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International Journal of Mechanical Sciences 44 (2002) 305318

Development of a WittrickWilliams algorithm for the spectralelement model of elasticpiezoelectric two-layer active beams

Usik Leea ;, Joohong Kima, Jinho Shina, A.Y.T. Leungb

aDepartment of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-ku,

Inchon 402-751, South KoreabDivision of Mechanical Engineering, University of Manchester, Manchester M13 9PL, UK

Received 22 August 2000; received in revised form 22 October 2001

Abstract

In this paper, a WittrickWilliams algorithm is developed for the elasticpiezoelectric two-layer active

beams. The exact dynamic stiness matrix (or spectral element matrix) is used for the development. This

algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency,

without the possibility of missing any due to close grouping or due to the sign change of the determinant of

spectral element matrix via innity instead of via zero. The uniform and partially patched active beams are

considered as the illustrative examples to conrm the present algorithm. ? 2002 Elsevier Science Ltd. Allrights reserved.

Keywords: WittrickWilliams algorithm; Spectral element model; Two-layer active beam; Natural frequency

1. Introduction

In structural dynamics analysis, one of the aims is to calculate accurate natural frequencies of

structures. In general, the natural frequencies are calculated by nding the roots of the characteristic

equation (or frequency equation), which can be obtained by substituting the harmonic representationsof general solutions into the governing structural dynamic equations and then by applying proper

boundary conditions [1].

When a structural system is approximated as a nite degrees of freedom (DOF) system model by

use of the conventional nite element formulation procedure, for instance, one may obtain a linear

or algebraic eigenvalue problem of which dynamic stiness matrix is the linear function of the square

Corresponding author. Tel.: +82-32-860-7318; fax: +82-32-866-1434.

E-mail address: [email protected] (U. Lee).

0020-7403/02/$ - see front matter? 2002 Elsevier Science Ltd. All rights reserved.P I I : S 0020- 7403( 01)00097- 2


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306 U. Lee et al. / International Journal of Mechanical Sciences 44 (2002) 305 318

of (circular) frequency !. In this case, powerful algorithms exist for determining all required natural

frequencies. This enables one to determine with ease as to how many natural frequencies lie below

any chosen frequency; thereby it makes it possible to converge on any particular natural frequency,

to any required accuracy, by an iterative procedure. However, when some DOF of a nite DOFsystem model are constrained to result in a condensed dynamic stiness matrix or when a system

originally has innite number of DOF, the eigenvalue problem is not the linear function of the

square of frequency ! any more. In these cases, the corresponding dynamic stiness matrix has a

transcendental dependence on the square of frequency !. It is not in general so easy to calculate

all the required natural frequencies from the transcendental form of characteristic equation: it may

require a trial and error procedure.

If one knows in advance the number of natural frequencies existing below any chosen frequency as

well as the frequencies at which the discontinuities of characteristic equation exist, it may be possible

to calculate all required natural frequencies by using a proper iterative procedure. In 1971, Wittrick

and Williams [2] developed an algorithm, which is known as WittrickWilliams algorithm, to cal-culate the number of natural frequencies that lie below any chosen frequency, without determining

them at all. In order to use the WittrickWilliams algorithm, the dynamic stiness matrix of a sys-

tem should be derived rst. The number of natural frequencies that lie below any chosen frequency

is then calculated by assuming all DOF are constrained. The WittrickWilliams algorithm has

been applied to various structures such as skeletal structures [3], marine structures [4], and helical

springs [5].

The dynamic stiness matrix spectrally formulated by using exact dynamic shape functions (or

wave solutions) is exact in nature and it is known as the exact dynamic stiness matrix or spectral

element matrix in the literature [615]. The spectral element method (SEM) is a structural dynamic

analysis method in which the spectral element matrix and the fast Fourier transform algorithms are

used to obtain very accurate frequency-domain and time-domain solutions in an extremely ecientway. The SEM has been applied to wave propagations in solids [6,7] and to various structural

dynamic problems [1116] as well.

For smart structures, it may be important to calculate all required natural frequencies to a

required accuracy for improved design of control systems. Very recently, Lee and Kim [14] derived

the spectral element matrix for elasticpiezoelectric two-layer beams (simply, active beams). As the

spectral element matrix for active beams is not a linear function of the square of frequency !, the

conventional root-nding methods in which the roots of the determinant of spectral element matrix

are searched with varying frequency ! in small steps may not be successful to calculate all required

natural frequencies to the required accuracy.

Hence, the purpose of this paper is to develop a WittrickWilliams algorithm to eciently calculateall required natural frequencies of an active beam, which lie below any chosen frequency, without

the possibility of missing any due to close grouping or due to the sign change of the determinant

of dynamic stiness matrix via innity instead of via zero.

2. Spectral element matrix for active beam

The active beam considered herein consists of two uniform layers: the elastic base-beam and the

piezoelectric (actuator) layer. The homogeneous forms of equations of motion for the active beam


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U. Lee et al. / International Journal of Mechanical Sciences 44 (2002) 305 318 307

are given by [14]

EI WIV !2 A W = !2(U W) + U;

EA U + !2 A U = !2W + W; (1)

where W(x; !) and U(x; !) are spectral components (or Fourier coecients) of the transverse

deection and axial displacement of the base beam, respectively, and ! is the (circular) frequency.

EI and A are the eective bending stiness and mass density per length of active beam, respectively.

The symbols and represent the axial-bending couplings and the eective rotatory inertia eect,

all induced by piezoelectric layer. They are all dened in Ref. [14] together with other symbols used

in above equations.

The spectral element matrix of the active beam has been derived by Lee and Kim [14] by using

the general solutions formulated in terms of exponential functions. However, in the present study,

we assume the general solutions of W(x; !) and U(x; !) as the functions of trigonometric andhyperbolic functions: this change makes the formulations given in the following simple:

W(x; !) = [(x; !)]{A}; U(x; !) = [(x; !)]{B}; (2)

where

[(x; !)] = [cos(k1x) sin(k1x) cos(k2x) sin(k2x) cosh(k3x) sinh(k3x)];

{A} = [A1 A2 A3 A4 A5 A6]T;

{B} = [B1 B2 B3 B4 B5 B6]T;

(3)

where ki (i = 1; 2; 3) are wavenumbers calculated from the dispersion relation given by

(2 EA EI)k6 + !2L2(AEI+ EA2)k4 + !2(2!2 A!2 + AEA)k2 !4A2 = 0:

(4)

In Eq. (3), Ai and Bi (i = 1; 2; : : : ; 6) are the constants to be determined from boundary conditions

and they are related to each other as follows:

B1 = A11 tan(k1x); B2 = A21 cot(k1x); B3 = A32 tan(k2x);

B4 = A42 cot(k2x); B5 = A53 tanh(k3x); B6 = A63 coth(k3x);(5)

where i

is given by

1 = k1(k21 !

2)=(A !2 EA k21 );

2 = k2(k22 !

2)=(A !2 EA k22 );

3 = k3(k23 + !

2)=(A !2 + EA k23 ):

(6)

Substituting Eq. (2) into the forcedisplacement relations [14], the (spectral) nodal forces and

moments dened in Fig. 1 can be expressed in terms of constants Ai as

{f} = [P(!)]{A}; (7)


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U2

W112W2

1

N

U1

1M1Q 2N

2M2Q

Fig. 1. Spectral element of active beam and its sign convention.

where

{f} = [N1 Q1 M1 N2 Q2 M2]T: (8)

Similarly, the (spectral) nodal DOF dened in Fig. 1 can be expressed in terms of Ai as

{y} = [Q(!)]{A}; (9)

where

{y} = [U1 W1 1 U2 W2 2]T: (10)

The matrices [P] and [Q] in Eqs. (7) and (9) are all frequency-dependent and they are listed in

Appendix A. By eliminating the constants Ai from Eqs. (7) and (9), one can obtain the nodal

forcenodal DOF relation as follows:

{f} = [P(!)][Q(!)]1{y} = [s(!)]{y}; (11)

where the symmetric matrix [s] is the spectral element matrix for active beams. Symbolic computation

softwares enable one to eciently compute the spectral element matrix from matrices [P] and [Q].

The spectral element matrices can be assembled in a completely analogous way to that used forthe conventional nite element method (FEM). After applying boundary conditions to the assembled

result, the global system dynamic equation can be obtained in the form as

[S(!)]{Y} = {F}; (12)

where [S] is the global spectral matrix, F the global force vector, and Y is the global nodal DOF

vector. The eigenvalue problem for the active beam can be obtained from Eq. (12) by enforcing

F= 0:

[S(!)]{Y} = {0}: (13)

3. Development of a WittrickWilliams algorithm

Spectral element model provides an innite number of natural frequencies by using only a nite

number of DOF, which is not possible for the nite DOF system represented by a FEM model, for

instance.

The natural frequencies of a system can be computed by nding the roots of the characteristic (or

frequency) equation of the system, which is derived by setting the determinant of global spectral

matrix [S] to zero. In general, the characteristic equation has a highly complicated transcendental

dependence on frequency !. It is not an easy task to calculate all required natural frequencies from


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the transcendental characteristic equation: it may require a proper trial and error procedure. Thus,

how to calculate all required natural frequencies to any required accuracy, without missing any,

should be an important issue when a dynamic system is represented by the spectral element model.

Wittrick and Williams [2] developed an algorithm to automatically count the number of naturalfrequencies below a chosen frequency. The WittrickWilliams algorithm is given as

J(!) = J0(!) + sgn[S(!)]; (14)

where J(!) represents the total number of natural frequencies of a system which are less than

a chosen frequency !, and J0(!) is the total number of natural frequencies which would still be

exceeded by the chosen frequency ! if constraints were imposed so as to make all the displacements

Y zero. In Eq. (14), sgn[S(!)] represents the sign count of matrix [S], which is equal to the number

of negative elements on the diagonal of the upper triangular form of [S]. The upper triangular form

can be reduced by using the simple Gaussian elimination procedure, without row interchanges [2].

In order to use Eq. (14), one should know the value of J0(!) in advance. For BernoulliEuler

beams, Williams and Wittrick [3] derived a simple formula for J0(!) as follows:

J0 = j 12{1 (1)j sign(1 cosh kL sinh kL)}; (15)

where k is the wavenumber, L the length of beam, and j is the largest integer kL=. In Eq. (15),

sign() has the value +1 or 1, depending on the sign of the argument. Eq. (15) gives the totalnumber of natural frequencies, between zero and any trial !, when all boundaries are clamped.

Since the spectral element matrix of an active beam is much more complicated than that of a

BernoulliEuler beam, derivation of J0(!) for the active beam will be much more complicated.

By the denition, J0(!) is the total number of natural frequencies (less than a chosen frequency!) obtained by constraining all boundary conditions, i.e., by applying clampedclamped boundary

conditions to Eq. (13). For uniform active beams, regardless of their lengths, the global spectral

matrix [S] in Eq. (13) can be reduced from a single spectral element matrix. In this case, simply

imposing clampedclamped boundary conditions (Y=0) on Eq. (13) will not be helpful for deriving

J0(!). To cope with the above diculty, following strategy is used in the present paper.

We rst calculate the total number of natural frequencies, JSS(!), and the sign count of the spectral

element matrix, sgn[SSS(!)], all for the case where the active beam is simply supported. Next, we

use JSS(!) and sgn[SSS(!)] to calculate J0(!) from

J0(!) = JSS(!) sgn[SSS(!)]: (16)

The sign count, sgn[SSS(!)], can be numerically calculated with ease. Once J0(!) is calculated from

Eq. (16), Eq. (14) can be nally used to compute J(!) for the real boundary conditions. The reason

we rst calculate JSS(!) is that it is much more easy and simple to derive the analytical expression

of J(!) for the simply supported boundary conditions (i.e., JSS(!)) than to derive that for other

boundary conditions. In the following, analytical derivation of JSS(!) will be detailed.

As shown in Eq. (13), the spectral element matrix [S] for a simply supported uniform active

beam is obtained by removing appropriate rows and columns to account for the freedoms supported

by the simple supports. The characteristic equation for the simply supported active beam can be

then derived by enforcing the determinant of spectral element matrix [S] as zero. After a lengthy


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mathematical manipulation, the characteristic equation can be expressed in a simple form as

sin(k1L)sin(k2L)fSS(!) = 0; (17)

where fSS(!) is the frequency-dependent function given in Appendix B. It is proved in Appendix Bthat fSS(!) is not zero at any frequency !, except for the zero frequency which corresponds to the

static states. As fSS(!) is not zero, Eq. (17) yields the following two conditions:

k1L = n or k2L = n: (18)

Therefore, JSS(!) in Eq. (16) can be calculated from

JSS(!) = J1(!) + J2(!); (19)

where

J1(!) = the largest integer k1L=;

J2(!) = the largest integer k2L=:

(20)

4. Illustrative examples and discussion

In this study, two active beams are considered as the illustrative examples. The rst one is that

fully covered with a uniform piezoelectric layer (simply, uniform active beam). The other one is that

partially covered with a single piezoelectric patch (simply, patched active beam). The geometries of

the example active beams are shown in Fig. 2. The material properties of active beam used in Lee

and Kim [14] are consistently used herein.

First, we conrm the accuracy of spectral element model by comparing the natural frequencies

obtained by SEM with the results obtained by using the conventional FEM. It is well known thatthe solutions by FEM converge to exact solutions when one uses more ne meshes in FEM. Thus,

the number of meshes used in FEM is increased gradually to observe that the natural frequencies

obtained by FEM converge to SEM results. This in turn proves the high accuracy of the spectral

element model. Tables 1 and 2 show the comparisons of the natural frequencies obtained by FEM

and SEM for the uniform and patched active beams, respectively. The single-spectral-element-model

is used for the uniform active beam, whereas three-spectral-element-model for the patched active

beam. The results clearly show that spectral element models indeed provide very accurate solutions

by using only a minimum number of nite elements.

For the case of simply supported uniform active beam, the total number of natural frequencies

JSS(!), which are less than a chosen trial frequency !, is given by Eq. (19). The wavenumbers k1and k2 in Eq. (19) can be computed from Eq. (4) for the axial and propagating bending modes,

respectively. Fig. 3 shows the change in the value of JSS(!) by varying the trial frequency ! from

zero value. It can be observed from Fig. 3 that, whenever a new natural frequency of bending mode

is added, the value of JSS(!) always increases by one on contact with the dispersion curve of the

bending mode, k2L=. However, the value of JSS(!) is found to be increased by one at 8041:4 Hz

without any contact with the dispersion curve of the bending mode. This is simply due to the

appearance of the rst axial mode with its natural frequency of 8041 :4 Hz.

Figs. 4 and 5 show the relationships between the frequency response function (FRF), the deter-

minant of spectral matrix (i.e., (!) = det[S]); and the value of J(!) for the cantilevered uniform


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PZT

PZT

Base Beam

27 mm 101.6 mm

Base Beam27 mm

0.762 mm

0.762 mm

2.286 mm

261.6 mm

261.6 mm

2.286 mm

101.6 mm

(a)

(b)

Fig. 2. Example active beams and their geometries: (a) uniform active beam (b) patched active beam.

Table 1

Comparison of the natural frequencies obtained by SEM and FEM: cantilevered uniform

active beam (n = total number of nite elements)

Mode no. !SEM (Hz) !FEM (Hz)

n = 1 n = 10 n = 20 n = 50 n = 100

1 30.038 30.046 30.040 30.038 30.038

2 188.217 188.584 188.308 188.232 188.221

3 526.888 529.398 527.498 526.985 526.9124 1032.131 1041.559 1034.358 1032.482 1032.219

5 1705.415 1731.478 1711.358 1706.340 1705.645

Table 2

Comparison of the natural frequencies obtained by SEM and FEM: cantilevered patched

active beam (n = total number of nite elements)

Mode no. !SEM (Hz) !FEM (Hz)

n = 3 n = 10 n = 20 n = 50 n = 100

1 32.944 32.948 32.945 32.944 32.9442 165.319 165.411 165.341 165.322 165.320

3 482.434 483.088 482.580 482.456 482.439

4 938.389 940.897 938.901 938.465 938.408

5 1604.396 1612.900 1605.952 1604.616 1604.450

and patched active beams, respectively. As expected, the two gures show that: (1) the frequency

locations at which the value of J(!) is increased by one coincide exactly with the resonance peaks

of FRF, and (2) the resonance peaks occur at the frequencies at which (!) becomes zero. The

function (!) has some discontinuities and they are indicated by glitches in Figs. 4 and 5. Very


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Fig. 3. Dispersion curves and the number of natural frequencies, J(!), for the simply supported uniform active beam.

Fig. 4. FRF, determinant of spectral matrix (!) (the circled numbers in the gure denote the types of discontinuity

illustrated in Fig. 6), and the number of natural frequencies J(!) for the cantilevered uniform active beam.

careful investigation shows that there exist three types of discontinuity; say Type-1, Type-2, and

Type-3. These discontinuities are indicated in Figs. 4 and 5 by the numbers 1; 2; and 3, depending

on the type of discontinuity. The characteristics of each discontinuity type are sketchily represented

in Fig. 6. When a conventional root-nding (numerical) algorithm is used to calculate the natural


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Fig. 5. FRF, determinant of spectral matrix (!) (the circled numbers in the gure denote the types of discontinuity

illustrated in Fig. 6), and the number of natural frequencies J(!) for the cantilevered patched active beam.

Frequency (Hz) Frequency (Hz) Frequency (Hz)

1 Type-1 2 Type-2 3 Type-3

()

()

()

Fig. 6. Types of discontinuity of the function (!), all indicated in Figs. 4 and 5.

frequencies from the characteristic equation (!)=0; Type-1 and Type-2 discontinuities often hinder

it from nding the resonance peaks (i.e., natural frequencies) which exist near the discontinuities:

this is simply due to the abrupt changes in the slope of the function (!) near the discontinuities.

There does not exist any resonance peak at the Type-3 discontinuity. However, Type-3 discontinuityoften mislead the conventional root-nding algorithms to conclude that there exists a resonance peak

at the Type-3 discontinuity. However, the WittrickWilliams algorithm developed herein may help

overcome these diculties.

The WittrickWilliams algorithm developed in the present paper is used to calculate the natural

frequencies of cantilevered active beams under three dierent states: the short-circuit, the open-circuit,

and the feedback control. The Youngs modulus of the piezoelectric material with open-circuit or with

feedback control, CD11, is related to that of piezoelectric material with short-circuit, Ec, as follows [17]:

CD11 = Ec + h231=

S33; (21)


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Table 3

Comparison of the natural frequencies of the cantilevered uniform active beams with short-circuit,

open-circuit and feedback controls

Mode no. Short-circuit (Hz) Open-circuit (Hz) Feedback control (Hz)

kp = 1 106; kd = 0 kp = 1 10

6; kd = 600

1 30.007 30.038 51.066 51.009

2 188.022 188.217 207.283 206.837

3 526.340 526.888 538.893 538.353

4 1031.058 1032.131 1040.854 1040.288

5 1703.642 1705.415 1712.260 1711.681

kp = proportional gain (V), kd = derivative gain (V s):

Table 4Comparison of the natural frequencies of the cantilevered patched active beams with short-circuit,

open-circuit and feedback controls

Mode no. Short-circuit (Hz) Open-circuit (Hz) Feedback control (Hz)

kp = 1 106; kd = 0 kp = 1 10

6; kd = 600

1 32.932 32.944 49.859 49.804

2 165.280 165.319 181.567 181.161

3 482.230 482.434 457.024 454.834

4 938.059 938.389 947.362 946.874

5 603.702 1604.396 1608.902 1608.668

kp = proportional gain (V), kd = derivative gain (V s):

where S33 and h31 are the dielectric constant and piezoelectric constant, respectively. As the second

term in the right-hand side of Eq. (21) has positive value, CD11 is larger than Ec.

The rst two columns of Tables 3 and 4 compare the natural frequencies of active beams with

short-circuit and open-circuit. It is shown that the natural frequencies with short circuit are in general

smaller than with open-circuit. This is because the piezoelectric eect disappears in the short-circuit

state [17].

The last two columns of Tables 3 and 4 compare the natural frequencies of active beams with

two dierent feedback controls: the proportional control and the proportional-derivative control. Thecontrol law considered in this study is given by

V(t) = kpw(t) kdw(t); (22)

where w(t) and w(t) are the bending displacement and velocity, respectively, and V(t) is the external

voltage applied to piezoelectric layer. kp is the proportional gain and kd is the derivative gain. The

proportional-control law can be reduced from Eq. (22) by simply forcing kd = 0. The Wittrick

Williams algorithm can be applied only to the linear undamped elastic systems [2]. Thus, a common

root-nding approach was used to calculate the natural frequencies of active beams with feedback

controls, but still using the present WittrickWilliams algorithm to start with the rst estimation of


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natural frequencies. Tables 3 and 4 show that the natural frequencies of active beams with feedback

controls are quite larger than with short-circuit or open-circuit. This is because the proportional gain

in general tends to increase the eective stiness of active beams. Comparing the last two columns

of Tables 3 and 4 to each other, one may nd that the derivative gain tends to lower the naturalfrequencies a little bit because it works as the eective viscous damping.

5. Conclusions

A WittrickWilliams algorithm is developed in the present paper for the elasticpiezoelectric

two-layer active beams. The exact dynamic stiness matrix (or spectral element matrix) is used for

the development. This algorithm may help calculate all the required natural frequencies, which lie

below any chosen frequency, without the possibility of missing any due to close grouping or due

to the sign change of the determinant of spectral element matrix via innity instead of via zero.The uniform active beams and the partially patched active beams are considered as the illustrative

examples. It is shown that there exist several types of discontinuity in the determinant of spectral

element matrix, which can make it dicult to calculate all required natural frequencies to the required

accuracy. The natural frequencies of active beams are compared when they are subject to dierent

states such as the short-circuit, the open-circuit, and the feedback controls.

Appendix A

[P] =

k1p11 0 k2p12 0 k3p13 0

0 p21 0 p22 0 p23

k1p31 0 k2p32 0 k3p33 0

C1k1p11 S1k1p11 C2k2p12 S2k2p12 Ch3k3p13 Sh3k3p13

S1p21 C1p21 S2p22 C2p22 Sh3p23 Ch3p23

C1k1p31 S1k1p31 C2k2p32 S2k2p32 Ch3k3p33 Sh3k3p33

;

[Q] =

0 1 0 2 0 3

1 0 1 0 1 0

0 k1 0 k2 0 k3

1S1 1C1 2S2 2C2 3Sh3 3Ch3

C1 S1 C2 S2 Ch3 Sh3

1S1 1C1 2S2 2C2 3Sh3 3Ch3

;

(A.1)


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where

p1n = EA n + kn;

p2n = EI k3n !

2kn + n(k2n !

2);

p3n = EI kn + n; (n = 1; 2);

Sn = sin(knL);

Cn = cos(knL);

(A.2)

and

p13 = EA 3 k3;

p23 = EI k3

3

+ !2k3 + 3(!2 + k2

3

);

p33 = EI k3 3;

Sh3 = sinh(k3L);

Ch3 = cosh(k3L):

(A.3)

Appendix B

The function fSS(!) in Eq. (17) is given by

fSS(!) = sinh(k3L)c2SSgSS(!)hSS(!); (B.1)

where

cSS = 2 EA EI;

gSS(!) =!8 A2(k21 k

22 )

2(k21 + k23 )

2(k22 + k23 )

2

(A !2 + EA k23 )2

;

hSS(!) =(EA A )2

(A !2 EA k21 )2(A !2 EA k22 )

2:

(B.2)

The symbols used in the above equations are all referred to Ref. [14].

(1) The hyperbolic function sinh(k3L) is not zero at any frequency !; except for the zero frequency.

(2) cSS is a non-zero constant because it is a real number determined by only material and geometric

properties.

(3) The function gSS(!) is not zero at any frequency !, except for the zero frequency, because

wavenumbers k1 and k2 always have dierent values.

Based on above discussion, one may conclude that the function fSS(!) will be zero only when the

function hSS(!) becomes zero.


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It can be shown that the function hSS(!) is zero only when the following conditions are fully

satised:

EA k

2

1 A !2

= 0;EA k22 A !

2 = 0;

EA = A:

(B.3)

Substituting the last condition of Eq. (B.3) into the dispersion relation, Eq. (4) in the text, one may

obtain the rst wavenumber k1 as follows:

k21 =

!2: (B.4)

Substituting Eq. (B.4) and the last condition of Eq. (B.3) into the left-hand side of the rst condition

of Eq. (B.4), it can be readily shown that the left-hand side of the rst condition becomes zero.

This violates the rst condition for the function fSS(!) to be zero at any frequency !, except forthe zero frequency. Thus, this proves that the function fSS(!) is not zero at any frequency !, except

for the zero frequency.

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