Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, 13 ICA 13 Montreal Montreal, Canada - 7 June 13 Structural Acoustics and Vibration Session 4aSA: Applications in Structural Acoustics and Vibration II 4aSA1. Analysis of the acoustic scattering from a submerged bilaminar plate Sabih Hayek* and Jeffrey E. Boisvert *Corresponding author's address: Engineering Science and Mechanics, Penn State University, 1 EES Bldg, University Park, PA , sihesm@engr.psu.edu The acoustic scattering from a submerged finite bilaminar rectangular elastic plate is modeled using the exact theory of three-dimensional elasticity. The two layers of the composite bilaminar plate have the same lateral dimensions, but have different thicknesses and material properties. The plate is set in an infinite rigid baffle and is coupled to a different acoustic medium on its two surfaces. The plate is insonified by an acoustic plane wave. The farfield backscattered and forward scattered waves are computed for various layer thicknesses and elastic material properties in contact with air or water. First, the scattering from a uniform finite, baffled, steel plate elastic plate was computed. A bilaminar plate was also analyzed with one layer made of steel and the other made of a damped elastomer. The backscattered and forward scattered acoustic farfield spectra versus frequency were computed at the on-axis point receiver due to a normally incident plane wave. The directivity functions for a normally incident plane wave insonifying the steel- or elastomer- side of the plate were computed for a range of frequencies, with either water or air backing. Published by the Acoustical Society of America through the American Institute of Physics 13 Acoustical Society of America [DOI: 1.111/ ] Received 17 Jan 13; published Jun 13 Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 1

2 INTRODUCTION The back- and forward-scattering from infinite multi-layered plates were discussed by many authors. In most of these, the elastic plates were modeled by the classical Kirchoff or the Mindlin plate theory [1-5]. The scattering from infinite elastic plates with a line discontinuity using Mindlin plate theory were discussed by few authors [6-8]. In this paper, a rectangular bilaminar plate consists of two perfectly bonded elastic plates, each having a different thickness and material properties. The bilaminar plate s two layers are modeled by elasticity theory. The bilaminar plate is set in an infinite rigid baffle and is in contact with a different fluid medium on each of its faces. The bilaminar plate is insonified by an acoustic plane wave applied to either of the two faces of the plate. The boundary conditions of the bilaminar plate are: stress-free as well as vanishing in-plane displacements on its periphery. This means that the entire bilaminar plate is free to move in the normal direction in response to acoustic surface pressures. 1. THEORY OF ELASTIC PLATES IN 3-D ELASTICITY Elastic waves propagating in 3-D elastic plates satisfy the vector PDE for wave propagation in an isotropic elastic medium, i.e. u u ( ) ( u ), (1) t where and are the Lame elastic constants, and is the plate material density. The vector wave solution can be obtained by using the Helmholtz decomposition in terms of three elastic wave potentials for the vector displacement field u N M, 1 () N ( ez ), M ez, where the direction z is chosen as the coordinate normal to the plate surface. Here represents the dilatational i t wave potential, and and represent the two shear wave potentials. Assuming e time dependence throughout this paper, the three potentials satisfy their respective scalar Helmholtz wave equations,,, (3) where and represent the dilatational and shear wave numbers, respectively, defined by / cd / cs (4) 1 1 cd [( ) / ] cs ( / ), where is the circular frequency, and c d and c s are the dilatational and shear wave speeds, respectively.. PLATE GEOMETRY AND BOUNDARY CONDITIONS Consider a rectangular plate, where the x and y coordinates lie in the plane of the plate, and the z coordinate being normal to the plate surface. A finite elastic bilaminar plate is made up of two plates whose lateral dimensions ( a and b ) are identical but their thicknesses ( h 1 and h ) are different, see Fig. 1. Each plate has a different set of physical properties, i.e. densities, s 1 and s, Lame constants 1, 1 and,. The two plates are totally bonded at their common interface, and the bilaminar plate is pinned in the plane of the plate, so that the displacements normal to the perimeter (either u x or u y ) vanish. The bilaminar plate is set in an infinite rigid baffle, and each plate is in contact with a different acoustic medium and is excited to vibration by an incident acoustic pressure wave as shown in Fig. 1. The two semi-infinite acoustic media #1 and # satisfy the scalar Helmholtz wave equation. Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page

3 FIGURE 1. Bilaminar plate geometry Furthermore, the bilaminar plate is free of shear stresses along the same perimeter. These boundary conditions can be enumerated as follows: 1, 1, u xy x, a yx y, b 1, 1, xz x, a yz y, b 1, 1, x u x, a y y, b The three elastic potentials for each plate satisfying the BC above have the form: nx my cos cos A sin k z z B cos k z z n m a b nx my cos cos C sin k 3 z D cos k 3 z (6) n m a b nx my sin sin E sin k 3 z F cos k 3 z, n m a b n m n m where k z, k 3. a b a b The Fourier solutions in the x and y coordinates constitute a set of orthogonal functions on the ranges a and b, respectively. From Eq. (6) each plate has six unknown Fourier constants. Since the two plates are totally bonded at their interface, complete continuity is enforced on the three displacements and the three stresses that are the components of the traction vector on that interface. These are stated as: zz1 1/ zz u z h z h/ z1 u zh1/ z zh/ u u (7) zx1 zh1/ zx zh/ x1 zh1/ x zh/ zy1 zh1/ zy zh/ y1 zh1/ y zh/ For an incident plane acoustic pressure wave, the backward and forward scattered waves are given as: u u (5) Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 3

4 p( xyz,, ) p ( xyz,, ) p( xyz,, ) p ( xyz,, ) p ( x, y, z) p e inc p ( x, y, z) p e r 1 ik( l1x m1y n1z ) it ik( lxmy nz ) it p ( x, y, z) p ( x, y, z) e sc it p ( x, y, z) p ( x, y, z) e, t inc r sc sc it (8) where l 1 l m 1 m n 1 n sin cos, sin sin cos. The boundary conditions at the interface of plate #1 and acoustic medium #1 enforces continuity of the plate s normal velocity to the normal acoustic particle velocity, the plate s normal stress to the acoustic pressure, and the that the shearing stresses vanish at the interface of the plate and acoustic medium u zx1 zh1/ zy1 zh1/ zz1 zh1/ 1 z z1 zh1/ p 1 p1 z xa, y b 1 z elsewhere, (9) where p 1 is the acoustic pressure on the surface of plate #1. Similarly, the boundary conditions at the interface of plate # and the acoustic medium # are expressible as u zx zh/ zy zh/ zz zh/ z z zh/ p 1 p z xa, y b z elsewhere. (1) Substitution of the wave potentials (6) into the six interface BCs (7) yields six homogeneous algebraic equations on the twelve Fourier coefficients. Substitution of the wave potentials of Eq. (6) into the first two BC of Eqs. (9) and (1) satisfying vanishing shear stresses results in yet four more homogeneous equations on the twelve unknown Fourier coefficients. 3. SEMI-INFINITE ACOUSTIC MEDIA Since the finite plate is in contact with semi-infinite acoustic media, one needs to perform two-dimensional Fourier transforms on the x and y coordinates. Applying the double Fourier transforms on the acoustic wave equations and the second and third boundary conditions of Eqs. (9) and (1) yields an expression for the BFS acoustic pressures in terms of the Fourier components of the respective plates normal displacements where ab ( ) ru sc rs t r s ' ik R s v e p ( x, y, z) Q cos cos dudv, ' a b R (11) Q rs are the Fourier coefficients of the plate normal surface displacements u z ' R ( xu) ( yv) z. Substituting the elastic potentials in Eq. (6) into the formula for the normal stress zz ( ) and the expression for p from Eq. (11) into the second BC of Eqs. (9) or (1), and using the orthogonality of the sc t eigenfunctions, results in the following equation for the normal stress components in terms of the Fourier component of the plate normal displacements: i ( ) cp c () R QrsZrs f. (1) a r s and Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 4

5 ( ) In Eq. (1) R represents the Fourier component of the normal stress zz, b/ a denotes the plate aspect ratio, () and f represents the Fourier component of the surface acoustic pressure n i l1 m i m1 () p h ( 1) e 1 ( 1) e 1 f n m lm 1 1, a (13) n l1 m / m 1 1/ where a/ cp, cp ( E / (1 )) s and cp / c. The expression Z rs in Eq. (1) represents the mutual modal acoustic impedance of the, mode with the rs, mode defined in an integral form 11 i R i ( ) e Zrsmn Frn ( u) Gsm( v) dudv, (14) R where R u v, and F rn and G sm are defined by 1u F ( u) cos r( uu )cosnu du rn 1v G () v cos s( vv )cos mvdv. sm These integrals were evaluated numerically, and it was shown by Sandman [9] that for the cross-coupled modes, ( ) ( ) Z rsmn is negligible when r n and s m. Thus, only the self impedance components Z mn were included in this paper. Finally, there results 1x 1 system of non-homogeneous equations on the twelve unknown Fourier constants of the plate potentials. Solving these equations for the set of twelve constants, one can substitute these for the th mn component of the three displacements and six stresses at any point in the plates. These constants then can be substituted in Eq. (11) to evaluate the radiated acoustic pressures in media #1 and #. 4. FARFIELD PRESSURE The expression in Eq. (11) gives the radiated acoustic pressures in acoustic media #1 and #. To obtain a formulation that is valid in the farfield, one can first convert the Cartesian coordinates to spherical ones, with an origin at (,, ). Performing approximations on the integrand, one obtains an expression for the farfield BFS acoustic pressures as: (15) ( ) psc R t e pa i R/ a a sin sincos sin sinsin f ( ) sincos sinsin Q s a f 4 ( ) sin sincos Q s n1,,.. n,1,.. m,1,.. m1,,.. n m 1 ( 1) exp( i sincos ) 1 ( 1) exp( i sinsin ) n sin cos m / sin sin (16) R () () p r p inc e ap i R/ a, where and are the spherical Eulerian angles, and R is the radial distance to an observer. The farfield expression is split so that the contribution of the mode is separated, it being the largest contributor to the farfield, and has a directivity function that is the same as that of a rigid piston i.e., two sinc functions. Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 5

6 5. NUMERICAL RESULTS AND DISCUSSION First, to examine the role the thickness of the plate plays in the scattering from a monolithic plate, numerical results were obtained for a bilaminar plate made of two identical steel plates with h 1 / ah / a.5 and.1, resulting in a monolithic plate with h/ a.1and.. These ratios span the range from a very thin plate to a fairly thick one. The damping factor used was.1 with the plate aspect ratio b/ a. Results are initially obtained for normal incidence only with water on both sides of the plate. For a thin plate, h/ a.1 the normalized scattering spectrum is shown vs. frequency in Fig. a. The spectra are almost identical for very high frequencies, indicating that the two plate surfaces were displaced equally, and hence the acoustic scattering from both surfaces of a thin plate is the same. The farfield directivities at 3 (Fig. b) and at =1 (Fig. c) show the diffraction caused by low order modes. FIGURE. Farfield response of a submerged monolithic steel plate with b/a= and h/a=.1, (a) on-axis farfield acoustic pressure, (b) directivity at 3, (c) directivity at 1 For a thick steel plate, h/ a.1, Fig. 3a shows the back- and forward-scattered pressures could be as much as 3 db different and the spectra have maxima and minima corresponding to thickness symmetric and anti-symmetric resonances and anti-resonances of the thick plate. The directivities at anti-resonance frequency =17.4 (Fig. 3b) and at resonance 34.8 (Fig. 3c) show the high number of modes contributing to those directivities. Consider now a bilaminar plate made up of an elastomer bonded to steel, with the elastomer layer being three times the thickness of the steel such that h/ a.1, with the elastomer having a high damping coefficient of.3. The frequency spectra for a bilaminar plate with the incident wave impinging on the steel side (Fig. 4a) shows that there are no resonances or anti-resonances. This is primarily due to the high polymer damping. It also shows that the steel plate drives the much softer polymer to high response, resulting in a much higher forward scattering levels. The directivities at =3, 1 (Figs. 4b, 4c) show almost an omni-directional or low frequency diffraction at that mode order. When the incident wave impinges on the polymer side (Fig. 5a), the soft polymer is driven to a high response resulting in a much higher backscattered pressure by as much as 11 db. The directivities for this case show diffraction effects at a higher polymer mode order (Figs. 5b, 5c). In conclusion, the thickness modes for thick plates have a very high influence on the BFS spectra and the directivities of monolithic plates. For a steel-elastomer bilaminar plate, the BFS pressures are highest on the elastomer side, and the spectra exhibit no thickness resonances and anti-resonances. Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 6

7 FIGURE 3. Farfield response of a submerged monolithic steel plate with b/a= and h/a=.1, (a) on-axis farfield acoustic pressure, (b) directivity at 17.4, (c) directivity at 34.8 FIGURE 4. Farfield response of a submerged bilaminar steel-elastomer plate with b/ a, h / a.75 and h / a.5, 1 excitation on steel surface, (a) on-axis farfield acoustic pressure, (b) directivity at 3, (c) directivity at 1 Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 7

8 FIGURE 5. Farfield response of a submerged bilaminar steel-elastomer plate with b/ a, h / a.5 and h / a.75, 1 excitation on elastomer surface, (a) on-axis farfield acoustic pressure, (b) directivity at 3, (c) directivity at 1 ACKNOWLEDGEMENT Work supported by NAVSEA Division Newport under the ONR/ASEE Summer Faculty Program. REFERENCES 1. Ko, S. H., Flexural wave baffling by use of a viscoelastic material, J. Sound and Vib.75, (1981).. Stepanishen, P. R., and B. Strozeski, Reflection and transmission of acoustic wideband waves by layered viscoelastic media, J. Acoust. Soc. Am. 71 (1), 9-1 (198). 3. Rudgers, A.J., Equivalent networks for representing the two-dimensional propagation of dilatational and shear waves in infinite elastic plates and in stratified elastic media, J. Acoust. Soc. Am. 91 (1), 8-38 (199). 4. Strifors, H. C., and G. C. Gaunaurd, Selective reflectivity of viscoelastically coated plates in water, J. Acoust. Soc. Am. 88 (), (199). 5. Ko, S.H., Reduction of structure-borne noise using an air-voided elastomer, J. Acoust. Soc. Am. 11 (6), (1997). 6. Woolley, B.L."Acoustic scattering from a submerged plate I. One reinforcing rib," J. Acoust. Soc. Am. 67, (198). 7. Seren, C., and S. I. Hayek, Acoustic diffraction from an insonified elastic plate with a line discontinuity, J. Acoust. Am. 86 (1), (1989). 8. Pathak, A. G., and P. R. Stepanishen, Acoustic harmonic plane wave scattering from infinite elastic plates with a line impedance discontinuity, J. Acoust. Soc. Am. 94, , (1993) 9. Sandman, B. E., Motion of a three-layered elastic-viscoelastic plate under fluid loading, J. Acoust. Soc. Am. 57 (5), (1975). Proceedings of Meetings on Acoustics, Vol. 19, 6555 (13) Page 8