A NEW ANALYSIS APPROACH FOR MOTORCYCLE BRAKE SQUEAL NOISE AND ITS ADAPTATION

Transcription

1 SETC A NEW ANALYSIS APPROACH FOR MOTORCYCLE BRAKE SQUEAL NOISE AND ITS ADAPTATION Hiroyuki Nakata, Kiyoshi Kobayashi, Masashi Kajita - Honda R&D Co., Ltd. - JAPAN C.H.Jerry CHUNG - MTS Systems Co. - USA ABSTRACT Concerning low frequency brake-squeal-noises of motorcycles, the noise caused by structural instability of the system was solved with the analysis of the mode-coupling mechanism by the complex eigenvalue analysis. An equation of motion of the brake system was simplified by converting from physical domains to modal domains. Required parameters for the stability analysis were obtained from the normal mode solution of system model. Moreover, the time for a repeat calculation in different friction coefficients and an instabilitycauses analysis in a mode were enabled to decrease 70% or more by using a numerical-analysis solver. An analysis model with high accuracy was established and was applied to the new approach, and to specify a mode affecting squeal noises and to study countermeasures were performed. From the test result for modified brake system, it was verified that the new method is effective for predictions and countermeasures of noises in the designing stage. 1. Introduction Brake noise will adversely affect customer satisfaction measurement. A number of man-hours are required for the solution of it, so that it presents an important technical requirement for the development of motorcycles. Squeal noise, among others, is one of the harshest sounds to the ear. It is supposed that there are two occurrence mechanisms of a squeal noise. The first mechanism is a phenomenon resulting from the stick-slip of a friction side. The second mechanism is a phenomenon resulting from geometric instabilities of the brake assembly. Regarding the squeal noise caused by geometric instability of system, mode-coupling analysis of mechanism using complex eigenvalues analysis has been utilized recently as an approach. However, In the complex eigenvalue methods, complex eigenvalues is only plotted to a complex plane, the identification of unstable modes is very difficult when the density of the mode is high. Furthermore, complex eigenvalues will be solved by NASTRAN is common. Since system model computation under one condition is only possible once at a time, much time is needed in order to obtain a root locus by calculation of a different coefficient of friction. Therefore, phenomenon analysis and design change analysis are difficult. Authors have developed a new modeling and numerical analytical method, and this made it possible to obtain a root locus in a short time. A new evaluation scale was developed to express the coupling strength of mode, which is coupled with instability (1). Generally, squeal sound is defined to be in the range from 1 khz to about 15 khz. The range is divided into low frequency domain (1 ~ 3 khz) and high frequency domain (5 khz and over). In this paper, the new analytical method concerning the geometric instability of the brake system was applied to the lowfrequency squeal noise (1 khz level) of motorcycle brakes to prove the appropriateness of the method. 1

2 .New analysis approach.1. Simplify the equation of motion of the system. The equation of system motion has been very much simplified by shifting the analysis of stability from physical to modal domains. The general equation of system motion can be expressed, in matrix form, as shown in the formula (1): [ M ]{} u& + [ C]{} u& + [ K ]{} u = { F f} (1) where, M, C, and K are the mass, viscous damping, and stiffness matrices respectively for the non-friction system, u is the displacement vector of the system, F f is the friction force applied to the system. A dot over the displacement denotes differentiation with respect to time. The stability evaluation model represents the contact surface between the disc and pads in the direction normal to the surface only. The friction force is given by the following equation. F f = µ N () N is the force between the pad and disc normal to the contact surface. µ is the coefficient of friction. Brake instabilities can result if either the coefficient of friction or the normal force vary. The theory of brake squeal stability used by us means that the coefficient of friction is constant and the normal force will vary as a function of the vibration of the pad and disc. The friction equation is then modified as follows. { F f} ({ Nstatic} + { Ndynamic} ) = µ (3) While solving for eigenvalue, the static term will be dropped from the dynamic equations. The dynamic normal force is caused by the vibration of the pad and disc. The contact points are connected with a simple spring Ks. Dynamic friction force modeled by the following equation. u u { N dynamic} Ks( { N. disc} { N. pad} ) = (4) where u N,pad denotes the displacement of pad in normal direction. The resulting friction force F f is applied to degree of freedom (DOF) u T,pad and - F f is applied to DOF u T,disc, where the subscribe T denotes the tangential direction. The normal force is determined by the relative displacement (rel) between node pairs as shown (5). u F T. rel = µ Ks N. rel (5) From ()-(5), the equations of motion (1) become [ M g ]{} v& + [ C g]{} v& + [ K g]{} v = µ [ K f ]{} v & (6) where M g, C g and K g are mass, damping and stiffness matrices, which includes the Generalized DOF. The vector of the DOF is then (7) {} v u = u u N. rel T. rel and the friction force term is (8) [ f ] [] 0 [] 0 [] 0 [] 0 [] 0 [] 0 [] [ ] [] 0 KT.N 0 (7) K = (8)

3 Laplace transform (6) to get system equation (9) For the stability study, we omit the viscous damping term [C g ] from the system equation of motion, the purpose which makes a equation simple. A final stability analysis defines a damping line. s [ M ]{} v + [ K]{} v = µ [ K f ]{} v where { v } = Laplace{} v transform (9) to modal domain by, (9) { } = [ ψ ]{} γ v (10) Where [ψ ] is the mode shapes, {γ} is the modal displacement. Substitute (10) and pre-multiply (9) by [ψ ] T to obtain Finally, it is obtained the system equation of motion in the modal domain is shown in formula(9) s ω 1 0 L 0 0 ω L µ γ M M O M 0 0 L ω n [ Λ f ] {} = {} 0 (11) where the ω i is the i -th natural frequency of the non-friction system. From the definition of friction term of (8), [ Λ f ] T [ f ] = [ ψ ] [ K f ][ ψ ] = Ks f ([ ψt. rel],[ ψn. rel] ) Λ (1) ψ T,rel is the mode shape of contact point in tangential direction. ψ N,rel is the mode shape of contact point in normal direction. All parameters of the equation of motion (11) can be obtained from the normal mode analysis of a non-friction system. The necessary parameters are natural frequencies and the mode shapes of contact points. All parameters represent real values and the results are all reliable. The complex eigenvalue in the equation (11) can then be computed in the numerical analysis solver such as Matlab of Math Works company. To evaluate the complex eigenvalues under different coefficients of friction, there is no need to change the model set up, the result can be obtained immediately. Therefore, large computation-time compaction of 70% or more of ratios is enabled conventionally... New Evaluation Scale of Instability Mode The magnitude of the real part offers the possibility of occurrence of a squeal noise by the complex eigenvalue approach, there is no information on what noise of magnitude it is. However, the accuracy of a model, connecting conditions, boundary conditions, and frequency shifts due to the analytic model by the tolerance of components, It sometimes becomes larger than spacing in two modes, and may significantly affect the prediction of the complex eigenvalues. As the approach of evaluating the stability of a noise, It is necessary to evaluate other stability criteria to determine if the system is potentially unstable. When the coupling strength is strong, the brake system has high possibility to squeal. Physical meaning of the coupling strength is how fast the eigenvalues will converge when the coefficients of friction increase from zero. Therefore, we define the New Evaluation Scale of Instability Mode. It expresses qualitatively the coupling strength of the coupling mode. The coupling strength between two unstable modes which converges and split when a coefficient of friction 3

4 increases can be expressed with formula 13 (1). [ Λ f ] [ Λ f ] li CS il = (13) il ( ω ω ) l i We use the complex eigenvalue and the coupling strength ( ) CS to evaluate the possibility for brake squeal. 3. Actual Cases of Analysis of Brake Squeal in Motorcycles 3.1. Construction of Brake System of Motorcycles The brake system for motorcycles as used in the present study consists of floating type calipers of twin pistons as shown in Fig.1. The calipers are fastened to the front fork by bolts through the support. The brake discs are fastened to the wheel by bolts. 3.. Grasping the Phenomenon of Squeal Noise The brake squeal noise was verified with a brake noise tester using an actual motorcycle. The characteristics of squeal noise were judged in a matrix mode simulating the conditions of users applying the brake with speed, extent of deceleration, temperature and applied pressure as parameters. The test results reveal that squeal noise is generated in a low deceleration zone of about 1. khz as shown in Fig.. Fig.1 Brake system of motorcycles Fig. Histogram of squeal noises 3.3. Preparation of Analysis Model and Correlation Component models of analysis FE models have seen made for the disc, calipers, friction pads, support and the wheel. In order to obtain a reliable brake-system model, the frequency-response function (FRF) of an impact excitation test and FEM needs to have high functionality for every component. Therefore, the model geometry and the physicalproperties value are adjusted and used for the exact value. The instrumentation location of a disc is shown in Fig.3. The correlation of an excitation test and an FEM result is shown in Fig.4. 4

5 Fig.3 FRF measurement points Fig.4 Correlation sample of disc The correlation was similarly examined as regards other component parts. As other examples, It is the instrumentation location of a caliper to Fig.5.The correlation of an excitation test and an FEM result is shown in Fig.6. Fig.5 FRF measurement points Fig.6 Correlation sample of Caliper 5

6 3.3.. Building up of Sub-system and Correlation The connecting conditions of each parts attached in the brake system defined the subsystem for every component, and determined the aptitude value by the correlation. (1) A modeling of the subsystem of a brake disc and a wheel The brake disc is fastened to the wheel by bolts as shown in Fig.7. The connecting conditions of the disc and wheel were adjusted, using the correlation of FRF in the frequency domain in target. The correlation of the subsystem of a wheel and a disc is shown in Fig.8. () A modeling of the subsystem of a caliper and a front fork In the actual vehicles, the calipers are fastened to the front fork by bolts through the support. The boundary condition between the support and the front fork, and the connecting condition between the caliper and the support was determined similarly. Fig.7 Sub-system of wheel and disc Fig.8 Correlation of sub-system of wheel and disc Building up of System Model and Correlation The brake system model combined each subsystem, and a piston, the slide section, the friction part, etc. set up and built the connecting condition. The connecting conditions for the FE model have been set up while checking correlation, measuring FRF in the state where brake squeal occurrence. A brake system model with the FRF characteristic of real motorcycle conditions was obtained by this. A system model is shown in Fig.9. Correlation is shown in Fig.10. There is correlative in the characteristic of an analytic model and a real motorcycle in the frequency domain which the brake squeal occurrence, and the validity of a connecting condition was verified. The difference among the conditions which the brake squeal occurrence can be changed by adjusting a connecting condition. 6

7 Fig.9 System model Fig.10 Correlation in system model 4. Results and Identified the Mode 4.1. Results of Analysis The parameters for the equation of motion in modal domain were figured out from the normal mode solution of the system model. The root locus plots (Fig.11 Left) and coupling strength (Fig.11 Center) were computed by putting them into the numerical value analysis solver together with different coefficients of friction. This result expresses well the squeal noise phenomenon near 1.kHz of a real motorcycle (Fig.11 Right). It is shown that this approach is effective. Fig.11 Correlation between root locus plot, stability metric and squeal noise 4.. Identification of Instability Mode The two coupling instability modes (10Hz and 160Hz) can be easily identified by the new stability metric( CS ). The mode shape of the two instability modes was observed. The two coupling pair mode shape is shown in Fig.1. The mode of 1,0Hz is '.5 diameter node' mode of the disc. The mode of 1,60Hz is the twist mode of wheel hub. The modes attributable to squeal could have been identified from this approach. 7

8 Fig.1 Mode shape, pair mode 5. Examination of Countermeasures Specifications 5.1. Way of Countermeasure Specifications The strong coupling mode of system converges and splits as coefficients of friction increases. It indicates that there are two methods to solve the problem of squeal noise. The first method is to separate instability mode, i.e., to put away eigenvalue of two coupling modes so as not to cause high and positive damping even if affected by the coefficient of friction. The second method is to reduce the coupling strength of instability mode. The examination of shape from these two viewpoints will present countermeasures against brake squeal noises. 5.. Examination of Countermeasure Specifications This paper considered the separation in the two coupling pair mode. It is effective to cope with it for that purpose paying attention to the high part of strain energy. The strain energy of coupling pair mode is shown in Fig.13. The rate of distorted strain energy contribution for every each part of coupling pair mode is shown in Table 1. The mode of 1,0Hz has high strain energy in the disc attachment part. The mode of 1,60Hz has high strain energy in the wheel hub. The change of the shape of disc attachment part and the change of eigenvalue of mode enabled to find the design specifications for disc, which will increase the separation of mode by 3.3%. Fig.13 Strain energy, pair modes 8

9 6. Evaluation of Design Change Specifications 6.1. Stability Evaluation The root locus plots and stability metric of the base specifications and the specifications of design change are shown in Fig.14. The change in design specifications reduced positive damping and coupling strength. The brake specification which squeal noises were not generated easily was able to be predicted. 6.. Evaluation of Prototype The predicted FRF and the FRF in the experiments are correlated, and it was verified that the characteristics of design change specifications and prototypes were corresponding, as shown in Fig.15. The design change as a specification was made. Fig.14 Stability analysis of base specifications and specifications of design 6.3. Verification of Brake Squeal by Actual Vehicles The squeal noise was checked in the brake-noise tester using the motorcycle. The characteristics of squeal noise were judged in a matrix mode simulating the conditions of users applying brake with speed, extent of deceleration, temperature and applied pressure as parameters. The brake squeal occurrence frequency of the base specification and the countermeasure specification is shown in Fig.16. Brake squeal ceased to exist in 9

10 the vicinity of 1. khz, which has been taken up as problematical area, to confirm the effect of the countermeasure. A left diagrammatic chart shows the occurrence frequency of a noise. The right shows the occurrence frequency in each condition. Table 1 Strain energy table Mode number 14 Mode number 15 Frequency 1 0(Hz) Frequency 1 60(Hz) Parts group Strain energy (%) Pad A Pad B Bolt Support Caliper Backplate A Backplate B Disc Spring Ks Beam Wheel Fig.15 Correlation between experiments and predicted values Fig.16 Frequency of occurrence of brake squeals (bar height indicates the squeal generation 10

11 7. Conclusion A method to solve the phenomenon of low frequency brake squeal noises of motorcycles easily and in a short period of time has been developed: (1) The equation was simplified very much by changing the complicated equation of motion for the system from a physical domain to a modal domain. An approach of calculating a complex eigenvalue analysis based on the normal mode analysis for a short time was taken. () A new evaluation scale of Instability mode was defined by the strength of coupling. (3) A setting approach of the connecting condition which expresses the squeal noise occurrence state of a brake system was built. (4) It could be effectively coped with by improving the configuration of the high rate of strain energy contribution in squeal noise occurrence mode. As mentioned above, the specification and the cure approach of a mode which affects the noise become clear. It was confirmed that this approach is effective to predict the squeal noise in the design stage. References A New Analysis Method For Brake Squeal Part :Theory For Model Domain Formulation And Stability Analysis, SAE 01 nvc