Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 103809, 11 pages

http://dx.doi.org/10.1155/2015/103809

## A Reduced-Order Model for Complex Modes of Brake Squeal Model and Its Application to a Flexible Pin-on-Disc System

^{1}School of Automotive Engineering, Tongji University, Shanghai 201804, China^{2}Collaborative Innovation Center for Intelligent New Energy Vehicle, Tongji University, Shanghai 201804, China

Received 16 March 2015; Accepted 13 May 2015

Academic Editor: Roman Lewandowski

Copyright © 2015 Lijun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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].