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Complex eigenvalue analysis (CEA) has been widely used to predict unstable frequencies in brake systems models. The finite element model is correlated with experimental modal test. The input-output relationship between the brake squeal and the brake pad geometry is constructed for possible prediction of the squeal using various geometrical configurations of the disc brake. Influences of the various factors namely; Youngs modulus of back plate, back plate thickness, chamfer, distance between two slots, slot width and angle of slot are investigated using design of experiments (DOE) technique. A mathematical prediction model has been developed based on the most influencing factors and the validation simulation experiments proved its adequacy. The predicted results show that brake squeal propensity can be reduced by increasing Youngs modulus of the back plate and modifying the shape of friction material by adding chamfer on both sides of friction material and by introducing slot configurations. The combined approach of modeling brake squeal using CEA and DOE is found to be statistically adequate through verification trials. This combined approach will be useful in the design stage of the disc brake. Keywords: Disc brake squeal, finite element analysis, experimental modal analysis, design of experiments 1. Introduction Brake squeal is a noise problem caused by vibrations induced by friction forces that can induce a dynamic instability (Akay, 2002). During the braking operation, the friction between the pad and the disc can induce a dynamic instability in the system. Usually brake squeal occurs in the frequency range between 1 and 20 kHz. Squeal is a complex phenomenon, partly because of its strong dependence on many parameters and, partly, because of the mechanical interactions in the brake system. The mechanical interactions are considered to be very complicated because of nonlinear contact effects at the friction interface. The occurrence of squeal is intermittent or even random. Under certain conditions, even when the vehicle is brand new, it often generates squeal noise, which has been extensively studied with the goal of eliminating the noise. However, mechanistic details of squeal noise are not yet fully understood (Joe et al., 2008). Several theories have been formulated to explain the mechanisms of brake squeal, and numerous studies have been made with varied success to apply them to the dynamics of disc brakes (Kinkaid et al., 2003). The reason for the onset of instability has been attributed to different reasons. Some of the major reasons are the change of the friction characteristic with the speed of the contact points (Ibrahim, 1994; Ouyang et al., 1998; Shin et al., 2002) the change of the relative orientation of the disk and the friction pads leading to a modification of the friction force (Millner, 1978), and a flutter instability which is found even with a constant friction coefficient (Chowdhary et al., 2001; Chakrabotry et al., 2002; Von Wagner et al., 2003; Von Wagner et al., 2004). In fact, recent literature reviews (Kinkaid et al., 2003; Papinniemi et al., 2002) have reported on the complexity and lack of understanding of the brake squeal problem. Though much work was done on the issue of squeal, it requires continuous study and investigation to refine the prediction accuracy of finite element models of brake assemblies to give brake design engineers appropriate tools to design quiet brakes. There are two main categories of numerical methods that are used to study this problem: (1) transient dynamic analysis (Hu et al., Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 254-271 255 1999; AbuBakar et al., 2006) and (2) complex eigenvalue analysis. Currently, the complex eigenvalue method is preferred and widely used (Liles, 1989; Lee et al., 1998; Blaschke et al., 2000; Bajer et al., 2003; AbuBakar et al., 2006; Liu et al., 2007; Mario et al., 2008; Dai et al., 2008) in predicting the squeal propensity of the brake system including damping and contact due to the quickness with which it can be analysed and its usefulness in providing design guidance by analysing with different operating parameters virtually. Many researchers in their studies on the dynamics of brake system tried to reduce squeal by changing the factors associated with the brake squeal. For example (Liles, 1989) found that shorter pads, damping, softer disc and stiffer back plate could reduce squeal whilst in contrast, higher friction coefficient and wear of the friction material were prone to squeal. (Lee et al., 1998) reported that reducing back plate thickness led to less uniform of contact pressure distributions and consequently increasing the squeal propensity. (Hu et al., 1999) based on the DOE analysis found that the optimal design was the one that used the original finger length, the vertical slot, the chamfer pad, the 28mm thickness of disc, and the 10mm thickness of friction material. (Brooks et al., 1993) found that by shifting the pistons away from the leading edge of the pads the system could destabilise. They also reported that the predicted unstable system was due to the coupling of translational and rotational modes of the disc particularly at high values of pad stiffness. From the sensitivity studies, they suggested that the effective half length of the pad, the piston masses, the effective mass, inertia and grounding stiffness of the disc and the second circuit actuation stiffness also have potential on the disc brake instability. (Shin et al., 2002) have shown that the damping of the pad and the disc were important in reducing instability. Their analysis also has shown and confirmed that increasing damping of either the disc or the pad alone could potentially destabilise the system. (Liu et al., 2007) found that the squeal can be reduced by decreasing the friction coefficient, increasing the stiffness of the disc, using damping material on the back of the pads and modifying the shape of the brake pads. (Dai et al., 2008) have shown that the design of the pads with a radial chamfer possesses the least number of unstable modes, which implies lesser tendency towards squeal. In the present study, an investigation of disc brake squeal is done by performing complex eigenvalue analysis using finite element software ABAQUS /standard. A positive real part of a complex eigenvalue is being seen as an indication of instability. Though the FE simulations can provide guidance, it will rather be trial-and-error approach to arrive at an optimal configuration and also one may need to run many number of computationally intensive analyses to formulate the input-out relationships for possible prediction. Hence, in the present investigation, a novel approach is proposed by integrating the complex eigenvalue FE analyses with structured DOE. The proposed approach is aimed towards prediction of optimal pad design through the various factors of the brake pad geometrical construction. The paper is organised as follows, it presents a detailed literature survey in this field in the recent period. From the literature survey the main objectives were formed. Methodology to develop FE model of the disc brake is presented and it was subsequently validated using experimental modal analysis. The CEA approach is presented in order to predict brake squeal. Then DOE approach for soft computing is presented. Also a methodology to test the adequacy of the developed statistical model is discussed. 2. Finite element model and component correlation A disc brake system consists of a disc that rotates about the axis of a wheel, a caliperpiston assembly where the piston slides inside the caliper that is mounted to the vehicle suspension system, and a pair of brake pads. When hydraulic pressure is applied, the piston is pushed forward to press the inner pad against the disc and simultaneously the outer pad is pressed by the caliper against the disc. Figure 1 (a) shows the finite element model of the car front brake under consideration, built using the ABAQUS finite element software package. The brake model used in this study is a simplified model consisting of the two main components contributing to squeal: the disc and the pad (Figure 1(b). (a) (b) Figure 1. Finite element models of a (a) realistic (b) simplified disc brake model Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 254-271 256 A simplified model was used in this study for the following reasons: 1. For brake squeal analysis, the most important source of nonlinearity is the frictional sliding contact between the disc and the pads. 2. The simulation includes geometry simplifications to reduce CPU time, allowing far more configurations to be computed. The disc is made of cast iron. The pair of brake pads, which consist of friction material and back plates, are pressed against the disc in order to generate a friction torque to slow the disc rotation. The friction material is made of an organic friction material and the back plates are made of steel. The FE mesh is generated using 19,000 solid elements. The friction contact interactions are defined between both sides of the disc and the friction material of the pads. A constant friction coefficient and a constant angular velocity of the disc are used for simulation purposes. Figure 2 presents the constraints and loadings for the pads and disc assembly. The disc is completely fixed at the four counter-bolt holes and the ears of the pads are constrained to allow only axial movements. The caliperpiston assembly is not defined in the simplified model of the disc brake system, hence the hydraulic pressure is directly applied to the back plates at the contact regions between the inner pad and the piston and between the outer pad and the caliper, and it is assumed that an equal magnitude of force acts on each pad. Figure 2. Constraints and loading of the simplified brake system For the purpose of validation, the main brake components, Frequency Response Functions (FRFs) were measured at free-free boundary conditions by exciting each component with a small impact hammer with sensitivity of 10mV/N and a hard tip. The acceleration response was measured with a light small accelerometer with sensitivity of 10mV/g through Dynamic Signal Analyzer type DEWE-41-T-DSA. FRF measurements were recorded for each component using SISO configurations. Then, the FRFs were processed using DEWE FRF software in order to identify the modal parameters, namely; resonance frequencies, modal shapes and damping values. Figure 3 shows the experimental modal test components. Figure 3. Experimental Modal Analysis Components The frequencies measured on the disc and calculated by the simulated model for modes with free-free boundary conditions are shown in table 1. It can be observed that the measured and simulated frequencies are in good agreement. Figure 4 shows the mode shapes of rotor with nodal diameters. In a similar way, the parameters for the pads are estimated based on the measured data indicated in Table 2. The measured and simulated frequencies are in good agreement. Figure 5 shows the mode shapes of the pad. Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 254-271 257 Table 1. Modal results of the rotor at free-free boundary conditions Mode shape Experimental Frequency (Hz) FEA Frequency(Hz) Differences (%) 2 nd bending 1220 1303 6.8 3 rd bending 2551 2636 3.3 4 th bending 4003 4108 2.6 5 th bending 5774 5591 -3.1 6 th bending 7873 7790 -1 7 th bending 9008 9209 2.2 Table 2. Modal results of the pad at free-free boundary conditions Mode shape Experimental Frequency (Hz) FEA Frequency(Hz) Differences (%) 1 st bending 3051 3231 5.8 2 nd bending 8459 8381 -1 2 nd Nodal Diameter Mode (1303Hz) 3 rd Nodal Diameter Mode (2636 Hz) 4 th Nodal Diameter Mode (4108 Hz) 5 th Nodal Diameter Mode (5591 Hz) Figure 4. Mode shapes of the rotor at free-free boundary conditions Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 254-271 258 6 th Nodal Diameter Mode (7790 Hz) 7 th Nodal Diameter Mode (9209 Hz) Figure 4 (contd). Mode shapes of the rotor at free-free boundary conditions 1 st Bending Mode (3231 Hz) 2 nd Bending Mode (8381 Hz) Figure 5. Mode shapes of the pad at free-free boundary conditions. 3. Complex eigenvalue analysis The complex eigenvalue analysis (CEA) has been widely used by researchers. It deals with computation of system eigenvalues which are prove to be complex valuated functions in the general case because friction causes the stiffness matrix to be asymmetric. The real and imaginary parts of the complex eigenvalues are, respectively, responsible for the stability and for the frequency of the corresponding modes. This method was first used on lumped models (Kinkaid et al., 2003; Ibrahim, 1994). Then, improvements in computer systems have made it possible to perform analyses on finite element (FE) models (Liles, 1989; Lee et al., 1998; Blaschke et al., 2000; Bajer et al., 2003; AbuBakar et al., 2006; Liu et al., 2007; Mario et al., 2008; Dai et al., 2008). In order to perform the complex eigenvalue analysis using ABAQUS (Bajer et al, 2003), four main steps are required as follows: (1) nonlinear static analysis for the application of brake pressure; (2) nonlinear static analysis to impose a rotational velocity on the disc; (3) normal mode analysis to extract the natural frequency to find the projection subspace; and (4) complex eigenvalue analysis to incorporate the effect of friction coupling. The governing equation of the system is: 0=+ KuuCuM (1) Where M, C and K are respectively the mass, damping and stiffness matrices, and u is the displacement vector. Because of friction, the stiffness matrix has specific properties: FS KKK += (2) Where S K is the structural stiffness matrix, F K the asymmetrical friction induced stiffness matrix and the friction coefficient. The governing equation can be rewritten as: 0)( 2 =+ KCM (3) Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 254-271 259 Where, is the eigenvalue and is the corresponding eigenvector. Both eigenvalues and eigenvectors may be complex. In order to solve the complex eigenproblem, this system is symmetrized by ignoring the damping matrix C and the asymmetric contributions to the stiffness matrix K. Then this symmetric eigenvalue problem is solved to find the projection subspace. The N eigenvectors obtained from the symmetric eigenvalue problem are expressed in a matrix as ,., 21 N . Next, the original matrices are projected onto the subspace of N eigenvectors ,.,., 2121 * N T N MM = (4a) ,.,., 2121 * N T N CC = (4b) and ,.,., 2121 * N T N KK = (4c) Then the projected complex eigenproblem becomes 0)( *2 =+ KCM (5) Finally, the complex eigenvectors of the original system can be obtained by k N k * 21 ,., = (6) Detailed description of the formulation and the algorithm are presented by Sinou et al. (2003). The eigenvalues and the eigenvectors of Eq. (3) may be complex, consisting of both a real and imaginary part. For under damped systems the eigenvalues always occur in complex conjugate pairs. For a particular mode the eigenvalue pair is iii i = 2,1 (7) Where, i and i are the damping coefficient (the real part) and damped natural frequency (the imaginary part) describing damped sinusoidal motion. The motion for each mode can be described in terms of the complex conjugate eigenvalue and eigenvector. A positive damping coefficient causes the amplitude of oscillations to increase with time. Therefore the system is not stable when the damping coefficient is positive. By examining the real part of the system eigenvalues the modes that are unstable and likely to produce squeal are revealed. An extra term, damping ratio, is defined as /2 . If the damping ratio is negative, the system becomes unstable, and vice versa. 3.1. Complex eigenvalue analysis (CEA) results Since friction is the main cause of instability, which causes the stiffness matrix in Eq. (3) to be asymmetric, complex eigenvalue analysis has been undertaken to assess the brake stability as the friction coefficient values. It was observed that high values for this parameter tend to facilitate two modes merging to form an unstable complex mode. In addition, an increase in the friction coefficient leads to an increase in the unstable frequency. Figure 6 shows the results of a complex eigenvalue analysis with variation of the friction coefficient () between 0.2 and 0.6. As predicted in the complex eigenvalue analysis, as the friction coefficient further increases, real parts of eigenvalues, the values that can be used to gauge the degree of instability of a complex mode, increase further, as well, and more unstable modes may emerge. This is because the higher coefficient of friction causes the variable frictional forces to be higher resulting in the tendency to excite greater number of unstable modes. In the past, a friction coefficient of 0.35 was typical. However, brake compounds today possess coefficient of friction that is 0.45 or higher, which increases the likelihood of squeal. This poses a greater challenge for brake designer to develop a quiet brake system. In an earlier work (Nouby et al., 2009), an attempt was made using parametric study to reduce squeal at 12 kHz. Based on the earlier study, a decision was taken to understand the effects of influencing variables of squeal at 6.2 kHz using DOE. Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 254-271 260 Figure 6. Results of CEA with variation of , and unstable mode at interested Frequency (6.2 kHz) 4. Methodology Human made products or processes can be treated like a system, if it produces a set of responses for a given set of inputs. Disc brake system can also be treated like a system as shown in Figure 7. Some systems like disc brake system produce unwanted outputs namely squeal for a set of inputs parameters. The present study was aimed at establishing the input-output relationships for prediction of brake squeal. Disc brake system has numerous variables. In order to arrive at the most influential variables and its effects a two phase strategies were proposed. In the first phase, initial screening with various variables was taken up. Fractional factorial design (FFD) of experiments was conducted to identify the most influential variables. Subsequently, in the second phase, central composite design (CCD) based Response surface methodology (RSM) was deployed to develop a non-linear model for prediction of disk brake squeal. Figure 7. Disc brake squeal system 4.1 First phase: screening FFD of experiments If any experiment involves the study of the effects of two or more factors, then factorial designs are more efficient than one- factor-at-a-time experiments. Furthermore, a factorial design is necessary when interactions may be present to avoid misleading Nouby et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 20