Acoustic finite element formulation of a flexible porous material—a correction for inertial effects

Journal of Sound and Vibration (1995) 185(4), 559–580

ACOUSTIC FINITE ELEMENT FORMULATION OF A FLEXIBLE POROUS MATERIAL—A CORRECTION FOR INERTIAL EFFECTS P. G¨ The Aeronautical Research Institute of Sweden, S-161 11 Bromma, Sweden (Received 6 October 1992, and in final form 5 January 1994) A weighted residual statement of the partial differential equations of acoustic wave propagation through a porous material, with effects of flexibility taken into account through an inertial correction, is discussed and shown to result in a symmetric finite element formulation of the coupled ‘‘fluid in porous medium’’–structure interaction problem. The porous finite element equations are applied to the problem of low frequency transmission through a plane, flat double wall, and the effects of filling the cavity between the panels of the double wall with a porous absorbent and changing the mass density of this material are discussed. 7 1995 Academic Press Limited

1. INTRODUCTION

Porous materials, of both fibre and foam types, are common in many engineering applications, such as thermal insulation, e.g., in buildings, cars, aircraft, etc. Owing to the specific character of these materials, they may also have an additional advantage in terms of reducing noise. The basic property of these materials that is used to reduce noise is that the acoustically induced fluid flow through the material is resisted owing to the frictional drag created by the fibres or cell walls on the fluid. Through this mechanism, energy is absorbed from the acoustic wave and converted into heat. Traditionally, porous materials have been used to treat noise problems in the higher frequency regimes; i.e., for air from 200 Hz and upwards. The reason for this is that the absorption of energy in the porous material decreases with increasing wavelength of the fluid motion, unless the thickness of the porous layer is increased in the same proportion. There is, however, evidence of low frequency effects on noise transmission through double wall structures, even for materials with a rather low density. The study of porous materials and their acoustic properties started with the work of Rayleigh more than a hundred years ago [1]. Zwikker and Kosten wrote several papers (see, e.g., references [2, 3]) and a book [4] on the theory of porous materials. More recently, Mechel [5] has written a set of books giving a summary of contemporary knowledge. Interest in these materials and their properties is continuously growing. New materials as well as analytical methods capable of modelling their behaviour are being developed. One such method that has been the subject of several papers is the finite element method; i.e., the propagation of sound waves inside a porous material is modelled with discrete continuum models similarly to the ordinary modelling of structures, etc. Craggs [6] started a line of work with the finite element method, and has demonstrated its possibilities in the area of acoustic wave propagation in porous materials in several papers [7, 8]. Christiansen and Krenk demonstrated a recursive solution technique for problems with rigid 559 0022–460X/95/340559+22 $12.00/0

7 1995 Academic Press Limited

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. ¨

porous materials [9]. Craggs [10] studied a model of a porous foam material, including the frequency dependence of the material properties but excluding effects of fibre motion. Most of these works concerning the finite element method, however, have dealt with a rigid, incompressible material model; i.e., all flexibility effects of the material were neglected. Such models, which are relevant for heavy materials or at high frequencies, may have a limited validity for the types of materials found in aerospace applications. Since these materials must be as light as possible, there is a need to extend the porous material models to take into account the flexibility of the solid part of the material. A first attempt to do this with a finite element formulation would be to adopt a simple correction for the inertial effects corresponding to the motion of the solid frame of the material; i.e., to treat the material as limp. Such models have been investigated by several authors; e.g., Beranek [11], Kosten and Jansen [12] and Ingard [13]. The validity of this assumption is based on the fact that the wavelength of the motion of the fibres is larger than the thickness of the absorbent. For a fairly heavy porous material, 140 kg/m3, however, it was shown that the equivalent modulus of elasticity is, rather low based on the wave speed obtained in measurements; see reference [14]. In this investigation the wave speed was measured to about 60 m/s; i.e., around one-fifth of the speed of sound inside the porous material. For thick absorbents this will be a serious limitation to the use of the limp material model, which only may be overcome by a formulation describing the full coupled fluid–frame structure of a porous material. A finite element formulation of this extended theoretical treatment will be the subject of a forthcoming paper. In the present paper a finite element formulation of an extended porous material model is discussed and an application to a double wall system with a porous absorbent is discussed. The effects of the inertial correction will be shown and identified, for materials differing in mass density. The primary concern will be the behaviour of the porous material at low to medium frequencies. The finite elements developed from this theory have been applied to an aircraft double wall transmission problem [15], and to a plane double wall facing a reverberant chamber [16]. These particular applications, both having a discrete tone source, in the form of a propeller for the aircraft application, showed a clear influence of the porous filling on the transmitted noise. This indicated that the transmission through the double wall system was substantially changed. The background of this effect, appearing for low frequencies, a light material (10–19 kg/m3 ) and for a narrow cavity, was not explained in references [15] and [16]. In the present paper the discussion is continued with the emphasis placed on the theory behind the elements and on the behaviour of the porous material, within the assumptions used both here and in the applications discussed above. As an application, a similar but more simple example of a flat double wall with and without porous filling will be discussed. The objective of this application will be to explain the behaviour observed for the aircraft model and the flat double wall and, more generally, the effects of the bulk mass density of the porous absorbent. 2. BASIC THEORY

The finite element formulation derived in this paper is based on the partial differential equations formulated by Ingard [13]. The developed and implemented finite element formulation, which was used in the analyses discussed above, is based on the previous work by Sandberg and Go¨ransson [17], giving rise to a symmetric weighted residual statement for the damped coupled fluid–structure interaction problem. The unknown quantities for the porous medium are the acoustic pressure and the fluid particle displacement potential. Since the range of frequencies suited for the finite element method is low to medium frequencies, the model used for the porous medium is one of a homogeneous fluid, and the

     

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equations thus derived have to be understood as relations that give the wave propagation in a spatial mean value sense. This approach is justified provided that the sizes of the irregularities in the arrangement of the pores are of much smaller dimensions than the wavelength of sound inside the material. Since the upper frequency ranges are not likely to be analyzed by FEM, this restriction is justified in a formulation as proposed here. The theory discussed in this paper is valid under the usual assumptions concerning linear acoustic wave propagation; i.e., all dependent quantities represent small fluctuations around a static reference value. The stiffness of the porous material is neglected, the flexibility being introduced as an inertia term; i.e., the material is considered to be limp [13]. 2.1.   In the mathematical model of a porous absorbent some specific parameters are needed to describe the macroscopic properties. The frequency range of interest is stipulated as being low; i.e., the increase in temperature caused by the compression and the decrease in temperature during expansion is regulated by the surrounding solid material in such a way that the temperature is kept constant. This assumption is based on the fact that the convection of heat in the solid frame is much more rapid than in the fluid in the pores. Thus the speed of sound inside the pores may be calculated as [4] cp2=

dp p RT0 = = dr r0 m

(1)

where r0 is the ambient fluid density, p is the acoustic pressure disturbance, r is the fluid density disturbance, R is the universal gas constant (=8·314 J/mol K), m is the molecular weight (=28·96×10−3 kg for air) and T0 is the ambient absolute temperature (=t°C+273 K). For higher frequencies, not considered in this paper, the speed of sound would once again approach the adiabatic value for free air, because the change in temperature occurring during the rapid expansion and compression is no longer absorbed by the fibres. Since the model used for the wave propagation inside the material is based on a specific volume displacement u, it is useful to introduce the volume porosity h, which is the ratio of the volume of air inside the pores to the total volume of the porous material. Since the porosity is always less than (or approximately equal to) unity, the volume displacement has to be smaller than the displacement of the fluid particles in the pores. This may be seen as an apparent increase in the fluid density, since only a fraction of the volume is filled with fluid. To distinguish the volume displacement from the fluid particle displacement in the pores, u' will be used for the latter quantity. The relation between the two is simply u=hu'. As thoroughly explained in reference [4], the fact that the diminished volume available for the fluid may be seen as an increase in density is not sufficient to describe the inertial forces related to the motion of the fluid particles. Effects of viscosity, the possible appearance of pores that may be accessed only with difficulty, etc., has to be accounted for by a structure factor Ks . Thus the inertial force acting on the fluid in a small volume of a rigid porous medium is Finertial=Ks

r0 1 2u , h 1t 2

(2)

where t is the time. The concept of a structure factor is based on viscous flow phenomena, accounting for an increase in apparent fluid density by 33% at low frequencies, and the presence of inaccessible

562

. ¨

pores, which for a randomly oriented material accounts for an additional increase by a factor of 3. Apart from these two effects, it is extremely difficult to estimate its value from the material characteristics or from measurements on samples of a given material. According to reference [4], the structure factor may take on any value between 1 and 7. For higher frequencies and for materials with a high porosity, the structure factor is likely to be close to unity, since the particular volume effects discussed above will be less pronounced. Apart from the stiffness, represented by the compressibility of the fluid, and the inertial force, equation (2), the fluid propagating through the porous material is also influenced by the solid parts of the material owing to viscous drag at the fluid–solid interfaces. This effect, which is the basic mechanism behind the absorption of sound energy in the material, is modelled as a specific viscous flow resistance f (see, e.g., reference [4]), giving rise to a drag force dependent upon the relative velocity of volume displacement of the fluid. This will be discussed in some more detail below. For some derivations it will be useful to introduce the viscous flow resistance related to the velocity of the fluid particles in the pores. This will in analogy to the fluid particle displacement be denoted by f' and is related to the specific flow resistance defined above by f'=hf. The material parameters discussed so far are all standard quantities used in the description of a porous material. For the purpose of the present paper, however, an additional parameter has to be introduced [13]: M=hr0+(1−h)rs ,

(3)

where M is the bulk mass density of the material and rs is the mass density of the solid frame. The material parameters presented above are in some cases difficult to determine. This is discussed in detail in reference [4], and further work concerning the flow resistance and the fluid density is discussed in reference [8], where finite element calculations have been performed to determine the frequency dependence of these parameters. The particular parameter entering the description discussed in the present paper, i.e., the bulk mass density of the material, may, however, be easily obtained, if the density of the solid frame is known, by weighing a sample of the material. 2.2.      An acoustic wave entering a flexible material will, partially because of the friction induced in the interior, set the material itself in motion. The amplitude of this motion depends on the induced frictional forces and the dynamic properties of the material. This means qualitatively, that some portion of the energy contained in the acoustic wave will be converted into heat, due to the friction, and some portion will be stored as deformation energy in the porous material (which in fact may also be converted into heat if the porous material has internal structural damping). In the present paper the effects of flexibility will be restricted to be of an inertial nature. For a rigid material, or equivalently a flexible material at high frequency, the viscous force acting on the fluid particles moving past the solid frame is Fviscous drag=f'

1u' . 1t

(4)

If, however, the solid frame were free to move without any restoring forces, with a displacement uf , then, from the force balance of the frame shown in Figure 1, one can see

     

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Figure 1. (a) Velocities of fluid particles and fibres of porous material. (b) Force balance in the case of a limp porous material.

that the displacement of the fluid particles could be related to the displacement of the frame through uf 1 = , u' 1−ivM/f'

(5)

where v is the frequency in rad/s, i=z−1, and a harmonic time dependence e−ivt has been introduced and will be used from here on. In the derivation of equation (5) it is implicitly assumed that the displacement vectors are parallel. This may be justified by the basic assumption that the only forces exciting the solid frame are due to the viscous drag force induced by the fluid motion. For materials with a high flow resistance or a low mass density, the two displacement fields will be almost equal in magnitude and direction. The movement of the fibres will modify equation (4), and the drag force acting on the fluid will now depend on the relative motion, between the fluid particles, u', and the fibres displacement, uf , so that Fviscous drag=−ivf'(u'−uf ),

(6)

which, by using equation (5) may be rewritten as Fviscous drag=−iv

f' . 1+if'/vMm'

(7)

As the frequency increases, this expression of course simplifies to equation (4). It is interesting to note that the bulk mass density serves to decrease the magnitude of the friction force. The maximum force amplitude occurs for either heavy materials or high frequencies, in both cases being equal to the force arising from a completely rigid material; see equation (4). Owing to the assumptions concerning the motion of the solid frame being induced by the fluid motion, the simple model of a porous material used in this paper does not allow for studies of the effects of mechanical excitation applied to the fibres. It is therefore assumed in the forthcoming discussion and analysis that there is no contact between the porous material frame and an eventual flexible vibrating surface. To incorporate this into the model would require a complete description of the coupled solid frame–fluid problem, which is beyond the scope of the present paper. 2.3.   To describe the propagation of sound through the porous material, the equation of continuity, in the absence of any sources of added mass, may be written as, p+

r0 2 c 9 · u=0, h p

(8)

. ¨

564

where 9 is the gradient operator, and a harmonic time dependence has been assumed and equation (1) has been used in the derivation. The factor density/porosity appears owing to the definition of the fluid displacement vector; see above. The momentum equation for the fluid may be formally written as 9p+Finertial+Fviscous drag=0,

(9)

or, after some simplifications explicitly written in terms of the fluid unknowns, and by using equation (7) for the viscous drag force and equation (2) for the inertial force, 9p−

v 2Ks r0 /h+ivf u=0. 1+ihf/vM

(10)

The effect of the porous material on the fluid motion appears as a modified fluid inertia and an extra force, both resisting the motion induced by the pressure gradient and both being complex-valued and frequency-dependent. Owing to these effects, it is clear that the fluid particle movement will vary with varying properties of the porous material; e.g., it may be deduced from equation (10) that, for a given pressure gradient and a constant flow resistance, the fluid particle displacement will be larger in magnitude for a light weight material than for a heavy material. In the same way a material with high flow resistance will have a lower fluid particle displacement magnitude than a material with a low flow resistance, everything else being kept constant. 2.4.   The solution describing the acoustic wave propagation in a porous material has to satisfy equations (8) and (10), as well as the appropriate boundary conditions for a given problem. In this paper three different types of boundary conditions will be discussed; namely, those corresponding to either a rigid wall or a flexible wall facing parts of the porous absorbent and also the boundary conditions at a free surface of the material. For this purpose the force equation (10) must be projected onto the normal to a surface: 1p v 2Ks r0 /h+ivf = u, 1n 1+ihf/vM n

(11)

where subscript n identifies the normal components and n is the normal to the surface, with Cartesian components n 1, n 2, n 3. 2.4.1. Rigid wall At a rigid wall the normal volume displacement of the fluid has to be equal to zero, which, according to equation (11), is equivalent to, 1p/1n=0,

(12)

which is the same condition that would hold for a free fluid in a non-porous cavity. 2.4.2. Free surface At a surface facing free air, the pressure and the density must be continuous going from the porous material into the free air outside the material. Since the mass flow over this boundary must also be continuous, this leads to the requirement porous ufree n =un

1+if/Mv , 0guporous n 1+ihf/Mv

(13)

where ufree is the normal displacement of the fluid particles just outside the absorbent and n uporous is the normal displacement of the fluid particles just inside the absorbent. Most n

     

565

materials used in different situations today have a rather high porosity, i.e., h is almost equal to unity, and hence the displacements would be approximately equal; see equation (13). 2.4.3. Flexible wall If the flexible wall is assumed to be separated from the face of the porous material by a small (in relation to the wavelength of the waves in the fluid) fluid-filled gap, the displacement of the fluid particles just outside the porous medium must be equal to the displacement of the flexible wall itself; i.e., wall porous uflexible =ufree . n n =gun

(14)

Here equation (13) hase been used. An alternative boundary condition would be also to require continuity in the solid frame displacement. Such a condition would, e.g., be relevant in the case of a porous material firmly attached to a flexible surface. However, this would require the introduction of solid frame displacement as an unknown quantity, which is beyond the scope of the present paper.

3. WEAK FORMULATION

The first step towards the solution of the wave propagation problem in a porous material is to write the governing equations in a weak form. There are several ways of doing this, but here a special form will be discussed that has the distinct advantage of ending in a symmetric equation system. This formulation was described in reference [17] for the undamped problem, and has recently been formulated for the case of a rigid porous material in reference [18]. The first step is to introduce a displacement potential for the fluid particle displacement vector: (15)

u=9c,

where c is the scalar displacement potential. Through the introduction of equation (15) into equations (8) and (10) and the use of Galerkin’s method, taking gw2 as the trial function for the acoustic pressure and g9w1 as the same for the displacement potential, one finds the weak form as

g

g9w1 · 9p dV−v 2

V

V

g

−

g

gh w p dV+ 2 c r0 2 V p

g

V

g

Ks r0 /h+if/v 9w1 · 9c dV=0, 1+ihf/vM

g9w2 · 9c dV−

g

gw2 9c · n dG=0.

(16)

(17)

G

Here Gauss' theorem has been used to obtain the last integral of equation (17), and the domain of integration is shown in Figure 2 together with the types of boundaries described above. The surface integral in equation (17) vanishes identically along the rigid part of the boundary, G rigid , since here the fluid displacement normal to the wall must be zero and hence also should the gradient of the displacement potential. Along the boundary to the free fluid, G free , the condition in equation (13) is satisfied by the choice of the weighting functions, and hence the continuity in the mass flow across this boundary leads to a cancellation of any contribution to the integral. Hence only the last boundary type, i.e., that of a flexible surface,

. ¨

Γ

ial

ter

Ω

a sm

rou

Γ

Po

fle xi bl e

"P

ur e

fr a ee ir "

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Rigid wall Γ rigid Figure 2. Integration domain for weak formulation. Boundary conditions of surfaces to integration domain.

requires some attention before the final symmetric coupled porous material–structure interaction problem may be derived. The basic assumption that the fluid is essentially non-viscous (even though the effect of the porous material is of a viscous character), expressed as the continuity relation in equation (14), together with the definition of the displacement potential, allow the integral along the flexible boundary to be written as

g

gw2 us · n dG.

(18)

G flexible

4. FINITE ELEMENT FORMULATION

To solve the acoustic problem, with the appropriate boundary conditions, some kind of approximation to the acoustic pressure, the displacement potential and the structural displacements, is needed. In the finite element method this is achieved by expanding the unknown function sought in a series of known and usually fairly simple functions, i.e., polynomials of different degrees: p(x)=s N j(x)pj , j

c(x)=s N j(x)cj , j

uls (x)=s N j(x)ulsj .

(19)

j

Here N j(x) is the interpolation function, corresponding to the jth node, pj is the unknown pressure at the jth node, cj is the unknown displacement potential at the jth node and ulsj is the unknown displacement component at the jth node of the structure, l=1, 2, 3. A description of the nature of the interpolation functions and the procedures related to these may be found in any standard textbook on finite element methods, and will not be repeated here. The solution of equations (16) and (17), in the weak or the integral sense, is now finally obtained by choosing a set of weight functions that will satisfy the condition of yielding zero weighted residual. One particularly convenient choice is, according to the method of Galerkin to choose the weight functions w1 and w2 from the same set of functions as the actual solution: w(x)=s N j(x).

(20)

j

The base functions are all real-valued, while the unknown acoustic and structural quantities are all complex-valued because of the damping properties associated with the

     

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porous absorbent. The submatrices arising from the integrals of equations (16) and (17) become (Mf )ij=

(Mc )ij=

g

g

gh i j 2 N N dV, r 0 cp V

(21)

gN iNj (u1sj n1+u2sj n2+u3sj n3 ) dG,

(22)

G flexible

(B)ij=

g

g9N i · 9N j dV,

(23)

V

(Kf )ij=

g

g

V

Ks r0 /h+if/v 9N i · 9N j dV, 1+ihf/vM

(24)

which, together with the structural equations, (Ks−ivaKs−v 2Ms )us−MTc p=Fs ,

(25)

where Ks is the structural stiffness matrix, Ms is the structural mass matrix, Fs is the load vector applied to the structure and a is the constant of proportionality assumed for the viscous damping of the structural part of the system, leads to the final system of damped coupled equations

8&

Ks 0 −Mc

' &

0 −MTc aKs 0 B −iv 0 BT −Mf 0

0 0 0

' &

Ms 0 0 −v 2 0 0 0

0 Kf 0

'9& ' & '

0 0 0

us Fs (v) c = 0 . 0 p

(26)

Here the elements of the fluid matrices are given by equations (21)–(24). The damping term due to the porous material is apparent only in the equation for the displacement potential, which should be compared with the case of damping introduced via an impedance boundary, which appears only in the equations related to the pressure in the fluid; see reference [19]. In both cases the influence on the structural displacements is only indirect; i.e., a vibrating structure will cause pressure fluctuations and corresponding fluid particle displacements, which in turn finally are resisted by the absorbent. This is of course a direct result of the assumptions involved in the model of the flexible porous material. From equation (26) it is also obvious that the damping has the character of a global scaling of the fluid; i.e., seen from the fluid point of view, the damping is of a proportional type but the proportionality factor is frequency-dependent. It should also be noted that equation (26) is written in the special case of a porous material–flexible structure interaction problem, but is easily extended to including non-porous fluid parts by essentially adjusting the material parameters properly. 5. APPLICATION TO A DOUBLE WALL WITH POROUS MATERIAL FILLING

As an application of the developed porous elements presented in section 5, equation (26) can be applied to a two-dimensional model of a double wall, with only the fluid in the double wall cavity as a coupling mechanism and with a vacuum outside the double wall. One of

. ¨

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1N Fluid finite elements 1N Inner panel, beam elements Excited panel, beam elements y z

x

Figure 3. Finite element model of double wall. Boundary conditions for panels and applied forces in response calculations.

the panels of the wall is excited by two point forces, one at mid height and one at quarter height. For this simple case the influence of the flexibility of the porous material will be demonstrated for various sets of parameters. The quantities that will be evaluated and discussed are the average absolute amplitude of each of the flexible parts of the double wall. As an additional parametric variation, the depth of the cavity separating the walls will also be varied, to show how a change in the dynamic properties of the acoustic medium influences the behaviour of the porous material. As mentioned in the introduction, the finite element method is particularly well suited for low frequency problems, while the traditional usage of porous materials is aimed at higher frequencies. Despite this, it will be shown that even for low frequencies and narrow cavities the porous filling in the double wall has a significant effect on the transmission of energy from one of the walls to the other. 5.1.      The material parameters used for the two panels simulating the plane wall (see Figure 3) are given in Table 1. For the present study the two panels have been taken as identical and T 1 Data for panels and dimensions of double wall used in example Material property

Value

Young’s modulus Density Thickness of panel Height Width Damping constant a

7·2×1010 N/m2 2700 kg/m3 0·001 m 0·5 m 1·0 m 1·59×10−5 s/rad

     

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T 2 Resonance frequencies for panels (double frequencies) in vacuo Number of half-wavelengths

Frequency (Hz)

1 2 3 4 5

45·35 99·85 152·83 205·36 257·85

simply supported, with the in vacuo resonance frequencies given in Table 2 as calculated from the finite element model used throughout this paper. As the double wall cavity is filled with air (see Table 5 for the porous material properties and Figure 3 for finite element mesh), the above frequencies are likely to change due to the influence of the fluid coupling the two panels together. As mentioned above two different cavity depths will be studied, 0·1 m and 0·05 m. At the low frequencies of interest here, the fluid is expected to behave very much like a spring, since the acoustic wavelength is much larger than the depth of the double wall. For the first case, 0·1 m thickness, the coupled resonance frequencies are given in Table 3, while Table 4 shows the frequencies for the reduced depth cavity. The mode shapes of the panels will now be of two different types, although having the same number of half-wavelengths for the structural part of the coupled modes, with more or less separated frequencies: i.e., one low resonance that represents the case when the two walls are vibrating in phase, and one higher frequency when they are vibrating out of phase with each other. In the latter case the fluid medium will resist compression, thus giving rise to a stiffness type effect with an expected increase in the resonance as compared to the in vacuo case. The main difference lies in the second mode with one half-wavelength, which goes from 45·35 in vacuo to 83·81 for the 0·1 m cavity and up to 98·56 for the 0·05 m cavity. A simple double wall model (see, e.g., reference [20]), predicts the increase in the resonance to be about 1·41 times, while the finite element results yields a factor of 1·18. At the same time, the influence of the coupling is less significant for the higher mode numbers owing to the break-up of the panels into shorter wavelengths, resulting in lower net volume changes of the double wall cavity. In the response calculations the damping constant a in equation (26) has been chosen to yield a damping level equivalent to 0.5% of critical damping at 100 Hz.

T 3 Resonance frequencies for double wall with air, 0·1 m cavity depth Number of half-wavelengths

Frequency (Hz)

1 1 2 2 3 3

45·23 83·81 99·18 99·77 152·78 153·06

. ¨

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T 4 Resonance frequencies for double wall with air, 0·05 m cavity depth Number of half-wavelengths

Frequency (Hz)

1 1 2 2 3 3

45·29 98·56 99·81 108·96 152·80 153·63

5.2.       For the analyses presented here, the main interest has been to study the influence of the flexibility of the material, here represented by the bulk mass density, upon the transmission of vibratory energy through the fluid medium to the other side of the double wall. The data pertaining to the porous materials are shown in Table 5. The lighter material has properties similar to those of the kinds of materials found in aerospace applications. The heavier material is more typical of a material found in buildings, etc. The increase in mass density for the same type of fibres, without increasing the flow resistance, may be achieved by using a different fibre diameter in the two materials; see, e.g., reference [21]. The corresponding porosity h has been calculated from equation (3) and the structure factor Ks , has been taken to be equal to the 30% discussed in section 2.1. 5.3.    5.3.1. Panel vibrations For the two different depths of the double wall two materials with different bulk mass densities have been analyzed, one being a lightweight material found in aerospace applications, 10 kg/m3, and one being considerably more heavy of the kind used in buildings, 100 kg/m3.

T 5 Data for porous materials used in examples Material property Speed of sound in free air c0 Density of free air r0 Speed of sound in pores cp Density of fibres rf Flow resistance f Structure factor Ks Light material Mass density M Porosity Heavy material Mass density M Porosity

Value 343·1 m/s 1·205 kg/m3 290 m/s 2500 kg/m3 2×104 kg/(m3 s) 1.3 10 kg/m3 0·996 100 kg/m3 0·960

     

571

Figure 4. Influence of porous material and variation of mass density. Average of displacement amplitude of excited panel. Cavity depth 0·1 m. (a) No porous material; (b) 10 kg/m3; (c) 100 kg/m3.

Figure 4 shows the average displacement of the excited panel with 0·1 m depth, without and with a porous material having the two different mass densities. The first resonance peak is significantly influenced by the heavy absorbent, being both substantially reduced and shifted downwards. It seems that the effect at this peak is one of a mass character, lowering the resonance frequency considerably. The effects of the lightweight porous absorbent are most pronounced at the second resonance peak, around 84 Hz. For both mass densities the response at this peak is reduced by a factor of about 70. In both cases the reduction is due to a shift in the location of the peak by about 10 Hz, but for the light material the resulting response at the new frequency is a factor of about four less, while for the heavy material the difference is a factor of about 40 less. At the third undamped resonance peak the reduction is generally smaller for both materials, and the maximal reduction occurs once again for the heavy material with a factor of about four. This peak is also generally lower, owing to the nature of the applied loading. The corresponding influence on the inner panel is shown in Figure 5 for the same case as above. The general feature is the same except for the higher resonance, at about 99 Hz, where the light absorbent seems to have no significant influence on the response. Furthermore the response of this panel is increased slightly relative to the undamped case for frequencies above the third undamped resonance peak, probably owing to a shifting of the higher frequencies. For the double wall with the narrow cavity, having a second resonance peak at 98 and a third at 109 Hz, the latter peak is not possible to distinguish for the heavy material; see Figure 6. The influence of the light material, however, appears to result in a similar frequency shift and decrease as discussed above. The response at the second peak, which is less affected by the changes in the dimensions of the cavity, is not affected by the light material, while the heavy material results in a reduction by a factor of about three. For the inner panel the same general trends hold; see Figure 7. An interesting observation is that the mass effect on the first resonance is less pronounced in this case as compared with the wide cavity, as could be expected based on pure volume considerations.

572

. ¨

Figure 5. Influence of porous material and variation of mass density. Average of displacement amplitude of inner panel. Cavity depth 0·1 m (a) No porous material; (b) 10 kg/m3, (c) 100 kg/m3.

To investigate the effects of changing the properties for the air in the double wall cavity, from normal data to those found in the porous material, without taking into account the damping or inertial properties of the porous absorbent, an analysis was performed with the speed of sound taken equal to 290 m/s. The resulting panel displacement of the excited panel is shown in Figure 8. Similarly, to the previously discussed cases, a substantial frequency shift downwards is observed, and, as a consequence, the response at the frequency of the original second peak is decreased substantially. The resulting response at the new frequency

Figure 6. Influence of porous material and variation of mass density. Average of displacement amplitude of excited panel. Cavity depth 0·05 m. (a) No porous material; (b) 10 kg/m3; (c) 100 kg/m3.

     

573

Figure 7. Influence of porous material and variation of mass density. Average of displacement amplitude of inner panel. Cavity depth 0·05 m. (a) No porous material; (b) 10 kg/m3; (c) 100 kg/m3.

is, however, of the same magnitude as for the normal air case. Thus, by comparing the results of Figure 8 with those of Figure 4, it may be seen that the frequency shift of the second peak in the porous material case is slightly larger, as expected from equations (10) and (24), and the response at the new shifted frequency is considerably influenced by the inertial and viscous drag effects introduced by the porous material. A general observation that may be made is that the responses of the panels are lower for the heavy material than for the light material. The reasons for this have been briefly

Figure 8. Influence of change in speed of sound on average response of excited panel. Cavity depth 0·1 m. (a) 343·1 m/s; (b) 290 m/s.

574

. ¨

addressed in section 2.3, where the apparent effects of the porous material were discussed. The change in the fluid particle displacement magnitude, and hence also the panel displacements due to the relation in equation (14), may be understood in terms of changes in the bulk modulus of compression of the fluid inside the porous material. For a free fluid this modulus is given by E0=r0 c02 ,

(27)

while for the porous material the same quantity is given by Ep=r0 cp2

Ks /h+if/r0 v . 1+ihf/vM

(28)

The fluid response may be shown to be directly proportional to the bulk modulus of compression, and thus the lower response for the fluid inside the porous material is obviously related to changes in the magnitude of this modulus, which may be regarded as the dynamic stiffness of the fluid. It is interesting to note that the magnitude of the modulus of compression, from equation (28), may be of the same order as in more incompressible fluids at low frequencies, provided that the mass density of the porous material is high enough. 5.3.2. Sound pressure field inside double wall To show how the results discussed above manifest themselves in terms of the pressure pattern inside the double wall, with and without the porous absorbent, the response in the cavity will be investigated. Two frequencies will be studied in particular detail: the original fundamental double wall resonance frequency and the shifted double wall resonance. The first frequency is interesting for a pure tone source, while the shifted one is more of interest for a broadband source. Figure 9 gives the pressure pattern in the case of pure air and 0·1 m cavity depth, while Figures 10 and 11 show the pattern for the cases of the light and heavy porous materials respectively. In agreement with the results for the panel vibrations shown in Figures 4 and 5, the pressure levels in the cavity decrease for increasing mass density of the porous absorbent. But, apart from this expected behaviour, a significant change in the pressure

H

K J I G F E D C

B A

A B C D E F G H I

J K

Figure 9. Calculated acoustic pressure in double wall cavity without porous filling at 83·81 Hz. Cavity depth 0·1 m. Levels in dB SPL: A, 145·50; B, 145·52; C, 145·54; D, 145·56; E, 145·58; F, 145·60; G, 145·62; H, 145·64; I, 145·66; J, 145·68; K, 145·70.

     

575

K J I

H

G F E D C

A B

C D E F G H

Figure 10. Calculated acoustic pressure in double wall cavity with porous absorbent, mass density 10 kg/m3, at 83·81 Hz. Cavity depth 0·1 m. Levels in dB SPL: A, 104·90; B, 105·24; C, 105·58; D, 105·92; E, 106·26; F, 106·60; G, 106·94; H, 107·28; I, 107·62; J, 107·96; K, 108·30.

pattern occurs between the air-filled cavity response and the pressure field inside the porous absorbent. At the double wall resonance of the air-filled double wall, where the two panels are vibrating out of phase, the pressure in the cavity is almost constant (see Figure 9), with the highest sound pressure levels occurring at the upper and lower ends where the motion of the panels is the smallest. The variation in sound pressure level is within 0·2 dB, a result supported by experimental investigations on a similar problem [22]. At the regions of the highest panel vibrations the pressure is at its minimum. The pattern is furthermore almost symmetric, showing only minor effects of the unsymmetric loading. For the same frequency but with the light porous absorbent (see Figure 10), the pressure levels are in general reduced and the span between the maximum level and the minimum level is larger than the previous case. The maximum levels still occur at the upper and lower regions of the cavity, and there are clear signs of minor asymmetries in the response as opposed to the pure air case.

B

B

A

A C D E F G H I J

K I H G F G H

Figure 11. Calculated acoustic pressure in double wall cavity with porous absorbent, mass density 100 kg/m3, at 83·81 Hz. Cavity depth 0·1 m. Levels in dB SPL: A, 106·40; B, 106·68; C, 106·96; D, 107·24; E, 107·52; F, 107·80; G, 108·08; H, 108·36; I, 108·64; J, 108·92; K, 109·20.

576

. ¨

Inner panel

Excited panel

Figure 12. Vibration amplitude of panels at 83·81 Hz, with 100 kg/m3 porous material. No phase information included.

Increasing the mass density of the porous absorbent to 100 kg/m3 results in a small increase in the pressure levels and a substantial change in the pressure pattern. The maximum level now occurs close to the region of the highest vibration amplitudes of the panels. In fact, as may be seen from Figure 12, the highest vibration amplitude is found for the excited panel in the lower part, thus relating the highest pressure to the highest amplitude in this case. The acoustic field in this case is clearly influenced by other cavity modes than in the previous two cases, indicating the change in the forced response of the neighbouring modes. This may be attributed to the modulus of the complex-valued density, which for low frequencies increases with increasing mass density of the porous material (see equation (24), where the complex-valued density is the expression within the parentheses), shifting the cavity modes down in frequency. To compare the results obtained for the double wall with porous absorbents at the original undamped resonance with the results obtained at the new shifted resonance frequencies, the pressure patterns were calculated at 74·0 Hz for the light absorbent and at 76·6 Hz for the heavy. As may be seen from Figure 13, the change in pressure pattern for the light absorbent is such that a more distinct regularity may be identified, very much like that in the pure air case. The difference in maximum sound pressure level between the undamped double wall resonance and the shifted resonance is 11 dB, thus supporting the results shown in Figures 4 and 5. For the heavy absorbent (see Figure 14), the change in the pressure pattern is also towards a higher degree of regularity and symmetry, but it is not the same pattern as for the light absorbent. The difference in the maximum sound pressure levels is also larger, 31 dB, between the pure air and the heavy absorbent cases, which once again supports the results of Figures 4 and 5. From the results discussed in this section it is clear that the wave length of the pressure field decreases with increasing mass density. The reasons for this may be found from expressions for the wavelength in the case of a free fluid and the case of the fluid inside the porous material. For the free fluid the wavelength is given by l0=2pc0 /v,

(29)

     

577

K J I H G F E D C

A

A C D E F G H I J K

Figure 13. Calculated acoustic pressure in double wall cavity with porous absorbent, mass density 10 kg/m3, at 74·0 Hz. Cavity depth 0·1 m. Levels in dB SPL: A, 134·60; B, 134·77; C, 134·94; D, 135·11; E, 135·28; F, 135·45; G, 135·62; H, 135·72; I, 135·96; J, 136·13; K, 136·30.

while for the fluid inside the porous material it is given by lp=2p

cp Re v

0X

1

1+ihf/vM , Ks /h+if/r0 v

(30)

where Re (...) denotes the real part of the quantity within the parentheses. For the porous material with the higher mass density, the ratio between the wavelength inside the porous material and that in free air is about 14% at 100 Hz. Thus it is clear that the viscous drag forces produced by the fibres will have a stronger impact for the heavy material owing to the much shorter wavelength of the acoustic pressure disturbance. This is also the background of the ‘‘point source’’-like pattern of the acoustic pressure response in Figure 11. As a comparison, the ratio of the wavelength inside the porous material to that in free air is about 30% for the light material at the same frequency, 100 Hz.

D C

C B

B

A

D E F G

K J I H

G F

C E

E

F

B D

Figure 14. Calculated acoustic pressure in double wall cavity with porous absorbent, mass density 100 kg/m3, at 76·6 Hz. Cavity depth 0·1 m. Levels in dB SPL: A, 112·60; B, 112·79; C, 112·98; D, 113·17; E, 113·36; F, 113·55, G, 113·74; H, 113·93; I, 114·12; J, 114·31, K, 114·50.

578

. ¨

These observations are in correspondence with the remarks in reference [20] concerning the effects of heavy porous absorbers in double wall transmission. 6. CONCLUDING REMARKS

From the calculations performed for a plane double wall with and without a porous absorbent in the cavity, it may be concluded that, for the double wall resonances where the panels are vibrating out of phase, the porous absorbent is quite effective. The transmission of sound energy through the fluid medium is reduced substantially for heavy porous materials and slightly less so for lightweight materials. There are three major effects responsible for this behaviour: the change in the speed of sound due to the isothermal conditions maintained inside the material, the increase in the magnitude of the equivalent bulk modulus of compression, which is especially pronounced at low frequencies and for heavy materials, and finally the decrease in the wavelength of the acoustic waves inside the porous material, also especially pronounced at low frequencies and for heavy materials. In the results presented the change in the wavelength of the sound waves in the double wall cavity, resulting in a frequency shift downwards, accounts for part of the reduction observed of the original resonance frequency. An additional, but smaller, reduction arises when the other porous material effects are added. The response at the new downwards shifted frequency is, however, still high for the ‘‘slow air’’ case, even compared with that for the lightweight porous material used in the analysis. Therefore it may be concluded that the effect of the porous absorbent on the transmission of noise through the double wall at the undamped double wall resonance, is mainly due to the change in the fluid properties and not so much to the viscous damping induced by the fibres. However, the flow resistance forcing the fibres to move as the sound waves propagate through the medium has a significant effect on the reduction of the transmitted noise at the frequency of the shifted resonance peak. The lighter the porous material, the less reduction is achieved, as foreseen in the discussion in section 2.2, where it was stated that the flexibility of the material serves to reduce the viscous drag opposing the fluid particles’ motion. The calculated acoustic pressure fields inside the double wall gives a hint to the background of this effect. For the light material the pressure at either of the two panels is approximately equal, and the pattern is similar to the pattern obtained for the air-filled cavity. For the heavy material the pressure is higher at the excited panel and substantially different from that in the air-filled case. In this case the highest pressure occurs close to the highest vibration amplitude of the excited panel as opposed to the other cases, i.e., the light porous material and the air-filled cavity, where the highest pressures occur at the upper and lower ends of the cavity. For these latter cases the regions of the highest panel vibration amplitudes corespond to the lowest pressure amplitudes. At this point the effects of including the stiffness of the fibres must be considered. For the heavy porous material the fibres will not move, and hence the viscous drag force is at a maximum. For the light material the viscous drag force will set the fibres into a motion that is opposed only by the inertia of the fibres. Therefore, as has been demonstrated in the present paper, at the resonance of the double wall with porous filling the heavy material will be substantially more effective than the light owing to the motion of the fibres. If, however, the structural stiffness of the absorbent is taken into account then this force will act to oppose the motion of the fibres—an effect that will serve to lower the response of the material, especially at frequencies below the first structural resonance of the absorbent. Another aspect of the absence of the mechanical description of the fibres is the validity of the assumption, made at the beginning of the paper, concerning mechanical contact between the fibres and

     

579

the flexible panels. At resonance the movement of the fibres is restrained only by the inertial forces in the present formulation, while the large displacement amplitudes that are likely to occur at the resonances of the double wall will result in at least partial contact between the fibres and the flexible vibrating panels. To investigate these phenomena requires a full flexible porous material–acoustic formulation. From the results presented and discussed in the present paper it is clear that the reduction in interior noise of a propeller aircraft observed in reference [15] is to a large extent due to a shift in frequency of the double wall resonances and to some extent also due to the increase in the stiffness of the fluid and the decrease in the wavelength of the acoustic field. The nature of this dominating phenomenon, being localized at the resonance peaks where the panels are vibrating out of phase, diminishes the effect of the absorbent as measured over a broad frequency band. However, for cases where the excitation consists of discrete tones, such as a propeller-driven aircraft, the effects of the porous absorbent may have a major impact on the interior noise levels, especially if it is taken into account that this is a material, not primarily intended for noise attenuation purposes, which it is necessary to incorporate.

ACKNOWLEDGMENTS

This work has been sponsored by the Swedish National Board for Technological Development, NUTEK, under the Swedish Civil Aircraft Research Programme. The author also wishes to thank Professor Sven Lindblad, Lund Institute of Technology, Department of Technical Acoustics, Sweden, for support and interesting discussions concerning models of porous materials.

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15. P. G¨ 1989 Proceedings of the 3rd International Conference on Recent Advances in Structural Dynamics, Southampton. Analyses of the transmission of sound into the passenger compartment of a propeller aircraft using the finite element method. 16. P. G¨ 1988 Proceedings of the Nordic Acoustical Meeting 88, Tampere, Finland. Acoustic finite element calculation of sound transmission through a double wall and a comparison with experimental data. 17. G. S and P. G¨ 1988 Journal of Sound and Vibration 123, 507–515. A symmetric finite element formulation for fluid–structure interaction analysis. 18. H. C 1992 Doctoral Thesis. Lund Institute of Technology, Department of Structural Mechanics, Report TVSM-1005. Finite element analysis of structure–acoustic systems: Formulations and solution strategies. 19. P. G¨ 1992 Proceedings of DGLR/AIAA 14th Aeroacoustics Conference, Aachen Germany. On the representation of general damping properties in modal synthesis solutions of fluid–structure interaction problems. 20. F. J. F 1985 Sound and Structural Vibration, Radiation, Transmission and Response. London: Academic Press. See p. 169. 21. D. A. B 1971 In Noise and Vibration Control (editor L. L. Beranek). Chapter 10: Acoustical properties of porous materials.. New York: McGraw-Hill. 22. P. S, F. A and J. V  P 1992 Proceedings of DGLR/AIAA 14th Aeroacoustics Conference, Aachen, Germany. Modelling the vibro-acoustic behaviour of a double wall structure.