The vibrational response of a clamped rectangular porous plate

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INTRODUCTION
The vibrations of porous structures are important in problems of sound absorption and in the aeronautical industry. The vibration of porous beams and plates has been scarcely studied analytically to date. A recent and signi"cant result has been given by Theodorakopoulos and Beskos [1], who have extended the classical theory of thin rectangular plates to porous materials including Biot's stress}strain relations in porous media [2]. They have established two coupled governing equations and have given the solutions for a simply supported plate by extending Navier's algebraic solution [3] to the porous case. Making an additional assumption about the #uid}solid relative displacement and using an alternate form of Biot's relations [4], Leclaire et al. [5] have proposed two equations of equilibrium, which are valid in applications where the plate is thinner than any acoustic wavelength. The main advantages of the second model are the simplicity of the equations involved without a signi"cant loss of generality, and the possibility to interpret in a more physical manner the #uid}solid interactions associated with the vibration. While the study of boundary conditions other than simply supported from the equations of Theodorakopoulos and Beskos seems to be a di$cult task, the equations proposed by Leclaire et al. easily lend themselves to the generalization of the analytical and numerical methods usually applied in the classical theory of plates. An overview of the most common methods can be found in the book by Szilard [3]. The generalization of one of these methods is the main subject of this article. In the present study, Galerkin's variational method is applied to porous plates where a classical set of trial functions obtained from the linear combination of trigonometric and hyperbolic functions is chosen. The method can be applied for any boundary condition involving simply supported, clamped or free edges. The particular case of all four edges fully clamped is treated and a comparison, in this con"guration, between numerical and experimental results is made for three di!erent porous plates.
Another advantage of the proposed model is that its output parameters are the plate lateral de#ection w Q and the #uid}solid relative displacement w. It can be shown that the simple sum of the time derivatives of w Q and w provides the total velocity "eld in the vicinity of the surface of the plate. The total velocity "eld is needed in the evaluation of the radiated sound power and the global vibroacoustic indicators. In problems of sound radiation by non-porous plates and structures, recent and re"ned models include the surface acoustic pressure generated by the plate motion and the e!ect of the surrounding #uid. The present article is one of the "rst to study the problem of vibration of porous plates including Biot's equations of relative #uid}solid motion in the pores. Although the #uid}solid interaction is studied, the contribution of the radiation impedance on the solid displacement is neglected in a "rst approximation. As we are concerned with problems where the #uid is air, the #uid}solid interaction is not symmetrical in the sense that the radiated sound is entirely generated by the solid motion, while the contribution of the #uid on the solid vibration is neglected. This corresponds to the &&light'' #uid approximation. Obviously, in the case of an acoustical excitation, the plate can be excited acoustically or mechanically, and the #uid does generate the solid vibration. However, this action is included in the excitation term and no other external force due to the radiated acoustic "eld is taken into consideration. Although this assumption is made, a reasonably good agreement can be found on the plate de#ection between the experimental data for di!erent materials saturated by air and the calculated in vacuo response of the plate.
The equations proposed by Leclaire et al. [5] are brie#y recalled in section 2. In section 3, these equations are rewritten in an integral form, as required in Galerkin's variational method, the mathematical form of the solutions is chosen, the coe$cients involved in these solutions are calculated from the boundary conditions at the edges and explicit forms are given for w Q and w. The evaluation of the di!erent elastic coe$cients involved in the model is discussed in section 4. Finally, an experimental validation of the theoretical results is proposed in section 5.

EQUATIONS OF EQUILIBRIUM FOR A POROUS PLATE
Two simple equations describing the dynamic equilibrium of thin rectangular porous plates have been proposed by Leclaire et al. [5] for a plate of thickness h subjected to a load q and with a pressure di!erence of P"P(!h/2)!P(#h/2) in the #uid between the two surfaces: Here w Q is the plate de#ection, w is the #uid}solid relative displacement, D is the #exural rigidity, and M are Biot's elastic parameters [6], and D are, respectively, the densities of the #uid}solid mixture and of the #uid, m is the mass parameter introduced by Biot [4], given by D /, where is the tortuosity and is the porosity. The space derivatives are written with the help of the operators " ( ) and "*/*x#*/*y of the system of co-ordinates (x, y) while the double dots denote the second time derivative. In the "rst equation of equilibrium (or plate equation), (D# Mh/12) w Q represents the internal potential force (per unit surface) within the #uid-saturated plate, while the inertia terms h w K Q and h D w K , and the load q will be considered as external forces. Similarly, an internal force associated with the #uid}solid relative displacement may be de"ned and is given by Mh w Q , while the external forces can be taken to be hmw K , h D w K Q and P.

VARIATIONAL EQUATIONS
According to the virtual work principle, the condition of equilibrium of a system is that the total work performed by the internal and external forces during a small virtual displacement is zero: The virtual displacement and the virtual work are de"ned at constant time. They must be distinguished from the real displacement and from the corresponding work. The virtual work principle can be seen as a particular form of a variational principle formulated as According to this principle, a function F characterizing the evolution of a system from a perturbed state towards a stable equilibrium state is minimum (extremum in a more general manner) between z and z . In Galerkin's variational method, the function to be minimized is the total virtual work between the boundaries of the middle surface (S): The virtual work is calculated from a virtual displacement of the system and from forces obtained directly from the governing di!erential equation which is known a priori and not from the calculation of energies. Upon applying this method to a porous plate and referring to the forces de"ned in section 2, the internal work in the plate is given by the product of the potential force (D# Mh/12) w Q and the virtual displacement w Q : The external work is given by 3 Inserting the expressions for W G and W C into Equation (5) yields the variational equation of the plate: For a virtual displacement w, a second variational equation can be obtained for the #uid}solid relative displacement by making use of equations (2) and (5): In the following, sinusoidal dependence in time is assumed for all quantities so that the terms containing the time can be discarded, the second derivative introducing the squared angular frequency .

CHOICE OF THE MATHEMATICAL FORM OF THE SOLUTIONS
In the linear domain, the displacements are small and are assumed to be normal to the surface (a, b) de"ned by the lateral dimensions of the plate which coincide with the system of co-ordinates (x, y). The problem consists in "nding suitable expressions for w Q and w verifying the system of integral equations (8) and (9) as well as satisfying the boundary conditions at the edges of the plate. Various approximate functions have been tried in the case of non-porous plates where only one governing equation is considered, such as a linear combination of trigonometric and hyperbolic functions [7,8], polynomials (see reference [9] for example) or trigonometric functions [10]. The following approach is used in the present study: a possible solution for the solid de#ection is sought "rst. The general form of this solution is also applied to the other unknown w and to the excitation terms. All these terms are then inserted in equations (8) and (9) leading to new equations. Finding solutions to these is a veri"cation of their compatibility and provides the amplitude coe$cients of w Q and w. The form of the trial functions chosen is the combination of trigonometric and hyperbolic functions. As in the Rayleigh}Ritz method, the classical decomposition of the solutions on the basis of orthogonal eigenfunctions is used. The solid de#ection is written as Since the plate is rectangular, it is convenient to write each function PL (x, y) as the product of two independent functions called the beam function P (x) and L (y) of one variable only: The chosen form for the beam function is where P is a frequency parameter corresponding to the rth root of a characteristic equation. The mathematical form of this equation and the coe$cients C P , C P , C P , C P are imposed by the boundary conditions.

BOUNDARY CONDITIONS
The standard boundary conditions are applied to the porous solid which is considered as a homogeneous e!ective medium in Biot's approach of porous media. For a clamped beam of length a, the conditions at the edges are that the solid displacement and its "rst derivative are zero: w Q (0)"w Q (a)"0 and w Q (0)"w Q (a)"0, corresponding to edges without motion of translation or rotation. The determinant of the homogeneous boundary conditions is then obtained from the following equations: which are veri"ed for and for P satisfying the following characteristic equation or frequency equation: Numerical values for P can be found in reference [7]. The values of C P , C P , C P , C P and P are then inserted into the expression for the beam function P (x). The beam function in the y direction L (y) is deduced from the expression for P (x), where x, a and r are replaced by y, b and n.
If the beam is simply supported, C P "1, C P "C P "C P "0 and the frequency equation is sin( P )"0. The same generic expression can be used for P (x) in the case of other boundary conditions involving a combination of simply supported, free, clamped or guided edges. Each case is described by a di!erent set of coe$cients C P , C P , C P , C P and a frequency equation leading to di!erent values of P .

SOLUTIONS
In this study, one seeks solutions for w(x, y), w Q , w, q and P with the same mathematical form as w Q : w(x, y)" q(x, y)" 5 Here, the coe$cients =Q PL and = PL are the unknowns of the problem. After insertion into equations (8) and (9), the solutions, if they exist, must satisfy the following equations: The fourth and second order derivatives are, respectively, denoted by the exponents (IV) and (II). The eigenfunctions and their derivatives satisfy the orthogonality requirement (depending on the choice of eigenfunctions, this condition can be satis"ed approximately).
The terms with non-identical subscripts r, i and n, k are therefore zero or negligible and equations (15) and (16)  ?
Solving system (17, 18) for a couple (r, n) yields the coe$cients =Q PL , = PL and the responses are given by equations (10) and (11). For a simply supported plate, the integrals take a simple form and a solution similar to Navier's algebraic solution [3] for a porous plate [1] is obtained.
The coe$cients Q PL and P PL correspond to the external exciting forces applied to the porous plate and are determined by the way the plate is excited. Rigorously, the external forces are responsible for the motion of the plate, which in turn, contributes to the loading of the plate but this contribution is neglected in this study in a "rst approximation. The coe$cients Q PL and P PL can be written [3] as For a load F concentrated at one point (x , y ) of the solid surface (obtained with the help of a shaker for example), the forces Q PL I I and P PL I I appearing in equations (17) and (18) are equal to F P (x ) L (x ) and 0 respectively. If the plate is excited acoustically by a plane wave of amplitude P G , the pressure applied on the #uid}solid mixture and the pressure di!erence in the #uid are uniformly distributed. In this case, with (r, n) odd [3]. From reference [11], the values taken for P PL are Q PL . From previous theoretical works, di!erent mechanisms of energy dissipation can easily be included in the calculations. Forces of viscous frictions in the pores can be introduced in equations (1, 2) and (8,9). These forces are derived from a dissipation potential and their e!ect is accounted for by considering the tortuosity to be complex and frequency dependent, where is the #uid dynamic viscosity and is the solid permeability. The function F( ) is the viscosity correction function and can be evaluated from a model [12]. This function is close to 1 in the low-frequency domain where this study is carried out. The dynamic tortuosity ( ) is inserted instead of into equations (9) and (18). Structural damping is included in the model by considering complex elastic moduli. For air-saturated materials, the energy loss by thermal exchanges between the solid structure and the #uid can be included by considering a dynamic compressibility [13]. The #uid compressibility appears in Biot's elastic coe$cients.

A NOTE ON BIOT'S ELASTIC COEFFICIENTS
Biot's coe$cients of interest in this study are and M and are given by [6] "(1#Q/R), M"R/, (22,23) where the elastic coe$cients Q and R can be written as In these expressions, K Q , K D and K are, respectively, the bulk moduli of the solid constituting the frame, of the #uid and of the frame at constant #uid pressure. The coe$cient c can be seen as a coe$cient characterizing the consolidation state of the heterogeneous material. Since K is less than the Hashin}Strikman upper bound Q K Q [14], c is between 0 (non-consolidated materials) and 1 (consolidated materials). The following relation is further obtained: and the bounds for are (consolidated materials) and 1 (non-consolidated materials). It is interesting to note the similarity for c"0 between the expression for Q and R and the Hashin}Strikman lower bound. Biot's elastic coe$cients can be determined from hydrostatic experiments [15]. Since the #uid is air, the approximation K D K Q , K can be made. The materials are fairly rigid and so we shall also consider that c"1 and " in the numerical computation. As a result, the coe$cients of interest are approximated by It may be noticed that the #exural rigidity or bending sti!ness is a key parameter in the problems of plate vibration. It involves Young's modulus E and the Poisson ratio of the solid frame. The coe$cient K given as a function of E and by K"E/3(1!2 ) must therefore be evaluated in any case. As a consequence, the parameters c, and M can be determined without making approximations if the bulk modulus K Q of the solid constituting the porous frame is known.

PRELIMINARY TESTS
Three plates were studied. Their properties are summarized in Table 1. The plates were obtained by consolidating the particles of a material with a binder. The CoustoneR plate contains #int particles with a mean grain size of about 1 mm and an epoxy rubber binder. The other two plates designated as G foam and Y foam are fabricated from particles of plastic foam obtained from recycled car dashboards. Their grain size varies from a few tens of m to 5 mm and the binder is manufactured by Hyperlast LTD.S For each material, the permeability was measured from #ow resistivity tests. The tortuosity has not been measured but estimated from Berryman's formula [16] "(1#1/)/2 as a value between 1)2 and 1)8 depending on the porosity. It was noticed that this coe$cient does not have a strong in#uence on the response of the plates studied. The numerical value of Young's modulus used for the Coustone is the one provided by the manufacturer. For the dashboard foams, E was evaluated experimentally from a simple experiment of bending of a cantilever beam submitted to a known force at the free end and from the corresponding de#ection formula.

NUMERICAL RESULTS
The plate response to a uniform acoustical excitation was calculated from the variational method described in section 3. The Gauss integration method with 40 terms of Legendre polynomial was used in order to obtain a fast and accurate evaluation of the integrals (I )}(I ). The response was calculated at the centre for three plates. The values for the density and the dynamic viscosity of air are D "1)2 kg/m and "1)839;10\ kg/m/s respectively. A typical value of 0)35 is taken for the Poisson ratio for all materials. In a "rst step, the computation included only the energy loss by viscous friction between the solid and the #uid. From the experimental results, it appeared that the observed damping could not be explained by this mechanism alone. The viscoelastic damping in the solid structure plays an important role and complex Young's modulus and the Poisson ratio had to be considered. An imaginary part between 10 and 15% of the real part has been taken for E and for (Table 1). Since a detailed analysis of structural damping is not the subject of this article and for the sake of simplicity, these values were chosen arbitrarily and independent of frequency, in order to match the experimental results on the damping.

EXPERIMENTS AND COMPARISON
The experimental set-up is shown in Figure 1. A porous panel clamped at the four edges was excited by the low-frequency noise produced by a loudspeaker of 40 cm diameter.
9 Figure 2. Calculated and measured de#ection of the G foam: **, theory; ))))))))), experiment. Figure 3. Calculated and measured de#ection of the Y foam: **, theory; ))))))))), experiment. E!ective clamping was achieved by using two heavy steel frames of about 25 kg each such that the mass of the sample was only a few per cent of the mass of the frames. The plate de#ection was measured by using an accelerometer of negligible mass (a few grams). With the help of a microphone at 1 cm from the surface, the amplitude of the incident pressure P G at (x"a/2, y"b/2) was evaluated to a value of about 0)1 Pa. The theoretical and experimental plate de#ections are plotted in Figures 2}4 versus frequency for three di!erent porous plates. Since the absolute value of the experimental de#ection was not available, an arbitrary reference value was taken for the theoretical calculation chosen to enable the comparison. The experimental response was measured at (x"a/2, y"b/2). Since the plate is excited by a uniform pressure, only the odd modes with an even number of nodal lines are 10 Figure 4. Calculated and measured de#ection of the Coustone: **, theory; ))))))))), experiment.

TABLE 2
First theoretical and experimental resonance frequencies (in Hz) for the G foam plate  excited. A reasonable good agreement is found when comparing the experimental and theoretical frequencies for these modes (see Tables 2}4), with a maximum error around 10% in the range of samples studied. The resonance frequencies are strongly dependent on Young's modulus E, the Poisson ratio and the porosity. In the case of the G foam, the "rst resonance, located at about 10 Hz, was in the low-frequency limit sensitivity of the microphone so that the experimental results are not reliable below this frequency.
The general shape of the experimental curves compares well with the theoretical curves. The discrepancies can be due to the fact that frequency-independent elastic moduli have been considered in the model in this simple analysis. In addition, it has been assumed for the calculations that the incident pressure "eld was uniformly distributed over the surface of the plate in the whole frequency range, while it is not perfectly constant in the real experimental con"guration. 6. CONCLUSION The de#ection of a porous plate and the relative #uid}solid displacement are important in the study of the e!ect of resonances on the surface impedance of porous layers. In this article, a variational method for solving the plate equations has been used and the solutions have been given for di!erent porous materials with the four edges clamped. The e!ect of #uid loading has been neglected and no external force other than the excitation terms has been taken into consideration. This e!ect is closely related to the calculation of global vibroacoustic indicators: i.e., the mean square velocity, the radiated sound power and radiation e$ciency. Nevertheless, the "rst comparison between experimental and theoretical results on the plate defection is reasonably good if the viscoelastic damping in the solid is accounted for.