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A Poynting vector formulation for thin shells and plates, and its application

to structural intensity analysis and source localization. Part I: Theory

Article  in  The Journal of the Acoustical Society of America · March 1990

DOI: 10.1121/1.398790

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A Poynting vector formulation for thin shells and plates, and its application to structural intensity analysis and source localization. Part I: Theory

AnthonyJ. Romano, a) Phillip B. Abraham, and Earl G. Williams Naval Research Laboratory, Code 513 7, Washington, DC203 75

(Received 8 September 1988; accepted for publication 16 October 1989 )

This paper deals with a formulation of the Poynting vector (structural intensity) for thin shells and plates, and its application to structural intensity analysis and source localization. The procedure begins with the insertion of a Taylor series expansion of the displacement components (about the middle surface of the shell) directly into the three-dimensional representation of the Poynting vector. From this representation, an average power flow per unit length, or equivalently an intensity resultant, is derived, whose form permits expressibility in terms of force and moment resultants. The corresponding equations of continuity for energy are derived for both body and surface forces, and the time integral of the net outflow is developed, yielding a technique for source localization. This technique offers a method for the determination of the structural intensity of thin, elastic shells and plates, and is successful for source localization.

PACS numbers: 43.40.Dx, 43.40.Ey

LIST OF SYMBOLS

a

1/

E

F

F

h

H

l(t) (I(t))

cylinder radius Poisson's ratio

Young's modulus Lam(• elastic constant [Eq. (A3) ] Lam(• elastic constant [Eq. (A3) ] elastic constant [ Eq. (A4) ] energy

body force surface force

shell or plate thickness energy density volume integral of h• Poynting vector, instantaneous intensity Poynting vector, time-averaged intensity

O'i j

Im

Re

N,

P

arg •>

arg*

stress tensor

stress component i, j strain tensor

idemfactor or identity tensor imaginary part of complex argument real part of complex argument thickness variable

density of elastic material stress and moment resultants

average power per area

average power per unit length displacement vector time derivative of displacement vector k th derivative of arg with respect to r (or y) complex conjugate of arg

INTRODUCTION

The purposes of the present article are twofold; first, to present a generalized formulation of the Poynting vector, or structural intensity for thin, elastic shells and plates, and second, to establish criteria and a corresponding method for source localization using the aforementioned formulation. In a subsequent article (Part II), we then show how to ob- tain the displacements of the surface of a thin structure from its experimentally measured response to locally applied forces, and insert this information directly into our present formulations. These efforts stemmed from recent demands

for a theoretically consistent and experimentally feasible technique for the study and determination of power flow and source localization in thin, elastic structures.

Also at Sachs-Freeman Associates, Landover, MD 20785.

Although the expressions for the Poynting vector in three-dimensional elasticity are well known, their utilization for the study of power flow in any real, physically realizable situation is currently impossible in light of our inability to precisely measure or determine the displacements through- out the medium. If certain assumptions can be made con- cerning the physical system under study, then we may of course approximate these displacements. For example, if we assume that the medium is thin enough to permit a Taylor series representation of the displacements about its median surface, then we are left with the problem of finding the displacements and their derivatives on this median surface only. If we can determine this information experimentally, then there remain the details of the insertion of this Taylor series into the three-dimensional form of the Poynting vec- tor.

In the development of the equations of motion in shell

1166 J. Acoust. Soc. Am. 87 (3), March 1990 0001-4966/90/031166-10500.80 @ 1990 Acoustical Society of America 1166

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theory, however, it is often the convention to integrate such representations over the thickness of the medium, obtaining quantities such as force and moment resultants. Corre- spondingly, we may integrate the Taylor series representa- tion of the three-dimensional form of the Poynting vector over the thickness of the medium and obtain an intensity resultant, representative of power flow per unit length, which is consistent with the dimensions of the equations of motion obtained from shell theory. In addition, we may ob- tain corresponding forms of the equations of continuity for energy and develop a method for source localization.

Although the literature is abundant with developments of the equations of motion for shells, until recently, little attention has been given to other field quantities such as the Poynting vector, or structural intensity. Notable exceptions are the papers by Fuller • and Pavic. 2 In this article, we will present the theoretical aspects of a development of the Poynting vector for thin shells and plates, and a correspond- ing source localization technique, while restricting our at- tention to time-dependent motion that is real and transient or real and monochromatic.

In Sec. I, we present the fundamental definition of the Poynting vector (or structural intensity) in three-dimen- sional elasticity, and portray this quantity in Cartesian, cy- lindrical, and spherical coordinates.

In Sec. II, we begin with the assumption that the dis- placements s are expressible in terms of a Taylor series ex- pansion about the middle surface of the shell (r = a in cylin- drical and spherical coordinates, or median surface in Cartesian coordinates). This series is then inserted directly into the three-dimensional form of the Poynting vector (as given in Sec. I), and a corresponding quantity called an in- tensity resultant is developed. This latter quantity can then be recast in terms of force and moment resultants, such that its relationship to shell conventions is more obvious. The corresponding two-dimensional resultant forms of the Poynting vector are presented in the three coordinate sys- tems previously mentioned, and specific examples are dis- cussed using cylindrical and Cartesian coordinate systems. It is seen that in the limit as the radius a tends to infinity in the cylindrical representation of the resultant, the represen- tation of the structural intensity in a plate can be identically recovered.

In Sec. III, the time integration and time average of the intensity are developed for real transient and real monochro- matic time series, respectively. The corresponding equations of continuity for energy are developed in Sec. IV, and criteria are thereby established for source localization, which is ela- borated upon specifically in Sec. V.

Appendices A and B contain general expressions for the Taylor series representations of the displacements in cylin- drical and Cartesian coordinates, respectively, and Part II of this sequence will deal with the direct application of these theoretical developments to experimental situations.

I. THE POYNTING VECTOR (I): 3-D ELASTICITY

The instantaneous field intensity vector, or energy flow vector for an isotropic (lossless) elastic medium, is given explicitly by the relationship 3

\St/ (1) where

= particle velocity = •, bt

O r = stress tensor

+ [Vs + sV],

A. and/z are the Lam• elastic constants of the medium, and • is the idemfactor, or identity tensor. Clearly, I is a function of position and time, that is I (x, y, z, t). The various vector components of Eq. ( 1 ) can be simply expressed in terms of their respective stress and velocity components in our three coordinate systems as follows (see Fig. 1 )'

In Cartesian coordinates, • = (•, •y, • )'

= - + + ],

= - + + ].

In cylindrical coordinates, g = (•r, •, •z )' I r = __ [&rr•r + &r4•4 + &rz•z ],

1• = -- [ &•r•r + &OO•O + &Oz•z ], (3)

In spherical coordinates, • = (•r, •0, •0 )'

L = - [errOr + arO•O + erOS, ],

!0 = -- [aOr•r + a00•0 + a0,•, ], (4)

II. THE POYNTING VECTOR (P): 2-D SHELL THEORY

To develop the corresponding field intensity vector for thin shells and plates, we begin with an example in cylindri- cal coordinates, where the basis of unit vectors is the cylin- drical set (er, e•, e• ), and the displacements, s, can be repre- sented by (Sr, S•, S• ) = (W, V, U), as shown in Fig. 1. We then make the basic assumption that for a thin cylindrical shell, (since the ratio of the shell thickness to the radius, h/a, is small), we may expand the displacements, $, in a Taylor series about the middle surface (r = a); i.e.,

O'r•

FIG. 1. Stress orientation in cylindrical coordinates.

1167 J. Acoust. Soc. Am., Vol. 87, No. 3, March 1 gg0 Romano ot a/.' Poynting vector formulation 1167

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o(•-I-p] =/o(•) / u(a+p) \u(a)/

'(a P \u(•(a)

[w•2•(a)•

+ .... (5)

The first term on the right-hand side (rhs) of Eq. (9) is the force resultant Nt multiplied by fi (a), where

m• = tr=(a +p) • + dp, -- hi2

(lO)

and the second term on the rhs of Eq. (9) is seen to be the moment resultant, Mr, multiplied by fi(1)(a), where

It must be noted that Eqs. (5) are completely general for the following development.

To define average power flow per unit length in terms of stress and moment resultants (with radial dependence re- moved, as shell theory demands), we first introduce the se- ries [Eqs. (5) ] directly into the three-dimensional form of I [Eqs. (3) ]. Next, we define an average power per area, pt, p •, by integrating the respective vector components, It, I•, over an infinitesimal area perpendicular to these compo- nents, and divide by this same area as follows:

- n/= It (a + p, C/bo, z, t) (a + p)dc/bo dp

/2

fn_/• z + az /2 .fz I• (a + p, c/b, Zo, t)dzo d,o fh/• z + - /2 œz "tdzo dp

(6)

We then define quantities representative of power flow per unit length, Pt, P,, which we will label intensity resultants,

d -- h/2

h/2 P• =hp•= I•(a+p,c•,z,t)dp. -- h/2

(7)

Equations (7) are then seen to be immediately expressible in terms of familiar shell notation. For example, we .may ex- press the first of Eqs. (7) using the third of Eqs. (3),

Pt = -- [ O'zr•J -Jl- O'zo •I -Jl- O'zzi• ] 1 + dp, (8) -- h/2

where each of the stress and velocity components are func- tions of the arguments (a + p, •, z, t). Notationally omit- ting the (•, z, t) dependence for the moment, and evaluating explicity the a• h component in Eq. (8) with the aid of the expansion in Eq. (5) for h (a + p), we obtain

n/2 crtt(a +p)[/t(a) + a(')(a)p h/2

+ fi(2)(a ) (p2/2) + '" ] ( 1 + p/a ) dp

=/(a)J_ •,/2 crtt (a + p) 1 + dp + a(')(a) X trzt(a +p) 1 + pdp

-- h/2

h /2 + [...lap. -- h/2

(9)

Mz = Crzt(a +p) 1 + pdp. d -- h/2

(11)

In a similar fashion, the other eight resultants, N•, M•, Nt•, Mt•, N•t, M•t, Qt, and Q• (Kennard 4 and Naghdi 5) will be seen to emerge as a result of the explicit evaluation of Eqs. (7). The third term on the rhs of Eq. (9) represents those contributions to the integral whose integrated forms are not expressible in terms of known or standard resultant quantities such as Nt, Mr, etc. It must be noted that this in no way implies their insignificance, and these contributions can be seen to occur as a direct result of any second and higher-order terms in the series representation of Eq. (8). There are, however, two additional terms (which have not been presented in previous formulations), which appear as a direct result of our formulation and will be identified and

labeled. One of these occurs, for example, from the explicit evaluation of the Crzr • component in Eq. (8) when using the expansion in Eq. (5) for •(a -3- p), i.e.,

h/2 0'zr (a + p) [ •(a) + •(•)(a)p -- hi2

d- ['"](/22/2) d- '"] ( 1 d- p/a)dp

= W(a) O'zr (a + p) 1 + dp + W(•)(a) -- hi2

x a•r(a +p) • + pdp d -- hi2

+ [ lap. (12)

The first term on the rhs of Eq. (12) is seen to be the shear force resultant, Qt, multiplied by •(a), where

Qz -- O'zr(a +p) 1 + O. d -- h/2

(13)

The second term on the rhs of Eq. (12) is somewhat pecu- liar. Although it bears the same relationship to Qt as, for example, Mt does to Nt, the resulting quantity is not a "mo- ment" in the usual sense. For lack of a better description, it appears that the effect of this operation is to linearly weight the shear force, crtr, over the thickness of the shell. For sim- plicity, we shall label these two quantities which arise from the evaluation of Eqs. (7) [and which are multiplied by •(1)(a)] as Rt (relating to Qz) and R• (relating to Q• ), where

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and

R z = O'zr(a-+-p) 1 + pdp d -- h/2

(14)

h/2 R,• = O'•r(a +p)pdp. (15) -- h/2

The general form of Eqs. (7) may now be expressed as fol- lows:

P• = -- [ &Q• + tb(•)Rz + bNz• + b ( l)mzq b

+ fiNz + h(l)Mz + ['"1],

P• = -- [ tbQ• + tb( • )R • + bN• + •)(l)m•b + itN•z + fi(')M•z + ['"]],

(16)

where the velocities and resultants are functions of the argu- ments (a, •b, z, t). The terms in Eqs. (16) represented by [.-' ] are, once again, those integrated quantities that do not result in standard or known representations, such as the third terms on the rhs of Eqs. (9) and (12). It must be em- phasized that this in no way implies their insignificance, and also that these terms can be explicity evaluated.

For completeness and consistency of notation with ref- erence to Eqs. (2)-(4), we present the expressions for the intensity resultants in our three coordinate system as follows (where the 12 force, moment, and weighting resultants are presented explicitly, and the remaining contributions to the Eqs. (7) are presented generically as [-"] ).

In Cartesian coordinates (taking y perpendicular to the plate), the velocities and resultants are functions of the argu- ments (x, 0, z, t)

Pz = -- [ .• y Qz + '• 3 TM R z '-Jr- •x Szx '-Jr- ,• (x TM Mzx .(1) +.•zNz +Sz Mz + ['"]],

= - + + +

'(') M•,z -F ["'1 ] +.•zNxz +Sz ß

(17)

In cylindrical coordinates [as already shown in Eq. (16), where (w, v, u) = s l, the velocities and resultants are func- tions of the arguments (a, •b, z, t)

Pz = -- [ .•r Qz -Jr-.•(r ' ) R z -Jr-.•r)Szr) -Jr-.•" mzr) .(1) +.•zNz +sz Mz + ['"]],

P& = -- [•'rQ& -Jr-•'(r ') R& -Jr-•'oNo -Jr-

+ zS, z + + ['--1].

(18)

In spherical coordinates, the velocities and resultants are functions of the arguments (a, 0, •b, t)

'(1) Ro + •No• + •) Mo• Po = -- [.•rQo -+- Sr

+ •oNo + •(o"Mo +['"l], (19)

'(') R,• + •N• + •') M• e• = --[.•rQ• -[-Sr

+ •oN•o +•" M•o + ['"l]. It must be noted that a corresponding power per unit

length in the direction normal to the surface, Pr, can be de- veloped in a fashion similar to Eqs. (7). Although this quan- tity does not have immediate relevance from the perspective of shell theory, it will be referred to in a later section on source localization.

A. Examples: Cylindrical coordinates

For a specific example, let us utilize the general result of the Epstein6-Kennard 4 formulation for the series in Eq. ( 5 ). Using Eqs. (7), we obtain a form of the intensity resultant as shown in Eqs. (16) land (18) ]. The expressions for the series in Eqs. ( 5 ) are given in Eqs. (A5) and the correspond- ing forms of the resultants [represented in Eqs. (16) and (18) ] can be found in Ref. 4, or derived in a straightforward manner.

If we were to use the Kirchhoff assumptions 7 [Eqs. (A6) ] in the representation for the series in Eqs. (5), then we would obtain the following form of Eqs. (7)

Pz = -- tbQz + bNz• + (.b 1 a a &)

XMzo + i•Nz - • ] OzMz , (20)

po = _ •bQo + bNo + ( b a a &b

X M,• + fi N , z O Co 8z

where the velocities have been written explicity in terms of their definitions [ functions of the arguments (a, •b, z, t) ].

These two forms of the intensity differ from those pre- sented by Fuller I and Pavic. 2 This is not surprising, since various representations of Eqs. (5) will yield various forms of Eqs. (7). In fact, Eqs. (20) can be seen to follow directly from Eqs. (16) [ and (18) ] by noting that the Kirchhoff assumptions, or series representation for Eqs. (5), is identi- cally a simplification of the Epstein-Kennard series repre- sentation. Further, if we compare their corresponding equa- tions of motion, we will find that the Donnell-Mushtari shell equations (which are based on the Kirchhoff assumptions) follow directly from the Epstein-Kennard equations of mo- tion, as a result of these same simplifications. If we continue to simplify in this manner (by neglecting higher-order terms in both the series and resultants), we will eventually arrive at Love's membrane equations, 8 which would correspondingly -yield yet another form of Eqs. (7). Although each particular series representation of Eqs. (5) may differ, the procedure for obtaining the corresponding forms of Eqs. (7) is the same.

1169 J. Acoust. Soc. Am., Vol. 87, No. 3, March 1990 Romano et aL: Poynting vector formulation 1169

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B. Recovery of plate intensity: Cartesian coordinates

An interesting observation concerning the recovery of. the plate intensity from the representation of the intensity for the cylindrical shell can be made quite easily. For exam- ple, with reference to Eqs. (20) (derived from the Kirchhoff assumptions), if we let the radius a tend to infinity, and note that

then upon arranging terms, we obtain the following repre- sentation for the plate intensity in Cartesian coordinates

wtZ - - + + 8x 8z

(21)

= - - ow OW M= + + 1 . axM -- & The velocities have, once again, been written explicitly in terms of their definitions [functions of the arguments (x,O,z,t)].

The first three terms in each of Eqs. (21 ) involving • and its derivatives are identical to those presented by Noi- seux 9 and Pavic. •ø The last two terms in each of Eqs. (21) involving/• and b represent in-plane or rather tangential con- tributions that are uncoupled from the flexural motion of the plate. As can be seen in Appendix B, these four components are due to the zero-order components in the Taylor series for b(p) and/t (p) in the tr• b, tr=/t, trxx b, and tr=/t integrations over the thickness of the plate. The second and third terms in each of Eqs. (21 ) are correspondingly due to the first-order components of the Taylor series for b(p) and/t (p) in this same integration, since b •) = -- 8tb/Sx and//(1) = __ 8/•}/8Z. Therefore, the contributions of these four new terms to Eq. (21 ) are, in general, appreciable.

III. TIME INTEGRATION AND TIME AVERAGING

As stated earlier, quantities such as I (instantaneous intensity), and P (instantaneous intensity resultant) are functions of position and time. In establishing the necessary criteria for source localization, it is useful to develop time- integrated and time-averaged forms of Eqs. (1) and (7) such that we may speak of net energy flow and time-aver- aged power flow, respectively. In the following subsections, we will develop (a) the time-integrated forms of Eqs. (1) and (7) for real, transient time series, and (b) the time- averaged forms for real, monochromatic time series. As will be shown, the nature of the time series dictates which quanti- ties are determinable and which formulations may be used.

A. Transient signals

The time integral of the instantaneous intensity can be developed from Eq. ( 1 ) as

I(t)dt = -- •(t).dr(t)dt. (22)

In practice, it is occasionally more convenient to express Eq. (22) in the frequency domain. In this instance, we define the Fourier transforms of the velocity and stress tensor as

o• •(t)e'øt dt = _ ico•(co)=•(co), (23) _• d•'(t)d ø•t dt•-(co). (24)

Although •-(co) and • (co) are, in general, complex, when •(t) and •(t) represent purely real time series (as is the case with any real, physically realizable system) then •-(co) and •(co) are Hermitian, or rather, their real parts are even and their imaginary parts are odd. In this case, we may express Eq. (22) in the frequency domain, while noting that the odd, imaginary components integrate to zero yielding

I(t)dt = 1 (Re[•(co) ].Re[•-(co) ]

+ Im[•(co)].Im [•-(co)]•dco

1 Re[•*(co).•-(co) ]dco. (25) 2•r oo

Letting

I(co) -- -- Re[•*(co).Jr(co) ], (26)

then, Eq. (22) becomes

I(t)dt= 1 I(co)&o. (27) -• 2•r _•

Equation (27) is simply a restatement of Parseval's theorem TM for the determination of the energy within a real time-dependent (broadband) system, and represents the time integration of the instantaneous energy flow (in joules per area in SI units or in ergs per area in cgs units).

Correspondingly, we can develop the time integration of the intensity resultant, P (t), as

P(t)dt= 1 P(co)dw, (28) • 2•r •

where P( co ) - Pz ( co ) ez + P• ( co ) % . Using Eqs. (7), then,

Pz ( cO ) = Iz ( a + p,c•,z, co ) 1 + dp -- h/2

= -- Re[•*(cO)•z(CO) +/,•{l)*(o.))•z (o.)) +... + [...]],

-.. •.h/2 •. Po (co) = I, ( a + p,c•,z, co ) dp -- h/2

= -- Re[•*(co)• (co) + •{'•*(•)• (•)

(29)

+... + [...]],

This latter form [Eq. (28) ] is in joules per length in SI units or in ergs per length in cgs units.

B. Monochromatic, steady-state signals

In the case of a real, monochromatic time series (of peri- odic frequency COo), the previous equations [ (22) and (28) ] are no longer valid, since Eq. (22), for example, diverges. However it is often the convention in modern intensity tech- niques (Williams•2 and Elko•3) to define a time-averaged intensity; i.e.,

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(l(t)) lim 1 fr/2 -- • l(t)dt r-.oo T d-- T/2

= - +

= -- «Re[•*(coo)'•(coo)]

= « l(coo), (30)

which represents the average power flow per area per unit time (in watts per area in SI units or in ergs per second per area in cgs units).

Correspondingly, we can develop the time average of the intensity resultant, P(t), as

(P(t)) lim 1 fT/2 = • P(t)dt- (COo) (31) T--,oo TJ-T/2 2 '

where P(coo) = P• (coo)e• + P• (coo)%. Using Eqs. (7), then,

(P•(t)) = (Iz(a +p,c),z,t)) 1 + dp -- h/2

=

= - •Re[ •*(coo)•z (COo) +/•)(l)*(coo)k z (COO)

+ '''+ [''']]' (32) h/2 (P• (t) ) = (I• ( a + p,qb,z,t) )dp

d -- h/2

= (COo)

= -- -•Re [•c*(coo)• (coo) + •{')*(coo)• (coo) ] +... + [...]].

This latter form [ Eq. (31 ) ] is in watts per length in SI units or in ergs per second per length in cgs units.

which is the rate of change of energy inside the fixed volume V.

If the medium inside Vis subject to body forces, then the equations of motion are

02s

Pm 0t 2 = •7"•r' -3- F, (36) where F represents a force density (i.e., force per unit vol- ume). When Eq. (36) is substituted into Eq. ( 35 ), the result is

[(Oq V (Oq tGb' ' dV Os j

= --fr•7'ldV+f•F.•sdV. (37) 0t

On the other hand, if the forces act on the medium inside V only at its external boundaries, the terms containing F in Eqs. (36) and (37) must be deleted. These external surface forces (loads, tractions) enter through appropriate condi- tions imposed at the boundaries for the explicit determina- tion of the displacements s. For such a case, Eq. (37) be- comes

fv •3h• dV= -- fvV.l dV, Ot (38)

which means that the divergence of the instantaneous energy flow vector at any point equals minus the time rate of change of the energy density at that point.

IV. ENERGY BALANCE RELATIONSHIPS

To show how Eqs. (27), (28), (30), and (31 ) can be used as indicators for source localization, we must first de- velop the necessary criteria from the standpoint of energy balance. We begin our discussion with a formulation of the equation of continuity for energy. Defining the Hamilto- nian, or energy density, for an isotropic elastic medium 3 as follows,

Pm (O•$• 2 = + 5- ' (33)

where p,• is the density of the medium, • = • (Vs + sV) = the strain tensor and the term is the trace of the matric product of the two tensors •r and •. In Eq. (33), the first term represents the kinetic energy and the second the potential energy of the system. The total ener- gy H, within a volume V, is then simply

H = fvh dV. (34) The flow of energy across any given closed surface may be obtained by finding the rate of change of the total energy inside the surface. 3 Upon application of the time derivative to Eq. (34), we obtain

A. Transient signals

Subject to the condition that our time series is real and transient, let us take the time integral of Eq. (38); i.e.,

X7.I + dV dt = 0. (39)

For an isolated and lossless mechanical system such as we assume here, the initial and final energies are equal. Hence,

•ooOhEdt hE(o•)-hE(- o•)=0. (40) oo 0t

This demonstrates that the time integral of the rate of change of the energy density is zero. On the other hand, since the divergence operator commutes with the time integration in Eq. ( 39 ), then using Parseval's theorem [ Eq. (27) ], we may express our conclusions as follows.

( 1 ) If the medium is subject to body forces, as represent- ed in Eq. (36), then application of the time integral to Eq. (37) yields the relationship

V.l ( r,c),z, co ) d V dco

= Re[•(r,c),z, co).•*(r,c),z,co) ]dVdco. (41)

(2) If the force is applied externally to the volume ele-

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ment, then application of the time integral to Eq. (38) yields the relationship

V.I(r,c/3,z, co)dVdco = 0. (42)

We have thereby established the necessary criteria for source localization in the event that our time signal is real and tran- sient.

B. Steady-state signals

Since it follows from Eq. (40) that the time average of •h•/•t is zero, then for a real, monochromatic time series, time-averaged forms of Eqs. (37) and (38) may be obtained which represent the balance of time-averaged power per unit area, and we may express our conclusions as follows.

( 1 ) If the medium is subject to body forces, as represent- ed in Eq. (36), then application of the time average to Eq. ( 37) yields the relationship

•, V.i ( r,c•,z, coo ) d V - f •, Re [ • ( r,c•,z, coo ) '•* ( r,c•,z, coo ) ] d V. --• (43)

(2) If the force is applied externally to the volume V, then application of the time average to Eq. (38) yields the relationship

V.I ( r,c),z, coo)dV = O. (44)

We have thereby established the necessary criteria of source , I

localization in the event that our time signal is real and mo- nochromatic. In the case in which I (t) represents a transient time signal (therefore containing many frequency compo- nents), characterized by a finite amount of energy (as as- sumed previously), then the time-averaging integral of the form in Eq. (30) is zero, since a finite amount of energy is averaged over an infinite time length. Therefore, as stated above, the nature of the time series dictates which quantities are determinable and which formulations should be used.

V. APPLICATION TO SOURCE LOCALIZATION

It must be mentioned that the volume element V used

throughout the preceding section is completely arbitrary. Although most developments of the equations of continuity for energy imply that V represents the net volume (Vtot) within which the displacement field occurs, this net volume can be expressed as the sum of smaller volumes Vi, such that Vtot = Ei Vi. In the event that the body and surface forces are concentrated or located at specific points within and on Vtot then Eqs. (41 ) and (43) will be zero in the regions Vi where there are no forces, and equal to their respective values in the regions Vi where forces are present. Since integration is a linear operation, the integral over Vtot is the sum of the inte- grals over the set of disjoint volume elements, Vi, of which Vtot consists. For example, we can recast Eq. (43) in terms of the two possible cases which can occur in any volume element, Vi:

fv, V -]' ( r, c) , z, coo)dV= fv, Re[•(r, q3,z,coo)-•*(r,c/3,z, coo) ]dV, if F(r,•,z, coo) %0 in Vi;

otherwise.

(45)

It is with these smaller volume elements that Eqs. (41 )-(44) are calculated throughout the entire shell similar to a spatial sliding window, indicating the net divergence at each loca- tion. It can be seen, of course, that Eqs. (42) and (44) are zero for either case occurring in Eq. (45). Under various assumptions, however, we can rearrange their components to yield an indicator as well.

For example, if our time series is real and transient, and if we once again choose an element of volume V, enclosed by a corresponding surface S upon which surface forces are present (as shown in Fig. 2 with a driver on the interior surface and fluid loading on the exterior surface), the use of Green's theorem yields

V-I d V dt = I-n dS dt = 0. (46)

Using the results from Sec. III, then

I(t)-n dSdt = 1 •(co)-n dSdco. (47) -oo 2•r

Further, if we assume that the surface forces are normal to the cylinder surfaces, then the energy entering this volume element does so through faces S1 and 5'2 of the surface enclos- ing V; i.e.,

I

= -- [•'fluid "JI- •'driver ], (48) where enuia is the energy delivered to the cylinder by the fluid loading, and Edriver is the energy delivered to the cylinder by the driver. Expansion of Eq. (47) therefore yields

S,• S:• f P S•

FIG. 2. Volume element enclosed by six surfaces, Si. Fluid loading on exte- rior surface ($1) represented by p, driving surface force on interior surface (S2) represented by F. Closed contour, •5 [as in Eq. (52) ], indicated within volume element.

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= [ •'fiuid •- •'driver ]' (49) Equation (49) states that the net outflow of energy from the volume element due to that energy which passed through the surfaces S3, S4, S5, and S6, equals the energy injected through surfaces S• and S2. Then, making use of Sec. II, we can note, for example, that

fs • (w)dS3 = a •z (co) 1 + d$ dp • h/2

= a Pz (•)d•, (50)

and

(o.,)d$s = (o.,)dz d,o .s -- h /2

=

then

(51)

1 f• (••(w)'ndl)dw=[e, uia+edrive•.]. (52) 2• _•

Or, equivalently from Green's theorem,

If. © (fz••72D'•(o))dA )do): [efluid -•- edriver ], (53) 2• -•

where A is the surface enclosed by the line integral in Eq. (52).

What we have just demonstrated [from Eqs. (46)- (53) ] is that if surface forces alone are responsible for the forced motion of the shell, the time-dependent motion is real and transient, and some reasonable assumptions about these unknown forces can be made, then Eq. (46) may be success- fully manipulated to yield an indicator as to the location of these forces. In particular, if the surface forces act normally to the surface of the cylinder, then one may reasonably locate them using the formulation of Sec. II in the form of Eq. (52) or (53) [where a typical closed contour for the line integral in Eq. (52) is shown in Fig. (2) ].

We may now observe the physical significance of the component corresponding to power per unit length in the direction normal to the surface, Pt, alluded to in the last paragraph of Sec. II. Specifically, Eq. (48) may be separated to identify the following relationships:

3 ) ] = • a + ,w dl do = - enuid , (54) 2• -•

2• •

) h o)] . _ 1 • • a--• dl do= +fidriver (55) • • 2' Therefore, we see that the quantity Pr has explicit meaning

from the perspective of energy balance. To be thoroughly consistent, of course, the representation for P should have contained the Pr (cO) component. However, we have not in- cuded its development here since its use is not generally fa- miliar.

In the experimental application of these developments, e•uid was assumed negligible in comparison to edr i .... and integration over co was performed over a band Aw. In addi- tion, since we can let Ato t = •iAi, then we can recast Eq. (53) in terms of its two possible cases as an indicator for source localization as

;A (•A '• ) IEdriver ' 1 V2o'P(w)dA doom 2zr •o , [0,

if F :•0 on otherwise.

(56)

The integration over a band of frequencies Aw was a practi- cal restriction, due to the fact that the experimental data are band limited. Further, since real systems are characterized by their eigenfrequencies, such that even if the driver sup- plies a broadband input, the output spectrum will contain various levels of energy in various, narrow frequency re- gions, we may approximate Eq. (53) with just such a narrow frequency band. This is, in general, a risky procedure, how- ever, since if we were to choose such a narrow frequency band a priori, we may find ourselves in a region containing little or no energy, which would yield results highly contami- nated by noise. Increasing the width of this band of frequen- cies will, of course, increase the accuracy of the integration. In the event that the intensity is monochromatic, there is obviously a single contribution in Eq. (56).

Vl. CONCLUSIONS

It was demonstrated in Secs. I and II how expressions for the structural intensity in shells and plates are easily de- rivable. For example, upon insertion of an appropriate Tay- lor series expansion for the displacements s [Eqs. (5) ], into the appropriate three-dimensional form of the intensity I [ Eqs. ( 1 )-(4) ], and application of the necessary spatial in- tegrations, one obtains an intensity resultant quantity, P [Eqs. (7) ], in terms of the well-known stress and moment resultants. In general, this approach is applicable for any coordinate system, and permits insight into the contribu- tions of the various resultant components in a straightfor- ward manner.

In Sec. III, the time-integrated and time-averaged forms of the structural intensity and structural intensity resultant were developed for both real transient and real monochro- matic time series, in preparation for the establishment of the relevant energy balance relationships in Sec. IV, which were applied in Sec. V to a particular source localization tech- nique.

Obviously, there are many points to be addressed con- cerning the accuracy of the final forms of our equations. However, it can be seen that ultimately, the burden of the validity of the final equations lies in the ability of the initial Taylor series [Eqs. (5)] to accurately represent the dis-

1173 J. Acoust. Soc. Am., Vol. 87, No. 3, March 1990 Romano et aL' Poynting vector formulation 1173

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placements within the shell or plate. Since the final form of the equations follows directly from this series in a simple manner, there is nothing particularly mysterious in this for- mation to be dealt with by the reader. In Part II of this se- quence, we will address this issue and present the results of the direct application of these methods to various synthetic and actual experimental situations, for the demonstration of source localization and structural power flow.

APPENDIX A: TAYLOR SERIES REPRESENTATIONS IN

CYLINDRICAL COORDINATES

( 1 ) In the Epstein-Kennard representation, the follow- ing quantities are defined that relate the external loads on the faces of the shell to series coefficients:

Zz + = o'•,. (a d- h/2) d- o'•,. (a -- h/2)

Zf = o',•,. (a d- h/2) d- o'&(a - h/2) , (A1)

Zf -- or,.,. (a d- h/2) d- or,.,. (a -- h/2) 2(A + 2p)

Crz,•(a + h/2) -Crz,•(a - h/2) Z _ • ,

•%r(a d- h/2) d- •%r(a - h/2) = , (A2)

•r.(a + h/2) - •r.(a - h/2)

h(,t + where the Lam6 elastic constants A and p are defined in terms of Young's modulus E and Poisson's ratio v as

A=Ev/(ld-v)(1--2v), ,a=E/2(ld-v). (A3) We define one more coefficient related to the Lam6 elastic

constants A and p as

a - X /(X + 2/t) = v/( • - v). (A4)

With these definitions, Eqs. (5) may be expressed, up to second order in p, as follows, where all displacement compo- nents on the right-hand side are functions of variables ( a,qb,z) :

w(a +p) - w(a) + w(•)(a)p + w(2)(a)(p2/2) d- '"

Ou(a) 1 Ov(a) = w(a) + -- a -4 Oz a

Ot 2 Ou(a) Ot 2 Ov(a) a(a + 1) t • • w(a) + Z i-

v(a +p) = v(a) + v(•)(a)p + v(2)(a)(t02/2) d- '"

= v(a) + (,v(a) lOw(a) )p (• 02u(a) a a O-•'•d- Z/ d- OzO•b

z; oz: + +'",

u(a +p) = u(a) + u(•)(a)p + u(2)(a)(p:'/2) + '"

3o( = u(a) d- _ Ow(a) + Zz + + a Oz Oz 2

w ( a ) d-Jr t9 d- a a Oz 2 1 02w(a) ) a 2 Oqb 2

o, z/-o,( øzz+ a Oz a 0 4 d-'"'

a O2/)(a) a Ow(a)

} a 2 042 • a 2 O-•• d-z•-

a O2v(a) a Ow(a) OZf ) to 2 a Oz&b • • +Z- , +'.'. a Oz z Oz T

(A5)

(2) The so-called Kirchhoff assumptions lead to a Tay- lor series representation as shown below:

w(ad-p) = w(a),

v(a + p) = v(a) + (v(a) 10w(a))p a a 0• ' (A6) ( Ow(a) )p u(a + p) = u(a) + - Oz ' '

It can be observed that Eqs. (A6) follow directly from Eqs. (A5) with various simplifications as can be verified by in- spection.

Explicit representations of the corresponding stress re- suitants were not provided here due to space limitations. However, they are easily derived from their fundamental definitions (Refs. 4-6, 8, 14, and 15), and can be inserted directly into Eqs. ( 17)-(19) to obtain the structural intensi-

I

ty--provided the appropriate and corresponding Taylor se- ries representation for the displacements is used in conjunc- tion with them. The derivation of the Taylor series shown in Eqs. (A5) is presented in Ref. 4 (in a somewhat ambiguous fashion), and, recently, it has come to the authors' attention that a similar and detailed development appears in Ref. 15. One of the authors of the current paper has developed the same Taylor series in a straightforward and simple manner, which will be the subject of a future publication.

APPENDIX B: TAYLOR REPRESENTATIONS IN CARTESIAN COORDINATES

( 1 ) In the Cartesian form of the Epstein-Kennard rep- resentation, the following quantities are defined that relate the external loads on the faces of the shell to series coeffi-

cients:

1174 J. Acoust. Soc. Am., Vol. 87, No. 3, March 1990 Romano eta/.: Poynting vector formulation 1174

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Z• + = trzy( d- h/2) d- trzy( - h/2) '

Z• + = tr•y(+h/2) + tr•y(-h/2) 2/• , (B1) + = ayy(+h/2)+ayy(-h/2) gy ,

2(2 + 2•)

a•y(+h/2)-a•y(-h/2) z;= ,

a•y( +h/2)- a•y( - h/2) Z; = h• , (B2)

_ ayy ( + h/2) - ayy ( - h/2) h(2 +2•)

With these definitions, Eqs. (5) may be expressed, up to second order inp, as follows, where all displacement compo- nents on the right-hand side are functions of the variables ( 0 ,x,z ):

W(fi) = W(0) + W(•)(0)fi + W(2)(0) (fi2/2) + '"

=w(o) + -a + + gf p •z •x

+ a •z • •x •

+Z--- + +'",

/.}(tO) = /.}(0) •'- /.}(1)(0)tO d- /;(2)(0)(/92/2) d- '''

=v(O) +(--8w(O)+Z:)p 8x

82u(0) 82v(0) + a•+a• Oz 8x 8x 2

+ Z ff •xx ' + ..., (B3) U(p) = U(0) d- U(1)(0)p d- U(2)(O)(p2/2) d- '"

=u(O) +(-8w(O)+Z,+)p 8z

82v(0) 82u(0) + a • + OOx

+Z- +'" • 8z

(2) The Cartesian form of the Kirchhoff assumptions leads to a Taylor series representation as shown below:

w(p) = w(0),

v(p) =v(0)+( 8w(O) ) p, (B4) • ,

8x

u(p) = u(O) + ( 8w(O) ) - 8z P' (3) The so-called thin plate assumptions 14 lead to a rep-

resentation as shown below:

w(p) = w(O),

v(p) =(--8w(O))p, (B5) - 8z P'

It can be observed that Eqs. (B4) and (B5) follow di- rectly from Eqs. (B3) with various simplifications as can be verified by inspection.

•C. R. Fuller, "The effects of wall discontinuities on the propagation of flexural waves in cylindrical shells," J. Sound Vib. 75(2), 207 ( 1981 ).

2G. Pavic, in Proceedings of the 12th International Congress on Acoustics, Paper D6-6 (1986).

3p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw- Hill, New York, 1953).

4E. H. Kennard, "The new approach to shell theory: Circular cylinders," J. Appl. Mech. 20, 33-40 (March 1953).

5p. M. Naghdi and J. G. Berry, "On the equations of motion of cylindrical shells," J. Appl. Mech. 21, 160-166 (1954).

6p. S. Epstein, "On the theory of elastic vibrations in plates and shells," J. Math. Phys. 21, 198 (1942).

7M. C. Junger and D. Feit, Sound, Structures, and Their Interaction ( MIT, Cambridge, MA, 1986).

8A. W. Leissa, Vibration of Shells, NASA SP-288 No. N73-26924. 99. U. Noiseaux, "Measurement of power flow in uniform beams and plates," J. Acoust. Soc. Am. 47, 238-247 (1970).

løG. Pavic, "Measurement of structure borne wave intensity, part I: Formu- lation of methods," J. Sound Vib. 49, 221-230 (1976).

l lR. N. Bracewell, The Fourier Transform and Its Applications (McGraw- Hill, New York, 1978).

•2E. G. Williams, H. D. Dardy, and R. G. Fink, "A technique for measure- ment of structure-borne intensity in plates," J. Acoust. Soc. Am. 78, 2061-2068 (1985).

•3G. W. Elko, "Frequency domain estimation of the complex acoustic in- tensity and acoustic energy density," Ph.D. thesis, Pennsylvania State University, University Park, PA (1984).

14K. F. Graff, Wave Motion in Elastic Solids (Ohio State U.P., Columbus, OH, 1975).

15L. Cremer, M. Heckl, and E. E. Ungar, Structure-Borne Sound (Springer- Verlag, Berlin, 1988), Chap. II.8.

1175 J. Acoust. Soc. Am., Vol. 87, No. 3, March 1990 Romano ot a/.' Poynting vector formulation 1175

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