Determination of sound radiation from a simplified disk-brake rotor by a semi-analytical method

Transcription

1 Determination of sound radiation from a simplified disk-brake rotor by a semi-analytical method Hyeongill Lee a) and Rajendra Singh b) (Received 24 May 27; revised 24 October 8; accepted: 24 October 15) This article describes a semi-analytical procedure to calculate the sound radiated from a simplified disk-brake rotor. First, analytical modal sound radiation solutions of an example of a thick annular disk (with free boundaries) are presented for flexural and radial modes. The analytical solutions were validated by means of experimental and computational methods. Second, the disk sound-radiation solutions were then applied to an example brake rotor. Natural frequencies and mode shapes were computed using a finite-element computer code. Sound-power and radiationefficiency spectra were obtained for both example disks in response to excitation by a multidimensional force. Predictions by the semi-analytical method matched well with predictions obtained using a commercially available computer code for the boundary-element method. The semi-analytical method is computationally efficient and analytically tractable while yielding insights into the contributions of various modes, and their interactions, to the sound radiated from a vibrating disk. 24 Institute of Noise Control Engineering Primary subject classification: 76; Secondary subject classification: 42 1 INTRODUCTION Thick annular bodies often radiate tonal sounds corresponding to their out-of-plane and in-plane structural vibration modes. Examples of mechanical or structural components that could be modeled as thick annular disks include brake rotors, clutches, flywheels, railway wheels, circular saws, and electrical machinery stators. Brake squeal is a classical example of such problems. Given the complex geometries and boundary conditions, one has to develop detailed analyses by the finite-element method and the boundary-element method to determine the vibro-acoustic responses. These analyses often require significant efforts and may not lead to a fundamental understanding of soundradiation properties. To overcome such problems, a semianalytical approach was developed and is described here. To illustrate the calculation procedure and underlying physics, we considered two example disks as illustrated in Figure 1. A generic thick annular disk (Example A) with free boundaries at the inner and outer edges was considered first. Sound radiation from a thick annular disk (without any baffle) is conceptually shown in Figure 2 in the context of source and receiver positions. Here, P is the amplitude of harmonic sound pressure at a receiver position (R,θ,φ) on the surface (S v ) of a sphere surrounding the annular disk radiator, R is the radius of the sphere, and ẇ and u. are the amplitudes of harmonic vibrational velocities at source positions on the normal surface (S sn ) and radial surface (S sr ) of the disk, respectively. In this study, both in-plane and out-of-plane vibrations of the disk were simultaneously considered to completely investigate the vibro-acoustic properties. Analytical, numerical, and experimental studies were carried out to provide benchmarks for validation of the proposed method. a) Acoustics and Dynamics Laboratory, Department of Mechanical Engineering and The Center for Automotive Research, The Ohio State University, Columbus, Ohio 4322, USA; osuadl@yahoo.com b) Acoustics and Dynamics Laboratory, Department of Mechanical Engineering and The Center for Automotive Research, The Ohio State University, Columbus, Ohio 4322, USA; singh3@osu.edu The chief objectives of this article are: (1) predict structural modal characteristics of Example disks A and B; (2) develop analytical or semi-analytical procedures to calculate sound radiation when the disk or rotor is excited by a multi-directional harmonic force; and (3) validate analytical or semi-analytical procedures using experimental (for the Example A disk only) and computational methods (for both example disks). r r b b ϕ ϕ a a Normal surface Normal surface Radial surface Disk h h Radial surface Fig. 1 Two example disks. Example A: a thick annular disk with freefree boundaries, Example B: a simplified disk-brake rotor with fixed (r=b) and free (r=a) boundaries. See Table 1 for principal dimensions and material properties. z Hat t h H z Noise Control Eng. J. 52 (5), 24 SeptOct 225

2 S v ẇ S sr u. S sn R P(R,θ,ϕ) A flow chart is given in Figure 3 for the calculation and benchmarking procedures used for analysis of a simplified disk-brake rotor. As an example, structural vibration characteristics and sound radiation of a simplified automotive brake rotor disk were investigated by first modeling the vibroacoustic characteristics of a thick annular disk. In particular, both radial and flexural vibration modes of the brake rotor (as experienced during squeal) were examined in detail. Structural vibration modes of a simplified, undamped rotor (Example B) were obtained by the finite-element method and compared with the structural vibration modes of a thick annular disk (Example A). The distribution of sound pressure over the surface of sphere S v as a result of the modal vibrations of the rotor was determined from analytical solutions to the modal response of a disk. Velocity responses of the structure of a rotor or disk to excitation by a multi-modal force, a multi directional force, or a combination of the two forces were obtained by the modal expansion technique. In particular, acoustic responses of both example disks to multi-modal excitations were investigated. Fig. 2 Sound radiation from a vibrating thick annular disk. The scope of the study was strictly limited to the frequencydomain analysis of a linear time-invariant system over the first few elastic modes of deformation (up to 8 khz). Complicating effects such as fluid loading, scattering at the disk edges, and rotation-induced noise were beyond the scope of this study and therefore were not considered. The simplified disk-brake rotor of Example B was assumed to be fixed at the inner edge and free at the outer edge. Table 1 provides sample values of the principal dimensions and material properties used for all studies of the example disks shown in Figure 1. TABLE 1 Principal dimensions and material properties of example disks shown in Figure 1. Dimension or Property Example A Example B Outer radius, (mm) Inner radius, (mm) Radii ratio (β = b/a) Disk thickness (h), (mm) Hat thickness (t h ), (mm) - 6 Hat height (H), (mm) - 24 Density (ρ d ), (kg/m 3 ) Youngʼs modulus (E), (GPa) Poisson s ratio (ν) LITERATURE REVIEW The structural dynamic characteristics of annular disks including the out-of-plane [1-5] * and in-plane [6-9] vibration modes have been studied by many researchers. Conversely, sound radiation from thin circular and annular disks has been examined by only a few investigators [1-14]. Lee and Singh [15,16] described sound radiation from pure in-plane or out-ofplane vibration modes of a thick annular disk. Sound radiation from the multi-modal vibration (including the coupling effect) of ideal one- or two-dimensional rectangular structures such as beams or rectangular plates have been investigated by analytical methods [17-19]. Also, the exterior acoustic radiation modes of simple beams and baffled finite plates have been defined based on the expressions for far-field sound power [2,21]. The acoustic-radiation-mode concept has been applied to an active structural-acoustic control problem as well as used to estimate radiated sound power [22,23]. Multi-modal sound radiation from a thin annular disk and the resulting modal coupling effects on sound radiation has also been investigated using the modal expansion technique [11]. Structural dynamic models have been used to explain the mechanisms that generate brake squeal; theories include self-excited vibration [24,25] or modal coupling phenomena [26-28]. Recently, non-linear transient analysis [29,3] and the complex eigenvalue method [31-35] have been implemented using commercially available finite-element-method software. In particular, it was observed that separation of natural frequencies between coupled flexural and tangential modes seemed to dictate the onset of high-frequency squeal noise [36,37]. The underlying acoustic radiation mechanism has not been adequately examined though an attempt has been made to address this problem using measured structural vibration velocities [38]. The previous study, however, considered sound radiation only from the flexural vibration modes of a disk-brake rotor while ignoring radiation from in-plane modes. * Numbers within brackets refer to citations in the list of references. 226 Noise Control Eng. J. 52 (5), 24 Sept-Oct

3 Undamped structural modal analysis FEM Velocities on radial surfaces Decompose modes (m, n, l, q) Velocities on normal surfaces Cylindrical radiator model (l, q) modes Annular plate model (m, n) modes Sinc function approach 2-D Fourier transform Modal sound radiation Modal participation factors Multi-dimensional harmonic force Modal expansion technique Modal damping ratios Modal acoustic power Π m,n,q Acoustic response function P/f Acoustic power spectrum Π(ω) Modal radiation efficiency σ m,n,q Comparison with BEM Radiation efficiency spectrum σ(ω) Fig. 3 Flow chart for the calculation and benchmarking procedure used for analysis of a simplified brake-rotor disk. Noise Control Eng. J. 52 (5), 24 SeptOct 227

4 3 STRUCTURAL MODAL ANALYSIS 3.1 Example A: Thick annular disk (freefree) Out-of-plane modes Based on thick plate theory [5,15], the equations of motion in terms of stress resultants (moments M and shear forces Q) in the polar coordinates (r,ϕ) as shown in Figure 1, and assuming harmonic motion at angular frequency ω, are as follows: [1] vector, and {} is a null vector. The eigenvalue λ m,n and mode shape Φ m,n for the (m,n) th structural mode were obtained from Equation (4) where m is the number of nodal circles and n is the number of nodal lines. As demonstrated in this paper, validation of this method was confirmed through comparisons with numerical analyses and modal measurements In-plane modes According to the approach proposed by Irie et al. [7] and Lee and Singh [16], the in-plane modes of a thick annular disk were obtained by the transfer matrix method where the displacements and forces at a radial position are related to those at the inner radial edge by transfer matrix [T(ξ)] as (5) [1] [1(c)] where M r, M ϕ, and M rϕ are the moments, Q r and Q ϕ are the shear forces, ρ d is the mass density of the disk, W is the transverse displacement, and Ψ r and Ψ θ are the bending rotations of the normal line to the mid-plane of a disk in the radial and circumferential directions, respectively. Specifying W, ψ r, and ψ ϕ in terms of three potential functions φ 1, φ 2, and φ 3 and necessary parameters along with a series of subsequent manipulations yielded three Laplacian differential equations for φ 1, φ 2, and φ 3 [5]. From the resulting equations, one can determine potential functions as follows, where n is typically a positive integer. Using Equations (1) and (2), we get the expression [2] [2] [2(c)] General solutions to Equation (3) involve ordinary and modified Bessel functions of the first and second kinds. There are six constants that are determined from the prescribed boundary conditions. For instance, when the free boundary conditions are imposed at the inner and outer radial edges, the following equation in matrix form was used to determine the modal frequencies of vibration: Here, [M CH ] is the characteristic matrix with elements of various Bessel functions, {C} is an arbitrary coefficients (3) (4) where, ξ = r/a and β = b/a represent normalized radial coordinates at an arbitrary radial location (r) and at the inner edge (r=b) respectively; see Figure 1 and Table 1. Furthermore, is the state vector that expresses normalized displacements and forces at ξ and satisfying following relation Details of the procedure and a description of [U(ξ)] can be found in [7]. Using Equations (5) and (6), the transfer matrix at frequency ω was obtained from the following relation By applying the boundary conditions at ξ = β and ξ = 1, the modal data set (λ q,φ q ) was determined from the transfer matrix [T(ξ)]. 3.2 Example B: Simplified brake rotor (fixedfree) The structural vibration modes of the disk-brake rotor of Figure 1 were investigated by means of a commercially available finite-element model with 21 solid elements and 396 nodes [39]. To simulate realistic boundary conditions, clamped nodal restraints were applied at the locations of mounting bolts; see Figure 4. Although results are presented only up to 8 khz in this paper, as many as 44 structural modes, up to 16 khz, were identified in the analysis. Selected natural frequencies and mode shapes are listed in Table 2. Structural vibration modes are described using four modal indices: m = number of nodal circles, n = number of nodal diameters, q = radial mode index, and l = tangential mode index. Structural modes of the Example B brake rotor were categorized in three groups: (1) pure out-of-plane modes described by (m,n), see Table 2; (2) pure in-plane modes given by (l,q), see Table 2; and (3) combined modes (6) (7) 228 Noise Control Eng. J. 52 (5), 24 Sept-Oct

5 f n f r identified by (m,n,q), see Table 2(c). As shown in the table, the q th radial modes are always coupled with out-of-plane modes having the same number of nodal diameters (n) as q, as a result of the hat structure and clamped boundary conditions. Mode 3 (at 151 Hz) is not illustrated in Table 2 because this is a rigid-body (disk-rotor) translation mode that has no nodal line or diameter corresponding to the disk of Example A, but the rotor vibrates because of the elastic deformation in the hat structure. Nonetheless, mode 3 was included in the modal expansion approach for the multi-modal vibro-acoustic analyses. Fig. 4 Finite element model of the Example B disk-brake rotor used for the normal mode and forced vibration analyses with excitation by a multi-directional harmonic force. The inner edge of the disk was clamped at the mounting bolts. TABLE 2 Selected structural modes of the simplified disk-brake rotor of Example B. 4 MODAL SOUND RADIATION CALCULATION FOR AN ANNULAR DISK 4.1 Out-of-plane structural vibration modes Lee and Singh [15] proposed an analytical solution for sound radiation from the out-of-plane modes of a thick annular disk that specifically considered the thickness effect. According to the analytical solution, the amplitude P at distance r p and angular frequency ω for the far-field time-harmonic sound pressure can be expressed by the following equation with reference to the configuration shown in Figure 5: Out-of-Plane Modes (m,n) Mode number 1, 2 4, 5 9,1 13 Frequency (Hz) Modal indices (, 1) (, 2) (, 3) (1, ) Mode Shape - In-Plane Modes (l,q) Mode number 6 2 Frequency (Hz) Modal indices l = q = Mode Shape (c) Combined Modes (m,n,q) Mode no. 7, 8 11, 12 18, 19 Frequency (Hz) Mode Shapes m =, n = 1 m = 1, n = 2 m = 1, n = 3 - q = 1 q = 2 q = Note: In Table 3(c), the + sign between the set of two mode shapes indicates the combination of modes for the 3-dimensional vibration of the disk. (8) In Equation (8), ρ is the mass density of air, c is the speed of sound, k is the acoustic wave number, is the amplitude of vibratory velocity of the disk in the z direction, r s and r s ' are the position vectors of source points on the normal surfaces that face toward or away from the field point at (R,θ,φ), S s is the surface area of the vibrating annular disk as shown in Figure 5, and i = -1. Employing the Hankel transform and the far-field approximation to Equation (8), sound pressure resulting from the (m,n) th structural vibration mode was expressed as a sum of sound radiations from two normal surfaces as follows: where and [9] [9] [9(c)] Here, B n [ẇ (r)] = ẇ (r)j n (k,r)rdr, k r = k sinθ, P s m,n and P m,n are the sound-pressure amplitudes from vibration of the two normal surfaces, and ẇ(r) is the variation of the surface velocity on the normal surfaces in the radial direction. Further, J n is the Bessel function of order n. Noise Control Eng. J. 52 (5), 24 SeptOct 229

6 ( ) ( ) S v W. mn γ z r p -r s r p z p r p r s S s R z h Fig. 5 Spherical-coordinate-system conventions for calculation of sound radiation from out-of-plane vibration of a thick annular disk. With Equations (8) and (9), sound pressures were calculated at predetermined observation points on the surface of the sphere of S V that surrounds the disk and is centered at the disk center as shown in Figure 5. The modal radiation solution for the (m,n) th out-of-plane mode (Γ m,n ) can be determined from the sound pressure distribution over the surface of the sphere; see Lee and Singh [15]. R z θ r p ( ϕ) 4.2 In-plane structural vibration modes Far-field sound pressures resulting from radial vibration modes of a thick annular disk can be calculated using the cylindrical radiator approach [16]. For a cylindrical radiator having the q th sinusoidal variation in the ϕ direction, the farfield amplitude P of the time-harmonic sound pressure at angular frequency ω can be determined from the following equation in the cylindrical coordinate system with reference to the geometric conventions shown in Figure 6. h x Fig. 6 Conventions for calculation of sound radiation from the radial vibration of a thick annular disk: cylindrical coordinate system; spherical coordinate system. y where [1] [1] Here, H q is the q th order Hankel function of the first or second kind depending on the direction of radiation, u.. r is the amplitude of harmonic acceleration in the radial direction, k z is the structural wave number in the z direction, Z(z) is the variation of acceleration in the z direction, Z ~ (z) is the complex acceleration in the z direction as determined from the Fourier transform of the acceleration I[Z(z)]. The symbol H' q in the denominator of Equation 1 represents the derivative of the Hankel function in the numerator. After transforming Equation (1) into spherical coordinates [see Figure 6] and employing the stationary phase approximation, the total sound-pressure amplitude P q at a receiver position resulting from the q th radial mode of vibration for a thick annular disk can be expressed as a sum of P qo soundpressure amplitudes from vibration of the outer radial surface and P qi sound-pressure amplitudes from vibration of the inner radial surface according to the following equations. 23 Noise Control Eng. J. 52 (5), 24 Sept-Oct

7 where for the outer surface, and [11] [11] For instance, Φ V is the velocity mode shape of the pure,2,-1 (,2) out-of-plane mode and Φ -1,-1,2 is the velocity mode shape of the pure q = 2 radial mode. Structural modal participation factors resulting from a harmonic force, with angular frequency ω, applied to a given disk were obtained from the modal data set (consisting of natural frequencies, mode shapes, and damping ratios) according to the following equation: for the inner surface, and [11(c)] [11(d)] Here, u.. qo and u.. qi are the absolute values of the amplitudes of harmonic acceleration on outer and inner radial surfaces, respectively. As in the case of out-of-plane modal vibration, modal radiation solutions for the q th in-plane modes (Γ q ) can be determined from the calculated sound pressures on the surface of sphere S v surrounding the disk; see Lee and Singh [16]. 5 VIBRO-ACOUSTIC RESPONSE TO A GIVEN MULTI-MODAL EXCITATION If an annular disk is excited by a multi-modal harmonic force at an arbitrary frequency ω and from an arbitrary direction, several modes are simultaneously excited including both in-plane and out-of-plane modes. Based on the modalexpansion technique, velocity distribution (v) on the disk surfaces can be expressed in terms of structural velocity modal vectors of the disk as shown by the following equations. where and [12] [12] (13) In Equation (13), ω is the excitation frequency. Symbols ω m,n,q and ζ m,n,q represent the natural frequency and modal damping ratio of mode (m,n,q), respectively. Symbols r f and ϕ f represent the polar coordinates of the position of the excitation force. Earlier, Lee and Singh [11] expressed the far-field sound pressure radiated from a thin annular vibrating disk excited by a multi-modal force. The expression used the structural modal participation factors and modal sound pressures. Applying the same procedure to an acoustical problem, the far-field sound pressure on the surface of the sphere surrounding the thick disk, given the surface velocity of Equation (12), is expressed as: where [14] [14] and where Γ m,n,q is the modal sound-pressure matrix obtained from Equations (6) and (11). The sound power Π(ω) radiated by the vibrating disk and the corresponding radiation efficiency σ(ω) resulting from an arbitrary harmonic excitation f(t) with an angular frequency ω were also calculated from the distribution of the far-field sound pressures on the surface of the sphere surrounding the disk. Sound power was determined from the product of the surface-average of the sound intensity and the surface area of the sphere. The following expressions describe the procedure. [12(c)] In Equations (12), Φ V is the velocity modal vector of the m,n,q disk expressed as the displacement modal vector multiplied by the corresponding natural frequency (in rad/s) and η m,n,q is the corresponding modal participation factor. In Equations (12), a new modal index notation (m,n,q) was introduced that is the combination of the out-of-plane mode index (m,n) and radial mode index q. The value of 1 in these indices is used to represent a null that means the corresponding velocity mode shape does not contribute to the newly defined modal velocity vector. and [15] [15] where A s is the total area of all radiating surfaces and subscripts t and s indicate time and spatial averaging. The time-space average of the surface velocity on area A s is given by [15(c)] Noise Control Eng. J. 52 (5), 24 SeptOct 231

8 where dimensions a, b, and h are as given in Figure 1 and U and W are the surface velocities in the r and z directions, respectively. 6 VALIDATION STUDIES FOR THE THICK DISK OF EXAMPLE A 6.1 Structural vibration response Utilizing the procedure described in Section 5, structural vibration response to a specific multi-modal harmonic force was calculated on the basis of the analytical modal data sets from the structural modal analysis described in Section 3 or by numerical eigen-solutions. For each method, the measured (or assumed) modal damping ratios were used in the modal expansion procedure. Analytical responses of the Example A disk, with force excitation in the normal and radial directions, were validated by means of numerical analyses and structural vibration experiments. In the experiments, an impulsive force f(t) was applied to a normal surface in the z direction at the outer edge of the disk (out-of-plane modes) or on the mid-plane at the outer radial edge in the radial direction (in-plane modes). The forces were applied using a PCB model GK291C impulse hammer at ϕ =. The setup for the experiments is shown in Figure 7. The upper frequency limit and resolution of the Fast Fourier Transform analyzer used for this experiment were set to 16 khz and 8 Hz, respectively. Natural frequencies (ω m,n and ω q ) and modal damping ratios (ς m,n and ζ q ) were extracted from measured structural frequency response function ẇ. /f n (ω) or u.. /f r (ω) using the half-power-point method where ẇ. and u.. are the flexural and radial accelerations, respectively, and f is the applied force along the corresponding direction. Results of the experiments are summarized in Table 3 along with predictions for the natural frequencies by the finite-element method and by the analytical method described in this paper. In the finite-element method, the frequency response functions ẇ. /f n (ω) or u.. /f r (ω) were obtained over the frequency range from 1 khz to 8 khz by means of a commercially available computer code [4]. The forced harmonic accelerations ẇ. (r,φ) and u.. (r,φ), given a unit harmonic force in a single direction at arbitrary frequency ω, were calculated based on the modal data set determined for the finite-element method and by use of the modal expansion method. Measured damping ratios from Table 3 were used assuming a proportionally damped structure. Figure 8 shows ẇ. /f n (ω) spectra for a normal force and.. u /fr (ω) spectra for a radial force at selected excitation and response positions. The structural response functions shown in Figures 8 and 8 were calculated over the range of frequencies from 1 khz to 8 khz. The measured normal structural response data in Figure 8 cover the same frequency range from 1 khz to 8 khz. For Figure 8, however, the measured structural response data were only available for the frequency range from 1.6 khz to 8 khz. Fairly good agreement among the results from the three approaches was observed over the frequency ranges for the data shown in Figures 8 and 8. Dominant peaks in these spectra corresponded to the natural frequencies for pure outof-plane or in-plane modes of the Example A disk. However, some discrepancies occurred in off-resonance regions. Anechoic chamber φ θ TABLE 3 Natural frequencies and modal damping ratios for the thick annular disk of Example A Signal conditioning unit Rotor FFT Analyzer FFT Analyzer Signal conditioning unit Type Out- of- Plane Mode Natural frequencies (Hz) Measured Indices modal Measured FEM Analytical m n q damping ratios (%) Fig. 7 Experimental setup for vibro-acoustic measurements of the structural response functions ẇ. /f(ω) or u.. /f(ω) and the acoustic frequency-response function P/f(ω). FFT represents a Fast Fourier Transform spectrum analyzer. See Fig. 6 for angle conventions. In- Plane Note: FEM = finite-element method; Analytical = the analytical method of Section Noise Control Eng. J. 52 (5), 24 Sept-Oct

9 ẅ/f [db re (2 µm/s 2 )/N] ü/f [db re (2 µm/s 2 )/N] (,2) q = 2 (1,) (,3) 6.2 Acoustic response Radiation properties of a vibrating disk such as the spectra of the acoustic frequency-response function P/f(ω), sound power Π(ω), and radiation efficiency σ(ω) for the Example A disk were obtained by the analytical method of Section 5. The acoustic frequency-response function is the complex quotient of the Fourier transform of the sound pressure amplitude P in pascals by the Fourier transform of the amplitude of the applied force f in newtons (N) at a given angular frequency ω. The same properties were then calculated by means of a commercially available boundary-element method computer code [4]; uncoupled, direct, exterior analysis was conducted without any baffle. The field point model for modal soundradiation solutions was used for this calculation along with the finite-element method that had been used for the numerical analyses of structural vibration by the finite-element method. Excitations to the boundary-element model were the velocity distributions in the direction normal to the external surfaces as obtained from the forced vibration analysis described in Section 6.1. Analytical and computational determinations of the acoustic frequency-response function P/f(ω) at two r p locations (R = 33 mm, φ =, θ = and θ = π/2) resulting (1,1) (,4) q = q = 3 Fig. 8 Structural frequency response functions for the thick annular disk of Example A: ẇ. /f(ω) at r = mm and φ = 18 ; u.. /f(ω) at φ =. Key: measured; computed using the finite-element method; computed using the analytical method. The (m,n) or q modes are identified. from normal or radial force were verified by vibro-acoustic experiments conducted in an anechoic chamber as illustrated by the sketch in Figure 7. An impulsive force excitation was also used in the acoustical experiments. Far-field sound pressures were measured with a MTS model L13C1 6-mm-dia microphone mounted on a MTS model 13P1 pre-amplifier and located at predetermined field points on the surface of sphere S v. The P/f(ω) spectrum, for a unit harmonic force in a single direction at arbitrary frequency ω, was then calculated and compared with the measured spectrum. For two r p locations, Figure 9 shows the analytical acoustic frequency-response functions P/f(ω) resulting from a force applied in the radial direction, the results from a calculation using the boundary-element method, and measured data. [Note that the frequency scale in Figure 9 ranges from 1.6 khz to 8 khz as in Figure 8.] For the same r p locations, Figure 1 shows the same data resulting from application of a force applied in the normal direction; the frequency scale here is from 1 khz to 8 khz. Dominant peaks in these figures correspond to (m,n) or q modes of the Example disk A. Good agreement among analytical, experimental, and numerical results was observed over the frequency range at each measurement position, especially in the vicinity of the disk resonances. The spectrum of the sound-pressure amplitude P(ω), sound power Π(ω), and radiation efficiency σ(ω) were calculated for a multi-directional harmonic force of magnitude ƒ n = ƒ r = 1 N at a single excitation location using the analytical procedure described in this paper and the boundary-element method. The combined force was applied at ϕ = on the mid- P/f [db re (2 µpa)/n] P/f [db re (2 µpa)/n] 5 5 q = 2 q = q = q = 3 Fig. 9 Acoustic frequency response functions P/f(ω) for the thick annular disk of Example A when excited in the radial direction: θ = π/2 and φ = ; θ = and φ =. Key: measured; computed using the boundary-element method; computed using the analytical method. Noise Control Eng. J. 52 (5), 24 SeptOct 233

10 P/f [db re (2 µpa)/n] P/f [db re (2 µpa)/n] 5 5 (,2) Fig. 1 Acoustic frequency response function P/f(ω) for the thick annular disk of Example A excited at r=a in the z direction: θ = π/2 and φ = ; θ = and φ =. Key: measured; computed using the boundary-element method; computed using the analytical method. The (m,n) modal indices are identified. (1, ) (, 3) (1, 1) (, 4) (1, ) q = 5 5 (,2) (,3) (1,1) 1) (,4) (1, (1,) ) q = q=2 q = q = 2 q = q = 3 3 q = 2 q = q = 3 Fig. 11 Calculated far-field sound-pressure level spectra P(ω) for the thick annular disk of Example A resulting from multimodal force excitation: θ = π/4 and φ = ; θ = and φ =. Key: computed using the analytical method; computed using the boundary-element method. The (m,n) modal indices are identified. plane of the Example A disk. Measured data were not obtained for comparison with the calculations because of practical difficulties encountered in implementing simultaneous excitations. Analytical P(ω) spectral data for two receiver locations are shown in Figure 11 along with the results obtained from the boundary-element method. Dominant peaks in the spectra in Figure 11 at certain (m,n) or q modes for the combined force excitation were also present in the uni-directional P/f(ω) spectra of Figures 9 and 1. Some discrepancies between analytical and computational results were found in the offresonance regimes. Spectra of sound power level and radiation efficiency calculated by the two computational methods are shown in Figure 12. For convenience, the spectra in Figure 12 use the symbol Π(ω), representing sound powers in watts in previous text, to represent sound power levels in decibels relative to the standard reference power of 1 pw. Sound power levels and radiation efficiencies determined by the two computational methods matched relatively well. The off-resonance-regime discrepancies seen earlier in the P/f(ω) and σ/f(ω) spectra now seemed to disappear because of the effect of spatial averaging. 7 SOUND RADIATION FROM A SIMPLIFIED BRAKE ROTOR 7.1 Modal sound radiation calculations As evident from Table 2, mode shapes of pure out-of-plane (flexural) and pure in-plane modes of a disk-brake rotor can be expressed in terms of the modes of a generic annular disk with virtually identical geometry. Consequently, sound radiation from such pure modes of the rotor can be expressed in terms of the analytical solutions for the corresponding annular disk. In these cases, far-field modal sound-pressure amplitudes, P m,n and P q, can only be calculated by applying Equations (9) and (11). Modal acoustic powers, Π m,n and Π q, and modal radiation efficiencies, σ m,n and σ q, resulting from the (m,n) th out-of-plane or q th in-plane mode of the brake rotor were calculated by means of Equation (15). Sound radiation from the hat structure [shown in Figure 1] was ignored. However, as discussed next, the effect of the hat structure seemed to be to couple the radial and out-of-plane modes. Next, consider excitation by combined modes. As evident from Table 2, the q th radial modes were always coupled with the (m =, n = q) or (m = 1, n = q) out-of-plane modes, except for the q = radial mode. Because the thickness, h, of the disk was beyond the limit of thin-plate theory, sound radiation from in-plane (radial) modes had to be included in a calculation of total sound radiation from such modes. Modal velocity distributions on the normal and radial surfaces were specified in terms of natural frequencies and modes shapes estimated by the boundary-element method. Consequently, the total farfield sound pressure at a given receiver position was expressed 234 Noise Control Eng. J. 52 (5), 24 Sept-Oct

11 σ 8 (,2) as a sum of the sound pressure radiated from normal surfaces caused by the out-of-plane vibration and that radiated from radial surfaces caused by radial vibration. Sound pressures radiated from all surfaces (except the hat structure) were calculated by equations given for the annular disk in Section 3. Modal acoustic power for a combined (m,n,q) mode was calculated using an equation similar to Equation (15) but based on the total sound pressure as the sum of the modal soundpressure amplitudes P m,n and P q. Modal radiation efficiencies for the combined mode were also calculated by means of Equation (15). In this case, the time and space average of the square of the magnitude of modal velocity < v m,n,q2 > t,s was calculated at each mode by averaging the surface velocity over the entire radiating surface. 7.2 Validation of modal radiation solution The validity of the analytical procedure described in this paper was confirmed by comparison with the results obtained from a purely numerical boundary-element analysis in terms of sound-pressure directivity patterns, sound powers and radiation efficiencies. For the boundary-element method, velocities on the rotor surface were calculated by the modal expansion method based on predictions by the finite-element method of natural frequencies and modes. Sound radiation was calculated using a commercially available boundary-element method computer code [4]. (,3) (1,1) (,4) (1,) q = 2 q = q = 3 Fig. 12 Calculated acoustic radiation functions for the thick annular disk of Example A resulting from combined harmonic excitations: spectrum of sound power level; spectrum of radiation efficiency. Key: computed using analytical method; computed using the boundary-element method. The (m,n) modal indices are identified. Uncoupled, direct, exterior, and unbaffled analysis was employed. Directivity patterns from the semi-analytical method of this paper for selected modes are compared in Figure 13 with the results from predictions by the boundary-element method. The analytical and numerical directivity patterns for three modes were consistent. Modal sound power levels and radiation efficiencies are shown in Table 4. Like the directivity patterns, modal sound radiation results from the semi-analytical method matched quite well with the predictions by the boundary-element method. 7.3 Multi-modal vibro-acoustic responses If a rotor is excited by a multi-directional harmonic force at angular frequency ω, several in-plane and out-of-plane modes are simultaneously excited. Based on the procedure introduced in Section 5, the spectrum of the velocity distribution, v(ω), on the rotor surfaces was expressed in terms of the structural modes according to the following expressions: where and [16] [16] [16(c)] and where Φ v j is the jth velocity modal vector of the rotor and η i is the corresponding modal participation factor. The far-field sound-pressure amplitude, P, over the surface, S v, of the sphere surrounding the rotor, as excited by the rotorsurface velocity distribution of Equation (16), was expressed as follows: where [17] [17] where Γ j is the modal sound pressure for the j th mode obtained from the procedure given in Section 7.1. Sound power radiated from the rotor and the corresponding radiation efficiency resulting from an arbitrary harmonic force were also calculated from the far-field sound pressures on the surface of a sphere surrounding the disk by means of an equation similar to Equation (15). 7.4 Validation of multi-modal sound radiation solutions As in the case of modal sound radiation, vibro-acoustic responses to a multi-directional force were obtained using the semi-analytical procedure and compared with predictions obtained from the boundary-element method. For the analysis by the boundary-element method, acoustic responses were obtained from the forced vibration analysis using the finiteelement method. Unit harmonic forces in the normal and tangential directions were applied simultaneously in the mid-plane of the rotor disk at ϕ = π/2 as shown in Figure 4 by the solid black arrows on Noise Control Eng. J. 52 (5), 24 SeptOct 235

12 Mode BEM Analytical m = n = 2 m = 1 n = 2 q = 2 q = Fig. 13 For selected modes, predictions of directivity patterns of sound-pressure amplitudes over the surface of a sphere surrounding the simplified brake rotor of Example B. BEM = boundary element method; Analytical = the semi-analytical method. 236 Noise Control Eng. J. 52 (5), 24 Sept-Oct

13 TABLE 4 Sound power levels and radiation efficiencies of the simplified brake rotor of Example B at selected modes. Mode Power level (db) Efficiency Type Indices m n q l Semi-Analytical BEM Semi-Analytical BEM Out-of-Plane In-Plane Combined Note: Sound power level in decibels re 1 pw; BEM = boundary-element method. P/f [db re (2 µpa)/n] P/f [db re (2 µpa)/n] (,1,-1) 4 2 (,,-1) (,2,-1) (1,,-1) (,1,1) (1,2,2) (1,,-1) (,4,-1) (,5,-1) Fig. 14 Calculated spectra of the far-field acoustic response function P/f(ω) for the simplified brake rotor of Example B at selected receiver positions: rp1 and rp2. Key: the semi-analytical method ; computed using the boundary-element method The (m,n,q) modal indices are identified. 1 (,,-1)(,2,-1) (,3,-1) (,4,-1) (,5,-1) (,1,-1) (,1,1) (1,2,2) (1,1,-1) (1,,-1) Fig. 15 Calculated spectra of sound power level and radiation efficiency for the simplified brake rotor of Example B. Key: computed by the semi-analytical method ; computed by the boundary-element method Noise Control Eng. J. 52 (5), 24 SeptOct 237

14 the top of the rim of the finite-element model. Far-field spectra of the acoustic response function P/f(ω) were obtained using the above procedure for frequencies from.8 khz to 8 khz. Results of the calculations by the semi-analytical method at two receiver positions r p1 (R = 1 m, θ =, φ = ) and Γ p2 (R = 1 m, θ = π/4, φ = ) are shown in Figure 14 compared with results calculated by the boundary-element method. For both cases, the spectra from the semi-analytical approach matched quite well the spectra from the boundary-element computation. The results depended on the receiver position indicating, as expected, that the source was highly directive. The spectra of sound power level and radiation efficiency were also determined. Figure 15 shows that analytical sound power and radiation efficiency spectra were in relatively good agreement with the results from the boundary-element method especially at frequencies less than 5 khz. Some discrepancies were found, however, in the prediction of the spectrum of the radiation efficiency. Overall, the proposed semi-analytical procedure was determined to be sufficient for predicting the sound radiation from a brake rotor when excited by an arbitrary harmonic force. 8 CONCLUSIONS This article has introduced a new semi-analytical procedure for calculating sound radiation from a simplified model of a disk-brake rotor given the modal vibrations of the disk. Benchmark analytical solutions for sound radiation from a thick annular disk were presented and validated using experimental and computational investigations. As in the case of the thick annular disk, the modal expansion technique was used to calculate the total sound radiation caused by radial and flexural modes when excited by multi-modal, multi-directional, or a combination of multi-modal and multidirectional forces. Predictions by the semi-analytical method were confirmed by comparison with predictions by application of the boundary-element method. The semi-analytical method is computationally efficient and analytically tractable and can easily accommodate mixed calculation methods or even predeveloped results from analytical, numerical, or experimental approaches. 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