Extension of a modal instability theory to real brake systems.
F. Massi, O. Giannini
University of Rome Department of mechanics and aeronautics
Via Eudossiana 18 00186 Rome Italy
e-mail: [email protected]
Abstract
Brake squeal emission is still one of the major problems in brake design, because there is not yet a clear
understanding of the phenomenon. A crucial problem in studying squeal is the complexity of brake systems.
For this reason many researchers in this field prefer to develop simplified, easy to model, experimental setups.
In the past years, some researches were addressed to modeling such simplified brake systems that brought
to the “lock-in” theory, able to explain the squeal occurrence on the developed experimental set-ups.
This paper presents a parallel experimental and numerical analysis, aimed to extend the “lock-in” theory from
simplified set-ups to real brake systems. The analysis of the dynamic behavior of the set-up is described and
tied to the squeal occurrence.
1 Introduction
During the last forty years several researches on the sound radiated by brake systems were performed with
the aim of understanding the mechanism leading to the squeal noise. Akay et al. [1], Tucinda et al. [2], Allgaier
et al. [3], developed their squeal studies on the beam-on-disc set-up. Their researches characterize the
conditions necessary to obtain the squeal instability and propose the “lock-in” theory to explain the squeal
mechanism.
Since these studies were performed on simplified set-up, it is necessary to verify the possibility of extending
the “lock-in” theory to more complex brakes.
This paper, developed on the Laboratory Brake set-up [4;5], is aimed to be a “trait d’union” between the
beam-on-disc and a commercial brake.
The new configuration of the “Laboratory Brake”, presented in this paper, is closer than previous ones to a
real brake system, because commercial pads with their real dimensions are added to the original apparatus.
After a description of the set-up, experimental data on squeal phenomenon are presented and correlated to
experimental and numerical dynamic analyses of the system.
2 Description of the set-up
The set-up (Figure 1) consists of a commercial brake disc with diameter 240mm and thickness 11 mm. The
disc is held by two bearings that are attached to an aluminum frame fixed to a table. The disc is rotated by an
electric motor (2.2 KW of power).
MOTOR
PAD
DISC
BEAMS
Figure 1. The laboratory brake set-up
The caliper consists of two steal beams that hold the brake pads. Four bolts, connected to the table apply and
change the normal load. Two thin aluminum plates hold the beams in the vertical direction by applying a
negligible force in the horizontal direction.
DISC
BEAM
THIN PLATE
Figure 2. Detail of the simplified caliper
The set-up was designed with the main objective to be easily modeled, in order to investigate the squeal
phenomenon and its dependence on many parameters. Its simplicity permits to obtain the dynamic
characterization of the system by experimental and numerical analyses performed separately on the single
parts (beams and disc) and on the coupled system. Such analyses are necessary to establish the relationship
between dynamics and squeal behaviour.
In order to extend the results obtained in [4] to a more complex system, closer to a real brake apparatus,
commercial brake pads are introduced. To investigate the influence of the pads’ shape and dimensions, the
pads were cut to have different extensions (Figure 3 ) of the contact surface.
Figure 3. Brake pads used for the analysis
3 Experimental squeal instabilities
The first step of this work consists of a set of squeal measurements. The tests are performed under the
following operative conditions: the disc rotates in the range between 5rpm and 25rpm and the normal load and
the pad dimension are adjusted until different squeal conditions arise. Once these tests are concluded, the
configuration of the set-up is changed to search for different squeal frequencies.
Table 1 shows the squeal frequencies obtained for different normal load conditions, with different pad
dimensions. Higher squeal frequencies occur when lager normal loads are applied
Pad extension
Frequency (Hz)
25 mm
40 mm
50 mm
70 mm
4550
3200
6700
3800
6000
6400
7100
3900
6100
6500
4150
6300
6600
4200
6400
6700
6800
7300
12900
7400
13000
13200
13400
14200
Tab. 1. Squeal frequencies obtained experimentally
Figure 4 shows the operative deformed shape measured with the laser scanner vibrometer during squeal
conditions at 6000 Hz and 4150 Hz. The deformed shapes obtained using pads with extended dimensions are
no longer axial symmetric. The vibrating circumferential section across the pads is larger than the others
sections. We will refer to these deformed shapes as unsymmetrical shapes.
During squeal experiments, changing the normal load, squeal instability having different frequencies and
different deformed shapes occur. Nevertheless, we obtain the same deformed shape, shown in figure 4_a, in
a wide range of frequencies from 6000 to 6800 Hz, and the deformed shape shown in figure 4_b in a range of
frequencies between 3800 e 4200 Hz. In Tab. 1 squeal frequencies having the same deformed shape are
highlighted.
a)
b)
Figure 4. a) Deformed shape for squeal frequency at 6000 Hz; b)Deformed shape for
squeal frequency at 4150 Hz
4 Dynamic investigation of the set-up
A dynamic analysis of the set-up is performed to understand which is the influence of the pads on the system
dynamics. The importance of such study is related to the coupling between the disc and the beam, that
increases when the contact area or the contact pressure increases. This explains the asymmetry of the
measured mode shapes presented above.
4.1 Experimental data
In order to establish a correlation between the squeal deformed shape and the dynamical behavior of the
system, we focus on the squeals occurring around 3900Hz. The following measurements are performed:
− the configuration of the set-up is adjusted until a squeal develops;
− the squeal deformed shape is measured with the laser scanner vibrometer;
− the electric motor is arrested and a dynamic test is performed by exciting the system with a frequency
sweep in the range between 3800 and 4200 Hz, which includes the squeal frequencies, in order to
measure the operative deformed shapes;
− several FRF measurements and deformed shapes measurements are performed for several values of
the normal load applied on the pad. In this way we retrieve the dynamic behavior of the system from
the condition of free disc (no normal load applied) until 400N, the maximum normal load. (After 400N
the eigenvalue does not change anymore if the normal load increases).
Eigenfrequencies Vs Normal load
4200
4100
4000
Frequency (Hz)
3900
3800
3700
3600
3500
3400
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
Normal load (N)
Figure 5. Eigenfrequencies of the system as a function of the applied normal load
We use the following notation to denote the coupled modes:
− (m,n) the mode with n nodal diameters, m nodal circumferences, and one nodal diameter passing
through the middle of the contact surface;
− (m,n+) the mode with n nodal diameters, m nodal circumferences, and an antinode passing through
the middle of the contact surface
Figure 5 shows the results of the analysis. The horizontal line represents the eigenfrequency of the disc in
free conditions. The triangles represent the frequency of the mode (0,4) and the squares represent the
frequency of the mode (0,4+). When pad and disc are put in contact, the frequency of the mode (0,4) (figure
6_a) suddenly decreases, to begin to increase again when a given value of the normal load is reached. The
results show the variations of the eigenfrequency ranging from 3600 to 4200 Hz.
The initial decrease of the frequency can be explained by the mass addition of the pad and the beam.
Later on, by increasing the normal load, the modal stiffness increases because of the increase of the contact
stiffness between the disc and the pad. The mode (0,4+), with the antinode in the middle of the contact
surface, is more influenced by the increase of the contact stiffness and its eigenfrequency wipes a large range
of frequencies (from 3550Hz to 4200Hz) (figure 5).
As the eigenfrequency increases the corresponding deformed shape looses its symmetry: the vibrating
section of the deformed shape in the contact zone increases its dimensions (figure 6_b).
It is important to notice that the deformed shape obtained during the dynamic tests is similar to the squeal
deformed shape obtained with the same normal load (figure 4_b).
a)
b)
Figure 6. a) Deformed shape for the mode (0,4) at 6000 Hz; b)Deformed shape for
the mode (0,4+) at 4150Hz
4.2 Numerical results
To better understand the dynamic behavior of the system obtained experimentally, an FEM analysis aimed at
reproducing its dynamical behavior is performed. The disc is modeled with solid elements defined by eight
nodes with three degrees of freedom per node. The same elements are used to model the pad and the
backplate. In order to simplify the model, the beam is modeled by springs placed on the back of the backplate. The contact between pad and disc is reproduced by a bed of springs connecting the nodes of the pad
with the nodes of the disc, perpendicular to the contact area. (In fact we are interested to study the influence
of the contact on the bending modes of the disc).
The major effect produced by increasing the normal load in the experimental analysis can be modeled as an
increase of the contact stiffness between the beams and their supports. i.e. by increasing the normal load the
contact stiffness increases, causing an overall increase in the stiffness of the pad’s constraints.
Figure 7 shows the eigenfrequencies (y axis) of the model in function of the total stiffness of the springs (x
axis). We find a good agreement with the experimental data (figure 5). The numerical results do not simulate
the initial decrease of the (0,4) mode frequency, because of the non simulated addition of mass. In fact the
analysis doesn’t start from the uncoupled condition, and the (0,4) mode frequency is already lower than the
(0,4+) mode frequency.
Contact Stiffness
Figure 7. Numerical study on the eigenfrequencies of the system as a function of the contact stiffness
Figure 8_a and figure 8_b show the deformed shapes of the mode (0,4+) for Kn = 105N/m and Kn = 1010N/m
respectively. As showed by the experimental deformed shapes, the vibrating section of the disc in
correspondence to the contact surface increases with the contact stiffness.
a)
b)
Figure 8. a) Numerical deformed shape for mode (0,4+) for Kn = 105;
b) Numerical deformed shape for mode (0,4+) for Kn = 1010
4.3 Conclusion
The paper shows an experimental and numerical investigation aimed at characterizing the dynamic behavior
and the squeal behavior of the set-up.
The results show that the squealing frequencies and deformed shapes correspond always to a mode of the
system. This confirms the possibility to predict squeal frequencies by a complex eigenvalue analysis and the
lock-in theory.
The extension of the contact surface causes an asymmetry on the deformed shape that increases when the
normal load increases.
The large range of squeal frequencies having the same deformed shape is explained by the large range of
eigenfrequencies excursion (for a given mode) obtained by changing the applied normal load. This permits to
identify the squeal as a modal instability, even when the same squeal deformed shape occurs at different
frequencies.
A further step will be addressed to perform a complex eigenvalue analysis with the new configuration, aimed
to predict possible squeal frequencies, as already obtained in [5].
References
[1] - Akay, Wickert J., Xu Z., Investigation of Mode Lock-In and friction interface, Final Report,
Department of mechanical engeneering, Carnegie Mellon University 2000
[2] - Tuchinda A., Hoffmann N. P., Ewins D. J. and Keiper W., Effect of Pin Finite Width on
Instability of Pin-On-Disc Systems., Proceedings of the International Modal Analysis Conference –
IMAC, Vol. 1, pp. 552–557,2002.
[3] – Allgaier R., Gaul L., Keiper W., Willnery K., Hoffmann N., A study on brake squeal using a beam
on disc, Proceedings of the International Modal Analysis Conference – IMAC, Vol. 1, pp. 528–
534,2002.
[4] - Giannini O., Massi F., “An experimental study on the brake squeal noise,”, Proc. International
Conference on Noise and Vibration Engineering-ISMA, Leuven, Belgium (2004)
[5] - Giannini O., Experiments and modeling of squeal noise on a laboratory disc brake, PhD thesis,
University of Rome “La sapienza” 2004.