This paper discusses two simplified analytical models for automotive disc brake vibration which can be used to complement more complex finite element methods. The first model approximates the brake disc as a simple beam structure with cyclosymmetric boundary conditions. Since the beam model is a onedimensional approach, modelling of the inner boundary condition of the brake disc, at the interface of the brake rotor and the central hat, is not possible. The second model, which is established based upon Kirchhoff’s thin plate theory, is presented in this paper in order to incorporate the vibrational deformation at the hatdisc interface. The mode shapes, natural frequencies, and forced response of a static disc are calculated using different inner boundary conditions. Among others, the springsupported boundary condition is proposed and applied in this paper to make appropriate predictions. The predicted results are compared with measurements of the vibration characteristics of a solid brake disc mounted upon a static test rig. These comparisons demonstrate that the most appropriate model for the inner boundary condition of the measured brake disc is the proposed springsupported inner boundary condition.
The investigation of brake squeal is of continuing interest to the automotive industry due to the high warranty costs that have to be paid every year to replace complete or individual parts of the brake systems. References [
Finite element models are widely used to investigate brake system instability and have given considerable insight to the problem. Specific examples of how the geometric complexity of a braking system can be modelled using finite elements is discussed by Trichês Jr et al. [
In this paper two analytical models for automotive disc brake vibration are presented. Of course, simplified analytical models will not be able to represent the complexity of mechanisms involved in practical brake system vibration. However, as noted by Papinniemi and his coauthors [
Since the beam model is a onedimensional approach and, therefore, the modelling of the inner boundary conditions of the brake disc is impossible, a second model based upon Kirchhoff’s thin plate theory is also presented. The mode shapes and natural frequencies of a static disc with clamped, simplysupported, and free inner boundary conditions are calculated. A novel approach of simulating the inner boundary by using an infinite number of springs with a spring stiffness calculated using the geometry of the central hat is also developed. The purpose of this investigation is to discuss the appropriateness of an alternative boundary condition in comparison to the commonly applied clamped inner boundary condition. It is hoped that this boundary condition may approximate some of the deformation at the hatdisc interface while avoiding the complexity of a fully coupled hatdisc analytical model. Predicted natural frequencies are compared to experimentally measured resonant frequencies of a brake disc rigidly mounted upon a static test rig. Using the springsupported inner boundary condition a forced vibration model is developed and compared to the experimentally measured forced response of the structure. As noted above, inplane vibrations become more significant at higher frequencies and for relatively thick discs. In this paper only the outofplane vibrations of the disc are modelled, and, thus, the investigation is relevant to the lowerfrequency modes of a brake disc.
In this approach the brake disc is modelled as a ring, which is assumed to be unwrapped into a straight EulerBernoulli beam with cyclosymmetric boundary conditions at each end. The derivation below follows the approach of Flint and Hultén [
Consider a straight beam of finite length
A cyclosymmetric beam of length
In an alternative to the wavebased approach, the cyclosymmetric beam was also modelled using a modal summation approach [
Predicted modulus of the point mobility of the large diameter experimental ring modelled as a cyclosymmetric beam with
The scope of this section is to develop free and forced vibration models for a solid, nonvented, and brake disc by using annular plate theory. Four different cases for the inner boundary are considered: (i) free, (ii) simply supported; (iii) clamped, and (iv) spring supported. In each case the outer boundary is assumed to be free. The standard boundary conditions for a brake disc model assume that the displacement at the inner boundary is zero. This assumption has been applied successfully by many researchers [
In the following section a brief summary of annular plate theory relevant to a disc with different boundary conditions is given. This is followed by a presentation of the existing models of disc vibrations used by various authors [
The theory used, following Kirchhoff, assumes small deflections and neglects the influence of rotary inertia and the additional deflections caused by shear deformation. In Figure
Sketch of the freefree disc, sectional view.
Expressions for the shear force, the bending moment, and the twisting moment are needed to define the boundary conditions. The bending moment in radial direction can be expressed as
Boundary conditions for the brake disc model.
Freefree  Simply supported free  Clamped free  

Inner boundary, 







Outer boundary, 



In the following approach a novel inner boundary is developed whereby the hat is modelled as springs with an equivalent stiffness relating the geometry of the hat and its Young’s modulus. At the inner boundary of the disc a spring force is assumed to act as a distributed force per unit length along a tiny element of the circumference of length
Sketch of the brake disc including the hat dimensions: inner radius
Figure
Dimensions and material properties of the experimental test structures.
Test structure  Young’s modulus, 
Density, 
Poisson’s ratio, 
Outer diameter, 
Inner diameter, 
Thickness, 

Large diameter ring  2.1 × 10^{11}  7850  0.3  950 × 10^{−3}  850 × 10^{−3}  5 × 10^{−3} 
Small diameter annular plate  1.2 × 10^{11}  7250  0.26  262.3 × 10^{−3}  160 × 10^{−3}  14 × 10^{−3} 
Standard production nonvented brake disc  1.2 × 10^{11}  7250  0.26  262.3 × 10^{−3}  152/140 × 10^{−3} excluding/including contact area with hat  14 × 10^{−3} 
Effect of the length of hat,
For the forced vibration model the modal summation approach of Section
To verify the predictions made from the cyclosymmetric beam model described in Section
The large diameter steel ring was selected because of its mild curvature and small ring width of 50 mm. Hence, this structure would be expected to exhibit the vibrational characteristics of a straight beam with cyclosymmetric boundary conditions over a wide frequency range. To measure the frequency response function the ring was suspended vertically on thin wires from a large frame and excited by an electrodynamic exciter using a random noise input over the frequency range from 0 to 3200 Hz. The point response of the ring was measured by using a force transducer and a lightweight accelerometer at excitation location. To identify the relevant mode shapes accelerometer measurements were made at sixteen equidistant locations around the centre line of the ring midradius distance between the inner and outer edges.
The small diameter annular plate was obtained by taking a nonvented brake disc and removing its central hat, thus, forming an annular plate with the dimensions and material properties similar to that of a standard production brake disc. The frequency response functions and mode shapes were measured in the same manner as for the large diameter ring.
In order to compare the simply supportedfree, clampedfree and spring supportedfree annular plate models, developed in Section
Figure
Modulus of the point mobility of the large diameter experimental steel ring: (solid line) theoretical prediction from the cyclosymmetric beam model; (dashed line) measured results; and (dotted line) equivalent “infinite” beam.
Difference in percentage between the measured resonant frequencies of the large diameter steel ring and the predicted natural frequencies of the cyclosymmetric beam model.
At low frequencies it is likely that the effect of curvature will become more significant on the measured data. Previous work on curved beams has shown that curvature effects are important for predominantly flexural waves below approximately 10% of the ring frequency [
Figure
Comparison of the predicted natural frequencies with the measured resonant frequencies of the small diameter annular plate.
Diametral mode  Cyclosymmetric beam model, 
Freefree plate model, 
Measured, 


Hz  Hz  Hz 
2  780  620  624 
3  1750  1700  1696 
4  3200  3200  3140 
5  4950  5000  4912 
Modulus of the point mobility of small diameter annular plate: (solid line) theoretical prediction from the cyclosymmetric beam model with
Also apparent in Figure
The natural frequencies predicted by the freefree annular plate model are also shown in Table
Figure
Modulus of the point mobility of the brake disc over the frequency range 1000 to 6400 Hz, logarithmic scale: (solid line) measured result; (dotted line) equivalent “infinite” beam.
The main resonant frequencies and associated diametral modes of the brake disc are labelled in Figure
Comparison of the predicted natural frequencies of the annular plate with different inner boundary conditions with the measured resonant frequencies of the brake disc.
Diametral mode  Clamped, 
Simply supported, 
Spring supported, 
Measured, 


Hz  Hz  Hz  Hz 
2  3198  1535  1325  1220 
3  3784  2520  2248  2196 
4  4798  3895  3599  3508 
5  6273  5655  5367  5120 
Comparing the predictions of the annular plate models, it can be seen in Table
The closest agreement with the measured resonant frequencies of the disc shown in Table
In Figure
Comparison of the modulus of the point mobility of the rigidly mounted brake disc over the frequency range from 1000 to 6400 Hz, logarithmic scale: (dashed line) theoretical prediction for the annular plate with a spring supported inner boundary; (solid line) measured results.
The aim of this paper has been to investigate the effectiveness of two simplified mathematical models that predict the outofplane vibrational behaviour of an automotive brake disc. In the first approach the disc vibration is assumed to act in only one dimension. Thus, the disc is modelled as a straight beam in flexure with cyclosymmetric boundary conditions. Predictions of the point mobility made using a wavebased approach and a modal summation approach gave identical results. Comparison of the predicted results with measurements made on both a large diameter ring and a small diameter annular plate formed by removing the central hat of a nonvented brake disc showed good agreement over the frequency range covered by the
In the second approach the brake disc was modelled as an annular plate. The inner boundary of the disc at its central hat location was modelled using four different boundary conditions: (i) spring supported, (ii) simply supported, (iii) clamped, and (iv) free. The inclusion of a nonfree boundary condition at the central hat location changed the character of the vibration from beamlike to platelike. Thus, the assumption of a free boundary condition at the central hat location gave least agreement with the measured results from a standard production brake disc. Comparison of the calculated natural frequencies predicted using these models and measurements made upon a standard production nonvented brake disc showed the best agreement with the spring supported boundary condition. The springsupported boundary condition was shown to vary in effect between a free boundary and a simply supported boundary. The clamped boundary condition was the least satisfactory. A comparison of the measured point mobility of a rigidly mounted disc and a prediction of the forced response of the disc made using the spring supported boundary condition showed good agreement over the frequency range covered by the main diametral modes of interest,