A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

Journal of Sound and Vibration (1973) 27(2), 235-260

A GENERAL THEORY OF HARMONIC WAVE PROPAGATION LINEAR PERIODIC SYSTEMS WITH MULTIPLE COUPLING

IN

D. J. MEAD Department of Aeronautics and Astronautics, University of Southampton, Southampton, SO9 SNH, England

(Received 5 October

1972, and in revisedform 5 January 1973)

A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleigh’s Quotient and Rayleigh’s Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system. 1. INTRODUCTION

A periodic system consists of a number of identical elements, coupled together in identical ways to form the whole system. Examples of such a system are a tall apartment block having a uniform structure and identical storeys, and an aeroplane fuselage structure consisting of a uniform shell reinforced at regular intervals by an orthogonal set of identical stiffeners. The identical elements composing the whole system are known as “periodic elements”. In the case of the aeroplane fuselage this would be one plate element between adjacent pairs of stiffeners together with the stiffeners along two adjacent sides. Analyses of the audio-frequency vibrations of such structures are greatly simplified by utilizing the periodic property. Principal modes and frequencies and responses to steady-state excitation are thereby calculated with greater facility. Indeed, it is almost unnecessary to determine principal modes at all, as the periodic structure has vibration characteristics which are best understood in different terms-those of propagating and non-propagating free wave motion. Wave motion in simple periodic systems has been studied for nearly 300 years, as Brillouin [l] has pointed out in his classic work. Physicists and electrical engineers have developed the studies over the years in relation to crystals, optics, electrical transmission lines, etc. It is only relatively recently that wave motion in engineering periodic structures (consisting of beams, plates, etc.) has been investigated. Cremer and Leilich [2] studied flexural motion in periodic beam structures, and showed that waves can propagate in some frequency bands but not in others. Heck1 [3] considered a system of beams coupled together to form a regular grillage and demonstrated the same property. Mead [4] included the effects of damping in the wave 235

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D. J. MEAD

propagation theory for periodic beams, and later [5] discussed the nature of the propagating waves and their possible interaction with acoustic waves. This has been followed [6] by a study of the harmonic and random responses of periodic beams on elastic supports and subjected to convected loading. Mead and Sen Gupta [7] have treated the “rib-skin” periodic structure by wave propagation methods, and Sen Gupta [8] has shown how natural frequencies of finite beam-type periodic structures can be found from the wave propagation characteristics. In all the work so far, exact harmonic solutions to the governing differential equations of motion have been found. This has been possible since the periodic element has been uniform, and its differential equation of motion reducible to an ordinary linear differential equation with a simple type of solution. Particular solutions have been found for each problem, in each of which the analysis has shown a very important property of the free wave motion. At any frequency, the number of distinctly different free waves which can exist in the whole periodic structure is twice the number of freedoms at the end of each periodic element. For instance, a periodic beam element on an elastic support at its end has two freedoms-translation and rotation. There are then four different free flexural waves which can exist at any frequency. If bending of the beam element is coupled with torsion so that twisting of the beam is also possible at the support, then three freedoms exist at the end of the element. There are then six possible free waves of motion. Each of these waves is associated with a particular “propagation constant”, p. When the system vibrates in just one of the free waves, the harmonic motion at any point in one element is equal to e’ times the motion at the corresponding point in the next element. /J may be real, imaginary or generally complex, and its values always occur in positive and negative pairs. These correspond to identical, but opposite-going waves. The real part of p is known as the “attenuation constant”, and the imaginary part as the “phase constant”. The purpose of the present paper is to present a more general theory of wave-propagation which is not restricted (as in references [I] to [8]) to flexural motion. The theory allows multiple freedoms at the ends of the periodic elements, or, in other words, it allows adjacent elements to be coupled through any number of coordinates. Whereas former studies have considered uniform elements, no such restriction is made in this paper. It is shown in a general manner that the number of possible propagation constants (and corresponding waves) is equal to twice the number of coupling coordinates between adjacent elements. The energy involved in the wave motion is also studied. Purely imaginary propagation constants are known to be associated with waves which propagate energy, whereas purely real propagation constants belong to waves with no energy flow. This is formally proved. For some undamped systems, propagation constants have been found which are generally complex, having both real and imaginary parts. At first sight, this seems anomalous. The existence of the imaginary part suggests a progressive, energy-carrying wave motion, whereas the real part suggests a loss or growth of energy in the wave as it progresses. With no damping or external forcing on the system, this is impossible. Accordingly, this paper also examines the energy flow in undamped systems in waves with generally complex propagation constants. It is formally shown to be zero. The potential and kinetic energies of the energy-propagating waves are studied, and it is found that the powerful energy methods of vibration analysis (Rayleigh’s method and the Rayleigh-Ritz technique in particular) may be extended to permit study of the propagation constants and the corresponding waves. The advantage of this lies in the fact that systems with non-uniform elements can be studied as easily as those with uniform elements. The restrictions of the former investigations are therefore bypassed. When damping is introduced into the periodic system all the propagation constants become complex. This is necessary if energy is to flow from the source of vibration to the energy dissipating sinks in each element. Damping terms are admitted into the general formulation

WAVE PROPAGATION IN PERIODIC! SYSTEMS

237

of the free wave problem and the relevant equations for the complex propagation constants are established. If the damping is hysteretic, a special form of forced wave motion may be postulated whereby a wave may propagate without attenuation. Such waves are complementary to the “damped forced normal modes” formerly postulated by the author [9] for finite dynamical systems with hysteretic damping. The equations for the propagation constants of these waves are derived and consideration is given to the “loss factor” of the propagating wave motion. The greater part of this paper deals with steady-state harmonic wave propagation through one-dimensional periodic systems. “One-dimensional” implies that the elements are connected end-to-end, albeit with multiple coordinate coupling between them. A final section to the paper deals with two-dimensional systems in which the elements are assembled both end-to-end and side-by-side. Multiple coupling between adjacent elements is allowed on all sides. The wave motion investigated is a simple generalization of that of the one-dimensional system, and is analogous to plane wave motion propagating over a flat surface in a particular direction. The more difficult (but more realistic) problem of wave motion spreading out in all directions from a point source is not considered.

Dlsplocements

Forces P

Figure 1. (a) Schematic diagram of a one-dimensional system.

2. THE GENERALIZED PERIODIC ELEMENT;

EQUATIONS EQUATIONS

periodic system. (b) The exterior force and coordinate

OF MOTION OF A ONE-DIMENSIONAL FOR THE PROPAGATION CONSTANTS

Consider a one-dimensional periodic system (Figure l(a)), each periodic element of which is coupled to its neighbours on each side by n coordinates and forces (Figure l(b)). The complete displacement pattern of an element is defined by these “exterior” or “coupling” coordinates together with N “interior” coordinates which define the displacements of all points other than the coupled points. The exterior coordinates at the left-hand side of each element will be denoted by the matrix column {q,}e imr,those at the right-hand side by (e> e’“‘, and the interior coordinates by {q,}e Irnt. The matrix column {q} (with no suffices) will denote the whole set of 2n + N coordinates: i.e.,

(1)

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D. J. MEAD

q, and q, will be taken to be the actual coordinates of displacement at the coupling points, whereas qi may represent an actual or more generalized interior coordinate system. Corresponding to each coordinate is a force Feiot. The force vector (F}e*~t defines the forces corresponding to q,, qi, q, :

(F} = i;

.

(2)

(1F*

The linear equations of harmonic motion of all coordinates of an undamped element are given in the standard reduced form by [K - w’Ml 00

= PI,

(3)

in which K and M are symmetric matrices of generalized stiffnesses and masses. Free wave motion through an infinite periodic system occurs when {F,} = 0 and is characterized by the displacements in one element being equal to the corresponding displacements in the adjacent element times e“, where k is the propagation constant. The exterior forces {F,} and {Fr} are not zero, since they are the means whereby wave motion is transmitted from one element to the next. Let {qi>,, {qlfp+l be the left-hand exterior coordinate vectors for elements p, p + 1. Then {qJpft = eYqJ,. Since the elementsp and p + 1 are connected {a),+ 1= {QJP so Ok>, = e%Jp.

This is true for any element, so the suffixp can be dropped leaving (4) Likewise, it is shown that Pi),+ I = e”{Kh. For equilibrium of the inter-connecting forces between elements p and p + 1

Combining these last two equations, and dropping the suffix p yields {F,} = -e*{F,}.

(3

Equations (4) and (5) can now be substituted into equation (3). With {F,) set to zero, it becomes K--‘Ml(ee~=

[_ileJ. (6)

It is convenient to partition the matrices K and M into the following forms to correspond with the three parts q,, qi, qr of the column of coordinates : K= E:i

i!!

gi:]and

M=[$

$

$1.

(7)

WAVE PROPAGATION

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Now, since qI and qr represent the actual displacements of the connecting points between the elements, the partitioned square matrices M,, and M,, must be diagonal, and Mli, M,,, Mm Mi,, M,,, M,i are all zero. Mii is square and symmetric. Further, since K is symmetric, it follows that K,, = K,T;,Ki, = K:, Ki, = KT,, where the superscript T denotes transposition. The 2n + N equations of (6) can now be reduced to n + N equations, since equation (4) reduces the number of independent coordinates from 2n + N to n + N. In the same reduction process the unknown force vector {F,} may be eliminated. The resulting matrix equation is

which may be written in the abbreviated form

(8b) These equations are those of an eigenvalue problem. For any given value of p, the frequencies can be found at which wave motion can occur with that propagation constant. The frequencies are obtained from the determinantal equation

(8~)

)[K PI - w’[MlI= 0,

which, when expanded, yields a polynomial equation in o2 of order equal to the rank of M. This is equal to the rank, Rii, of Mii plus the number of non-zero elements, n’, on the diagonal of [M,, + M,,]. There are, therefore, Rii -t n’different values of w2 at which waves with a given propagation constant can occur. Appendix A shows the circumstances under which n’ is less than n. When M,i is non-singular, its rank is N and there are N + n’frequencies for a given propagation constant. From now on it will be assumed that the internal coordinate system has been chosen to make Mi, non-singular. Section 4 shows that, when the given propagation constant is purely imaginary, all these N + n’ frequencies are real. When the propagation constant is real, the frequencies may be real or complex. Equation (8) may be re-formed to create an eigenvalue problem for finding p, given o. Let K,, - m2M,, = D1,, K,, - m2M,, = Dir, etc. Then

This is a quadratic eigenvalue problem for e”. Write this in the condensed form [e’D, + D, + e-“D,]

“,1 =-0. 0 I The determinant of the matrix in this equation must be zero for non-trivial solutions, so Ie’D, +D2+e-‘D,I

=O.

(11)

When this is expanded, and the terms in eUand e-” are collected together, it yields a polynomial equation in coshp. (This follows since D, = DT.) The order of the polynomial equation cannot exceed n, the number of coupling coordinates between adjacent elements.

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D. J. MEAD

Now the value of n may vary, depending on where the junction between the periodic elements is taken. For instance, the periodic system of Figure 2 would have three exterior coupling coordinates if the periodic element AA’were taken, but only one if element BB’were taken. The order of the polynomial equation in coshp is actually found to be equal to the minimum number, n,,,i,,, of coupling coordinates obtainable by appropriate choice of the junction. This is formally proved in Appendix B. When the periodic system is undamped, K and M, D1, D, and D, are all real. The coefficients of the polynomial equation in coshp are also all real. The values of coshp, however, may be real (positive or negative) or may occur in complex conjugate pairs. If coshp lies between +1 and -1, p is purely imaginary (= ic, say). The factor e@= e*’then represents a simple phase difference of E between the corresponding displacements of adjacent elements, but the amplitudes of the displacements are the same. If cash p is greater than + 1,e’ represents an exponential decay or growth of the displacement from element to element, with corresponding displacements in adjacent elements being in-phase. If coshp is less than -1, p has an imaginary part of rc and a non-zero real part. eU then represents exponential decay or growth of the displacement but with corresponding displacements in adjacent elements being in counter-phase.

L------_-_~.I

E’

E

Figure 2. A particular ordinates.

system to show how the choice of boundary affects the number of coupling CO-

If coshp is generally complex, there is both exponential decay (or growth) together with phase difference (# 0 or II) between corresponding displacements in adjacent elements. For a given purely imaginary value of p (isR, say) there is a particular complex wave vector {q) = {qR}. The corresponding negative value of ~1(-icR) belongs to the complex wave vector {d}, where the asterisk denotes “complex conjugate”. This is shown by equation (10) in which the D-matrices are all real. The same applies to the two wave vectors corresponding to the conjugate pair of propagation constants 6, + kR, 6, - iER,if such a pair exists. 3. ENERGY

PROPAGATION

BY FREE WAVES PERIODIC SYSTEMS

THROUGH

UNDAMPED

Associated with the natural free waves described by the eigenvectors of the previous section is a certain energy flow, and this will now be investigated. It is more easily studied by reference to the receptances of each periodic element, rather than by reference to the mass and stiffness matrices. Hence, another set of equations will be set up for the eigenvalues, e”, and the eigenvectors, {q}. In this formulation, only the exterior coordinates, {q,) and {qr>, and the exterior forces, {F,) and {F,}, need be considered. Since free waves only are to be considered, {F,} will be set to zero. The exterior coordinates are related to the exterior forces through a matrix of receptances:

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241

where alI and a,, are symmetrical, and a,, = airT. Since the system is undamped, the a’s are entirely real. Each of the a sub-matrices is of order n x n. Put q, = e”q,, F, = -&PI, as in equations (4) and (5). Equation (12) becomes {qJ = [ali - e”ad @,I

(134

and e%> = [a,, - e’a,,l {F, f.

(13b)

[ali + a,, - e” air - eVpa,,] {F,} = 0.

(14)

Hence This equation constitutes a quadratic eigenvalue problem for e” similar to equation (lo), but the eigenvector is now the exterior force vector, {F,}, instead of the displacement vector, (q}. It is convenient to introduce two matrices which are half the sum and difference, respectively, of air and a,i : i.e., Bs, = +(a,, + a,,)

(15a)

br,, = !&ii, - a,,).

(15h)

and

Since a,] = a;,, it follows that&. is symmetric, and & in skew-symmetric (hence the suffices). Let the Rth eigenvalue be e“~, and the corresponding eigenvector be {FIR}. Associated with these is the exterior coordinate vector (qlR}, given by GIiR)= [ai, - e@%,l (FIRI.

(16)

Now the time-averaged energy crossing the junction between two elements is given by E = #RelF&I{Q,,}, where IF&J is the transposed conjugate of (FIR}. {QIR}= ia{qlR} is the exterior velocity vector for the Rth wave. Then

[a,, - ePw,I P,,}) = 3Re(iwLF,%J [alI- ePRVsy + As)1{FIR)).

E = 3Re(iwLF,?A

(17)

Now, since ai, and bs, are real and symmetric, the scalar products

LF?d[all1WA = A

(184

lF:RJUJs,l(FmI = B

(18b)

and

are entirely real. Since /Jssis real and skew-symmetric, the scalar product LCJ

UU {FIR)= iC

(18~)

is entirely imaginary. C, as defined here, is therefore real, e”n may be complex, depending On PR. The time-averaged energy flux is therefore given by E = 4 Re(e”R(C - iB)).

(19)

Now e“nitself is related to A, Band Cand may be expressed in terms of these. Its dependence upon these terms will therefore be investigated in order to find, ultimately, how E depends on the nature of p: i.e., whether p is real, imaginary or complex. To do this, put ,U= pR, {F,} = {FIR} into equation (l4), pre-multiply it by LFTRJand put LF?RJ[Qll+ %I {FIR) = D.

(lgd)

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D. J. MEAD

Since both a,, and arr are real and symmetric, D is entirely real. Equation (14), thus manipulated, becomes eZDR(B+ iC) - eORD + (B - iC) = 0.

(20)

Hence e%=

D dDz

- 4(B2 + C”)

2(B + iC) (D &l/D’ - 4(B2 + C’))(B - iC)

(21)

2(B2 + C’)

When this is substituted into equation (19) and use is made of the fact that B, C and D are entirely real, it is found that E = $ Re(dD’

- 4(BZ + C’)).

(22)

Now consider the energy flow associated with a real value of c(~. In the first place, it will be noticed from equation (14) that a real value of pa yields a real vector {FIR}.This means that C (equation (18~)) must be zero. Equation (21) then shows that a real value of pR can only be obtained provided D2 > 4B2. This, in turn, shows that E in equation (22) is zero. Thus, there is no energy flow associated with a real propagation constant in an undamped system. Physically, this is an obvious feature. The amplitudes of the forces (FLR}and displacements {qIR} are growing (or decaying) from one junction to the other. If the forces at one junction were doing work in the course of one cycle, so also would the forces at the next junction. However, the ratio of the energies flowing across the junction would be e2@R: 1. The differences between the energy flows cannot occur in an undamped system undergoing steady-state harmonic vibration, for there is no source or sink of energy between the junctions. Hence, no energy can flow in this case. A purely imaginary value of pI; is obtained when Dz c 4(B2 + C2). Equation (21) shows that e“R is then complex, and further examination shows that leUR(is unity. Hence pLRis purely imaginary, and has equal positive and negative values, &i.+ The energy flow (equation (22)) is then given by the real expression E=+4(B2+C2)-

D2.

The positive energy flow is associated with the negative value of kR. Thus, when the propagation constant is purely imaginary, energy flows through the system. There is an exception to this when pR = Qn, for then D2 = 4(B2 + C’), C = 0 and E = 0. When D2 > 4(B2 + C’) and B and C are non-zero, e@Ris more generally complex (its modulus is no longer unity). pR is then complex. At the same time, il/D2 - 4(B2 + C’) in equation (22) is purely imaginary, so the energy flow is zero. Thus, when pLRhas a general complex value (6, + it&), no energy flows along the system. That this must be so is again obvious from physical considerations, since the amplitudes of force and displacement in adjacent elements differ by the factor esR. Since the system is undamped, there can be no energy flow under these conditions. The displacements {qR} and the forces {FR) are not in phase or in counter-phase when pR is generally complex. (This appears from equation (13a), in which e0 will be complex.) It is evidently possible for some of the forces in the vector (F,} to do work on the element as they move through their corresponding displacements in {qr}, but the other forces in {F,} together must extract the work put in, so that the net energy flow is zero.

WAVE PROPAGATION

4. PROPERTIES

IN PERIODIC

OF THE PROPAGATING

WAVES;

243

SYSTEMS

THE RAYLEIGH

QUOTIENT

Consider the free wave motion which is propagating with the purely imaginary propagation constant ,u = ia,. Equation (6) shows that the displacement and force vectors are then related by (23)

[K--zMiiile,~d_(_~,ei~.l.

It has been shown that there are (n’+ N) discrete frequencies at which waves with p= ia, can occur and (n’+ N) corresponding displacement and force vectors. Let the frequencies be denoted by WRj(j = 1 to n’ + N) and the corresponding force and displacement vectors by {qRj}, (FR,}. In general, the vectors will be complex, but it will be shown that the frequencies are all real. Equation (23), for thejth wave and frequency, becomes [K - mij

Ml {QRjIz {F,, I

PaI

[K - a;,

Ml ha) = {Fid.

(24b)

and, for the kth wave,

Premultiply equations (24a) and (24b) by Lq$l and LqzTI,respectively. It is found immediately that L&J {FRj} = 0

and

Lq*,;J:F,,} = 0.

(25)

Since K and M are both symmetrical, and oRj # tiRk, the following orthogonal relationships are then proved in the usual way: LGJ [Kl {qRj) = 0;

LqSIJ[M]{QRj/

=O

(k #iI*

(26)

Now pre-multiply equation (24a) by L.qX:J.Once again Lq*R:J {FR,}= 0 so WJ

WI{QRjI= oiij Lq*RTI[Ml {GjI.

(27)

Equation (27) is a most important relationship. K and M are both real, symmetric and positive definite, so these two quadratic forms are both real and positive. The left-hand expression represents twice the sum of the maximum amounts of potential energy stored in each elastic component of the periodic element in the course of one cycle of the motion. The elastic components will not be displaced in phase with one another since {qRj) is complex, so the components will not all be in their maximum displaced position (with maximum amounts of stored energy) at the same instant. The left-hand expression is therefore twice the sum of the maximum energies stored at different instants of time. The right-hand expression represents the sum of the maximum kinetic energies possessed by the masses of the element in the course of the cycle. Since the velocities of the masses are not in phase, the maximum kinetic energies of different masses will occur at different instants of time. Equation (27) simply states that when a wave is propagating freely, the sum of the maximum potential energies in the element is equal to the sum of the maximum kinetic energies. It can be re-written in the form of Rayleigh’s Quotient as

(29

244

D. J. MEAD

This is always a positive quantity, even though the displacement vectors are complex. There are n’+ N of these complex vectors, so it follows that there are ,’ + N real values of o. This Rayleigh’s Quotient can be used to find good approximations for wRj for a given value of aR by inserting approximate (guessed) values for (qR,} in the right-hand side. Denote the approximate wave displacement vector by {q_,,,), in which &,,> = E,j

= ~XJ.

The suffices 1, i, r have the same meaning as in equation (l), etc. Now the approximate wave vector {qapp}may be expressed as a linear sum of the exact eigenvectors (qRlr} (k = 1 to n’+ N), which are the solutions to equation (23) : i.e., y'1 hW,)

=

hRlrqRZ?qR3.

..qRn'+NI

i !Pn'+N II

=

[Ql {‘PI.

(29)

The columns of Q are the exact vectors qRK,and the column vector {Y} denotes the contributions of the different exact vectors to {qapp}.Put this into equations (28) to yield the approximate value of the frequency, o, : L’p*TJ [Q*TlPI

[Ql VP)

(30)

tY”*‘J[Q*‘][M][Q]{~)’

O’=

By virtue of the orthogonal properties of the wave vectors (equations (26)) both numerator and denominator of equation (30) reduce to expressions involving only 1Yfl, 1Yz], etc. Put L’I::~ El

hRk)

=&,

KJ[Ml{qRA

(31)

= M/o

to yield jY:l K1 + IYfl& + *.’ ly,z,+N( w’

=

1Y:l

hf,

+

1!?;I

bfz

+

. *. 1‘y,z,+NI

&+N h&+N’

(32)

From this it is easy to show that the frequency has a stationary property when all but one of the Y.‘s vanish: i.e., when the approximate mode is an exact mode. It follows, as in principal mode theory, that a small first-order error in the choice of the wave-vector {q,,,) which apprOXimateS t0 the true WaVe-V&Or (qRj} Will redt in 0, differing from the true c&I by a second-order quantity only. This stationary property may be used, as in the classical Rayleigh-Ritz method, to find more accurate values of oR, from the energy method by using more than one pre-assigned approximate wave vector. Let motion take place in the m pre-assigned approximate wavevectors (~a~)@~(j = 1 to m). Qj is the generalized coordinate corresponding to the jth approximate wave. Each of the pre-assigned wave vectors must satisfy the essential “wave/ boundary condition”, which is that the left-hand exterior coordinates must be efsRtimes the left-hand exterior coordinates: i.e., +J,,> = [ii;:] = (%,.,.). where the suffices r, i, 1 have the same meaning as in equation (l), etc. All other kinematic constraints on the periodic element must be correctly represented in the wave-vector. The

WAVE PROPAGATION

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245

total motion, {qapp} is therefore given by [Q,j]{4j}, where the columns of Qa, are the qaj vectors. This approximate vector is used in equation (28) instead of (q&, and wz is differentiated appropriately with respect to the complex 4j elements and equated to zero, thus using the stationary property. As in the usual Rayleigh-Ritz method, this leads to the set of equations for the frequencies which has the form [R - IX2R] {r#Jj}= 0. The elements of R and l@ are now complex, given by Ei;il= t$?iJ WI {qaj>;

@ij = Lqa*i I [Ml {qaj>.

The matrices K and %I are evidently Hermitian, and, in consequence, all the m frequencies found from the determinantal equation IR-&RI

=o

are found to be real and positive. An application of this extended Rayleigh-Ritz method is described in the next section. 5. APPLICATION OF THE BAYLEIGH QUOTIENT TO WAVE PROPAGATION IN MULTI-SUPPORTED BEAMS The Rayleigh Quotient of equation (28) has been expressed in terms of stiffness and mass matrices. For continuous systems, such as beams undergoing flexural vibration, the two energy expressions involved may be expressed as integrals of approximate displacement functions. In this section this method is used to determine the curve of propagation constant vs. frequency for a uniform beam on periodic simple supports. The exact solution for this is already known, and the exact numerical values will be used to assess the accuracy of the approximate method. Let the transverse flexural displacement of one bay of the beam (the periodic element) be W&)eiot when flexural wave motion occurs in the beam with propagation constant icR. Let the origin of x be at the centre of the bay of span 1. The transverse displacement and slope at end x = +I[2 are related to those at x = -Z/2 by W,(+1/2) = eieRW&l/2),

(33a)

Wi(+Z/2) = eieRWi(-//2),

(33b)

The maximum potential energy in an element 6x of the beam in the course of the cycle is given by 6 vtll,, = _)@Z)I WIi I2 6x, where (EZ) is the flexural rigidity of the beam, and W,”= d2 W,Jdx*. The total maximum potential energy in the whole beam is +112

Vmax= 3 j- (E/Z) [W;l’ dn-.

(34a)

-l/Z

Likewise, the total maximum kinetic energy is given by +1/z T,,,=3

1

mlwR(2dx

-l/2 +1/z cu2 =-

2

s

-l/2

where m is the mass per unit length of the beam.

m( W,(‘dx,

WW

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D. J. MEAD

Equating these two energies, I’,,,,, and T,,,, one obtains +1/2 [ (El) 1W;;(‘dx (32 =

-1;2 (35)

+1/2

J m\W,l’dx l/2 This differs from the usual energy expression for the natural frequency of a beam in that the modulus signs are present in the integrals and that the wave displacement, W,, is a complex function of x satisfying the equations (33). As usual in using Rayleigh’s method, it is necessary to have some idea of the mode of deflection of the system whose frequency is required. As an example of choosing a suitable mode the waves and propagation constants in the lowest frequency propagation band of the periodic beam will be investigated. At the low frequency end of this band, the mode of deflection is known to be that of the fundamental mode of a simply supported beam. At the high frequency end, it is the fundamental mode of an encastre beam. Between these frequencies the wave motion resembles a combination of these two modes together with an antisymmetric component. The modes at the ends of the propagation band can be represented approximately by W,(x) = a + Cc2 +fx”,

(36a)

with an appropriate choice of a, c andf. The antisymmetric component can be represented approximately by Was(x) = bx + dx3.

(36b)

The whole approximate displacement function is therefore W,(d) = W,(x) + Was(x) = a + bx + cx2 + dx3 +fx”.

(36~)

If the supports of the beam are rigid, W, = 0 at x = *l/2.

(37a)

If the wave motion is propagating with p = ie,, then the wave boundary condition equation (33b) is WA(1/2)= eieAWA(-l/2). (37b) Furthermore, the bending moments, and hence the curvatures, at each end of the beam must be related in the same way, so that Wi(1/2) = eieRWi(--l/2). These boundary conditions at x = +1/2 permit a, b, c andfto 5 = 2x/1 it is found that W,(t)

= +-id(-4 + 5”) - % ((3 cot CR/2+ 5 tanaJ2)

(37c) be found in terms of d. With

-

- (6 cot aR/2 + 6 tan e,J2) 5’ + (3 cot aR/2 + tan aR/2) 5”>.

1

(38)

Finally, this approximate complex mode is substituted into equation (35), to yield o2= 16

7.2 cot2(eR/2) + 4.8 tan2(a,/2) + 16.8 EI O-229cot2(aR/2) + 0.7875 tan2(e,/2) + 0.9146 2’

(39)

WAVE

PROPAGATION

IN PERIODIC

SYSTEMS

247

Notice that this result has been obtained by means of a “single wave approximation” in Rayleigh’s Quotient. Only one wave (equation (38)) satisfying certain boundary conditions, has been used to represent the actual wave displacement. Figure 3 shows the frequencies calculated from this expression for various given values of Ed, and compares them with the exact values calculated from previous exact theory. The abscissa is a non-dimensional frequency Q, defined by a = om. Very close agreement is seen to exist. When E, = 7cadjacent beam bays vibrate in anti-phase, each bay vibrating naturally as a simply supported beam. The exact frequency for this is D = 71’= 9.87, whereas the calculated approximate frequency is 10.3. When cR = 0, adjacent beam bays vibrate in phase, each vibrating as an encastrC beam. The exact frequency for this is 52= 22.373 . . ., whereas the calculated approximate frequency is 22.42. These latter comparisons simply bear out the accuracy of ordinary natural frequencies calculated by Rayleigh’s Method, for with Ed = 0 or R the above method degenerates to a straightforward application of Rayleigh’s Method with real modes of a standing wave system. Comparison of the frequencies in the

-4 0

I 5

I 15

I IO Frequency

Figure 3. Approximate(O--O) and exact (-)

parameter,

1 2c

I 25

J 30

.fJ

values of .sn us. Q for a uniform beam on periodic simple

supports.

intermediate region (0 < eR < 71)shows the accuracy of the Rayleigh Method when generalized to deal with complex waves of the propagating wave system. Approximate curves such as those of Figure 3 can be used to obtain the natural frequencies of finite, periodically supported beams, by the method proposed by Sen Gupta [8]. It is now obvious how the Rayleigh-Ritz technique can be applied to the present problem to obtain improved values of o for each .sR.Suppose the approximate beam displacement is represented by the sum of r linearly independent complex displacement functions, f,(i) (s = 1 to r), where thef,‘s satisfy the kinematic relationships f,(+l) = eieRfs(-I), fJ+l)

= e’““fs(-1).

(40)

(It is not necessary forfl(+l) to be equal to eiERfd’(-I), but it is desirable if the beam has no rotational constraints at the supports. If rotational constraints are present, it would be incorrect to impose this condition.) Now write (41)

248

D. J. MEAD

where the &‘s are complex generalized coordinates. Substitute this into equation (35) to yield

(424 or

Differentiatingt both sides with respect to 4: and using the stationary o2 (k.P/&#$ = 0) one obtains +1

property

of

+1

s

mfsfTd5 =

sI$A ;

-1

s

El--f :* d<

(43)

-1

for all values oft (1 to r). The whole set of r such equations can be expressed in the matrix form

[R-W2sI](f$}=0

(4)

where R and R are complex Hermitian matrices having the elements

W)

6. FREE WAVE MOTION

IN DAMPED

PERIODIC

SYSTEMS

When linear damping is allowed in the periodic system, an imaginary term is introduced into the governing equations of motion (equations (6), (8), (9), etc.). In particular, if the damping is viscous the sub-matrices of equation (9) become D,, = K,, - 02 M,, + wClr

(464

and if the damping is hysteretic they become D,, = K,, - 02 M,, + rH,,.

WW

C,, is a matrix of real viscous damping coefficients, and H,, is a matrix of real hysteretic damping coefficients. Either way, the matrices D1, Dz and D, of equation (11) become complex, and with a given frequency o the determinantal equation yields values of P which are all complex. None of them are purely imaginary any more, as the presence of damping enforces decay of free wave motion as it progresses from element to element. If the damping is light, propagation bands can still be identified in the curve of the attenuation constant. 6, is t Differentiating directly with respect to )r poses the problem of what is &6,/&# This is avoided, in practice, by differentiating equations (42b) with respect to Re(#:) and using &?/aRe(#)= 0, and also by differentiating with respect to Im(&and using W/aim(#)= 0. Combining these two results yields equation (43).

WAVE PROPAGATION

IN PERIODIC

249

SYSTEMS

no longer zero in these bands, but it is much smaller than in the adjacent attenuation bands. The phase constant is no longer zero or 71in the attenuation bands, but is close to these values if the damping is light. Figure 4 shows how b, varies with frequency for a periodically-supported beam with internal hysteretic damping. The beam flexural loss factor, qb, is assigned the values 0.25 and 1.0 and rotational constraints exist at the beam supports. Notice how the curve dips down into the frequency regions which are the propagation bands for the undamped beam. The dips become progressively less obvious as the frequency and/or the damping increases.

I

200

300

I

I

I

I

400

Frequency parameter, R 4. l/+0=

Figure ----,

The attenuation constant for a periodic uniform beam with heavy damping (-, 1.0).

;rlb=

0.25;

When damping is present, energy must flow along the system at all frequencies of wave motion in order to supply the energy being dissipated in the vibrating elements. The complex propagation constants must still occur in positive and negative pairs, for the same reasons as presented after equation (11). 7. “FORCED-NORMAL

WAVE” MOTION

IN DAMPED

PERIODIC

SYSTEMS

In a paper on the normal modes of finite, hysteretically-damped systems, the author [9] has shown that there exists a set of forced orthogonal modes, between which there is no damping coupling. These are convenient for analysing general forced response problems for the system, and also for providing a means of uniquely specifying the damping. Such “damped forced modes” must exist in finite periodic systems, where they can be regarded as superpositions of certain equal and opposite-going damped forced waves. The damped forced waves of an infinite periodic system are readily found by using a simple extension of the foregoing theory for undamped systems and this extension is set out in this section. The study of the damped forced waves enables resonant frequencies and loss factors of the finite periodic system to be determined, and this is the motivation for this section. The damped forced modes [9] are excited by external forces which are proportional to but in quadrature with the local inertia forces. They can only exist at discrete frequencies (the resonant frequencies of the modes) at which the unique constant of proportionality (external force i inertia force) can be identified with the loss factor of the mode. It will be supposed that damped forced waves can exist under similar conditions: i.e., when external forces act on the whole periodic system, proportional to but in quadrature with the local inertia forces. A propagating wave motion will be sought having the purely imaginary propagation constant ,u = A$.

250

D. J. MEAD

Although damping is present in the system, the wave can propagate without attenuation since the external forces supply energy to each periodic element and prevent decay of the progressive wave motion. The equations of motion of a hysteretically-damped periodic element have the same form as equations (3) for the undamped element, but an imaginary matrix of hysteretic damping coefficients, i[H], must be included. H itself is a symmetric real matrix, and in general will not be proportional to K. When the system is excited by the particular type of external forces for the damped forced waves, the force vector F can be expressed in the form (47) where the first column on the right-hand side represents the forces on the element from the adjacent elements, and the second column (the matrix product) represents forces which are proportional to and in quadrature with all the local inertia forces. The whole equation of harmonic motion is [K f iH - o2 M] {q} = o”l + io2 tj[M] {q}

(4W

ilF,

or [K+iH-W2(l

+iq)M]{q}=

:I

.

(@b)

0F,

This equation can be manipulated in precisely the same way as equation (3), to eliminate {F1) and p,> and to reduce its order. The following equation is then obtained:

in which

[IQ]

=

i;: ___-_HI, + e-‘H,i !!!!?_!%.~~~.!!.;~~_~:_!!!! , __________ II

II

Hii

Wb)

I

and [M,p] is defined in a similar way. Now consider the special case when p = &kg, where aR can have any assigned real value. The wave motion is now progressive and unattenuated. Equation (50) allows (n’+ N) values of cY( 1 + iv) to be found for this particular value of sR. This is a complex eigenvalue problem, and corresponding to each value of d(i + iv) so found will be a particular complex eigenvector QI representing the forced wave motion. ( qi 1 Taking any one of the complex eigenvalues 02(1 + iv), and substituting it back into equation (50), one can find 2(r1,,,~,- 1) other values of p which will satisfy the equations besides the assigned values of &CR. Altogether, then, there are 2n,,,*,,different “forced” propagation constants for each of the complex eigenvalues, and these occur in positive and negative pairs. Corresponding to each propagation constant, of course, is a different damped

WAVE PROPAGATION

IN PERIODIC

251

SYSTEMS

forced wave, represented by the vector 4”: . Apart from the purely imaginary propagation 0

constants p =&is,, the propagation constants will be generally complex and their corresponding waves will decay as they propagate. Consider now the energy involved in the wave corresponding to p = ia,. Denote its total wave vector by {qR) which, in general, is complex. The exterior forces and displacements at the ends of the periodic element are related through the expressions {PrR1= -eie{FjR1, {qrR)= eie&l

(52)

(as in equations (4) and (5)), so equation (49b) becomes [K + iH - w2(1 + iv) M] {qR} = o”l.

.

i . .ei”RP,, i

(53)

Now pre-multiply this by Lqfl. The right-hand side becomes zero by virtue of equations (52). The left-hand side yields the quadratic forms LqiJ[K] {qR}, LGJ [H] {qR} and tqtl [M] {qR}. All of these are entirely real and positive since the matrices K, H and M are all symmetric and positive definite. The pre-multiplied equation (53) can then be separated into its real and imaginary parts to yield LstJ

PI hd = 02ts3 [Ml{qd

(54)

and LsXJ[HIh,d

= w*rlts*,l [Ml bd.

(55)

By combining these to eliminate w2, it is found that

is3 WI h) ‘I= LWKI {qR)’

(56)

Now LqR*_l[H]{qR}can be shown to be equal to l/rc times the total amount of energy dissipated by the whole damping system in the periodic element in the course of one cycle of [K] (QR} has already been identified with motion in the wave-displacement given by (qR}. Lep*i the total maximum potential energy in the periodic element during the cycle. The factor r~, then, which was initially introduced as a constant of proportionality between inertia force and local external exerting force, can now be recognised as a “loss factor” defined by

q=’

Energy dissipated in the course of a cycle 27~Total maximum potential energy stored in the cycle *

(57)

Notice that this has been derived only for the propagating forced wave which is characterized by the propagation constant ,u = kiER. It is not true for the non-propagating forced waves with the generally complex $s. Equation (54) shows that the frequency at which this particular forced progressive wave motion can take place can be found from Rayleigh’s Quotient, as for the undamped system. It has the physical significance that in this forced wave motion of the damped system, the total maximum potential energy in a periodic element is equal to the total maximum kinetic energy. An approximate column vector {qapp},when substituted into equation (54), will yield a good approximation to o, but the same approximate vector substituted into equation (56) will not necessarily give a good approximation to q. The Rayleigh Principle only applies to thefrequency of vibration, and not to the damping characteristics.

252

D. J. MEAD 8. TWO-DIMENSIONAL

PERIODIC

SYSTEMS

A two-dimensional periodic system consists of elements which are coupled on all sides in the same plane to adjacent identical elements. Figure 5 shows diagrammatically such a system consisting of rectangular elements, each one of which is coupled on its four sides to four adjacent elements. Let there be n, coupling coordinates on the left- and right-hand sides of the element, and denote these by {q,} and {qr}. Let there be n, coupling coordinates on the other two edges (the “bottom” and “top” edges) to be denoted by {q,,} and {qt}. The forces acting at these coordinates from the adjacent elements are {F,}, {Fr}, {F,,) and {F,}. Ninternal coordinates and forces {qi} and {Fi} exist as before. Let the periodic length in the x-direction be I, and in the y-direction be I,,. Free wave motion in the two-dimensional system is, in general, much more complicated than in the one-dimensional system. Waves can spread out in a manner analogous to circular waves over a flat surface, or in the simpler form analogous to plane waves over a flat surface.

U

~---L--L_Ju Figure 5. Schematic diagram of part of a two-dimensional

periodic system.

Wave motion of the first type could be caused by a single point source acting on one of the periodic elements, and sending out wave energy in all directions. The wave intensity may be greater in some directions than others (depending on the geometry and structural characteristics) and the attenuation and phase constants will be different in different directions. The analysis of such wave motion has not yet been attempted. Wave motion of the second type (analogous to plane waves) could be generated by a certain distribution of force along a straight line at any angle across the system. All the periodic elements vibrate in identical complex modes, but there is an identical phase difference or attenuation factor between the motions of corresponding points in any pair of adjacent elements. Such “plane wave motion” is analysed below. Suppose the general direction of the wave motion is inclined at angle 0 to the x-axis (Figure 6). Let the propagation constant per unit length in the wave direction be k. The propagation constant per length 1, across the x-wise length of the periodic element is k x the projection of 1, on the wave-direction vector: i.e., pL,= kl, cos 8.

W)

Likewise, the propagation constant per length 1, across the y-wise width of the periodic element is k x the projection of 1,: i.e., p,, = kl, sin 8.

(59)

WAVE PROPAGATION IN PERIODIC SYSTEMS

253

Evidently then (60) for this type of “plane” wave motion across the two-dimensional periodic system. Wave

dlrectcon

/

nx forces and couplmg coordtnates

nr forces and couphg

coordmates

Figure 6. The periodic element of a two-dimensional periodic system; the exterior coordinates and the wave-direction vector.

The displacements {q,), (qr} and the forces (F,), {F,} are related to one another and ,uxin the same way as those of the one-dimensional system: i.e., W = eu=Oh>, PA = -e”={F,I.

(61)

The displacements (q,,}, (qt} and the forces (Fb), (F,) are similarly related through b-h) = eJ%A {F,} = -e”y{F,,}.

(62)

The whole set of q’s and F’s are related through the equations of motion:

(63)

II and M are the square and symmetric generalized stiffness and mass matrices of order 2(rr, c nY)-I-N. These K and M matrices can be partitioned in the same way as equations (7):

Klb _-___

Krb _____ Klb ---WV

K bb

&i

Ktb

254

D. J. MEAD M,, i M,, i Mu i M,b i MI, -____ M,, i -;;, i -&y; / -$i;; i -$;; ----_ ;_____ I_____: _____:_ es-_E!!!M,jMii , -____,iMib\Mit _____, _____ .

M=

(Mb)

Since both K and M are symmetrical, it follows that Kbl = K&, M,, = MTt, etc. The 2(n, + n,) + iV equations in equation (63) may be reduced in number to (n, + nY+ N) by using equations (61) and (62) and by eliminating the F’s, just as equation (3) was reduced. An equation corresponding to equation (8) can then be derived, having the form [K; + eflxK\ + e_flaK’z + e”YK; + e-fi*Ky + e@y-@xK;+ . . a . . . + e-fly +flzK’z]

q1 qi (1qb

-

w’[M;

+

&‘lM;

+

e-pxM;T

+

&‘YMj

+

e-UyM;T

+

eWu_NxMk

+a..

91 . . . +

e-h+~xM’~]

qi

= 0.

(65)

0 qb

The matrices K;, K; . . ., M;, M; . . . are composed of the parts of K and M and are defined in Appendix C. It is not easy to see immediately how the number of possible propagation constants at any frequency and direction of wave motion, 19,is related to the number of coupling coordinates along the sides of the two-dimensional element. It obviously depends on the direction of the wave motion, as shown by two simple extreme cases. Suppose that each element is symmetrical about its centre-lines in the x- and y-directions, and that the wave travels in the direction 6’= 0 (i.e., parallel to the x-axis). There will be no motion of the n,, coupling coordinates on the top or bottom edges of an element, and the whole system reduces in effect to a onedimensional system extending in the x-direction with n, coupling coordinates at the end of each element. There are then 2n, propagation constants and corresponding waves at any frequency.? Equation (65) can be developed to show this when pLy= 0 and the determinant of the matrix is expanded and set to zero. On the other hand, suppose the wave travels in the direction 0 = 7c/2(parallel to the y-axis). The system now reduces to one dimension extending in the y-direction, with n, coupling coordinates at the end of each element. There are then 2n, propagation constants, as equation (65) can show when pX= 0. A more comprehensive study of the number of propagation constants associated with a given frequency and any other direction 19is beset with certain difficulties and remains yet to be done. The most practicable method of solving equation (65) for pX,p,, and w is the inverse method of prescribing pL, and pY, and finding o2 as a straightforward eigenvalue problem. The direction, 8, of the wave motion must first be assigned, after which a value of pL,(real or imaginary) should be chosen. Equation (60) then yields p,,. With these values of pXand P,,, o2 can be found from equation (65). It will not always be possible for real frequencies to be found for real values of pL,.It is conceivable that real values of pX above a certain figure do not occur with the particular system under investigation. In such a case, complex or negative values of o2 would be found. t It is assumed in this that n, and ny are the minimum number of coupling coordinates.

WAVE

PROPAGATION

IN PERIODIC

SYSTEMS

255

On the other hand, it will always be possible to find real frequencies at which any purely imaginary value of pXcan occur. The order of equation (65) suggests that there should be at the most (n, + n,, + N) such frequencies for a given ,uX= k. All of these should be real and positive according to the arguments of a previous section. The inverse method cannot find complex values of pX and the corresponding frequencies. Other computational methods must be investigated for this purpose. The Rayleigh, or Rayleigh-Ritz, method may obviously be used’to determine the frequencies of propagating waves with known imaginary values of pX= i& and cl,,= k(l,./l,) tan6. Equation (63) may be used to show that the Rayleigh Quotient (equation (28)) applies to the two-dimensional system as well as to the one-dimensional system. This fact is self-evident when the physical significance of the Rayleigh Quotient is considered. In applying the Rayleigh or Rayleigh-Ritz method to the two-dimensional system, one must choose approximate complex modes which satisfy the appropriate wave-conditions on both opposite pairs of edges: i.e., the coordinates on the right-hand edge must be eie times those on the left-hand edge, and those on the top edge must be eie(iJlx)tanetimes those on the bottom edge. It must be emphasized that the above analysis applies to the simplest possible form of wave motion in the simplest two-dimensional periodic system. There are questions about the waves (particularly the non-propagating waves) which could be asked, and which cannot be answered at the present time. The two-dimensional theory is still at an elementary stage. Some computational work has, however, been done for a uniform flat plate on orthogonal simple supports [lo] and the results justify the general ideas presented above. Further work is in progress. 9. CONCLUSIONS

The generalized theory of wave propagation in multiply connected periodic systems has confirmed the results previously derived from particular flexural theories. In a one-dimensional periodic system the number of distinctly different free waves that can exist at any frequency is twice the number of coupling coordinates at the end of one of the periodic elements. This is true whether the system is damped or undamped, or forced in a particular manner to create “damped forced waves”. Even in the undamped system, the propagation constants corresponding to the waves may be real or complex, besides pure imaginary. Only the waves with pure imaginary propagation constants can transmit energy through an undamped system. If the periodic element of the system has a finite number of freedoms, the number of different frequencies at which waves with a given imaginary propagation constant can exist is equal to the total number of freedoms less the number of coupling coordinates at one end. A propagating wave in an undamped periodic system has a total maximum kinetic energy of all mass elements equal to the total maximum potential energy of all stiffness elements. This feature leads to a “Rayleigh Quotient” which permits the propagation constant vs. frequency relationship to be found in terms of an approximate complex wave displacement function. Rayleigh’s Principle for principal modes of a finite system has been found to apply to the propagating waves of an infinite periodic system. Furthermore, the Rayleigh-Ritz method can be used to obtain improved relationships between the propagation constant and frequency. The significance of this is very great, for it opens the wa>’for the powerful finite element methods to be applied to wave propagation in periodic systems. This will be of greatest use in connection with non-uniform periodic structures, to which all previous methods of wave propagation analysis could not be applied. Free wave motion in damped periodic systems is always characterized by complex propagation constants and energy transmission. When the damping is hysteretic, a special class of

256

D. J. MEAD

propagating “damped forced waves” can be shown to exist, for which the forcing has a special form. The Rayleigh Quotient applies to these waves, and a “loss factor” of the wave motion can conveniently be defined. Two-dimensional periodic systems can undergo a form of “plane wave motion” in which the waves propagate across the system at a characteristic angle. The number of propagation constants and distinct waves which can exist at any frequency appears to depend on the direction of propagation. The energy method of determining frequencies for a given propagation constant, as developed for one-dimensional systems, applies directly to two-dimensional systems. Indeed, the physical interpretation of the Rayleigh Quotient means that it must apply to three-dimensional systems as well. A knowledge of the free waves and their propagation constants greatly facilitates the computation of the response of the periodic system to steady-state impressed forces and pressure distributions. The analysis of certain sonically-induced vibrations of aeroplane stiuctures, which was virtually intractable by the classical normal mode methods, has been readily solved by using exact wave propagation methods. The approximate energy methods opened up by this paper should still further widen the scope of problems which can be handled by wave-type analyses.

REFERENCES 1. L. BRILLOUIN1946 Wavepropagation in periodic structures. New York: Dover Publications, 2. L. CREMERand H. 0. LEILICH 1953 Archiv der Elektrischen ijbertragung, 7, 261-270.

Inc. Zur

Theorie der Biegekettenleiter. 3. M. A. HECKL 1964 Journalof the Acoustical Society of America 36, 1335-1343. Investigations on the vibrations of grillages and other simple beam structures. 4. D. J. MEAD 1966 Shock and Vibration Bulletin 35, Part 3, 45-54. The random vibrations of a multi-supported heavily damped beam. 5. D. J. MEAD 1970 Journal of Soundand Vibration 11, 181-197. Free wave propagation in periodically supported infinite beams. 6. D. J. MEAD 1971 Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 93, Ser. B, 783-792. Vibration response and wave propagation in periodic structures. 7. D. J. MEAD and G. SEN GUPTA 1970 U.S.A.F. Report No. AFML-TR-70-13. Propagation of

flexural waves in infinite, damped rib-skin structures. 8. G. SEN GUPTA 1970 Journal of Sound and Vibration 13,89-101.

Natural flexural waves and the normal modes of periodically supported beams and plates. 9. D. J. MEAD 1970 Proceedings of the Symposium on Structural Dynamics, Paper CS, Loughborough University of Technology. The existence of normal modes of linear systems with arbitrary damping. 10. G. SEN GUPTA 1970 Ph.D. Thesis, University of Southampton. Dynamics of periodically stiffened structures using a wave approach.

APPENDIX

A

THE ZERO DIAGONAL ELEMENTS OF THE MASS MATRIX M The particular system of Figure 7 will be used to illustrate some general points. This represents a periodic element which has four exterior coupling coordinates at both ends and N., interior coordinates. As in section 2, M,, will be assumed to be non-singular. At the

WAVE PROPAGATION

257

IN PERIODIC SYSTEMS

left-hand end the masses associated with the coordinates are Mtl, 0, Ml3 and 0, respectively. At the right-hand end they are Mfl, MrZ,0 and 0. In this case the matrix [Mu f M,,] is given by MI1 + M,I,

0,

0,

0

0,

M,2 + M,z,

0,

0

0,

0,

M,,,

0

0,

0,

0,

0

*

This has three non-zero diagonal elements (i.e., n’= 3) so there will be Nj + 3 frequencies at which a given propagation constant can occur. Left- hand exterior coordinates

Mrl

Ml, g(l

e-VW-+

-

+--VW

M,,=O 62

--WV--+

-

+-VW-. IV,i internal coordinates

MC4 414

-

9r1

--,

4r2

M,3=0 +--ve

WM

-

w2

ML3 413

Right-hand exterior coordlnotes

Masses

Masses

-

Ml4

-0 -VW

-++I*

-

4r4

A

?

R r4

*,4

Figure 7. A

Qr3

Mr4=0

particularsystem to illustrate Appendix A.

Notice that the mass M,, at qrl could be transferred to qll without changing the matrix elements and hence without changing the frequencies of propagation. This is to be expected, as the dynamic characteristics of the whole periodic system must not depend on whether the total mass M,, + M,, is assigned to one end of the periodic element or to the other, or is shared between the two ends. In the same way, spring elements can be moved from one end of the system to the other. Suppose the whole spring k,4 is transferred to the right hand end of spring kr4, changing the total stiffness of the right hand spring to k,,k,,/(k,, + k,). The displacement of mass Mi4 must now be regarded as an exterior coordinate and is the new q14. Thus, the number of interior coordinates is reduced from NJ to Nj - 1 and the order of Mii is reduced correspondingly. Now that M,, is associated with an exterior coordinate, the fourth diagonal element of [MI1 f M,,] becomes M,,. Hence, all four diagonal elements of this matrix are non-zero, and the total number of propagation frequencies is (N, - 1) + 4 = Nj + 3. This, of course, is the same as before. With this latter choice of exterior coordinates, the mass matrix M becomes non-singular and is more suitable for computational purposes. The number of propagation frequencies is then N, + n, where Ni is the number of interior coordinates of the modified arrangement.

APPENDIX ON THE NUMBER OF PROPAGATION

B

CONSTANTS AT A GIVEN FREQUENCY

Consider a periodic element which is split up into two sub-systems A and B (see Figure 8). The boundary between A and B is selected to minimize the number of coupling coordinates

D. J. MEAD

258

between the sub-systems: i.e., the periodic element cannot be split in any other way and produce fewer coupling coordinates. Let this number of coordinates be n,,,,,,. At the left-hand end of A and the right-hand end of B the element is coupled to the adjacent periodic elements through n coordinates (n > nmin). The receptance method of section 3 will be used to find the number of possible propagation constants at a given frequency.

n coordmates

and

forces

nmn coordmtes

n cwrdmotes forces

and

forces

ol?d

\

(hl IF,)

Figure 8. An element of a periodic system consisting of two sub-systems.

Let the n forces acting on the left of system A be denoted by {F,}, and those on the right of system B by {F,}. These are the “inter-element” coupling forces, and are related as in equation (5) by (F,} = -e”(F,}.

WI

Let the nmlnforces acting between the systems A and B be denoted by {FAB}. The actual forces acting on A will be +(FAB} and on B will be -{FAB}. Denote the n exterior coupling coordinates at the left-hand end of A by {~a~>,and those at the right-hand end of B by (qBr}. Since these are the two sets of exterior coordinates of the whole periodic element, they are related in the same way as in equation (4): i.e.,

The nmincoordinates at the right-hand side of system A will be denoted by {qAr}, and those at the left-hand side of system B by {qB,}. For continuity of displacement between systems A and B, (B3) Now the displacements of system A are related to the forces acting on it through a receptance matrix, 034) and similarly, for those of system B, 035)

WAVE PROPAGATION

IN PERIODIC

259

SYSTEMS

In these, the sub-matrices have the following orders: zcA,, and

ag,,: n x n;

a *rr and

agil:

nmin x nmin;

and and

aBr,:

n x nmin;

aAlr aArl

aBlr: nmin x n.

Further, due to the symmetry of the whole receptance matrix, @Arl

=

T aAlrt

.

aBlr

=

a&l.

When the relationships of equation (Bl), (B2) and (B3) are used in equation (B5), and the e” terms are re-arranged, equation (B5) becomes i aBrIe-”

aBrr _______ _ , _________ aB,,e”

j

W)

asll

The right-hand sides of equations (B4) and (B6) can now be equated, to yield =

, __________________

________________

o,

(B7)

The propagation constants are given by the determinantal equation iaAll

+

IaArl

aBlr

@Brrlr

IaAlr

eC19 b&r

+

aBrl e-Pl

+

aBlll

0.

(Bg)

=

The orders of the sub-matrices in this are such that when the determinant is expanded, it yields a polynomial equation in coshp of order nmln. Thus, there are 2nmindifferent values of p (in n,,,i,, positive and negative pairs): i.e., the number of propagation constants is equal to twice the minimum number of coupling coordinates.

APPENDIX C DEFINITIONS

OF THE

K;

AND

M;

MATRICES

OF EQUATION

(65)

These are defined in terms of the partitioned elements of K and M in equations (64a) and (64b) as follows :

____--______ -___ I

_-_____

___-_

--__--___ ___ ,---;---______-_1

i; ___ I I D. J. MEAD

260

0

K; =

0

0

i-i;

___ : ---

K,,

i -g;;

,

’ -----

0 ; 0 ; K,,

M;, M;, Mj, N are defined in the same way, but with M replacing K.