1 Introduction. 2 Boundary Integral Equations

Transcription

1 Analysis of sound transmission through a thin elastic plate by using boundary integral equations T. Terai*, Y. Kawai* "Department ofarchitecture, Faculty ofengineering, Kinki University, 1 Umenobe Takaya, Higashi-Hiroshima, Hiroshima , Japan *Department ofarchitecture, Osaka Institute of Technology,16-1, Omiya-5, Asahi-ku, Osaka 535, Japan Abstract Sound transmission through a thin elastic plate is a fluid-loading problem which one encounters in room acoustics. In this paper, a system of boundary integral equations which describe both wave motion in air and the vibration of the plate is used. In order to demonstrate the effectiveness and accuracy of the method, some numerical examples are shown. 1 Introduction There are many studies on sound transmission through or reflection from a thin elastic plate under the condition of plane wave incidence. Theoretical treatments are for the infinite geometry and the periodic structure [1-6]. For thefinitestructure, some numerical methods such as the combination of the acoustic BEM and FEM for the vibration offiniteplates have been presented [7]. In this paper the method which apply BEM both fluid and plate vibrations is presented. 2 Boundary Integral Equations Let us consider a cavity formed with rigid walls, a thin elastic plate and an absorbent material on the bottom as shown in Fig. 1. The whole space is divided into the outer space 0^ and the cavity H, by the plate B and the enclosure; the outer boundary is 8^(8^, S^e, Swae) and the inner boundary 5$ Si(Sbi,Swi,Sai). Boundary conditions are = 0 on 5^, S^ae, &w and

2 Computational Acoustics Point Source p.? P Receiving Point Thin Elastic Plate Thin Rigid Wall Absorbent Material Figure 1: A cavity formed with a thin elastic plate, thin rigid walls andan absorbent material. fa = ika$ on S,i where A is the specific admittance. In order to obtain the solution, the wave equation in the air and that for the plate have to be solved simultaneously. 2.1 Equations for the outer space and cavity The velocity potential $ at point P e JX,fl, is denoted by w i ~, P. exp(-ikr) nere Or(f, q) = - - denotes the fundamental solution in air, k = u/c wave number, and <& is the direct wave, $j,(p,,p) = G(P P)- P is situated in H. orfl,-.the time factor exp(iurf) is omitted here. Throughout this paper, upper and lower case letters denote points in the air and on the boundaries, respectively. By taking the directional derivative of Eq.(l) with respect to P to the

3 normal direction n^, one obtains Computational Acoustics 5 drip JJse+St ( dripdriq driq drip } drip /,, (2) Let P converge to a point p on the boundary along the normal rip and also the thickness of both the plate and the walls be negligible compared to the wavelength. Taking into account the jump of the double-layer and normal derivative of the single-layer on both the outer and inner surfaces and changing the normals at points on inner surfaces SM and S^i to reverse directions, one can obtain <94>n(-P«, D\,,y/_j_ P] II ff * $(<?) \ V(P'V)ja dsq +, // ff $+(<?) * f \ drip JJB+W dn^dnn JJSwae -f,,3g(p,g) &%p UW-(p \,. _,,.,. + -^ =-%H, PG#, (3) 2 \ drip drip = 0, PEM/,^, (4) where 0 = 0+ 0_, and #_^_ and <&_ represent potentials on the outer and inner surfaces, and B and W refer to Sbe( Sbi) and 5^e( 5^), respectively (See Fig. 1). On the plate on velocity of the plate. 2.2 Equations for the plate = - -^ = v holds, where v is the vibration on Let the thin square elastic plate be located on X Y plane. The governing equation of bending vibration is denoted by DV% mh- = /, (6) where / is a force to produce deflection in z direction, and D = Eh^/'12(1 i/2) the bending stiffness in terms of Young's modulas je", the thickness of the plate /%, Posson's ratio z/ and the density of the plate ra [8]. The relations among variables, $, /, (, sound pressure p, and vibration velocity v, are as follows: P = /> = 2W/)0, U =, /= -(p+ - p_) = -2W/)#, (7) where /? is air density. The pressure difference p+ p_ acts along negative Z direction. At the plate's surface it is noted that #* 3(,g* v = -grad 0 = - =, ( = - / -z-dt = - = v. (8) 02 (% V &Z ZW0Z ZW

4 6 Computational Acoustics By using these relations, Eq. (6) is transformed into V»-A = A*; 0' = in', A = «' = /i^. W The fundamental solution g(p,q) of the equation V*g-n*g = -6(x)6(y) (10) is obtained from the two-dimensional Fourier transform: i.e., where HQ and KQ are the Hankel and modified Bessel functions, respectively. Note that 47T. + ^o(h}=a(p,9), (13) and a > 0 and V^0» -- Inr for a small r. It is also noted that the.. %7T laplacian operation V^ with respect to p is equivalent to V^. Substituting Eqs.(9) and (11) into the Rayleigh-Green identity [9] J. _ V, (14) where barred points p, q denote points on the edge db of the plate B and Vq inward-drawn normal to the edge. Here He is a small circle of center p and small radius c (See Fig.l). Provided that each edge of the plate is simply supported and parallel to the X- or Y-axis, then the boundary conditions are ( = 0 and V^( = 0 on <~v ' v7 O ' J <~\ O OVg OVq OV-q OVn,^,15, Integrals over f^ and 90^ in Eq.(15) are LHS // ds > -6(p) and JJQf n RHS / ds» u(p), therefore one can obtain J 9f2 ^ \ / / *" //^ 77^ ' ^ V^B I ' 9i/s dva \

5 Computational Acoustics This equation contains and - on the edge db of the plate, there-. 9z/ <9z/ fore we have to derive other equations by taking the directional derivative and laplacian Vj; of Eq. (15). When taking the limit of p > p. one can obtain the following integral equations, // VVB-n, / J8B OVp o OVq o OVp ~ _^,,, /,_ 9> p db, (17), V9. ^ g p g The estimation of the singular term, the integral over f^ in the above equations, is explained in Section 3. On the plate the particle velocity is equal to the vibration velocity of the plate (Eq. (16)). Therefore the integral equation for the air can be expressed as = 0, pew,s«,«, (4) = i»a$_(p), P e &,. (5) Unknown variables to be solved in the above equations are $ on B -f W, <9u 9V^ 0, on 5^ae, ^- on 5^, ^ and - on db. By solving Eqs. (19), (4). av ov (5) for the air, and (17) and (18) for the plate, one can obtain potential distributions on the boundaries. The sound field can be calculated from 3 Numerical Procedure In the numerical calculation, the boundary surfaces. B, W, S^ae and 5^. are divided into rectangular elements and the boundary of the plate db

6 8 Computational Acoustics into line segments. The node points are taken to be in their middle of each element or segment. The potential distribution is assumed to be uniform inside each element. For the evaluation of the coefficient, the surface integral over mth divided element ASVn, the following representations are used. YA o = 0?u "p :b a = (a) Eq.(20) (b) Eqs.(21),(22) (c) Eqs.(22),(23) Figure 2: Geometry of element A5^ relative to point p. (20) IL = i f [?^ V) - (22) where a takes the value 2?r or zero according to whether the point p lies inside or outside the divided element AS^. If the point p is located on one of the edges of the element A5^, Eq.(21) has no singularity, whereas the approximation lim f {^-H^(iix) - Ko(nx) dx ~ 2(alog(^a) - a) c-*u J c { Z (23) is used for the evaluation of Eq.(22). _,, _., _ dg Ihe surface integrals ol G, -^3, ^^, dri formed into line integrals [10]. and,, ^ can also be trans- 4 Numerical Examples As numerical examples of this method, sound fields for a glass plate with back cavity are calculated. Fig. 3 shows the sound fields inside and outside the cavity which is enclosed by (a)rigid walls and an absorbent surface on the

7 Computational Acoustics Source f Source t *>; Oi Plate \ 0.25d t j 25Hz ' \ / Figure 3: Sound pressure amplitude for a thin glass plate (3x3[m*]) with a back cavity (a) enclosed by rigid walls and an absorbent surface, and (b) by rigid walls entirely: the depth of the cavity 0.25[m], m 2500 [kg/rn^], #=7.0x10* [Pa], y=0.22, A=0.003 [m], A=l, 1536 elements, 15 C; eigenfrequencies of the plate without fluid loading is 26.1, 52.2, [Hz], the structural wave number //=5.78 (25Hz), 8.18(50Hz), 8.57(55Hz), and wave number A:=0.92

8 10 Computational Acoustics bottom and by (b)rigid walls entirely. A point source generates a sinusoidal wave and is located at a distance of 49.5[m] above the center of the plate. The specific admittance on the surface Sai in Fig. 3(a) is assumed to be unity; i.e., no reflection occurs from the absorbent surface. Node points are collocated according to the structural wave number \i which is larger than that in air. Calculation of sound transmission without the term DV*( in Eq.(6) for Fig. 3(a), which means no bending vibration occurs, yields close results to those from the mass law. It is clearly shown that the sound pressure amplitudes in the cavity around eigen-frequencies of the plate are considerably larger than those in mass law. 5 Conclusions To solve the coupled problem of sound and the vibration of a thin elastic plate, boundary integral equations for the plate and the normal derivative of the boundary integral equation in the air are used. Numerical examples show this method is effective to solve practical situations of finite size. References (1) F. G. Leppington, Scattering of Sound Waves by Finite Membrane and Plates near Resonance, Q. JL Mech. Appl. Math, Vol. XXIX, Pt. 4, , (2) F. G. Leppington, Acoustic Scattering by Membranes and Plates with line Constraints, Journal of Sound and Vibration, 58, , (3) D. G. Crighton, The 1988 Rayleigh Medal Lecture: Fluid Loading - The Interaction between sound and Vibration, Journal of Sound and Vibration, 133, 1-27, (4) M. C. Junger and D. Feit, Sound, Structures and Their Interaction, 2nd Ed., MIT Press, (5) D. Lee, R. L. Sternberg and M. H. Schultz (Eds.), 1988 Proceedings of the 1st IMACS Symposium on Computational Acoustics: Algorithms and Applications, Elsevier. (6) T. L. Geers and C. A. Felippa, Doubly Asymptotic Approximations for Vibration Analysis of Submerged Structures, J. A. S. A., 73, , (7) I. C. Mathews, Numerical techniques for three- dimensional steady-state fluid-structure interaction, J. A. S. A., 79, , (8) K. F. Graff, Wave Motion in Elastic Sloids, Oxford University Press, (9) M. A. Jaswon and G. T. Symm. Integral Equations in Potential Theory and Elastostatics, Academic Press, (10) T. Terai, On calculation of sound fields around three dimensional objects by integral equation methods, Journal of Sound and Vibration, 69, , 1980.