LETTER TO THE EDITOR: COMMENTS ON SIMPLE MODELS OF THE ENERGY FLOW IN VIBRATING MEMBRANES AND ON SIMPLE MODELS OF THE ENERGETICS OF TRANSVERSELY VIBRATING PLATES

Journal of Sound and Vibration (1996) 195(4), 679–685

COMMENTS ON SIMPLE MODELS OF THE ENERGY FLOW IN VIBRATING MEMBRANES AND ON SIMPLE MODELS OF THE ENERGETICS OF TRANSVERSELY VIBRATING PLATES M. N. I  L. J Laboratoire de Tribologie et de Dynamique des Systemes, UMR CNRS 5513, Ecole Centrale de Lyon, 69131 Ecully Cedex, France (Received 24 July 1995, and in final form 7 March 1996)

1.  The results published by Bouthier and Bernhard [1, 2] concerning the energy models of vibrating membranes and plates are generalized and discussed. For this purpose, a general propagative approach is employed to build the energy description of some systems. The propagative approach uses a random wave field description instead of the space average concept employed by Bouthier and Bernhard. Moreover, the proposed method is only group velocity (energy speed) dependent, thus allowing the study of classes of propagative systems rather than classical vibrating systems as presented in references [1, 2]. It is also shown that the propagative approach used is equally valid for one-dimensional or multi-dimensional vibrating systems. In their recent papers, Bouthier and Bernhard [1, 2] presented some interesting results concerning the energy model of vibrating membranes and transversely vibrating plates up to medium and high frequencies. In fact, for each vibrating system studied, they constructed a fan energy flow balance involving the active energy flow divergence as a function of the energy loss pdiss and the input power density pin . The energy flow balance when written for the analysis of a steady state case takes the form pin = pdiss + 9 · I .

(1)

For the sake of simplicity, the notation used in this letter are almost identical to those used in references [1, 2]. For a hysteretic damping model, the time averaged energy flow dissipated density is [3, 4] (a list of notation is provided in the Appendix) pdiss  = hve.

(2)

Accordingly, by introducing the energy dissipated density in the energy flow balance, a local equation relating the evolution of the energy flow to the total energy density is obtained. The required simplified energy flow model will be obtained from the elimination of the energy flow I  in the resulting equation. In fact, the main goal by far in energy flow model developments is the exhibition of an energy operator I  = J e) which permits an analytical differential equation to be written in terms of the total energy density alone. In their study, Bouthier and Bernhard used a complete and heavy method to establish the energy operator form. They treated separately the cases of membranes [1] and plates [2]. For each case, the authors exhibited a plane wave approximate solution of the displacement equation, and thus, having the heat conduction model in mind, they calculated explicitly the total energy density, the active energy flow and finally the gradient 679 0022–460X/96/340679 + 07 $18.00/0

7 1996 Academic Press Limited

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   

of the active energy flow. They verified that the space averaged total energy density is proportional to the space averaged active energy flow for finite membranes and plates:  = −(cg2 /hv)(i 1e¯ /1x + j e¯ /1y). I

(3)

The expressions I  and e¯  refer, respectively, to the space averaged active energy flow and energy density. The space average operator used by the authors is H  =

kx ky p2

g g p/2kx

p/2ky

−p/2kx

−p/2ky

H dy dx

(4)

Many comments can be formulated about the space average concept. Firstly, it has been shown [3], that the study of complicated vibrating systems such as Timoshenko beams or curved wave guides [4] introduces more than one group velocity or wavelength. The space average technique therefore cannot formally be used and generalized for those cases. Secondly, the use of such a concept for multi-dimensional vibrating systems such as in references [1] and [2] involves an obscure problem, which is the choice of the averaging surface. Indeed, the choice of integral bounds made by Bouthier and Bernhard is not unique and is difficult to justify. The main objective of this letter is to give a consistent demonstration of some energetic properties of vibrating systems. The demonstration given here is solely group velocity dependent and can thus be applied equally to some one-dimensional vibrating systems such as strings, rods, Euler–Bernouilli beams etc. or multi-dimensional systems such as membranes, plates, acoustics etc. 2.     Before establishing some energetic properties, it is appropriate to clarify the fundamental assumptions of the simplified energy models. As mentioned in references [3, 4], the set of assumptions required to derive the energy models can be summarized as follows: (i) linear, elastic, dissipative and isotropic systems; (ii) small hysteretic damping loss factor (h1); (iii) steady state conditions with harmonic excitation of frequency v; (iv) far from singularities, evanescent waves are neglected; (v) the interferences between propagative waves are not considered. The assumptions (i), (ii) and (iii) define the general context of the study. It should be noted that the hysteretic damping factor is assumed to be very slight (iii). This implies that the wave numbers of vibrating systems are approximately equal to those of the undamped systems. The hypothesis (iv) implies that no energy is transported by evanescent waves. This assumption is easily justifiable since the near field vanishes strongly far from singularities as the frequency increases. The assumptions mentioned are commonly used in the literature for medium and high frequencies. The new assumption (v) has been introduced by Ichchou and Jezequel [3, 4] in order to generalize the space average concept. In references [3, 4] the energy model construction is made for one-dimensional systems having more than one energy speed (Timoshenko beams, curved wave guides, etc.). The assumption (v) is that the interferences between different propagative waves are not taken into account, it being assumed that in the mid and high-frequency ranges no correlation remains between propagative waves. Hence, representing the mid-frequency structural/ acoustical dynamics by a random wave field, one has E[ai a* j ] = ai a* i dij .

(5)

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Here ai and aj represent any sort of propagating waves, dij is the Kronecker delta symbol and E represents the mathematical expectation. In a multi-dimensional configuration, the interferences are not considered between any sorts of propagative waves and which spread in any direction. Expression (5) then becomes E[ai (uik )a* j (ujh )] = ai (uik )a* i (uik )dij d(uh − uk ),

(6)

where uik is the kth direction of the ith kind of wave. It should be noted that a similar assumption was introduced by Langley [5] in order to develop the so-called wave intensity technique, WIA, which is a directive version of the SEA. As mentioned by Langley [6], this assumption can viewed in a number of ways. For random loading the mathematical expectation given in equation (6) represents some realizations of the loading. In a deterministic case, equation (6) can be viewed as an average over different realizations of the system. For the present purposes, partial energy variables denoted I i and ei , and representing the active energy flow vector and the energy density associated with each kind of wave, respectively will be employed. Under the set of the assumptions (i)–(v) required for the energy model, and especially the assumption (v), the resulting global time energy parameters denoted here as I and e  are obtained from a linear superposition of the partial energies: 0 :

0 :

I = s I i ,

e˜  = s ei .

i

(7)

i

For some one-dimensional vibrating systems, and especially those having only one propagative mode, it has been shown [3] that the space average concept over a half wavelength is reduced to the linear energetic superposition (7) as involved in assumption (v). Obviously, an energy flow balance for each kind of wave can be written in terms of I i and ei as 9 · I i  + hvei  = 0.

(8)

At this stage, an interesting relationship relating to the partial active energy flow and the partial energy density is introduced. This law is often used in wave propagating phenomenon and is, according to reference [7], I i  = cg ei n i ,

(9)

where cg is the real group velocity. It corresponds to the group velocity of the undamped system studied as the damping loss factors are assumed to be very slight. n i is a normalized vector which defines the wave direction. Expressions (7)–(9) will be used in what follows to derive the energetic operator and hence the energy model. Two classes of multi-dimensional systems are considered. The first class is that of systems with cylindrical or spherical waves, while the second class is a classical system with plane wave dynamics. 3.          In the case of cylindrical or spherical symmetry, the multi-dimensional systems studied can be represented readily by a one-dimensional differential operator which depends on the radial position solely. For such systems, the differential energy flow balance becomes 1 d(r aIi ) + hvei  = 0, dr ra

(10)

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where the parameter a is equal to 1 for cylindrical symmetry and to 2 for spherical symmetry. In this case, the constitutive partial law says that Ii  = cg ei .

(11)

Upon putting expression (11) into the partial energy flow balance, one obtains an interesting relationship relating the partial time averaged active energy flow and energy density: Ii  = −

cg2 1 d(r aei ) . hv r a dr

(12)

Here, the energy flow carried by each cylindrical/spherical wave is obtained as the linear superposition of the total energy density contained in such a wave and its first derivative. From the linear energetic superposition given in equation (7) and the partial constitutive law (12), a constitutive global law is obtained when the group velocity is assumed to be the same for all waves considered:

I  = s Ii  = − i

0

d r as ei 

2 g

c 1 hv r a

i

dr

1

=−

cg2 1 d(r ae˜ ) . dr hv r a

(13)

The relationship (13) has been proposed elsewhere [8, 9]. In reference [9], circular membranes were considered, while in reference [8] circular plates were studied. In both studies, the energy operator (13), was obtained from a completely developed calculation. The expression (13) used here generalizes those results. Now, upon putting expression (13) in the global energy balance written in the cylindrical or spherical case, this latter becomes cg2 1 d2(r ae˜ ) + hve˜  = 0. w−\ hv r a dr 2

(14)

T 1 Energy flow models for some cylindrical/spherical cases Configuration

Energy equation −

Circular membranes

0

1

0

1

Group velocity

cg2 d2e˜  2d de˜  + + hv e˜  = 0 r dr hv dr 2

cg2 d2e 2r de˜  − + + hve˜  = 0 hv dr 2 dr Circular plates

0

1

cg2 d2ye˜  4 de˜  2 + 2 e˜  + hve˜  = 0 − + r r dr hv dr 2 3-D spherical acoustic field

cg = zT/r

4 2 cg = 2vzrhv /D

cg = zgp/r

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683

Figure 1. A plane wave propagating in three dimensions.

This expression constitutes the energy model for cylindrical/spherical symmetry when written in a more general form. It is solely group velocity dependent. In Table 1, some results relating to those vibrating systems are summarized. Bouthier and Bernhard [2] presented some energetic properties of infinite plates. It should be noted that the infinite case they considered is a simple particular case of the cylindrical model presented here. The energy model given by equation (14) is the closed form of the energy models for cylindrical/spherical vibrating systems. 4.        As mentioned by Bouthier and Bernhard [1, 2], the complete study of energetic behaviour of finite non-symmetric systems is complicated due to the reflections which occur at the system boundaries. Hence, a propagative state of such systems has a large number of wave types, making the differential energy model very difficult to construct. A propagative plane wave field is selected here to represent some vibrating systems. Obviously, a plane wave field solution is just an approximation of a real dynamical state. In Figure 1 is shown a plane wave propagating in three dimensions. For such propagating disturbances, the motion of every particle in the planes perpendicular to the direction of propagation is the same. The vector n i in Figure 1 represents the normal to the plane. It can readily shown that the gradient of the energy density associated with each plane wave is :

grad(ei ) =

1ei  n . 1ni i

(15)

Having those plane wave characteristics in mind, one can introduce the intrinsic partial energy relationship (7) in the energy balance (8): 9 · (cg ei n i ) + hvei  = 0. Thus, since

(16)

:

9 · ( f n i ) = f 9 · n i + n i · grad (f),

(17)

where f is a scalar function representing here the partial energy density. Upon introducing the relationship (17) into expression (16), the latter becomes :

cg ei 9 · n i + cg n i · grad(ei ) + hvei  = 0.

(18)

At this stage, one can simplify relationship (18) and introduce the plane wave characteristic given by equation (15), to obtain cg 1ei /1ni + hvei  = 0.

(19)

   

684

T 2 Energy flow model for some non-symmetrical cases Configuration

Energy equation

Rectangular membranes

Group velocity

0 0

1 1

0

1

cg = zE/r(1 − v 2)

0

1

cg = zE/2r(1 + v)

cg2 1 2e˜  1 2e˜  + + hve˜  = 0 − 1y 2 hv 1x 2

cg = zT/r

cg2 1 2e˜  1 2e˜  − + + hve˜  = 0 1y 2 hv 1x 2

4 2 cg = 2vzrhv /D

Out-of-plane motion of plates cg2 1 2e˜  1 2e˜  − + + hve˜  = 0 1y 2 hv 1x 2 Extensional motion cg2 1 2e˜  1 2e˜  + + hve˜  = 0 − 1y 2 hv 1x 2 Shearing motion

0

1

cg2 1 2e˜  1 2e˜  1 2e˜  − + + + hve˜  = 0 1y 2 1z 2 hv 1x 2

cg = zgp/r

3-D acoustic field

By multiplying expression (19) by the normal vector n i and introducing the intrinsic constitutive law given expression (9), an interesting result is obtained, as :

cg grad (ei ) + hvI i /cg = 0

:

I i  = −(cg2 /hv) grad (ei ).

(20)

The active flow carried by a plane wave is therefore proportional to the gradient of the energy density associated with such a plane wave. If one considers a unique propagative mode (one group velocity), the linear energetic superpostion given in equation (7) guarantees that a similar relationship can be written in terms of the global energy flow and energy density as 0

:

I  = −(cg2 /hv) grad (e˜i).

(21)

The resulting energy model is obtained by introducing the energy operator (21) in the energy flow balance. It becomes −(cg2 /hv) De˜  + hve˜  = 0

(22)

This is identical to the energy model proposed by Bouthier and Bernhard for membranes [1] and for vibrating plates [2]. The energy model (23) comprises both those cases and more. Indeed, this equation is valid for all vibrating one-dimensional [3, 4] or multi-dimensional systems having one mode of propagation and whose dynamics can be approximated by a plane wave field. In Table 2, some vibrating multi-dimensional systems are summarized. They obey the required hypothesis. 5.   The propagative approach proposed in this study is a simple demonstration of some energetic properties of vibrating systems. It is valid for both one-dimensional and multi-dimensional cases and permits the energy operator J to be exhibited for each

   

685

configuration. The partial version of the energy operator seems to be mainly dependent on the wave nature. Three waveforms have been studied here: cylindrical, spherical and plane. For each case, an intrinsic constitutive law has been developed. The energy operator J relating to the global energy variables is therefore wave nature and number dependent. The energy model proposed here is solely group velocity dependent, which allows a concurrent study of classes of vibrating systems. ACKNOWLEDGMENT

The authors gratefully acknowledge Mr R. Aquilina of the Centre d’Etude et Recherche de la Discretion Acoustique des Navires for his kind advice and suggestions. REFERENCES 1. O. M. B and R. J. B 1995 Journal of Sound and Vibration 182, 129–147. Simple models of energy flow in vibrating membranes. 2. O. M. B and R. J. B 1995 Journal of Sound and Vibration 182, 129–164. Simple models of energetics of transversely vibrating plates. 3. M. N. I and L. J 1995 Proceedings of Inter Noise ’95, Newport Beach. A general propagative approach for the energy flow models and the heat conduction analogy of one-dimensional systems. 4. M. N. I, A. L B and L. J 1995 Proceedings of Vibration and Noise ’95, Venice, Italy. Radial and tangential energy flow models for curved wave guides. 5. R. S. L 1992 Journal of Sound and Vibration 159, 483–502. Wave intensity technique for the analysis of high frequency vibration. 6. R. S. L 1995 Journal of Sound and Vibration 182, 637–657. On the vibrational conductivity approach to high frequency dynamics for two-dimensional structural components. 7. L. B 1956 Propagation des Ondes dans les Milieux Periodiques. Dunod: Masson. 8. H. S. K, H. J. K and J. S. K 1994 Journal of Sound and Vibration 174, 493–504. A vibration analysis of plates at high frequencies by the power flow method. 9. A. L B and L. J 1993 Proceedings of the Institute of Acoustics 15(3), 561–568. Energy formulation for one dimensional problems.

:  9·: grad D  H  E

divergence operator gradient operator Laplacian operator, time average time and space average mathematical expectation