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THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 34, NUMBER 3 MARCH, 1962

Vibration of Circular Plates

Tm;• WAtt

Southwest esearchnstitute, San Antonio, Texas

(Received uly 24, 1961)

The vibration of circularplateswith a large nitial tension or compression)s studied or the caseof

(a) simply supported nd (b) clampededges.The basicequationused s that of the Poisson-Kirchhoff

theory. Numerical resultsare given.

1. INTRODUCTION

HE ibrationf ircularlatesccordingo he

oisson-Kirchhoff heory has been investigated

by several writers. The case of free edgeswas studied

by Kirchhoff) Lamb,2 and Rayleigh, and the caseof

clampededges y Rayleigh?Timoshenkohas used he

energy method for solving the case of the plate with

clampededges.

The caseof the circularplate with simply supported

edgeshas, however,apparently receivedscant mention

in the literature. The reason for this seems to be that

suchedgeconditionsare practically difficult of realiza-

tion. It is well known, on the other hand, that the

theoretical "clamped edge" conditionsare almost im-

possibleo simulate n the laboratory. n general,plates

behaveas f they had boundary conditions ntermediate

between he theoretical"simplesupport"and "clamped

edge" conditions.For this reasonalone, if for no other,

it is desirable o have the solution or the caseof simply

supported ircularplates.

It is often necessary, n applications, o take some

accountof the effectof in-plane ensions r compressions

that may develop in the plate. A theoretically exact

treatment of this problem requires the solution of a

nonlinear problem and necessitatesapproximate or

numericalprocedures.

The caseof a circular plate taking into account he

effectof tensionsn its middleplane has alsobeensolved

by Timoshenko by an energyprocedure.Although his

procedure epresentsan attempt to approximate the

solution or large amplitude vibrations, the method of

solution essentially restricts it to relatively small

deflections.

If, however, it is assumed that a uniform radial

tension or compression)s initially applied o the plate,

and that this tension s large enoughso that its fluctua-

tions rom its initial value duringvibrationarenegligible,

then the problem is a linear one and may readily be

solved. Admittedly the solution is invalid for large

amplitudesof vibration.

In this paper theoretically exact solutionsare given

for the caseof a circularplate with a large nitial tension

• G. R. Kirchhoff, J. Math. (Crelle) 40, (1850).

2 H. Lamb, ?roc. Roy. Soc. (London) 98, 205 (1921).

8 Lord Rayleigh, Theory of Sound (Dover Publications, New

York, 1945), Vol. I, p. 359.

4 S. Timoshenko, Vibration Problems n Engineering, D. Van

Nostrand[Company, rinceton,New Jersey,1954),3rd ed.

or compressionf varying magnitudeand (a) simply

supported t the circumferencer (b) clampedat the

circumference.

Somenumericalvaluesof the roots of the frequency

equationsare given in the tables.

2. NOTATION

a radiusof circularplate (in.)

D - Eh•/12 (1 -- v2) flexuralrigidity of plate

0b

E modulusof elasticity of plate material

(psi)

J,,, Y•, I•, K• Bessel unctionsof order n

p natural frequencyof plate (sec•)

R function of r

r,O cylindrical coordinates

T membrane tension (or compression)

(lb/in.)

lateral deflection f plate (in.)

function of 0

Poisson's ratio

mass per unit area of the plate (lb

sec in. a)

4 nondimensional arameter

t time (sec)

V4 = (Vs)s=biharmonicoperator...

w

thickness f plate (in.)

3. EQUATION OF MOTION

If a large nitial radial tensionTr is applied o a plate

without a hole, it is readily shownby elementaryelas-

ticity theory that the tangential tension To= Tr= T.

The differentialequationof motion of such a circular

plate may thereforebe written in the form

T p O•'w

W(w)--W(w)q .... 0, (1)

D D

in which w is the lateral deflection, T is the uniform

radial tension,D= Eha/12(1 •) is the flexural igidity

of the plate, and p the densityper unit area of the plate.

Let

w=R(r)O(O) sinpt, (2)

275

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276 THEIN WAH

TABLE . Simplysupported late.

rib=0 nb=l

n b= 2

1.50

1.00

0.50

0.25

-0.25

-0.50

- 1.00

2.31 3.71 8.55 3.71 4.71 17.47 5.07

5.46 6.18 33.75 6.93 7.51 52.05 8.30

8.61 9.09 78.28 10.17 10.57 107.54 11.60

2.30 3.40 7.81 3.70 4.47 16.55 5.07

5.46 6.01 32.79 6.93 7.37 51.07 8.30

8.61 8.97 77.27 10.17 10.47 106.52 11.60

2.28 3.07 6.99 3.69 4.22 15.57 5.06

5.46 5.83 31.80 6.92 7.22 49.94 8.30

8.61 8.85 76.24 10.17 10.37 105.50 11.60

2.26 2.68 6.05 3.68 3.96 14.55 5.06

3.45 5.64 30.78 6.92 7.07 48.92 8.30

8.61 8.73 75.21 10.17 10.27 104.49 11.60

2.24 2.46 5.52 3.67 3.81 13.98 5.06

5.45 5.55 30.25 6.92 7.00 48.41 8.30

8.61 8.67 74.69 10.17 10.22 103.94 11.60

2.22 2.22 4.94 3.67 3.67 13.47 5.06

5.45 5.45 29.72 6.92 6.92 47.89 8.30

8.61 8.61 74.15 10.17 10.17 103.43 11.60

2.20 1.94 4.27 3.66 3.51 12.86 5.06

5.45 5.35 29.17 6.92 6.84 47.36 8.30

8.61 8.55 73.62 10.17 10.12 102.90 11.60

2.16 1.60 3.46 3.65 3.35 12.23 5.06

5.45 5.25 28.62 6.91 6.76 46.78 8.30

8.61 8.49 73.09 10.17 10.07 102.37 11.60

2.05 0 0 3.64 3.01 10.95 5.06

5.45 5.05 27.49 6.91 6.60 45.60 8.30

8.61 8.36 72.00 10.17 9.96 101.30 11.60

Jn+l o0 In+l (lg) lg2"lt-ot-

Roots f equation' ---+/5 .... ; v=0.3; /52--M=4.24)

Jn (ot) In (•) 1-- v

5.84

8.79

11.95

5.66

8.67

11.87

5.46

8.55

11.78

5.26

8.43

11.69

5.16

8.36

11.64

5.06

8.30

11.60

4.96

8.24

11.56

4.85

8.17

11.51

29.55

72.97

138.62

28.62

71.97

137.67

27.62

70.96

136.65

26.64

69.93

135.60

26.12

69.39

135.02

25.60

68.89

134.56

25.07

68.36

134.16

24.53

67.83

133.52

4.63 23.4I

8.04 67.76

11.41 132.36

=nodal circles.

=nodal diameters.

where R is a function of r alone and O a function of

0 alone.

Substituting2) into (1), onemaywrite he resulting

equation in the form

where

(V•'q-M/a) (V•'--•'/a•')RO O,

],

= 1 . 1

2'DE\ T ' /

.ai( ]

1+.. +1 .

2D \ T •, /

(3)

(4)

Equation 3) showshat the completeolutionmaybe

obtained y adding ogether,with appropriaterbitrary

constants, he solutions f the two equations

(V2+a2/a2)RO=O,

(V2_i•./a•.)RO=O. (5)

* N. W. McLachlan, BesselFunctionsfor Engineers, Oxford

UniversityPress,London,1955),2nd ed., p. 122.

On letting

O= COSr•0 - 'yn) (6)

Eqs. (5) yield wo equationsor the determinationf R'

dr' dr • --n•'•R=O'

•.

W'RdR

-t- ..... R=0.

d?' r dr \a •'

(7)

The solution f the first equationof (7) is

= :i d.(.r/a)+ (.r/a)

and that of the second quationof (7) is

R= B,J,, (•r/ a) q-D•K,, (•r/ a)

in which J• and Yn denote Bessel functions of order

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VIBRATION OF CIRCULAR PLATES 277

of the first and secondkinds, respectively, N and K,

are the modified Bessel functions of the first and second

kinds, and A, B, C, D are arbitrary constants. The

notation used s that of reference .)

In the caseof a solidplate, Y. and K. are nadmissible

because f the singularityat r=0 and the generalsolu-

tion of (7) may be representedn the form

R= A nJ,• aria)-+-B.I. (15r/a). (8)

The generalsolutionof (1) may now be written as

w_-

X (cosn0+X•sinn0)sinpt, (9)

whereA., B,, X, are arbitrary constants nd

Ta2[+_4p2oD•],

=•-•[(1 --1

2 /

• TaX p•D\],

/5ø---ø = Ta•/D.

4. PARTICULAR CASES

4.1. Simply SupportedCircular Plate

Substituting 9) into the boundaryconditions

w--O,

at r = a,

O•w10w0•w

-[-v •+-- • =0,

Or*' ; Or r2O0

TABLE I. Clamped plate.

(lO)

(11)

q0 sb a O

he=2

1.50

1.00

0.50

0.25

-0.25

-0.50

- 1.00

0 2.84 6.12

1 6.16 8.21

2 9.39 10.84

0 2.89 5.51

1 6.19 7.78

2 9.40 10.51

0 2.95 4.84

1 6.23 7.31

2 9.40 10.15

0 3.05 4.08

1 6.26 6.82

2 9.42 9.80

0 3.11 3.66

1 6.28 6.57

2 9.43 9.61

0 3.20 3.20

1 6.31 6.31

2 9.44 9.44

0 3.30 2.70

1 6.33 6.03

2 9.45 9.25

0 3.44 2.12

1 6.36 5.75

2 9.46 9.06

0 3.83 0

1 6.42 5.15

2 9.48 8.68

17.37 4.39 6.97 30.61

50.60 7.66 9.38 71.87

101.81 10.71 12.00 128.52

15.92 4.43 6.45 28.59

48.17 7.69 9.01 69.27

98.77 10.71 11.69 125.20

14.30 4.48 5.89 26.41

45.52 7.71 8.61 66.38

95.44 10.72 11.38 121.99

12.44 4.54 5.29 24.00

42.75 7.74 8.20 63.47

92.33 10.74 11.07 118.89

11.39 4.57 4.99 22.81

41.29 7.76 7.99 62.02

90.59 10.75 10.92 117.39

10.21 4.61 4.61 21.25

(10.21) (21.22)

39.77 7.77 7.77 60.37

(39.78)

89.09 10.76 10.76 115.78

(88.9)

8.91 4.64 4.23 19.61

38.19 7.79 7.55 58.81

87.45 10.77 10.60 114.16

7.28 4.69 3.83 17.94

36.55 7.81 7.33 57.21

85.76 10.78 10.43 112.48

0 4.84 2.96 14.31

33.03 7.85 6.85 53.79

82.28 10.79 10.09 108.82

Rootsof equation: a +O =0; 02--a2=14.68•

5.77

9.14

12.33

5.81

9.15

12.34

5.85

9.18

12.35

5.88

9.19

12.36

5.90

9.20

12.37

5.92

9.21

12.38

5.95

9.23

12.39

5.97

9.25

12.40

6.03

9.27

12.41

7.92

10.63

13.47

7.47

10.28

13.20

6.99

9.95

12.93

6.47

9.58

12.65

6.22

9.39

12.52

5.92

9.21

12.38

5.64

9.03

12.24

5.32

8.84

12.10

4.66

8.44

11.80

45.67

97.11

166.06

43.39

94.09

162.93

40.91

91.31

159.70

38.07

88.04

156.39

36.72

86.39

154.84

35.05

(34.84)

84.82

153.26

33.53

83.34

151.65

31.75

81.81

150.04

28.08

78.25

146.48

Note:Figures n parentheses re thosegiven in reference .

s =nodal circles.

n ==nodal diameters.

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VIBRATION OF CIRCULAR PLATES 279

o5

Fro. 3.

pression nd, n particular, he valueof T corresponding

to •=- 1 representshe buckling oad of a radially

compressed late.

From Eqs. (10) it is readily deduced hat

p= (D/p) •(at•/a2). (16)

In the particularcaseof T=0, a=/• and (16) assumes

the form

p= (D/p)«(a2/aS). (17)

Using he first equation f (12), Eq. (9) may be written

X (cosn0+Xn innO)(sinpt+C,,ospt), (18)

in whichA •, C•, and X• are to be determinedrom the

initial conditions.t is to be noted that except or n-0,

one obtains two sets of modal functions

[ (ar)J,•(a)(•)]

- - In COSrt0•

J•-• In(t•)

and a similarone with sinn0 n placeof cosn0.

L•

o

-0.5

-L

FIG.

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280 THEIN WAH

4.2. Clamped Plate

The boundary conditions or this caseare

w•O•

at r=a. (19)

Ow/Or=O,

Substitution f (9) into (19) yields

A ,•[-nJ,•a) -- J,•+xa)_-]+,•[-nI,• •) q- I,•+• (•)-I = O.

(20)

From (20) one inds he frequency quation o be

• q-g . --0. (21)

Introducing the parameter

T/ T*= •k,

where T*=14.68D/a 2, one obtains from the third

equationof (10)

•2--a•= 14.68½. (22)

The natural frequencies f the plate may now be deter-

mined rom Eqs. (21) and (22) for variousvaluesof the

parameter •, negative values of • representing adial

compression.he value of T correspondingo q•= -- 1 is

the buckling oadof a radially compressedlate clamped

at the boundaries.

Equations 16), (17), and (18) apply to this case lso

with the proviso that a and • are determined from

Eqs. (21)and (22).

5. COMPARISON WITH MEMBRANES

It is readily shown hat the natural frequencies f a

circularmembrane tretchedby a tensionT are givenby

p= (u/a)(T/p) ,•, (23)

where the t• are the roots of the equation

Jn (/g)=0. (24)

The rootsof Eq. (24) have been airly well tabulated,

and, in any event, can be determinedwith relative ease.

It is thereforepertinent to inquire under what circum-

stances he natural frequenciesof a stretchedplate may

be approximated by those of a geometrically similar

membrane.

Using he last of Eqs. (10), Eq. (16) may be written

or

(25)

Comparing (23) and (25), it is seen hat the plate

frequencyapproacheshat of the membraneas a-•/•

and if

« (a•D/ aST)<<. (26)

Substitutingu for a and using (14) and (22) the

inequality (26) may be written

/•<<(8.4½)• or a simplysupported late,

<<(29.36•)• for a clamped late.

(27)

20--

o

5o

-•=/

FIO. 5.

CLAIdPED PLATE

nodal

nodal c•rcle$

T/r ~

:•= 14.G•9

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VIBRATION OF CIRCULAR PLATES 281

L•

IO

•.5

Fro. 6.

For given • and 4, the inequalities 27) may be used

to establish he accuracy of using the membrane fre-

quencyas an approximation o the plate frequency. t

is noteworthy hat while a given tension T may permit

suchan approximation o one of the lower frequencies

it may completelyail to do so or the higher requencies.

A larger value of the tensinn t r•quired far

degreeof approximation n the higher frequencies.

As an example, f it is required o determine he value

of the tension T that would permit the fundamental

frequency of a simply supported plate to be approxi-

mated, within about 5%, by that of the membrane,

one may use he first of (27) and set

u/(8.4•)«_< 0.05.

Letting tm = 2.405, this gives

4>275.4,

and since T= (4.2D/a2)½k,he tensionmay be readily

calculated f the propertiesof the plate are known.

The above results actually answer the question

"When can the behaviorof a plate be approximatedby

that of a membrane?" In fact, when the inequalities

(27) are satisfied within the desired imits) the modal

functionsof the membranemay be usedas approxima-

tions to those of the plate. However, these results are

not very useful n determining he frequencies f a plate

with rnnderaIely large val• of •, the values of • re-

quired for the validity of the above results being, in

general, too high.

A more useful result for the determination of the

natural frequencies f a stretchedplate emergeswhen

one comparesTables I and II with a table of uns (in

reference4, for example). It is found that the values

of ansare very close o the corresponding alues of t•,,

even or relatively smallvaluesof the tensionparameter

q•and obviouslyapproach he values of t•,, asymptoti-

cally (see Figs. 1 through 6). Thus, a fairly good

approximation o the frequencies f a plate with q•> 2

may be obtained by using the values of t•,, in place of

the corresponding,,, determining ns from Eqs. (15)

or (22) as the casemay be, and then using he general

formula (16) for the plate frequency.

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