'TWIXT EARTH AND SKY WITH ROD AND TUBE 1153

(Text continued from page 1119)

having it associated with the letter s. Mechanical admittance is indigenous to a tube-connected system and the aforementioned series of structural elements

would turn out to be a parallel combination of elements in a tubing diagram; the mechanical admittance of the parallel combination of these elements would with perfect analogy be the sum of the individual mechanical admittances: Y•= Ymq- Y•.q- Y•a.

3m-10. Couplers: Transformers and Trans- ducers.--A coupler introduces a constraint between circuits or between different portions of the same circuit. It specifies relationships between variables at different places, a more complicated type of relationship than the 1 to 1 specified by connectors, which might be considered as simplified couplers. Ideal passive couplers transmit energy but store none. Direct couplers couple through variables to through variables and across to across, while inverse couplers couple through variables to across variables. Transformers and transducers couple

like and unlike system types, respectively. In general, a transformer consists of a subtractor operated by the difference between the values of a variable at two

points, actuating a multiplier, the multiplied value then being impressed as a difference between the values of a variable at two other points.

3m-11. Units.--Although any consistent system of units may be used in computations with analogies, the mks system is particularly advantageous since the watt will then be the unit of power in both the acoustic, mechanical, and electric portions of a transducer. Thus velocity v is in m/sec, force F in newtons, mobility z in m/sec per newton; angular velocity vR is in radians/ sec, torque FR in newton m, rotational mobility z• in radians/sec per newton m; volume velocity U is in mS/sec, sound pressure p in newtons/m •', acoustic mobility zA in mS/sec per newton/m•'; voltage E is in volts, current I in amperes, impedance Z in ohms. Power P is in watts and energy W in joules, throughout.

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 28, NUMBER 6 NOVEMBER, 1956

Equivalent Circuit of a Resonant, Finite, Isotropic Elastic Circular Disk* ROSHAN LAn SI-IARIIA i

Collins Radio Company, Western Division., Burbank, California (Received February 21, 1956)

The equivalent circuit of a finite circular disk resonant at any frequency corresponding to the first three axially symmetric flexural modes is obtained from a function of two dynamic variables; (1) the kinetic energy in the disk, (2) velocity of the mechanical motion. This function, called the equivalent mass, is com- puted over a domain of arguments which is of practical interest. The results indicate fairly good agreement between theory and practice.

INTRODUCTION

OME mechanical filters • which possess excellent selectivity characteristics use relatively thick cir-

cular disks. Proper design requires complete under- standing regarding the free axially symmetric flexural vibrations of such disks. The work of Deresiewicz and

Mindlin •' has made it possible to obtain the frequency spectrum of a thick circular disk for an extended region of frequencies? Such an analysis assists in predicting accurate disk dimensions for any frequency. Never- theless a thorough understanding of the equivalent circuit of a circular disk was needed for proper band width design. This paper discusses a method whereby the equivalent mass of a resonant circular disk at any radial distance is computed as a function of the kinetic energy due to flexural motion of the disk and the velocity of the flexural motion at the same radial distance. Once

* Part of a thesis for the Degree of Master of Science at the University of Southern California, 1955 (unpublished).

$ Now temporarily located at Collins Radio Company of Canada Ltd., Toronto, Ontario.

• M. L. Doelz and J. C. Hathaway, Electronics 26, 138 (1953). 2 H. Deresiewicz and R. D. Mindlin, J. Appl. Mech. 75, 86

(1953). 8 R. L. Sharma, thesis, University of Southern California, 1955

(unpublished).

the value of the equivalent mass of a disk is known, the value of the equivalent compliance can be easily computed by using the resonance condition. For prac- tical reasons this analysis was limited to the first three axially symmetric modes and considers only those fre- quencies which are smaller than that of the first thickness shear mode. The material considered is a

constant modulus nickel alloy commercially known as Ni-Span C. A fair degree of agreement between theory and practice exists thereby proving the validity of this analysis.

DISCUSSION OF THE METHOD FOR OBTAINING

THE EQUIVALENT MASS

The equivalent circuit of a resonant continuous system is generally obtained from the ratio of force F to velocity • both measured in the same direction and at the same point x, as follows 4

=Zx'- jooMeqq- - near resonance, (1)

4 W. P. Mason, Electromechanical Transducers and Wave Filters (D. Van Nostrand Company, Inc., New York, 1948), second edi- tion, pp. 86-92.

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1154 ROSHAN LAL SHARMA

L• C•

(A)

FIO. 1. Equivalent circuit of a reed clamped on one end.

where Meq and C•q are the equivalent mass and equiva- lent compliance, respectively.

Such a technique requires the exact knowledge of the force function F in addition to the velocity function •. Also, in order to break up the impedance function Z into its equivalent parts, one must assume frequencies very close to resonance. This in general demands laborious mathematical manipulations related in par- ticular to obtaining the approximate values of the various algebraic series which express the force and velocity functions. This procedure can become ex- tremely difficult in cases where the force function is not readily available and the algebraic series do not lend themselves to approximations. In such cases it will be highly desirable to obtain the equivalent circuit by employing a method based on energy considerations alone. This method requires only a knowledge of the velocity function and the tedious algebra is eliminated.

The equivalent mass of a resonant continuous system can be evaluated through

p/2fff½'d•l•d• kinetic energy (M•q)•= - , (2)

where o is the density and the triple integration is performed over the whole volume V of the system.

This approach seems to be legitimate in view of the fact that the term "equivalent" as applied to a network as shown in Fig. 1 (A) is not restricted by any require- ments on physical realizability except those based on energy considerations. The usefulness and the validity

Fxo. 2. The axial and the radial components of displacement present in a circular disk of finite thickness.

of this method can be easily tested by considering the flexural vibrations of a straight thin bar of uniform cross section S, whose one end is clamped and the other end is free as shown in Fig. 1 (B). In this case the ampli- tude function • depends only on distance x and time t. Therefore the expression for the equivalent mass of the rod at the free end (x-l where l is the total length of the bar) becomes

PSf • (M•q)•= . (3)

The value of the integrated square appearing in Eq. (3) can be either actually evaluated or found in Lord Rayleigh's theory of sound. 5 Using Rayleigh's value, we obtain

(M•,) •= 0.25•$•= 0.25•,•,•, (4)

which agrees with Mason'½ value, which he obtained by means of the method as defined in Eq. (1).

Once (Meq)• is known, the value of the equivalent compliance at x= l can be directly calculated through

(C•),= 1/p•'(M•)t, (5) C t

LV, 2C, L• C, Ly, 2C,

FR•UœNC¾

(A') ('B)

FIO. 3. Mechanical filters constructed from thick circular disks.

where p is the resonant frequency in radians per second. In order to include the effects of rotatory inertia

and shear, we must resort to an improved theory which yields a proper amplitude function 7. One such applica- tion deals with the evaluation of the equivalent circuit of a resonant, finite, isotropic elastic circular disk. In such a case, 7 it becomes necessary to take account of shear in addition to rotatory inertia.

DISPLACEMENT EQUATIONS FOR THE CIRCULAR PLATE

If account is taken of rotatory inertia and shear deformation, the equations of motion for a finite, isotropic, free circular disk for the axially symmetric flexural vibrations are •'

r•= A •Jo(•i•r)nLA do(•.r), (6)

•p,= --A ½•(•-- 1)J•(•i•r)--A •.•.(•.- 1)J•(•.r), (7) 5 Lord Rayleigh, Theory of Sound (Dover Publications, New

York, 1945), second edition, Vol. I, pp. 262-265. 6 See reference 4, p. 91. It is interesting to note that the more

correct value 0.236 M,t•ti. as obtained by Mason in Appendix A, reference 4, was also obtained by the present author by actually solving the right-hand side of Eq. (3).

? R. D. Mindlin, J. Appl. Mech. 73, 31 (1951).

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EQUIVALENT CIRCUIT OF RESONANT DISK 1155

, , ,

I.(• t , o.• ----

0.7

FIRS1 MOD•

O.&

Fro. 4. (Meq),,/M,t vs d/h curves for the first three modes of axially symmetric flexural modes. [(Meq),,/M,t=ratio of edge equivalent mass to static mass; d/h= ratio of diameter to thickness.]

where

804 8x 2, 822 =--{Rq - T-q-[-(R-- T)2+ 480-4-] • }

2

T-D/K2uh; 8o4=pp2h/D; R=h2/12; K2=•r2/12

Eh a 822, 8• • D= ; •r•, •r2=

12 (1 -- r2) (RS04_ S-•)

and E= Young's modulus, u=shear modulus, •= Pois- son's ratio, h= thickness, and Jo(x) and J•(x) are Bessel functions of the first kind.

We must also note that n and • are related to the axial and radial components of displacement, respec- tively, according to the approximations

U•y.•(r,t)

(8) where r and t are the radial distance and time respec- tively, and the thickness h is in the y direction (see Fig. 2).

For practical reasons, mechanical filter designs re- quire equivalent circuit analysis for the first three axially symmetric modes for frequencies which are less

than that of the first thickness shear mode of an infinite

plate of thickness h. We can express this by saying that p<p where fi=•r(•/p)•/h.

For the case of p<fi,/b. is always imaginary. For this reason we must rewrite Eqs. (6) and (7) as

n = A •J0 (8 •r) +A 210 (82r) (9)

•b,=--A•Sx(•,--1)J•(8,r)q-A282(•r2--1)I•(82r), (10)

where Io(x) and I• (x) are the modified Bessel functions of the first kind.

CALCULATION OF THE EQUIVALENT MASS

The structural design of most mechanical I.F. filters as shown in Fig. 3(A), consists of thick circular disks joined together by coupling rods spot welded around the disk peripheries. The equivalent circuit of this combination is that shown in Fig. 3(B). The disks represent the series resonant circuits while the coupling compliances are represented by the capacitance C2. The geometrical configuration suggests that only the kinetic energy associated with the flexural motion of two ad- jacent resonant disks is directly coupled through the coupling rods. Therefore according to Eq. (2) the

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1156 ROSHAN LAL SHARMA

70.1

I00.0

20.0

I0.0

"', 6OK( EQUIVALENT MASS AT "',J THE EDGE OF THE DISK

. ,

............ FIRST MODE

SECOND MODE

',, so KC • .... THIRD M ODE -,,•o,o • , •, ,

'.,,,",,,, \•,o, .c•,,. ,, X,. ',,.

\', \ \ ', ",•. ',. \',\ \.• ',. -,•.,. --,

',•,_ ' • _ •.% -,, ..%•-,. '.,.,,. _ '•. X, ",•? ',. •"•

-,•. -,

•oo,• •• " , ,

"', •,,

140 KC ' • '

,

5.0

q..O

ø09

.07

.05

.05

,0•

.02

IO.O

I,O

O,I

.o'• .01

FIO. 5. The edge equivalent mass •s d/h for some representative frequencies associated with the first three modes of axially symmetric flexural modes.

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EQUIVALENT CIRCUIT OF RESONANT DISK 1157

equivalent mass as seen by the coupling rod at the edge of the disk is

(M•) •=•=

Eq. (11) as

By using the second part of Eq. (8) and noting that there is no azimuthal dependence, we can rewrite

If we now insert Eq. (9) in Eq. (12) and perform the required integration, we obtain

where [(A•) •' 2A• q Z Jø•'('v)+Iø•'ø5'v)+'•: Jø('v)Iø(fi'v)]

•=(/i•a), •=-, and

(13)

and Ms•ati.=static mass of the disk=rpa2h. Using Eq. (5) we thus can obtain (C.q)a directly, once (M•q)a is known.

According to direct electromechanical analogy, the inductance and capacitance are analogous to mass and compliance, respectively. Thus an equivalent circuit of a circular disk at its edge can be expressed once Eq. (13) is solved. In Fig. 4 the ratio (M,q)a/-/l//sta•i• is plotted against a useful range of values of d/h for the first three modes. Once the choice of dimensions and mode of

vibrations corresponding to any frequency has been made, the equivalent mass at the edge of a circular disk can be easily evaluated. One such set of calculations performed on a desk calculator, is graphically shown in Fig. 5. Knowing the function 7, the equivalent mass at any radial point can be easily computed due to the fact that in Eq. (1) the denominator is the only variable.

The form of coupling shown in Fig. 3(A) is found to be the most efficient for such mechanical filters. If any other form of coupling, such as coupling between the radial modes is used, proper choice of the wave function must be made in regard to Eqs. (2) or (11).

COMPARISON BETWEEN THEORY AND PRACTICE

For the purpose of comparing the theory with experi- ment, information concerning a large number of such filters was compiled. Using some well-known network principles, 8 we get from Figs. 3 (A) and 3(B)

(14)

where (f•'-f•) is determined by the center frequency and band width of the filter.

s For example, see reference 4, pp. 52-53.

The value of C•. is shown to be •

t/s,

where l and S are the length and the total cross section of the coupling rods employed, respectively, as shown in Fig. 3(A).

By substituting the measured values of (f•'--f•') and computed values of C•. in Eq. (14), values of were derived. These experimental values were then compared with theoretical values. Figure 6 shows a comparison between theory and practice for the first mode. The large discrepancy is due to the fact that in welding the coupling rods on the peripheries of the disks, the condition for free vibrations is disturbed. This results in some distortion in the nodal pattern on the plane face of the disk as shown in Fig. 7. Because of a resultant decrease in velocity at the weld points, the value of (Me•), increases. The increase is proportional

FIRST MODE

COUPLING WlRrS ONLY')

/

/ /

/

I I I i , i I i [ I o,• 0)4 0,6 o,6 I.o i,z x io- z

TOTAL CROSS-SECTION OF THE COUPLING RODS (Sq CENTIMETERS)

Fro. 6. Deviation from theory associated with the first mode.

• R. L. Sharma, "Analysis of coupling as employed in Collins mechanical filters," Collins Radio Company (unpublished).

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1158 ROSHAN LAL SHARMA

/•LLY SYMMETRIC NODAL PATTERN

DISTORTED

NODAL PATTERN

COUPLING ROD

FIG. 7. The distor- tion in the axially symmetric nodal pat- tern associated with the first mode pro- duced as a result of welding coupling rods at the edges of the disks.

to the total cross section of the coupling rods. From Fig. 6 it also becomes apparent that the theoretical values of (M•q)• are approached with decreasing cross sections of the rods. In addition it has been observed

that by welding a larger number of coupling rods sym- metrically around the peripheries of the disks, the deviation is successively reduced. Nevertheless in all such cases deviation can be known, and satisfactory design information obtained.

No such deviation is noticeable regarding filter designs which make use of the second or higher mode of disk vibrations. It is believed that the distortion in the

nodal patterns corresponding to the second (and higher) mode, is negligible. The comparison between theory and experiment for the second mode, shown in Fig. 8, clearly points towards such a conclusion. A closer

SECOND MODE A •50.4(C ANALYSIS

ß 465,-,KC ANALYSIS

x SO0',"KC ANAlySIS

I I I I I I

I • S 4 S S

BANDWIDTH IN KILOCYCLES

FTO. 8.

agreement between theory and experiment will result, once one notes that most of the deviation shown in Fig. 8 could be due to errors in evaluating C•. correctly. Such errors are due to losses and reflections in the coupling rods caused by sharp discontinuities produced as a result of the spot welding techniques.

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Collins Radio Company for making this study possible and permitting its publication. His sincere thanks are also due Dr. R. D. Mindlin, Dr. H. Deresiewicz, and R. L. Peters for some valuable suggestions.

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 28, NUMBER 6 NOVEMBER, 1956

Contour Vibrations of Isotropic Circular Plates MoRio ONOE

Institute of Industrial Science, University of Tokyo, Chiba-City, Japan (Received December 12, 1955)

A mathematical analysis has been done according to Love's theory. The results are: (a) The coefficients of the dilatation and the rotation are equal in amount regardless of Poisson's ratio only for the modes whose circumferential order is one. (b) The first mode whose order is two has the lowest natural frequency. This mode is practically equivoluminal, while the second mode is dilatational. (c) Locations of nodes are considered. Circular nodal lines can exist only for the radial and the tangential modes. (d) Frequencies and displacements of the most important modes are calculated.

Applying the above results to barium titanate vibrators, we find that only the radial modes can be excited with full electrodes and the compound modes with split or partial electrodes, while the tangential modes can never be excited. Such an excitation of compound modes provides a convenient means for measuring Poisson's ratio of the material. Equivalent series inductances of the radial modes are calculated.

Vibrations of thick circular plates are also considered. It is shown that vibrations of certain compound modes which are practically equivoluminal can be approximated by the rotation only. Comparisons between these solutions and the above results are made to reveal that the accuracy of approximation is satisfactory.

INTRODUCTION

ONTOUR vibrations, which involve no transverse displacements, of isotropic elastic plates were exactly solved by Love • for the case of an infinitely thin circular plate with free edges. He performed, however, neither investigations of characters of modes nor numerical estimates of natural frequencies and displace-

• A. E. H. Love, Mathematical Theory of œlasticity (Cambridge University Press, New York, 1927), fourth edition, p. 497 if.

merits, because it seemed to him "not to be of sufficient physical importance." Within the last few years, however, these •dbrations have become of practical importance because of the discovery of piezoelectric vibrators. Petr•ilka's extensive study on circular z-cut quartz and tourmaline plates supported Love's theory quite well? .a Although he calculated natural frequencies

• V. Petr•.ilka, Ann. Physik 15, 881-902 (1932). 3 V. Petr•ilka, Ann. Physik 23, 156-168 (1935).

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