Transactions on Modelling and Simulation vol 7, 1994 WIT Press, ISSN X

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1 Vibration and sound radiation by baffled and constrained shells P.-O. Mattel Laboratoire de Mecanique et d'acoustique, C.N.R.S. 31 Chemin Joseph Aiguier, Marseille cedex 20, France ABSTRACT A cylindrical thin elastic shell is embedded in a fluid extending up to infinity. The interior of the shell is a vacuum. Two different problems are considered : the constrained shell, which is free to vibrate within and outside a finite region but is clamped on the boundary between them ; and the baffled shell, which is free to vibrate within the finite region but is rigidly clamped outside this region. The displacement of the baffled shell can be approximated by that of the constrained shell, restricted to the domain limited by the constraints. The formulae for the acoustic far fields are proposed. Numerical examples show that, for a steel shell embedded in water, the exact and approximate solutions agree. I-INTRODUCTION The sound field radiated by an embedded structure in a fluid can be calculated if the normal velocity of the external boundary of the structure is known. When the fluid surrounding the structure is a gas, like air, the structure response can be, more or less, approximated by its "in vacua" one for which powerfull methods are available [3]. When the density of the fluid increases, the loading cannot be described by a simple added mass or by radiation impedance [1]. The energy transfered into the fluid is not well taken into account. To give an exact description of the physical phenomenon, it is necessary to solve the complete problem : one differential equation for the structure displacement, one for the acoustic pressure and a condition to ensure the continuity of both the normal velocities of the structure and the fluid near it. Because there are no exact analytical technics for solving the problem of a baffled structure, numerical methods, like Finite or Boundary Element Methods, are required to describe accurately the solution. With these methods, the computational effort increase rapidly with increasing frequency. To avoid this

2 56 Boundary Element Method XVI problem, an approximate solution of the baffled fluid-loaded structure problem is proposed. This method, which is frequency independent and well adapted to parametric studies, gives accurate results even with "strong" loading. Section two is devoted to the governing equations. In the third paragraph, the exact equations are solved by a Boundary Element Method. This solution is used as a reference to point out the efficiency of the approximate method. The fourth section is devoted to an analytical method for the constrained shell. Then, the various expressions for the far field sound pressure are given (constrained infinite shell, exact and approximately baffled shell). A concluding section discusses the results. H-GOVERNING EQUATIONS Let ft be a bounded cylindrical surface, defined in cylindrical co-ordinate (x^, x^, x^ ) by -L<x^<L, 0<x^<2ir, x^=r. The boundary 6ft, of ft is defined by x^=±l, 0<x^<2ir, x^=r. Let us denote cylinders. by ft its complement, ft is composed of two half Two cases are considered. For the baffled shell, ft is occupied by a thin elastic vibrating shell and ft is assumed to be perfectly rigid while for the constrained shell, all the cylindrical surface ftuft is occupied by the shell. In the two cases, the shell is assumed to be clamped along 6ft. The exterior Z of the surface ft is occupied by a fluid extending up to infinity, the interior of the shell is a vacuum. The shell displacement is denoted by u = Mu^u^/u^] and the acoustic sound pressure by p. These functions are solutions of : Cu=F-P 0,0, lim p(q) Q-WEft on ftjft for the constrained shell on ft for the baffled shell (1-a) (A + k^)p(q)=0 V QEZ (1-b) lim 6^ p(q)=o^p0u Q-tMEft V M ftjft for the constrained shell (1-c) lim 6^ p(q) = 0 V M E ft for f the baffled shell V M E ft (l-d) Conditions for u on 6ft (1-e) Condition at infinity (1-f) where F is the shell excitation, harmonic, located in ft and

3 Boundary Element Method XVI 57 denoted by F = ^ [F^F^/F^]. For the sake of simplicity, the time dependence is omitted. C is the Donne 11-Mushtari operator and k=w/cg is the acoustic wavenumber The uniqueness of the solution is ensured by the boundary conditions and the condition at infinity. The mechanical parameters of the shell are : v the Poisson ratio, E the Young modulus and p, the density. The radius of the shell is denoted by R and its thickness by h. The mechanical parameters of the fluid are : c^ the sound speed and x^ the density. There is no source in the fluid. E- BOUNDARY ELEMENT METHOD FOR THE BAFFLED SHELL Let us denote by u^ and p^ the displacement and the acoustic sound pressure which satisfy the system constituted by the equations 1-a, 1-b, 1-d, 1-e and 1-f. This system is transformed into a system of boundary integral equations by using the Green's representation theorem. Because two different differential equations govern the pressure and the displacement, two Green's representations are used. ffl-1 -Green's representation of the pressure Let G be the Green function for the Neumann problem in Z. It satisfies the equations : (A + k" )G lim d X (0) = G(Q) = &(Q) 0 Q--(MEH Co ndition at iiif inity It is classical that the Green's representation of the sound pressure becomes : V V QGZ M on p = (M')G(M-M')ds(M' (2) hjn where, if (x^, x^,x^) are the co-ordinate of M, G is given by. im (X_ ) the Fourier series G = G^ (x^, x^ )e ^, the harmonics m = - oo are given by the inverse Fourier transform of (KR) and, Im ( K) >0. is the Hankel function of first kind ffi- 2 -Greenes representation of the displacement Because u^ is a discontinuous function of x^, the equation 1-a is written in the distribution sense. One has Eu, = F-p -S +S

4 58 Boundary Element Method XVI where S and S are sources located along the shell boundaries x = ±L. Theses sources are related to u and its successive derivatives with respect to x^. Their expression depend on the boundary condition for u. Now, let F^ be the Green's tensor of the Donne11-Mushtari operator which satisfy a limit absorption principle,- this tensor gives the response of an infinite in vacuo elastic shell to a Dirac excitation. F^ satisfy the equation CT^(M,M') = 8^, I in HUH and is easily obtained by direct and inverse Fourier transform. Each of its component is a sum of complex exponentials. The Green's representation of the displacement then becomes : u. b = f, b * (F-P. \ b -S +S l / (3) where * is a matrix convolution product. Each boundary source S introduce four unknowns. To determine them it is sufficient to write the boundary conditions for u. E-3-Fourier series expansion of the solution The problem is 2ir-periodic in x^, then one can expand the shell displacement and the pressure in Fourier series with respect to x : The component u^ are given by Ubm (^1 ) = ^bm (=1 ) " (xj :he angular components of F^, F, P and S. Then one obtains an infinite set of integral equations u (x ) = f (x. ) * (i- (x. ) -5 (x ) -S (x ) +s_ (x. ) } bm 1 Jam 1 \ m 1 ml ml miy -L = 0, u(±l) = 0, u^(±l) = 0, u^(±l) = 0 E- 4 - Boundary Element Method For each angular harmonic, the unknown functions u^ and p, are calculated by the simplest Boundary Element Method, say the collocation method [2]. Roughly speaking, the unknown functions are approximated by piecewise constant ones, the coefficients of the developments are the new unknowns and are estimated by requiring that the corresponding equations are

5 Boundary Element Method XVI 59 satisfied only at the so called collocation points. It is well known that the number of points depend on the wavenumber. The lenght of each interval must be less than the sixth of the wavelenght. And important point of this method lies in the fact that the acoustic and structural wavelenght are differents, this allows the use of two different meshes, one for the displacement and one for the acoustic pressure. Our numerical experience have shown that, out of the eigenf requencies, choose the tenth of the wavelenghts gives a good precision (less than one db), while close to an eigenf requency it is necessary to use a more precise discretisation. EZ-AN ANALYTICAL METHOD FOR THE CONSTRAINED SHELL Let us denote by u^ am p^ the displacement and the acoustic pressure which satisfy the system constituted by the equations 1-a, 1-b, 1-c, 1-e and l-f. As done for the baffled shell, this system is transformed into a system of boundary integral equations by using the Green's representation theorem. EZ-1-Green's representation of the pressure G is always the Green kernel for the Neumann problem in Z, defined in M-l. The Green's representation of the sound pressure is : ^ (M' )G(M-M' )ds(m' ) = (if ^u^^ *G. * Using this result in _equation l-a leads to an integrodiffential equation for u : ( -KR)u^ = F on OJQ (5) (R is a matrix operator defined by JR. =0 if (i,j) ^ (3,3) and f^ = uv^g * ( ) EZ- 2 -Green's representation of the displacement As done in paragraph E-2, equation (5) is written in the distribution sense. One has : (E+fR)u^ = F-S +S 4- where S and S are_ the sources located along the shell boundaries. Now, let F, be the Green's tensor of the operator (C+lR) which satisfies a radiation condition. This tensor, which gives the response of an infinite_ fluid- loaded elastic shell to a Dirac excitation, satisfy (DMR)]^ = & I in f2uq. The components of this tensor are only known by Fourier transform. The Green's representation of the displacement is : u\ = f, * (F-S +S ) where * is always a matrix convolution product. In order to

6 60 Boundary Element Method XVI evaluate the eight unknowns introduced by the boundary sources, it is sufficient to write the boundary conditions for u. Thus, for each frequency, one obtains a system of eight integral equations with eight unknowns. EZ- 3 -Fourier series expansion of the solution As done for the baffled shell, one can expand the shell displacement in Fourier series with respect to x^ : m=+oo r-n = 2- the component u^,^ are given by (X,) The expressions of F^ are obtained by inverse numerical Fourier transform. It is to be noticed that this quadrature is the only numerical difficulty of this method. It is to_be noticed that around the poles of the Fourier transform of F^,, an analytic integration must be performed. Then one obtains an infinite set of linear algebraic equations of order eight. 2- FAR FIELD SOUND PRESSURE The directivity diagram is defined as the ratio of the first terms of the asymptotic expansion of the pressure radiated by the shell and an unit point source. If (r,@,<p) are the spherical co-ordinate of a point of Z, the directivity diagram is n(0,<p) -- ;- / u^ I - where u^ 3m (C) ^ is the Fourier transform of u, 3ml (x. ). The Fourier transform of the displacement of the baffled shell is estimated from the discrete form by a sum of complex exponentials while these of the constrained shell is known analytically since F is known by Fourier transform. An approximate displacement of the baffled shell is proposed as the restriction of the displacement of the infinite constrained shell to the domain limited by constraints. The displacement of this finite constrained shell is denoted by -* u. t m (x, 1 ). It is defined, if Xn (*i ) is the characteristic function the domain D, by ^f m ( *! ) - XQ (*i)ucm (*i )- Then, the directivity diagram of the approximate solution for the baffled shell is calculated by estimating the Fourier transform of ^3 f m (^i) It is easy to shoow that it is given by a convolution product, evaluated numerically without any particular difficulty, between the sine function and u(^) : cl sin(2ttl(r-cr) ) u. (a)- dcr. I::

7 2T-NUMERICAL RESULTS Boundary Element Method XVI 61 Figures 1 and 2 present two directivity diagrams of a cylindrical steel shell of lenght 4 m., radius 3 m., thickness 0.06 m., E=2.261(r*Pa, v=0.28 and x=7850 Kg/rrr. The shell is immersed in water with c^=1500 m/s and ^=1000Kg/rrr The excitation force is a Dirac delta distribution located on the shell at^ the origine, 0=90 and <p=0. The curve plotted is 20 Log For the two figures, the directivity diagrams are plotted for <p fixed and 0 variable and are in the plane of the excitation. For the figure 1, the frequency is 60 Hz and coorespond to a nondimensional frequency ka=l, a=2l, the maximum level is 47 db while for the figure 2, the frequency is 600 Hz (ka=10) and the maximum level is 57 db. On each curve, the continuous curve is the direvtivity diagram of the baffled shell, the discontinuous curve is that of the infinite constrained shell and the dot curve is the approximation of the baffled shell, the finite constrained shell Figure 1 : directivity diagram for ka=l continous curve : baffled shell discontinous curve : infinite constrained shell dot curve finite constrained shell

8 62 Boundary Element Method XVI Figure 2 : directivity diagram for ka 10 As figure 1 CONCLUDING COMMENTS here The results presented show that the method proposed gives accurate results. The main advantage of this method is to reduce the amout of numerical computation. In the cases presented here, for ka=l, the calculation time for the exact and approximate solution are comparable while for ka-10, for the baffled shell, the computational effort is considerable ; the computation of the drawing requires about four hours for the exact solution and less than an hour for the approximate one. REFERENCES 1. Crighton, D.J. 'The 1988 Rayleigh medal lecture : fluid loading - the interaction between sound and vibration' Journal of Sound and yibratzion, Voi 133(1), pp. 1-27, Kress, R. 'Linear Integral Equations' Chapter 13, Projection Methods, pp , Springer-Verlag, Berlin, Lesueur, C. Rayonnement acoustique de5 structures Eyrolles, Paris, 1988.