr1 vr2 · · · vrn
]
(8)
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vol. 60 no. 1/2020 Eigenvalues evaluation of generally damped elastic disc brake model. . .
and the square matrix eλt with terms eλit on the
leading diagonal. Define
Rr = Vr−1
rr1
rr2
· · ·
rrn
(9)
the homogeneous response of the system can be ex-
pressed as a weighted superposition of the system
modes eλitvri where an initial condition y(0) affects
the strength of an excitation of each mode
yh(t) = (Vreλt)(Rry(0))
=
n∑
i=1
eλitvri(rriy(0))
=
n∑
i=1
cie
λitvri
(10)
where the scalar coefficient ci = rriy(0).
The yh1 coordinate in Figure 4 represents the ho-
mogeneous response of the mass displacement x1 in
Figure 1. If the friction is absent in the system, both
modes are attenuated over the time (Figure 4 a)) with
the rate given by their damping ratios ζi derived from
the corresponding eigenvalues λi. Damped eigenfre-
quencies ωD1 = 5000 rad·s−1 and ωD2 = 7071 rad·s−1
can be read from a Single Side Amplitude Spectrum
(SSAS) and must correspond to given eigenvalues. If
the friction coefficient reaches stability threshold, the
negative real part of the corresponding eigenvalue
changes its sign - the mode with a negative real part
of λi is attenuated over the time, while the mode with
a positive real part becomes unstable and is increas-
ing or maintaining (if real part is 0) its amplitude
(Figure 4 b)). It can be seen that the mode with the
lower frequency is attenuated while the second one
with the higher frequency becomes unstable. Damped
eigenfrequencies ωD1,2 become closer to each other
with increasing coefficient µ. A last plot (Figure 4
c)) shows the response for the friction coefficient set
over the stability threshold value. From eigenvalues
λi, it can be seen that the stable mode is attenuated
faster (compared to b) case) due to the lower real part
value. Oppositely, the unstable mode is increasing its
amplitude faster over the time due to the higher value
of the positive real part. The SSAS shows that the
unstable mode fully dominates the stable one over the
analysed time interval.
6. Experimental results and
discussion
To investigate the existence of described effects in
a real application, a simple experimental device has
been set up. The device consists of a simplified pad-
on-disc system where a friction material is pressed
onto a rotating disc. Elastic properties of a pad are
achieved through a thin plate attached between the
Figure 5. Experimental pad-on-disc set up.
friction material and a piston of a linear motor which
generates a normal force.
Dynamical instability and self-excited vibrations of
the system lead to a brake squeal. This effects can be
measured both using accelerometer sensors attached
to the flexible structure and a microphone, which
directly measures the sound pressure as a side effect
of an extensive vibration of the system. The time
evaluation of the sound pressure (Figure 6) carries out
the information about an intensity of a the vibration
and also a spectral content of the vibration. The
spectral content is observed through a single side
amplitude spectrum by the Fourier transformation of
the sound pressure time signal (Figure 7).
In a real physical environment, more physical as-
pects are involved in the process of dynamic destabi-
lization. It can be demonstrated that nonlinear effects
in friction contact, such as stick slip [10] or contact
surfaces detaching [11], lead into limit cycles of vibra-
tions. These effects bring perturbations to the system.
If the system is perturbed, both stable and unstable
eigenmodes should be excited over the time. This be-
haviour can be seen from the experimental observed
data from the real physical system. The experimental
data have been studied over the time interval of 0.06s.
The variation of the vibration amplitude can be seen
in Figure 6. The variation of the amplitude shows
that one or more modes consisting in the response are
changing their vibration propensity. This can be due
to the combination of nonlinear effects, such as arising
the amplitudes of unstable modes into limit cycles
along with the attenuation of the repeatedly excited
stable modes. Strong nonlinear effects on the signal
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J. Úradníček, M. Musil, M. Bachraty Acta Polytechnica
Figure 6. Sound pressure evaluation over the time interval of 0.06s.
Figure 7. Single side amplitude spectrum of the sound pressure over the studied time interval of the length 0.06s.
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vol. 60 no. 1/2020 Eigenvalues evaluation of generally damped elastic disc brake model. . .
can be concluded from the poly-harmonic response of
the signal in Figure 7 consisting of the fundamental
frequency of 5700 Hz and its higher harmonics. A
detailed observation of the signal spectrum around
5700 Hz shows more harmonics with relatively close
frequencies contained in the signal. This can be due to
the upper mentioned bifurcation effects. Coexistence
of more than one eigenmode in the unstable state
points out the effect of non-proportional damping in
the system, which has been described in the previous
chapter.
7. Conclusions
The destabilization paradox due to the non-
conservative friction force in the system with non-
proportional damping can play a significant role, since
the evolution of system eigenvalues is different when
compared to the proportionally damped or undamped
systems. The stability threshold value of the fric-
tion decreases with the non-proportional damping.
Damped eigenfrequencies don’t couple, more precisely,
do not cross each other in the bifurcation point in com-
parison to the undamped system and proportionally
damped systems. This behaviour is known, in vari-
ous physical problems, as the destabilization paradox
or the dissipation induced destabilization. Even the
described model represents only the mode coupling
mechanism of the linear system where an unstable
mode cannot coexists with a stable one in the steady
solution, the experimental observation proved the co-
existence of more than one mode in the response of the
unstable (squealing) system. This coexistence of dou-
ble modes points out on the general (non-proportional)
damping or a system non-linearity. The cause of
double modes in experimental analyses should be
closely examined by the measurement of damping
properties of the system to determine how signifi-
cantly non-proportional the damping is and how this
non-proportionality can reduce the stability threshold
value. The non-proportional damping is expected due
to significantly different material properties of the
friction material and the rest of the system [12]. This
discussion leads to the question how much important
role a general damping plays in a brake squeal. The
significance of these effects in brake systems will be
further investigated in a next research.
Acknowledgements
The research in this paper was supported by the grant
agency VEGA 1/0227/19 of Ministry of Education, Sci-
ence and Sport of the Slovak Republic and the author
also appreciates the financial support provided by Slo-
vak Research and Development Agency for the project
APVV-0857-12.
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http://dx.doi.org/10.1016/j.jsv.2006.11.023
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Acta Polytechnica 60(1):81–87, 2020
1 Introduction
2 Physical problem description
3 Mathematical problem description
4 System eigenvalues evaluation
5 Modal decomposition
6 Experimental results and discussion
7 Conclusions
Acknowledgements
References