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1、400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A.Tel: (724) 776-4841 Fax: (724) 776-5760 SAE TECHNICAL PAPER SERIES 1999-01-0144 Study of a Method for Reducing Drum Brake Squeal T. Hamabe, I. Yamazaki, K. Yamada, H. Matsui, S. Nakagawa and M. Kawamura Nissan Motor Co., Ltd. Reprinted From: Br
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9、bility Database 1 1999-01-0144 Study of a Method for Reducing Drum Brake Squeal T. Hamabe, I. Yamazaki, K. Yamada, H. Matsui, S. Nakagawa and M. Kawamura Nissan Motor Co., Ltd. Copyright 1999 Society of Automotive Engineers, Inc. ABSTRACT Since the modal density of a drum brake system is higher than
10、 that of a disc brake system, it is very difficult to iden- tify the cause of brake squeal. The causes of squeal were examined by both complex eigenvalue analysis and experimental analysis. It was confirmed that a complex eigenvalue analysis of a finite element model, a tech- nique so far generally
11、applied to disc brake squeal stud- ies, was effective in analyzing squeal problems of drum brakes. INTRODUCTION Brake squeal is an unpleasant sound that detracts from the passenger comfort of vehicles. There are thought to be two general causes of brake squeal. One cause is attributed to self-excite
12、d vibration that is induced when the friction material has a negative slope in relation to the relative velocity 1. The other cause is ascribed to self- excited vibration due to the coupling of two eigenvalues that are present in parts experiencing friction 2. It is dif- ficult to conceive that brak
13、e squeal, which occurs at high frequencies of several kHz, would be caused by either one of these factors operating alone. Rather, it is pre- sumed that brake squeal is caused by the compound effect of both factors. One means used to reduce the former type of squeal is to improve the friction charac
14、ter- istics of the friction material. This generally involves reducing the velocity dependence of the friction coeffi- cient of the friction material as much as possible. The lat- ter type of squeal has been reduced by optimizing the vibration characteristics of the brake assembly that cause self-ex
15、cited vibration 3. This was accomplished by con- ducting a complex eigenvalue analysis of a brake assem- bly model in which the friction coefficient was set to a certain constant value. In the case of a drum brake system, the eigenvalue den- sity that expresses the vibration characteristics is vastl
16、y higher than that of a disc brake system, making it difficult to optimize the vibration characteristics. Most of the studies reported in the literature have attempted to eluci- date the vibration characteristics of a drum brake system by using the finite element method to conduct a real eigenvalue
17、analysis in an effort to reproduce the complex vibration characteristics as faithfully as possible 4. In this study, a complex eigenvalue analysis was con- ducted with a finite element model of a drum brake sys- tem that produced two types of squeal in squeal bench tests. The results revealed the vi
18、bration characteristics that caused the squeal sounds. Moreover, experimental measurements of the vibration modes when squeal occurred confirmed that complex eigenvalue analysis can also be applied to drum brakes as an effective design tool for reducing squeal. FUNDAMENTAL STUDY OF SQUEAL MECHANISM
19、Drum brake squeal is caused by the friction force gener- ated between the rotating brake drum and the stationary friction material. The mechanism causing squeal can be considered with a simple two-degree-of-freedom model like that shown in Figure 1. Figure 1. 2 DOF model with friction force. In this
20、 model, a mass, , is connected to a spring having two spring constants, and , which are orthogonal to each other . Friction force acts on the mass in the direction indicated by the arrow. The equation of motion of this model can be written as (1) k1 N k2 N m V y x m k1k2 m m x y kk kk x y ky0 00 111
21、2 2122 22 + = as the friction coefficient is increased, the two frequencies con- verge and couple to form one frequency. The fact that the real parts of the eigenvalues become positive at that point indicates the occurrence of self-excited vibration. It is also seen that under a condition of no fric
22、tion (), self-excited vibration tends to occur even at small friction coefficients, the closer the two eigenvalues are to one another. These results confirm that the separation of the two eigenvalues that couple is an important factor in designing a brake assembly with little propensity to squeal. F
23、igure 2. Eigenvalues as a friction of the friction coefficient. kkxky kkxky kkkxky 11 22 22 22 1221 =+ =+ = cossin sincos ()sin cos k1 =( &) Vx =( &) ( )( )&VxVV x ( )( ) m m x y V kyx y kkV k kk x y 0 0 0 00 0 0 22111222 2122 + + = ? ? ? ? &xy T ( )=v0 xy T ( )=v0 ( ) m m x y kkV k kk x y 0 0 0 0 1
24、11222 2122 + = & & x y x y est = 0 sbsc 42 0+= b m kk=+ 1 1122 () c m k kkkk= 1 2 1122121222 () bc 2 40 s s =/ 4 kk 21 = 0 = 0 1600 1800 2000 Frequency (Hz) 0 400 800 00.10.20.30.4 Friction coefficient (rad/sec) ()=/ 4 Real part of eigenvalue kk 21 0 7= . kk 21 05= . kk 21 0 9=. kk 21 0 7= . kk 21 0
25、5= . kk 21 09= . 3 FREQUENCY IDENTIFICATION IN SQUEAL TESTS Squeal tests were conducted in which braking was per- formed under conditions that simulated various real-world braking environments. It was found that two types of squeal occurred at different frequencies under the differ- ent driving cond
26、itions used. One squeal occurred at a frequency of 4.5 kHz and the other squeal occurred at a frequency of 3 kHz. FACTOR ANALYSIS OF 4.5 KHZ SQUEAL The brake drum, which has the largest surface area among the parts making up a drum brake system, was presumed to be a principal sound source for conver
27、ting self-excited vibration to squeal noise. Accordingly, the vibration characteristics that occurred when the brake drum alone was subjected to impact excitation were examined, focusing on the frequencies around 4.5 kHz where squeal had been recorded. As seen in Figure 3, it was found that two reso
28、nance points occurred in close proximity. Based on the results of the fundamental study explained earlier, it was hypothesized that the coupling of these two resonance points gave rise to squeal at 4.5 kHz. Figure 3. Frequency response function of the brake drum. An experimental modal analysis and a
29、n eigenvalue anal- ysis of a finite element model were then conducted to identify the mode shapes of these two resonance points. It was confirmed that one of the peaks was the 6 nodal diameter (ND) mode of drum vibration (Figure 4 (a) induced only by radial vibration of the friction surface. The oth
30、er resonance point was the 2 ND mode of drum vibration (Figure 4 (b) induced by radial vibration of the friction surface accompanied by vibration of the drum side wall. Figure 4. Mode shapes of drum without shoes. APPLICATION OF COMPLEX EIGENVALUE ANALYSIS A brake assembly model was created by addin
31、g a finite element model of the brake shoes to the finite element model of the brake drum and by applying friction force at the friction interface between the two components. This assembly model was then used in conducting a complex eigenvalue analysis. As the calcu- lated results in Figure 5 indica
32、te, the real part of the eigenvalue became positive at 4491 Hz near the squeal frequency. In other words, an unstable eigenvalue that became self-excited vibration was calculated. The com- plex eigenvector of this unstable eigenvalue is shown in Figure 6. Figure 5. Complex eigenvalues of drum and sh
33、oe model. Figure 6. Complex eigenvector at 4491 Hz. The modal participation factor (MPF) 5 was then calcu- lated in order to examine this unstable eigenvalue more closely. Calculating the MPF makes it possible to sepa- rate a coupled complex eigenvalue into its several con- stituent eigenvalues befo
34、re the coupling took place. The results of the MPF calculation are presented in Figure 7. It was found that this unstable eigenvalue at 4491 Hz consisted of a 4485 Hz eigenvalue (55%), a 4488 Hz eigenvalue (37%) and other eigenvalues that accounted for the remaining 8%. Figure 8 shows the calculated
35、 modal assurance criteria (MAC) values for these constitu- ent eigenvalues versus the eigenvalues of the brake drum alone. The results indicate that the principal con- tributors to both of these eigenvalues of the brake assem- bly were the 6 ND mode of drum vibration involving radial vibration of th
36、e friction surface and the 2 ND mode of drum vibration involving radial vibration of the friction 10dB 4200430044004500460047004800 Frequency (Hz) Inertance (dB) 6 ND mode 2 ND mode with side wall vibration. (a) 6 ND mode (b) 2 ND mode with side wall vibration 4200 4300 4400 4500 4600 4700 4800 4900
37、 5000 0100200 Real part of eigenvalue (rad/sec) Frequency (Hz) Squeal frequency 4 surface accompanied by vibration of the drum side wall. These analysis results suggested that it was highly likely that the 4.5kHz squeal was caused by the coupling of these two eigenvalues. EXPERIMENTAL VALIDATION OF
38、HYPOTHESIS In order to validate the foregoing hypothesis experimentally, the vibration modes that were present when squeal actu- ally occurred were measured. Accelerometers were positioned at equidistant intervals at 16 locations on the periphery of the friction surface of the brake drum so that rad
39、ial vibrations could be measured. Using the test setup employed in the squeal bench tests, the conditions gen- erating squeal were reproduced and measurements were made at all 16 locations simultaneously. The results measured when squeal occurred at 4.5 kHz were then fil- tered to obtain clear data
40、for the 4.5 kHz frequency alone. Figure 9 illustrates the vibration mode extracted from the filtered data at the moment the greatest acceleration was recorded. The brake drum at that point appeared to be undergoing deformation characterized by the mixed pres- ence of the 6 ND mode, in which six node
41、s occurred in the radial direction, and the 2 ND mode. Figure 7. Modal participation factor of the 4491 Hz unstable eigenvalue. Figure 8. MAC values of 4485 Hz and 4488 Hz eigenvalues of the assembly model vs. eigevalues of the drum model. Figure 9. Experimental vibration mode of the drum during 4.5
42、 kHz squeal. The accelerometer output signals sampled simulta- neously at the 16 locations at certain specified intervals were arranged in a series and a Fourier transform opera- tion, like that indicated in Eq. (7), was performed on the 16 signals. Figure 10 shows the number of nodes that passed th
43、rough the center of revolution of the brake drum vibration modes. These results also make it clear that the 6 ND and 2 ND modes of drum vibration were under- going large deformation, thereby validating the above- mentioned hypothesis. (7) where is the output of the i-th accelerometer and is the (k +
44、 1) ND mode. Figure 10.Acceleration of each vibration mode during squeal. EXAMINATION OF A MEASURE AGAINST SQUEAL With the aim of reducing the 4.5 kHz squeal, an attempt was made to separate the eigenvalues of the 6 ND and 2 ND modes of drum. First of all, an investigation was made of how the resona
45、nce points of the brake drum change during actual braking. Figure 11 compares the frequency response characteristics of the brake drum for Unstable eigenvalue 4491 Hz 4485 Hz 55% Others 8% 37% 4488 Hz 4485 Hz 6 ND 2 ND Others 4488 Hz 2 ND 6 ND Others EigenvalueEigenvalue Actual mode Measured point (
46、)1510 16 1 15 0 16 2 , ,kexC i kij ik L= = xiCk 0 20 40 60 80 01234 Time (msec) Acceleration (m/sec2) ?6 ND mode ?2 ND mode Others 5 two conditions, with and without the application of brake fluid pressure. It is seen that the 6 ND mode of drum vibration involving only radial vibration of the fricti
47、on sur- face shifted to a higher frequency under the application of brake fluid pressure, whereas the 2 ND mode involving radial vibration of the friction surface accompanied by side wall vibration remained unchanged. Since the fre- quency of the former resonance point increased during braking, it w
48、as reasoned that moving it to the high fre- quency side farther away from the resonance point that did not increase in frequency would produce a greater separation of the two resonance points as a result of braking. That would therefore be an effective means of preventing the coupling which causes brake squeal. Squeal tests were conducted again using an actual brake drum whose stiffness had been changed in places so as to raise only the frequency of the 6 ND mode, making it even higher than that of the 2 ND mode. As a result, since the frequency of the 6 ND mode rose