Vibro-Acoustics of a Brake Rotor with Focus on Squeal Noise. Abstract

Transcription

1 Te International Congress and Exposition on Noise Control Engineering Dearbor MI, USA. August 9-, Vibro-Acoustics of a Brake Rotor wit Focus on Sueal Noise H. Lee and R. Sing Acoustics and Dynamics Laboratory, Te Oio State University, Columbus, Oio 43-7, USA Abstract Brake sueal as been a major noise problem in many ground veicles and yet many issues remain poorly understood. One unresolved problem is te sound radiation mecanism. Tis article investigates tis problem via a generic rotor example and determines te contributions of out-of-plane (flexural) and in-plane (radial) modes of te rotor to total sound radiation given armonic excitations from to 8 khz. A new semi-analytical procedure is applied to solve tis problem. In te proposed procedure, te disk surface velocities at selected modes are first defined using a finite element model and ten acoustic radiation modes are obtained from te velocities via new analytical solutions tat employ modified circular and cylindrical radiator models. Sound radiation due to multi-modal excitations is calculated using te modal expansion tecniue. In additio acoustic power and radiation efficiency spectra corresponding to a specific force excitation are obtained from far-field sound pressure data. Analytical predictions are confirmed by comparisons wit numerical and/or experimental investigations. Introduction Te veicle brake sueal problem as been investigated by using te complex mode metod as te unstable elastic modes of deformation are likely to generate sueal noise from to 6 khz [-]. One of te ypoteses is tat te dynamic coupling between in-plane and out-ofplane modes of brake rotor leads to an instability tat produces intense sound at several ig freuencies. Most of te prior studies ave focused on te structural dynamics of brake rotors and related components and to te best of our knowledge, no one as adeuately examined te acoustic radiation mecanism. In tis article, sound radiation from a brake rotor, te main sueal radiator, is investigated using a tick annular disk example. A new semi-analytical approac, based on numerically obtained disk surface velocities and analytical modal sound radiation solutions, will be introduced. Te validity of tis procedure as been confirmed by computational analyses and vibro-acoustic experiments. Figure describes te geometry and material properties of te example disk. For a complete investigation of te vibro-acoustic caracteristics of a tick annular disk, we simultaneously consider in-plane and out-of-plane vibrations.

2 f r b a ϕ z f z Dimensions and Material roperties Outer radius (a) 5.5 mm Inner radius (b) 87.5 mm Tickness () 3.5 mm Density (ρ d ) Kg/m 3 Young s Modulus (E) 8 Ga oisson s Ratio (ν).35 Figure : Dimensions and material properties of a tick annular disk. Te scope of tis study is strictly limited to te freuency domain analysis of a linear timeinvariant (LTI) system wit free-free boundaries. Complicating effects suc as fluid loading and scattering at te disk edges are not considered. rimary assumptions are as follows: () For out-of-plane modes, structural velocities in te normal direction (z) vary sinusoidally in te ϕ direction. () For te radial modes, velocities on te two radial surfaces are uniform in te z direction but vary sinusoidally in te ϕ direction. Cief objectives of tis article are: () Introduce new sound radiation solutions for te normal modes of a tick annular disk. () Develop a semi-analytical procedure for calculating sound radiation from te disk given multi-modal armonic excitations. Te new metod sould allow us to examine te couplings between in-plane and out-of-plane modes as well as couplings witin te same type of modes as tey ave considerable effects on te total sound radiation.. Structural Analysis for a Tick Annular Disk As te first step, bot out-of-plane and in-plane modal vibrations of a tick annular disk are investigated using a finite element model (FEM) tat consists of 4,4 solid elements and 6,6 nodes [3]. Numerical results are cecked wit analytical solution based on te tick plate teory for out-of-plane modes [4-5] and te transfer matrix metod for radial modes [6-7] along wit experimental modal analyses. Te results are summarized in Table were λ mn and λ are dimensionless eigenvalues for te ( n) t out-of-plane and te t in-plane modes. Finite element based predictions matc uite well wit analytical and experimental results and tese can be used for forced vibration response and acoustic radiation calculations. Table : Comparison of natural freuencies for a tick disk wit free-free boundaries ω mn or ω (Hz) Mode Mode Type Indices Computed Measured Analytical (FEM) Out-of-lane ( n) ω mn = λ mn /a (ρ d /D b ) / In-lane () ω = λ /(ρ d a /D) /

3 3. Modal Acoustic Radiation Solutions Sound radiation from te out-of-plane vibrations of circular and annular plates as been examined using te Rayleig integral metod, te impedance approac, and te -D Fourier transform tecniue [8-]. Recently, we developed an analytical solution for sound radiation from te out-of-plane modes of a tick annular disk considering te tickness effect and expressed far-field sound pressure in terms of modal vibrations as follows [] (See Figure ). Here, m and n are te numbers of nodal circles and diameters, ρ is te mass density of air, c is te speed of sound, k mn is te acoustic wave number, W mn is te velocity amplitude at r s, and J n is te Bessel function of order n. ik mnr ρck e ikmn cosθ mn + θ φ = n, ) e cosnφ( i) W ( r) J ( k sin θr ) mn mn n mn rdr () R S v R W mn z r p -r s r p mn (R, S v R z θ r p (R, r s S s y Figure : Radiation from an out-of-plane mode in te sperical coordinate system. x ϕ a, U Figure 3: Radiation from an in-plane mode in te sperical coordinate system. Also, we ave developed expressions for far-field sound radiation from radial modes using a cylindrical radiator model [7]. Sound pressures are generated by two radial surfaces of te disk. Te total sound pressure is ten expressed as a summation of O from te outer radial surface and I from te inner radial surface (See Figure 3). Here, is te radial mode index, k is te corresponding acoustic wave number, H is te Hankel function of order, and a is te acceleration amplitude of te on te radial edges due to te t radial mode. Also, te Sinc function is defined as Sinc(ξ) = sin(ξ) / ξ. = O I I + ρe = a πk R sin θ ρe = a πk R sin θ ik R ik R O Sinc( k sin θ / )( i) H ( k a sin θ) Sinc( k sin θ / )( i) H ( k bsin θ) + + cos φ cos φ (a - c) Modal acoustic radiation solutions ave been confirmed by numerical and experimental investigations wit te sample disk in terms of modal acoustic power (Π), modal radiation efficiency (σ), and directivity patterns. Te computational boundary element (BEM) model is constructed using 6,46 acoustic field points and 6,44 elements tat are defined on te spere surrounding te disk tat is represented by te finite element model for structural dynamics []. Te center of tis spere coincides wit te disk center. Typical results are

4 summarized in Table and Figure 4. Analytical metods produce sufficiently accurate modal radiation solutions and tus tese are used for furter calculations. Table : Comparison of modal acoustic power and radiation efficiency for selected modes Mode Measured Computed (BEM) Analytical In-lane = Π (db re pw) σ Mode Type Measured Computed Analytical Measured Computed Analytical (BEM) (BEM) In lane Out-of-, lane, Out-of- lane Figure 4: Directivity patterns for selected structural modes. 4. Effect of Multi-Modal Excitations on Sound Radiation If a disk is excited by a armonic force vector consisting of one or more freuencies, several in-plane and out-of-plane modes are simultaneously excited. Based on te modal expansion tecniue, velocity distribution (v) on te normal and radial surfaces can be expressed in terms of te normalized elastic modes of tis disk. Also, te far-field sound pressure is assumed to be determined by te modal sound radiation solutions of Section 3. T T v = η and = η {} {}{ } {}{ } {} η = { η,, η,, η,3,, η η,, η, 3,, η, } { } = {,,,,,3,,, 3,, } { } = {,,,,,,,, },,,,,3, m, n, 3 (3a - e) were η is te modal participation factor, is a modal vector of te disk, and is te modal radiation solution for a specific mode. In tis expressio modal indices ( ) combine te out-of-plane mode indices ( n) and radial mode index. Te value in te indices is used to represents a null. For instance,,,- is te out-of-plane mode wit only two radial diameters and -,-, is te = pure radial mode. Modal participation factors due to a armonic excitation at freuency ω is calculated from te modal data set as follows: T ( rf, ϕ f ) ( r, ϕ) η = (4) ( ω / ω ) + iς ( ω/ ω )

5 Furter, Π and σ due to an arbitrary force f(t) is calculated from te far-field acoustic intensity (I) or sound pressures on a spere (S v ) surrounding te disk as follows: Π = IS v s = π π ρ Π H R sin θ dθ dφ; σ = c < v > t, s (5a - b) Here < v > t, s is te spatially averaged mean-suare velocity on te radiating surfaces and it is expressed as follows. / π( a+ b) a π { U dl dz + W dϕ dr} < v > t, s = (6) 4π( a + b) + π( a b ) / b Fundamental radiation properties suc as sound pressure spectra (ω), acoustic power spectra Π(ω), and radiation efficiency spectra σ(ω) of te sample annular disk are calculated given unit amplitude force. Also, te same radiation properties are obtained wit numerical (FEM and BEM) analyses and vibro-acoustic experiments. Figure 5 sows (ω) at two r p locations (R = 33 m φ =, θ = ; π/) due to an excitation in te z direction. In bot cases, excellent agreements between analytical and numerical results are found, especially in te vicinities of te peaks corresponding to natural freuencies of te disk. Figure 5b sows a significant coupling between in-plane and out-of-plane modes in measured spectrum. Tis is due to an imprecise force excitation in experiments. Furter, note tat (ω) depend on te receiver positions since te source is igly directive. In additio Π(ω) and σ(ω) for te combined (normal + radial) excitation are calculated using te proposed analytical procedure and compared wit corresponding numerical results in Figure 6. Every peak in Π(ω) corresponds to a specific natural freuency. As evident from Figures 5 and 6, all analytical results matc well wit numerical (FEM + BEM) results over te given freuency range. a (db re e-6) 5 (a) Π (db re e ) a (db re e-6) 5 (b) σ Freuency (khz) Figure 5: (ω) due to force f z = N. (a) θ = π/ and φ = ; (b) θ = and φ =. Key:, measured;, analytical calculation; ----, computed (FEM + BEM) Freuency (khz) Figure 6: Π(ω) and σ(ω) for a combined excitation. Key:, analytical calculation; ----, computed (FEM + BEM).

6 5. Conclusion Tis article as introduced a new semi-analytical procedure for te calculation of sound radiation from modal and multi-modal vibrations of a tick annular disk. Based on a comparison wit numerical and experimental results, it is evident tat te proposed procedure as sufficient accuracy. Our procedure provides an efficient metod of calculating modal and multi-modal sound radiations. Bot out-of-plane and in-plane components of te disk vibration are included in te total sound radiation. Using tis procedure, sueal noise radiation from brake rotor and te effect of modal interactions among adjacent components can be effectively analyzed. Tis subject is te focus of current work along wit a determination of sound radiation from coupled modes. Acknowledgements Tis project as been supported by te Center for Automotive Researc Industrial Consortium and te sponsors include Bosc, CRF Fiat, Delpi Cassis, Dow Automotive, Ford, Edison Welding Institute, General Motors and LuK over te 999 period. References. S. W. Kung, K. B. Dunlap, and R. S. Ballinger SAE Tecnical aper Complex eigenvalue analysis for reducing low freuency brake sueal.. C. H Cung, W. Steed, K. Kobayasi, and H. Nakada SAE Tecnical aper - -6 A new analysis metod for brake sueal art I: Teory for modal domain formulation and stability. 3. I-DEAS User s manual version 8.. SDRC, USA. 4. A. W. Leissa 987 Te Sock and Vibration Digest 9(3), -4. Recent Researc and plate vibratio art : Classical teory. 5. O. G. MCgee, C. S. Huang and A. W. Leissa 995 Int. J. Mec. Sci. 37(5), Compreensive exact solutions for free vibrations of tick annular plates wit simply supported radial edges. 6. T. Irie, G. Yamada and Y. Muramoto 984 Journal of Sound and Vibration 97(), 7-75 Natural freuencies of in-plane vibration of annular plates. 7. H. Lee and R. Sing submitted to Journal of Sound and Vibration Acoustic radiation from radial modes of a tick annular disk, submitted to Journal of Sound and Vibration. 8. M. R. Lee and R. Sing 994 Journal of te Acoustical Society of America 95(6), Analytical formulations for annular disk sound radiation using structural modes. 9. W.. Rdzanek Jr. and Z. Engel Applied Acoustics 6(5), Asymptotic formula for te acoustic power output of a clamped annular plate.. H. LEE and R. SINGH submitted to Journal of Sound and Vibration Acoustic radiation from out-of-plane modes of an annular disk based on tick plate teory.. SYSNOISE User s manual Version NIT, Belgium.