A model for a two dimensional distributed resonator

Journal of Sound and Vibration (1980) 68(l),

A MODEL FOR A TWO

45-58

DIMENSIONAL

DISTRIBUTED

RESONATOR J. F. W. BELL, J. Y. F. CHEN AND G. K. STEEL Department of Electrical and Electronic Engineering, University of Aston in Birmingham, Gosta Green, Birmingham B4 IPB, England AND

S. A. C. SANDERS Department of ElectricaI Engineering, City of Birmingham Polytechnic, Perry Barr, Birmingham, England (Received 21 July 1979)

In many distributed resonators the low order mode frequencies are sufficiently separated to permit satisfactory modelling in terms of only the single mode being used. It can be represented mechanically by mass, stiffness and energy loss by dissipation and coupling. The acoustically and electrically resonant frequency and the Q-factor are commonly used as parameters. A feature of high order modes not given by this model is the stepped nature of the transient response. It is also inadequate for the treatment of the interaction of adjacent frequency modes. The case of a low dissipation distributed resonator can be represented on the plane of the complex Laplace variable s, by a series of pairs of poles and zeros. Each pair corresponds to a mode of resonance and the positions in the plane are given by the frequency on the imaginary axis and the coupling on the real axis. A pair considered alone (the dominant pole approximation) is identical to the simple model. A comprehensive s-plane model would incorporate all the mode poles and zeros and the vibrational response could be found by performing a Bromwich contour integral round the points. In practice good results are obtained by using only a limited number of modes. The longitudinal vibrations of a rod driven at the end are well understood and this has been used as an example. A more difficult case, that of the in-plane vibrations of a thin disk, has been analyzed to obtain theoretical expressions for the coupling factors of the lower modes. The theoretical values have been checked experimentally and the resultant model used for material characterization experiments. When two adjacent resonances are excited by the drive signal a trial and error comparison between experimental and theoretical responses, by using computer graphics, enables the parameters of the two modes to be found. The s-plane mode can incorporate the effects of internal friction and by using it a convenient locus diagram representing the change in internal friction with temperature has been obtained.

1. INTRODUCTION The pulseecho technique due to Bell [l] can be used to determine the modes of a resonator which is typically a thin rod or disk. An accurate value of resonant frequency can be found by observing the first echo which results from a driving burst of sinusoidal oscillations. Sharp [2] has analyzed the response of a rod to such a forcing function by 45 0 1980Academic Press Inc. (London) Limited 0022460X/80/010045+ 14 $02.00/O

46

J. F. W. BELL ET AL.

considering it to be a one dimensional plane wave resonator. In this paper the analysis is extended to the case of a two dimensional resonator. Mechanical impedance, which is derived from the Laplace transformation of the equation of motion, enables the system to be described in terms of the complex variable s. This is referred to as the s-plane model. As used here, it embraces the effects of internal friction in the specimen and interaction between the resonator and its driving line: both are described by a “coupling” term. Thin disk resonators have been used for the precise measurement of elastic constants and ultrasonic characterization of materials [3]. An s-plane model can also be derived for this two dimensional system. The stored energy can be used to calculate the coupling terms and the eigenvalues can be obtained from consideration of the radial and tangential displacement. Each eigenvalue and its corresponding coupling can be represented by a pair of complex poles in the s-plane. Unlike those of the line resonator, the mode frequencies are not harmonically related and may be close together. A simplified form of this impedance model has been used to determine the response waveforms from a disk resonator. The results are obtained by numerical analysis in which a computer program is used to provide an output to a graphic display unit. The program is capable of generating the transient behaviour of a wide range of mechanical resonators in response to a sinusoidal drive. A comparison of experimental and computed responses can be used to resolve modes with frequencies which are close together and this provides an improved method of measuring Poisson’s ratio.

2. THE PULSE

ECHO TECHNIQUE

The essential features of the pulse-echo measurement system are shown in Figure 1. A magnetostrictive transducer is used to transmit a burst of sinusoidal stress waves along an acoustic line of nickel or telcoseal. The transducer comprises a broadly tuned coil whose

Pulse- burst Generator Figure

1. General

arrangement

of the pulse-echo

system.

position may be adjusted along the line which is biased by means of a permanent magnet. The driving oscillator is controlled to allow the number of oscillations in the burst to be preselected, as well as the burst repetition rate. The response of the resonator is observed in the signal induced in the driving coil by the return echo which is amplified and displayed on an oscilloscope. The typical echo shown in Figure 2 consists of two parts. One part is due to the reflection at the junction between the driving line and the resonator and the other is the return of energy which has been delivered to the resonator during the burst. When the transmitting frequency is exactly equal to the mode frequency of the resonator, the echo signal has a distinctive null or crossover. Mistuning causes the crossover to vanish and by observation of this null the resonant frequency can be resolved to an accuracy which is typically 0.1%.

A TWO

DIMENSIONAL

DISTRIBUTED

Echo signal

<

RESONATOR

47

Echo decrement

Figure 2. Typical echo pattern of a simple resonator. steady state echo signal equal to the initial signal.

3. s-PLANE

The burst number

MODEL

FOR

is 50. A lossless material

gives the

A DISK

The response of a resonator is often described in terms of its coupling, Q,, which determines the number of oscillations to crossover, N, (a list of principal symbols is given in Appendix C). A derivation of these terms is given in Appendix A, where they are related to the approximate model derived by Sharp [2]. The s-plane model can be extended to any other resonator provided the coupling term and the appropriate eigenvalue are known. The model can also be extended to high order modes by including more pairs of poles and zeros. For a rod resonator, the extra poles and zeros are harmonically related so that for mode number n, LO,= nw, and x is a constant, but this condition does not hold in general. The thin disk is a widely used example of a two dimensional resonator, the two displacements involved being the radial (u,) and tangential (Us). It would require a great amount of manipulation of the force and velocity functions to obtain the coupling values (b) by using the mechanical impedance approach. Alternatively, it can be shown that [4] where Z, is the acoustic impedance of the line and (M,,): is the equivalent mass of mode n on the edge of a disk radius a. The equivalent mass (M,& can be obtained by using the Rayleigh energy method [S]. The kinetic energy of an element is summed through the disk and is equated to the energy of a lumped equivalent mass:

(2) For the contour vibration of a thin disk which involves no transverse displacement, the radial and tangential components are given by [6]

[.:l=[Lt?][:I~ r 88

(3)

dr

which has solutions of the forms ICI,= AJ,,(Kr/a) cos nf9, the J” being Bessel functions.

$# = BJ,(KOr/a)

sin no,

48

J. F. W. BELL ET AL.

depend on the elastic constants of the material; K = ma/C, and From equation (2) the equivalent mass on the edge of the disk at 0 = 0” is

K and 0 0’ = 2/(1--o).

a (~P_,)a

=

2n [(i?~,/i?t)~

Ph

r dr d# ‘[(&,!‘C;t)‘+(Su,/c’-t)“],=, i

+(d~,/dt)~]

Is0 0 where h is the thickness obtained for (Me,),: (M

=l0 )

=

of the disk. From equations

(3) and (4) the following

,

(4)

expression

is

~,,~~,(K)+~2(K)+~2C~~(~~)+~~~K~)l+2wC~,W)+LW~1~ [K dJ,(K)/dK]* + [n,uJ,,(K@)]* + 2npJ,(KO) dJ,(K)/dK

’

(5)

where

B J,(K)

$(K@)*-,(,+l)+M,(K)

I* = 2 J,(KO)

n[M,,(KO)-(n+l)]

Z,(K) =j:y(f’“)yr Luke [7] gives integrals be made :

dr,

of Bessel functions

’ M,,(x) = ,xs.

Z,(K) =jis(Fdr, which enable

the following

.+zW)+3J,-,W)l)

Z,(K) + Z,(K) = $K2{J,2+1(K)+3J,2_1(K)-J,(K)[J Also, in equation

-

substitutions

to

-&J,‘(K).

(6)

(5) Z,(K) =jiJ”(G)$b,,E)}dr,

Z,(K) =j~J~~)$~~(~)}dr, from which

Z,(K) + Z,(K) = J,(K) J,(KO).

(7)

Table 1 is derived from equation (5) and it shows the ratio (I$,):/& with mode order n for a useful range of Poisson’s ratio in intervals of 0.1. The equivalent mass can be evaluated once the mass of the disk (MS,) and its mode of vibrations are known. The TABLE 1

Calculated M,,IM,, for the in-plane modes of circular disks as a function of Poisson’s ratio (M,,/M,, = ratio of edge equivalent mass to static mass) Poisson’s ratio,

M,IM, 0

0.1

0.2

0.3

0.4

0.5

1,3 194 135 136 1,7 1,8 139 1. 10

0.4778 0.3341 0.2764 0.2410 0.2154 0.1953 0.1788 0.1651 0.1534

0.4932 0.3532 0.2932 0.2563 0.2296 0.2086 0.1914 0.1770 0.1648

0.5056 0.3689 0.3082 0.2705 0.2429 0.2211 0.2033 0.1884 0.1756

0.5158 0.3823 0.3215 0.2831 0.2550 0.2327 0.2144 0.1990 0.1858

0.5242 0.3936 0.3331 0.2945 0.2659 0.2432 0.2246 0.2088 0.1952

0.5314 0.4037 0.3434 0.3046 0.2758 0.2528 0.2338 0.2177 0.2039

AR

0.7050

0.7317

0.7562

0.7832

0.8111

0.8401

\Mode

132

A TWO DIMENSIONAL

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49

RESONATOR

equivalent mass of a thin rod resonator for each mode is always half the static mass, but for a disk it changes from mode to mode. For the fundamental mode it might have a value of more than half the static mass because the amplitude component reaches a maximum near the edge of the disk but at the drive point (at the edge) it falls noticeably. Experimental observation showed that the coupling of the lowest five asymmetric modes is approximately constant and this is confirmed by using the equivalent mass concept. Figure 3 shows Q,, normalized to mode two, plotted against mode order for both a disk and a thin rod (which has Meg = M,,/2). It can be seen that the change in coupling with mode order is small in the case of the disk. The values of Q, have been calculated for the lowest 10 modes by using equation (1) and

(8) Equation (8) is derived eigenvalues.

in Appendix

OW

A. The calculation

’

5

1

,

7 6 Mode order

was based on tabulated

I

I

6

9

I

IO

Figure 3. Representation of the coupling Q values of a thin rod and disk. For comparison, the coupling Q, is normalized to the mode number 2 and Poisson’s ratio is assumed to be 0.3. The coupling values of the thin disk change slightly as compared to the thin rod. This means that the N, values of a thin disk vary insignificantly with mode order.

The distributed response r(x, r) evaluated at x = 0 gives the received echo signal r(0, r) of a disk. When subjected to an input sin(w,r), the Laplace transform of this response is given in general form in Appendix B as

(9) For the specific case of the disk, the fundamental mode having two nodal diameters is identified by n = 2 but the response is otherwise similar to that given in equation (9). The mode frequencies, o,,, can be obtained from the equation derived by Love [S] and the coupling term /?,, can be found from the equivalent mass by using equation (1). The inversion of equation (9) is of the form r(O,r) 2: sin(o,r+&)+

i

[U,exp(-/3,r)sin(o,r+&)].

II=1

The derivation of this expression is shown in Appendix B.

(10)

50

J. F. W. BELL ET AL.

The value of (M,,), can be derived for the asymmetric modes as shown previously; in addition values can be obtained for both the pure radial and the tangential modes. For the pure radial modes, (M,,/M,,) is M,,/M,,

= 1-

J,(K,,)J,(K,,)IJ:(K,,),

where K iR can be determined from the characteristic pure tangential case, it becomes M,,/M,,

= 1-

equation

(11) M (K ,R) = 1 - 0. For the

(12)

J,(K,,)J,(K,,)IJ:(K,,),

with K,, determined from the characteristic equation M,,(K,,) = 2. From the values of (MJM,,) have been calculated and are shown in Table 1. Frequency

equation

(1 1),

axis

three contour modes (having nodal diameters only) and the radial Figure 4. In this s-plane diagram, (breathing) modes are shown. The Poisson’s ratio is assumed to be 0.3. The vertical frequency axis is normalized unlike the thin rod coupling axis to f11,2. The frequencies here are not harmonic; to tr%, and horizontal resonator, the coupling increases slowly with mode order n.

If a disk is driven radially both the asymmetric and the pure radial modes can be excited, but not the tangential mode. An s-plane diagram for a disk with this drive is shown in Figure 4 with asymmetric modes up to n = 4.

4. EXPERIMENTS

To test the values of equivalent mass derived above, measurements a series of thin disks. From equations (1) and (4), (Me,):: = {ZJ2 log,(2)1 {NJf’} .

were carried

out on

(13)

Z, can be established by using a thin rod (driven longitudinally) since for such a resonator Meq = M,,/2. A brass rod was used to measure Z, for nickel lines of 0.7 mm and 1 mm diameter and the results are given in Table 2. The lines were then used to drive disks of aluminium, brass and steel. The equivalent masses were found by measuring N, and frequencies for modes up to n = 5.

A TWO DIMENSIONAL

DISTRIBUTED RESONATOR

51

TABLE 2

The experimental

line impedance

data from a brass rod of 100 mm long and static mass (M) of 25.047 gm

1 mm nickel line drive Frequency Mode 2 3

(kW

n,

33.081 49.549

20 30

Q,

0.7 mm nickel line drive Frequency

z, = 2+2)

91 136 Mean Z, =

e 28.7 28.6 28.7

W-W

n,

Q,

33.101 49.552

186 41 281 62 Mean Z, =

Z(Q) 14.0 13.9 13.95

In Table 3, the experimental and calculated results are compared. The experimental values compare favourably with calculated results and justify the method of evaluating the coupling terms of a two dimensional resonator. An exception is the result for the 1,3 and 1,R modes in the aluminium disk; these modes of equal frequency interact considerably and there are significant changes in the values of N, . Figure 5(a) and (b) shows computer plots associated with the 1,3 and 1,R resonances of a tantalum disk while Figure 5(c) and (d) shows corresponding oscilloscope displays obtained by direct measurement. One important feature that can be observed from Figure 5 is that the true resonant frequencies were slightly different from those which gave

Figure 5. (a) Oscilloscope display showing the echo pattern of a tantalum disk (a = 0,362) at the mode with the three nodal diameters (1,3); the nature of stepped decrement and interaction from the adjacent radial mode can be observed; signal frequency = 61.668 kHz. (b) The computer plot for the particular case shown above; this shows a combination of two mode frequencies at 61.730 and 63.070 kHz; it is the only one to give a crossover with a signal frequency of 61.668 kHz; the steps disappear if the 1,2 mode is omitted from the model. (c) As in Figure S(a), the tantalum disk resonating at 1, R mode with frequency 63.090 kHz. (d) The resulting computer plot with the experimental data; here the actual frequency separation is 2.17% rather than the experimental value of 2.31%; this enables an improved value of Poisson’s ratio to be obtained.

r Frequency

48.148 73.822 96.250 116.305 70.103

Mode

1,2 1.3 174 1,5 LR

tcalc)

0.5148 0.3810 0.3200 0.2819 0.7805

f&,/M,,

50.80 mm 1.50 mm 23.193 gm 0.287

tev)

= = = =

66.704 102.711 134.042 163.361 102.711

Frequency

0.3484 0.2859

tcatc) 0.5193 0.3871 0.3263 0.2878

M&f,,

35.50 mm 1.56 mm 3.880 gm 0.340

teW

0.5056

I&,/M,,

Aluminium disk diameter = thickness = static mass = Poisson’s ratio = \

30.506 46.967 61.412 74.904 48.591

I Frequency

0.5202 0.3882 0.3275 0.2890 0.7970

M,,IM,, tcalc)

53.848 mm 1.50 mm 35.15 gm 0.355 text)

= = = =

0.5208 0.3883 0.3449 0.2906 0.7265

M,,IM,,

Brass disk diameter thickness static mass Poisson’s ratio

\

values of equivalent mass associated with asymmetric and pure radial in-plane modes for steel, aluminium and brass disks: the equivalent mass (M,,) is normalized to the static mass (MS,)

0.4687 0.3704 0.3201 0.2612 0.6934

K,IW,

Steel disk diameter thickness static mass Poisson’s ratio

The calculated and experimental

TABLE 3

A TWO DIMENSIONAL

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53

RESONATOR

crossover. The mode frequencies at 61.730 and 63.070 kHz were found to be the only combination to give the crossovers with the signal frequencies measured from the experiment. The nature of the stepped decrement and the interaction from the adjacent frequency are well predicted by the model. In the pulseecho method of measuring Poisson’s ratio it is necessary to use the ratio (fiR/fi3 - 1) and the graphic plot for the tantalum disk enabled the value of this ratio to be corrected from 0.2306 to 0.2171 with a small improvement in Poisson’s ratio from 0.362 to 0.360.

5. RESONATOR

WITH LOSSES

The echo analysis described so far applies to a lossless resonator. Including losses inside the resonator results in dramatic changes in the shape of the echo. For material with loss, the real part of the s-plane poles and zeros will contain an additional term. For a rod resonator, the function F(0, s) has poles and zeros at s = -/I-n% fjno, and s = P-n2a?j,,1, where n = 1,2, . . . , co. Inversion by using the single dominant pole approximation (Appendix A) gives r(O,r) = -s[l

-&exp(-/?-nia)t]sinwt,

(14)

The internal friction of the material is often described in terms of l/Q, = n’u/nf. be combined with the coupling Q, to give a total coupling term

This may

(Q&l = (Qcl-' + (Qm)-'.

(15)

By setting equation (14) equal to zero, the number of oscillations to crossover is

Km =

2

f p+n2cllwe

(16)

/C3_n2a

In the absence of material loss, Q;’ = 0, equation (16) reduces to equation (AlO): i.e., N, = QJ4.53. Figure 6 shows the s-plane diagram for a resonator with material loss. It may be observed from equation (14) that Q, can be determined from the steady state value of r(0, t). For Q, > Q, the zeros are in the right-hand half of the s-plane but an increase in internal friction causes the poles and zeros to move to the left and when Q, = Q, the zeros are on the imaginary axis with the result that r(O,r) has a steady state value of zero. X

I

X

I

Figure 6. s-plane diagram of a single resonance symmetrical about the frequency axis.

0

0

including

material

losses. The data

points

are no longer

54

J. F. W. BELL

ET AL.

Temperature I

120

,

I

,

I

/

PC)

I

‘9’,

(b)

’temperature ’ ’IZO”C1 ’ ’ Room

Figure 7. (a) The variation of material losses for an aluminium disk with temperature; a peak is observed at 500°C. (b) s-plane zero locus diagram with temperature as a parameter; the corresponding pole has a similar locus but displaced by an amount dependent on the coupling term.

In general Q;’ varies with temperature and Figure 7(a) shows the variation for an aluminium disk. A locus plot showing only the zero corresponding to this case is also given in Figure 7(b).

6. CONCLUSION

The experimental results given confirm the analysis of the disk resonator in terms of its equivalent mass and an s-plane model for the disk has been derived from the analysis. The model can be generally used to determine the transient behaviour of any resonator and has been employed here to resolve the frequency of two modes which are sufficiently close to give a complex response. Internal friction also affects the response of a resonator in a way which is clearly demonstrated by corresponding variation in the s-plane diagram.

REFERENCES 1. J. F. W. BELL 1968 Ultrasonics 6, 11. A case acoustic thermometer. 2. J. C. K. SHARP 1974 Ph.D. Thesis, University of Aston. A theoretical and experimental investigation

into the spectra of selected resonators. 3. J. F. W. BELL, J. Y. F. CHEN and K. C. R. CHAPLAIN 1980 (to appear) The Journalofthe American Society for Non-destructive Testing. The elastic constants of refractory materials at high temperatures.

A TWO DIMENSIONAL

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55

4. J. Y. F. CHEN 1979 Ph.D. Thesis, University of Aston. Surface wave technique for the study of engineering structures and materials. 5. LORD RAYLEIGH 1877 Theory of Sound (two volumes). New York: Dover Publications, second edition, 1945, re-issue. See 1.262. 6. R. HOLLAND 1966 Journal of the Acoustical Society of America 40, 1051. Numerical studies of elastic-disk contour modes lacking axial symmetry. I. Y. LUKE 1962 Integrals of Bessel Functions. New York: McGraw-Hill Book Company. 8. A. E. H. LOVE 1927 A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, fourth edition. See pp. 497498.

APPENDIX Sharp has analyzed mechanical impedances sending end in response

A

a transmission line and line resonator using their respective Z, and Z,. The Laplace transform of the signal received at the to a forcing function F(r) is of the form R(0, s) = W(s) F(s).

641)

The signal is subjected to a delay because of the time needed to propagate along the line; however the form of the response is not affected and the delay may be excluded from w(s). The forcing function F(t) is a burst of sinusoidal oscillation so that F(s) = {o,/(s2+c0,2)}[1-exp(-ST)] and the overall response

(‘42)

is given by R(O,t) = r(O,r) - H(t-T)r(O,t-T),

where H (t - T) is a unit step function

at t = T and

r(0, s) = W(s) co,/(? + 0,‘).

(A3)

For the line resonator w(s) = {Z,(s)-Z,(s)

tanh(yl)Il{Zr(s)+Z,(s)

tanh($

where y is the propagation constant and 1 is the length of the resonator. infinite number of poles and zeros in the s-plane. A typical pole has the form

and the corresponding

(‘44) W(s) has an

s=--cr-/?+jj,

(A5)

s=--a+/?+j~,

(‘46)

zero is

where o is the resonant radian frequency. The parameter CIis associated with the energy lost due to internal friction in the material while /? depends on the mismatch at the junction between line and resonator. The factor /I is related to the coupling Q factor by

Q, = 433.

(A7)

Provided the resonant frequencies are widely separated, the analysis can be carried out by ignoring all but the resonance close in frequency to the signal frequency 0,. This dominant pole approximation gives

W)

56

J.

where for convenience (0, = o), the inversion

the material of ?(O, s) is r(0.t)

By setting equation (A9) equal crossover (zero amplitude) N, is

F. W.

BELL

is assumed

ET AL.

to be lossless so that CI= 0. At resonance

= -sinwt[l-2

exp(-bt)].

to zero, the number

N, = Q, log,(2)/n

(A9)

of oscillations

from

initiation

= QJ4.53.

to

(A9)

Also (All)

B = f log,(2)lN,, where f is the resonant

frequency.

APPENDIX

B

The simple model of the resonator response summarized in equation (A8) can be extended by including resonances adjacent to that being considered so that the response function is modified to

k (s - pn)z+ 0,” 0s s2+0,2 n=,(s+8”)2+d’

f(O,s) = ~

I-I

where the resonator is assumed lossless. In the case CJJ” = nw, but the general form of equation (Bl) can any geometry provided the appropriate values of response $0,~) contains only simple poles and can expansion. Thus

of a rod resonator, b,, is constant and be used as a model for a resonator of /I,, and w, can be determined. The be inverted by using partial fraction

L+~[lL+A:],

A ?(O,s) = ---+

(s-jq)

(s+jo,)

where 6, = -/I, +jo,, and * implies the complex

A, =

“=,

(Bl)

(s-4)

conjugate.

V-32)

(s-43

A, can be written

as

%exp[_i(k -:)I,

(B3)

so that A,* = iL;,exp[-j(H.

--;)I.

In the inversion A, and A,* can be combined into the term U, exp(-&t) sin(o,t+&). The factors U, and 0, are available from the single term A, associated with the positive frequency pole at s = 6,. In addition A,, is evaluated from equation (Bl) with the denominator term (s - 8,) deleted, so that A, =

-

23”0s

C- P, + j 64 - 41 C- B. + j @, + 41 k X

(A + PJ +_i(0, - ~J C- CP,+ B,) +_ib, + qA1 I-IC-C(P,-P,)+j(o,-w,)lC(B,-B,)+j(o,+o,)l .

mmT;

(B5)

,

A TWO

Normally

DIMENSIONAL

DISTRIBUTED

RESONATOR

/I. 4 CD,,and &,, G w, so that

From equation

(B6)

(B7) m#n

where Ll = C(B. + 8fn)2 + (%I -

%J’l/Wn - Bm12 + 6%- w?J21 9 rn = n+l,n+2

and by symmetry

l,, = I,, . Also from equation

n = 1,2 ,..., k,

,..., k, (B6)

038) where & “In

=

tan-’

( ) ~“-%I ~

[,,

A-Bm ’

= tan-’

( > ~%-%I B,+B,

& = tan-’

’

wn-“s ( P.

>’

and again from symmetry E mn =

Upon using the approximations of equation (Bl) can be written as

E nm -

71,

of equations

i,,

=

(B9, BlO)

-r,,.

(B7), (B8) and (B9), the complete

inversion

‘

r(O,t) = sin(w,r+Q

+ 2 [U,exp(-j$t)sin(w,t+8,)],

(Bll)

n=l

where (B12) n=l

APPENDIX

C: LIST OF PRINCIPAL

SYMBOLS

radius of disk thickness of disk density of disk static mass of disk edge equivalent mass displacement function dufdt radial and tangential components of displacement in disk corresponding potential components polar co-ordinates Bessel function of first kind, order n

dJ,CWx

J. F. W. BELL ET AL

xJn- ,(x)lJnW

rk t) R (2,t) F(s)

m

T

Z(s)

Poisson’s ratio [2/( 1 - a)] 1’2 wa/C, (C, is plate velocity) complex Laplace variable frequency radian frequency signal frequency resonant frequency of mode n (- 1)“2 coupling Q factor material Q factor total Q factor relaxation term coupling term number of oscillations to crossover time response to a sinusoidal forcing function response to a sinusoidal burst Laplace transform of forcing function F(t) transfer function duration of burst mechanical impedance