Abstract
Brake squeal is often analytically studied by a complex eigenvalue analysis of linearized models of the brake assembly that is usually quite large. In this paper, a method for determining those frequencies having the most effect on the pair of coupling frequencies that saves much time is put forward and a reduced-order model is presented based on the complex modes theory. The reduced-order model is proved to be effective when applied to a flexible pin-on-disc system; even damping and nonlinearity are taken into consideration. This reduced-order model can predict the onset of squeal as well as the squeal frequency with sufficient accuracy and largely reduced amount of calculation and gives us a practical guide to perform design optimization in order to reduce brake squeal.
1. Introduction
The problem of disc brake squeal has long been a major concern for the automobile industry. During the past few decades, brake squeal has been studied thoroughly by both theoretical and experimental approaches. Till now, four main frictional squeal mechanisms are highlighted; they are mode coupling instability theory, negative friction-velocity slope instability theory, sprag-slip motion theory, and stick-slip motion theory [1]. Among these theories, mode coupling instability theory is most widely accepted by the academic world, many researches have shown that mode coupling instability frequency is nearly consistent with the squeal frequency [2–5], which suggests a complex modal analysis of brake system models as a tool of investigation on squeal and this method is widely used in industry with the help of finite element software to predict the squeal frequency.
Typically, the order of the modal-based brake system model is quite large, which creates a formidable task of studying coupling among a large number of modes for the prediction of the onset of squeal and the instability frequency. However, it can be found that mode coupling is the coupling between a single pair of modes or coalescence of two eigenfrequencies of the system and a great majority of modes have little influence on the coupling between these two modes [6–10], so many scholars, such as Ouyang and Huang, have put forward their methods to simplify the system so as to predict the onset of squeal or squeal frequencies with largely reduced amount of calculation and sufficient accuracy, which is proved to be effective under certain circumstances. For example, Huang et al. [6] presented a perturbation method for determining the critical friction coefficient for disc brakes based on the eigenvalues and eigenvectors of the elastically coupled system. However, this method can only be effective under the condition of real mode and other modes have little effect on the two coupling modes. What is more, it does not work well when used to predict the squeal frequency when friction coefficient is large. Ouyang et al. [7] present a different and more efficient approach based on the receptance of the symmetric system, and the efficiency of this method relies on the knowledge of measured receptances. Unfortunately, these methods can only be applied to real modes. Nevertheless, in many engineering problems such as the system with nonproportional damping, dynamic system under nonconservative forces, analysis of aeroelastic flutter, the system matrices are not real symmetric and may be complex asymmetric. What is more, when the nonlinearity factors are taken into consideration, the linearized system will be more complicated so as to lead to the failure of these methods. In this case, a new method needs to be developed for the complex modes to simplify the system so as to predict the onset of squeal and squeal frequency with sufficient accuracy. As everyone knows, the matrix perturbation theory [11] is an efficient method when applied to small changes of the structure; Huang’s method is a convergent method in spite of inaccuracy in some situation.
In this paper, a method combining the matrix perturbation theory for complex modes with Huang’s method is developed to judge those eigenfrequencies having the greatest effect on the two coupling eigenfrequencies, which is time-saving and then a reduced-order method for complex modes system is put forward to predict the onset of squeal and squeal frequency. What is more, this method is applied to a nonlinear flexible pin-on-disc system and accurate results are obtained. This method provides us with an insight into the dependence of propensity to squeal on system parameters and thus performing design optimization.
2. Method to Determine the Influential Eigenfrequencies
2.1. Matrix Perturbation for Complex Modes
The free vibration equations of the linear or linearized systems with modes are as follows:where the matrix is assumed to be complex asymmetric because of, for example, the rotation of the disc and nonlinearity. Similarly, the matrix is assumed to be complex asymmetric because of, for example, the effect of the nonconservative forces—frictional force.
The governing equations in (1) are mapped into a state space by introducing ; (2) can be derived from (1) aswhere , .
The eigenvalue problem corresponding to the system in (2) is shown below:where and are the eigenvalue and right eigenvector of the system, respectively.
Consider the following power series expansions for with respect to the coefficient of friction :
It should be noted that is the matrix when the components of the system are coupled only by normal stiffness of the lining, or it can be said that there is no friction between components, or the coefficient of friction sets to be zero, under which situation the system is chosen to be the unperturbed system.
The eigenvalue problem corresponding to the unperturbed system of (2) is where and () are the eigenvalue and right eigenvector of the unperturbed system, respectively.
The left eigenvalue problem associated with the unperturbed system in (5) is where and () are the eigenvalue and left eigenvector of the unperturbed system, respectively. Note that the eigenvalues in (5) and (6) are the same since the matrixes and have the same eigenvalues in the complex domain.
It is assumed that the right and left eigenvectors are normalized to satisfy the biorthogonality condition
When there is friction with the coefficient of friction between the components of the system that is called the perturbation system, Equation (3), according to the matrix perturbation theory, there will be small changes both in eigenvalues and eigenvectors, so they can be expressed as the power series with respect to which is the perturbation parameter; that is, where , , and are th order perturbation of , , and , respectively.
Substituting (9)–(11) into perturbed system in (3), then
Equating the terms of the same orders in (12a) yields
It can be seen that (13a) is exactly (5). Let the perturbed eigenvectors be expressed as a linear combination of the unperturbed eigenvectors as
Substituting (14) into (13b), premultiplying by and using (7), (8) yields where and .
It can be obtained that
Similarly, equating the terms of the same orders in (12b) yields
It can be seen that (18a) is exactly (6). Let the perturbed eigenvectors be expressed as a linear combination of the unperturbed eigenvectors as
Substituting (19) into (18b), premultiplying by and using (7), (8) yields where .
It can be obtained that
and , (), can be obtained using the normalization condition of the eigenvectors; the right eigenvectors and the left eigenvectors should satisfy the condition
Substituting (10) and (11) into (23) yields
Equating the terms of the same orders in (24) yields
It can be seen that (25a) is exactly (7). Substituting (14) and (19) into (25b) using (7), (8) yieldsIt is obvious that in (17) and (22) should be the same, so that so it can be obtained that
It is obvious that (, ) can be obtained using the above equations.
2.2. Use of Huang’s Method
The above-mentioned method will be invalid when the mode coupling takes place. So Huang’s method is used here to obtain the influential eigenfrequencies. The characteristic equation for the two coupling eigenfrequencies is a 2nd order polynomial in that can be factored into the following form:where , , and , are sets of (the largest perturbation order ) yet-to-be-determined coefficients.
Assuming that and are the two coupling eigenfrequencies for perturbed system, the coefficients , can be determined by matching known values of , , and their derivatives , () aswhere and the same for .
The coefficients and can be solved explicitly from (30a) and (30b) to produce for
Using (31a) and (31b), the roots of (29) define the two eigenvalues and , which is called original perturbation in this paper.
2.3. Judge the Influential Eigenfrequencies
If only the eigenfrequencies belonging to the set are taken into account when calculating and using the above-mentioned method, then when calculating in (16), in (21), and in (17), should be replaced by . In this way, the corresponding and that rule out the eigenfrequencies not belonging to the set can also be obtained, which is called reduced-order perturbation in this paper. If ruling out the eigenfrequencies not belonging to the set , and change little (in the permissible range) compared with and ; then, the set is called the influential set in this paper. Note that this method will save much time compared with the complex eigenvalue analysis at every given friction coefficient.
3. The Reduced-Order Model
Substituting (4) into (2), the original model changes to
The coordinates are transformed to a new set of coordinates such that where .
Premultiplying by gives where .
Then, the reduced-order model is obtained aswhere .
Equation (35) can also be expressed as where .
Note that (the th order perturbation of corresponding to the reduced-order model) can be obtained just if is replaced by when calculating in (16), in (21), and in (17). After using the method mentioned in Section 2.2, the solutions are just and . In conclusion, when and are close enough to and , respectively, then the two coupling eigenfrequencies solved in the reduced-order model will get close enough to those solved in the original model. So this reduced-order model is useful when predicting the onset of squeal and the squeal frequency.
It is assumed that there are no repeated eigenvalues for the unperturbed system; that is, , which is typically true for brake system without friction. If there are repeated eigenvalues for the unperturbed system, other methods such as matrix perturbation theory for multiple eigenvalues [11] can be used to calculate (, ). Also note that the same approach can be used for system with one parameter other than the coefficient of friction and also for system with more than one parameter.
4. Example of Application of the Reduced-Order Model
4.1. Flexible Pin-on-Disc System
4.1.1. Pin-on-Disc Transient Model
The pin-on-disc system used in the present study consists of two components: (i) a flexible disc and (ii) a flexible pin, as shown in Figure 1. The disc is assumed to be clamped at the inner rim and free at the outer rim. The pin has a uniform round cross-sectional area and is assumed to be fully clamped at one end and in contact with the disc at the other end. It is assumed that the pin can vibrate in two principal directions: the axis direction, , and the transverse direction, , whereas the disc can vibrate only in the out-of-plane direction, . One end of the pin is contact with the disc, which rotates at the constant speed of , with a contact stiffness of . The pin inclines with an angle of along the direction of rotation of the disc. Since disc brake squeal tends to appear at low speeds [12] when centripetal and gyroscopic effects may be omitted. The effect of friction follower force is neglected on the basis that contact stiffness term is much larger than preload term in the stiffness matrix [5, 13].
As is depicted in Figure 1, the disc is rotating at the constant speed of , there is a preload between the pin and disc. There are three coordinate systems in the model, which are the absolute coordinate systems , the coordinate systems fixed on the disc, and the coordinate systems fixed on the pin, respectively. Assume that at the beginning, the coordinate of the free end of the pin in the coordinate systems is . At the moment , the rotational angle of the free end of the pin relative to the axis of the disc is , and the rotational angle of the free end of the pin relative to that of the disc is , so . Rotational effects and the effect of friction follower force are neglected and the equation of motion of the disc in the coordinate systems iswhere is the moment acting on the disc.
The transverse vibration of the pin in the coordinate systems can be written as
Similarly, the axial vibration of the pin in the coordinate system can be written as
Assuming that the free end of the pin is in constant contact with the disc with a contact stiffness of , the transverse displacement of the disc at the contact point between them can be derived as
To transform the deflection of the pin in the coordinate system to (shown in Figure 1), the transformation matrix can be derived. The deflection of the pin in the coordinate system is , so the normal force acting on the pin can be obtained as
The transverse vibration of the disc can be expressed by the summation of its natural modes and modal coordinates as where () which satisfy the orthonormality conditions,where and the bar over a symbol denotes the complex conjugation.
(Note that the mode shapes of the disc are in the form of nodal circles and nodal diameters denoted, resp., by the subscripts and .).
Similarly, the transverse vibration of the pin can be expressed by the summation of its natural modes and modal coordinates as which satisfy the orthonormality conditions:
The axial vibration of the pin can be expressed by the summation of its natural modes and modal coordinates aswhich satisfy the orthonormality conditions:
With regard to the disc, substituting (42), (44), and (46) into (37), multiplying (37) with , and then integrating with the help of (43) yields
Similarly, with regard to the transverse vibration of the pin, substituting (42), (44), and (46) into (38), multiplying (38) with , and then integrating with the help of (43) yields
Similarly, with regard to the axial vibration of the pin, substituting (42), (44), and (46) into (39), multiplying (39) with , and then integrating with the help of (43) yields
Considering that is marginal compared with , so for the sake of simplification, is neglected in the model so that .
To make use of the method of state space, new modal coordinates are introduced as
Equations (48), (49), and (50) then become
4.1.2. Friction Model
The velocity-dependent friction model used in this study is reasonably described with well-known exponential formulation. The coefficient of dynamic friction is introduced as [14]. Considerwhere governs the larger relative velocity behavior, denotes the static friction coefficient, and controls the gradient of relative to . and are provided in the Appendix.
4.1.3. Linearization of the Model
The onset of squeal is widely believed to be due to an unstable behavior occurring in linear conditions during the braking phase, linearization of the nonlinear system is conducted around the static equilibrium point for various system parameters, and only small perturbation around the static equilibrium point is considered. The friction coefficient in (53) is linearized via Taylor series as follows:where is the friction coefficient at the speed of the contact point between the pin and disc. is the slope of the friction coefficient at the same point.
By defining , , and , the static equilibrium point can be easily determined by forcing all time derivative terms to zero in (25a), (25b), (26), (27), (30a), and (30b) as follows:where , , , , , , and are also provided in the Appendix.
Assuming small perturbation around the static equilibrium point, that is, , and neglecting , leads to the following linearized system:where is the damping matrix, is the natural frequency matrix of the disc and pin, is the contact stiffness matrix resulting from the contact between the disc and pin, is the complex nonsymmetric nonconservative work matrix produced by friction.
4.2. Application to This System
Obviously, it can be obtained from (32) thatwhere is complex asymmetric because of the effect of the frictional force and the rotation of the disc, and :where is complex asymmetric because of the rotation of the disc and nonlinearity.
Choose the perturbation parameter as . In (56), can be expressed as the power series expansions with respect to :where .
Then, consider the following power series expansions for with respect to : where reduces to 1 since for small oscillations around the equilibrium point; the free end of the pin has such a low velocity that it does not reach the rotation speed of the disc.
In this section, the physical parameters used for the disc and pin are given in Tables 1 and 2, respectively. The problem of squeal is considered in a frequency range of 1–12 kHz so that only the first 13 modes of the disc, the 2nd and 3rd bending mode of the pin, and the first axial mode of the pin are used, so .
The eigenvalues for the unperturbed system can be sorted with the ascending order of the imaginary part such as . Firstly, the parameters are assumed to be , , , , , and so that a plot of predicted frequency (i.e., imaginary parts of eigenvalues) as a function of in a frequency range of 1–12 kHz can be derived, as is shown in Figures 2(a) and 2(b).
(a)
(b)
Figure 2(a) shows that, in the frequency range of 1–6 kHz, frequency 23 coalesces with frequency 24 when reaches 0.25, and there might be many frequencies having effect on this pair of coupling frequencies. Figure 2(b) shows that in the frequency range of 6–11 kHz, frequency 29 coalesces with frequency 30 when reaches 0.09; however, there might be few frequencies having influence on this pair of coupling frequencies.
Using the above-mentioned method, the influential set can be determined as corresponding to frequencies 23 and 24, and the effect of the reduced-order model is shown in Figure 3(a) that when the reduced-order perturbation gets close to the original perturbation, the error of reduced-order model with respect to the exact is always less than 20 Hz, which is a satisfactory result. Note that the frequencies solved from the original model are called “exact” solutions here. The same is for frequencies 29 and 30 when choosing ; as is shown in Figure 3(b), there is hardly any error between the result of reduced-order model and the exact.
(a)
(b)
A more complicated example is shown in Figures 4(a) and 4(b), when taking the damping and nonlinearity into account. The influential set can also be determined as corresponding to frequencies 23 and 24, and the effect of the reduced-order model is shown in Figure 4(a) that when the reduced-order perturbation gets close to the original perturbation, the error of reduced-order model with respect to the exact is always less than 40 Hz, which is also a satisfactory result. The same is for frequencies 29 and 30 when choosing ; as is shown in Figure 4(b), there is hardly any error between the result of reduced-order model and the exact.
(a)
(b)
Obviously, this reduced-order model is so effective that it can be used to the critical coefficient of friction that means the onset of squeal as well as the squeal frequency with sufficient accuracy and largely reduced amount of calculation, thus giving us a practical guide to perform design optimization in order to reduce brake squeal.
5. Conclusions
Brake squeal is often analytically studied by a complex eigenvalue analysis of linearized models of the brake assembly that is usually quite large. In this paper, a time-saving method for determining those frequencies having the most effect on the pair of coupling frequencies is put forward and a reduced-order model is presented based on the complex modes theory.
The method for determining the influential set is efficient, which only needs to solve the eigenvalues and their derivatives for the unperturbed system and the reduced-order system. The reduced-order model gives good estimations of the exact result when the influential set is determined and is proved to be effective when applied to a flexible pin-on-disc system, even damping and nonlinearity are taken into consideration.
This reduce-order model can predict the onset of squeal as well as the squeal frequency with sufficient accuracy and largely reduced amount of calculation and gives us a practical guide to perform design optimization in order to reduce brake squeal.
Appendix
Consider
Nomenclature
| ,: | Inner and outer radius of the disc |
| , : | Kelvin-type damping coefficient of the pin in the transverse and axial direction |
| : | The diameter of cross-sectional area of the pin |
| : | Flexible rigidity of the disc |
| : | Kelvin-type damping coefficient of the disc |
| , : | Young’s modulus of the disc and pin material |
| : | The friction force acting on the pin |
| : | Preload between the pin and disc |
| : | The normal force acting on the pin |
| : | Thickness of the disc |
| : | Second moment of the cross-sectional area of the pin |
| : | Contact stiffness between the pin and disc |
| : | The length of the pin |
| : | Radial coordinate in the cylindrical coordinate system |
| : | Cross-sectional area of the pin |
| : | Time |
| : | Axial displacement of the pin |
| : | Transverse displacement of the pin |
| : | Out-of-plane displacement of the disc |
| : | Angle of inclination of the pin |
| : | Circumferential coordinate in the cylindrical coordinate system |
| : | Coefficient of friction |
| , : | Poisson’s ratio of the disc and pin |
| , : | Density of the disc and pin |
| : | Rotational speed of the disc. |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is sponsored by the National Natural Science Foundation of China (NSFC), Grant no. 51175380. The authors greatly appreciate the support provided by NSFC during this research.