Random vibration of an elastically supported circular plate with an elastically restrained edge and an initial tension

Journal of Sound and Vibration (1978) 58(3), 443-454

RANDOM VIBRATION OF AN ELASTICALLY SUPPORTED CIRCULAR PLATE WITH AN ELASTICALLY RESTRAINED EDGE AND AN INITIAL TENSION S. CHONAN Department of gfechanical Engineering, Tohoku University, Sendal, Japan

(Received20 October 1977, and in revisedform 31 January 1978) This paper deals with the problem of the random vibration of an elastically supported circular plate with an elastically restrained edge and an initial radial tension. The random excitation is taken to be a distribution of spatially uncorrelated random forces. Analytical expressions for the mean square displacement and moment are derived by using the orthogonality property of the mode functions. Numerical results are then given for a range of parameters. The main result is that the mean square displacement of the plate, in general, takes on a maximum value at the center of the plate, while the mean square moment takes on a maximum value at the center of the plate or along the edge of the plate depending upon the relative values of the elastic edge constraint, the initial radial tension and the stiffness of the foundation. 1. INTRODUCTION Circular plates are commonly incorporated in the design of aerospace, marine and petrochemical structures. On that account, it is of technological importance to study the dynamic behaviours of circular plates with various types of boundary conditions, and many papers have been published during past years. The study of vibrations of circular plates with elastically restrained edges is of interest since ideal simple supports or clamps are not easy to obtain in practice. As a problem of this kind, Weiner [I] studied the forced axisymmetric motions of a circular plate. The boundary of the plate was assumed to be supported in a manner that prevents transverse edge motion and provides a restoring edge moment proportional to edge rotation. The displacement curves of the plate were obtained for a concentrated central load and for a uniform pressure loading. Schlack, Kessel and Dong [2] investigated the non-axisymmetric response of an elastically supported circular plate with an initial tension. The plate was considered to be supported by a Winkler elastic foundation and elastically constrained against rotation at the outer edge. General natural frequency equations were presented with numerical values included for the first four modes of vibration. The general solution for the displacement of the plate subjected to an arbitrary surface load was given in an integral form, but no numerical result was presented for the forced vibration. Laura, Pombo and Luisoni [3] dealt with an approximate analysis of the steady state axisymmetric vibration of a circular plate elastically restrained against rotation along the edge and subjected to a sinusoidal excitation (a) uniformly and (b) parabolically distributed in space. The solution for the displacement and the bending moment were expressed in terms of simple polynomials which identically satisfy the boundary conditions. The Galerkin method was used to evaluate the expansion coefficients. The author [4] studied the free non-axisymmetrie vibrations of elastically connected circular plate systems with elastically i'estrained edges and initial radial tensions. The plate system consisting of three and two identical plates with identical 443 0022-460X178/0608-0443 802.00/0 9 1978 Academic Press Inc. (London) Limited

444

s. CHONAN

boundary conditions and a uniform radial tension were treated in detail. The first nine eigenvalues and the corresponding natural frequencies of the systems were obtained. For the circular plate subjected to a general surface load, it appears that there is no analysis which has dealt with an exact, analytical calculation of the bending moment. This is due to the fact that the bending moment must be obtained from the second derivatives of the mode functions and the mode functions are not known with sufficient accuracy to give satisfactory results for the moment. In view of these circumstances, the present paper treats the problem of the random vibration of an elastically supported circular plate with an elastically restrained edge and an initial radial tension. The random excitation is taken to be a distribution of spatially uncorrelated random forces. Analytical expressions for the mean square displacement and moment are derived by using the orthogonality property of the mode functions. Numerical results are then given for a range of parameters. Attention is directed to the effects of the elastic edge constraint, the radial tension and the stiffness of the foundation on the statistical responses of the plate. Referring to this problem, Eringen [5] studied the vibration of a clamped circular plate subjected to a random load. The mean square displacement at the center of the plate was calculated with the use of the fundamental mode, one-term approximation. 2. GOVERNING EQUATIONS An elastic circular plate of radius a is shown in Figure 1. The plate is supported by an elastic medium of effective stiffness k and is subjected to an initial radial tension T and an applied normal force p(r,O, t). In addition, the edge of the plate is elastically constrained against rotation, as represented by the torsional springs of stiffness k,. The governing equation of motion for the displacement w(r,O,t) is D V + w - T V 2 w + k w + e OwlOt + ph 0 2 wlOt ~ = p(r, O, t),

(1)

where W = O*lOr 2 + (l/r) O/at + (l/r 2) O'/O0".

Here, D, h and p are the flexural rigidity, thickness and mass density ofthe plate; c and t are the external damping coefficient and time, respectively.

-

-

),

1 ///////////'~,'//I////I/

.

Figure I. Geometry of problem and co-ordinates.

RANDOM VIBRATION OF h CIRCULAR PLATE

445

The boundary conditions at the edge r = a are w = 0,

[ a"

va

(2)

v a2 1

aw

- D [0"~r2+ r ~ + ~ 0"~] w = k.--~r ,

(3)

where v is the Poisson's ratio. The homogeneous solution of equation (I) satisfying the boundary conditions (2) and (3) is [2, 4]

w = ~ ~. [A,".sinnO+ B,".cosnOlR...(r)e'%.', ,"-0 n-O

where

R.,.Cr)=J.

( a/ +-'-(r) ct,".

~ 1 .

fl""a "

(4)

(5)

Here, m and n are the number of nodal circles and diameters, respectively; A,". and B,.. are the constants; J. and I. are Bessel and modified Bessel functions of order n of the first kind; ct,". and tim. are the roots of the characteristic equation

J.+a(~,".) ~n,n

I.+x(fl,..)

~L + fl~. 1-v-(k.a/D)

J . ( ~ . , . ) + P,"" I.~,".)

2 fl,.. - ctL = Ta'ID.

(6)

Furthermore, the natural circular frequencies co,.. are determined from

(pha41D) to~. - (ica41D) to,.. = ~ . [1~. + ka+l D _= ?,,,..

(7)

3. SOLUTION FOR RANDOM EXCITATION The solution for the forced vibration may be written in the form M

N

w = ~. ~. [A,".(t) sin nO + B,".(t) cos nO] R,".(r).

(8)

m-O n.O

Substituting equation (8) into equation (I) and arranging terms appropriately give

tt-- aph

O, +p/-~a47"~A,"" sinnO

m-O n - O

+ [ a--~--'I" ph at +p/-~a3,""B"" cosn0 Rm.(r)=-~-~zp(r,O,t ).

(91

Multiplying both sides of equation (9) by rR~q(r)sinqOdrdO or rR~+(r)cosqOdrdO and integrating from r - - 0 ~ a and 0 = 0 ~ 2~, one obtains, by virtue of the orthogonality property of Rm.(r) [2], 2= a

02A,". c OAr.. D 1 f f p(r,O,t)rR...(r)sinnOdrdO, Ot 2 q-ph Ot k'p~za4?""A""=phN,".. 0

0

,f/ 2It

~-~ t ~

~

Fp~za4~,".B,".=phN.~. c

(I0)

tl

p(r,O,t)rR,".(r)cosnOdrdO,

(I1)

446

$.CHONAN

where 2Z

a

N... = f ~ rR~(r)sin2nOdrdO, 0

N--e = f f rR~.(r) c~

0

0

0

from which one has a

9

N,,~ = N,..~ = rr f rR~.(r) dr

for

n # 0,

(12)

for

n = O.

(13)

0

Nm., = 0

N,..~ = 2n f rR~.(r) dr

and

0

Here,

f rR~(r) dr = a2 [

/

f 2n 2a 2

]

J.(ct,,~)

+ a22J"2(ct'~)l~(fl~I~(fl,.,) -

l.(fl,,~) + l.+1(fl,..) l.+x(fl~)

The solutions of equations (10) and (11)may be written as [6] cO 2X

,4,..(t)

a

1

=

phN.., f f f h,..(t)p(r,O,t-OrR..(r)sinnOdrdOdz,

-

-

-too 2t

a

1

B..(t) = phN,..e f f f h..~(t)p(r, O, t - z) rR,u(r) cos nOdr dO dz, -~

(14)

0

0

(15)

0

where h..~(t) is the response of the single-degree-of-freedom system represented by each of equations (10) and (I 1) to a unit impulse and is given by

'f H,..(to) e'=t dto.

h~(t) = ~

(16)

Here, H,..(to) is the Fourier transform of hm.(t) represented by

n,..(to) =

h,,~(t) e-l~' dt =

r.. - to2 + -~ to .

(17)

Now it is assumed that the ensemble averages ofp(r,O,t) are given by

(p(r, O, t)> = 0, (.p(r. Oa,t + z)p(r2, 02, t)) = (rz r2) -s/2 di(rl - r2) 6(01 - 02) R(T),

(18)

where 6( ) is the Dirac delta function. While this applied force is not necessarily a typical form for real loadings it has the feature of ease in performing the ensuing integrations. Later a special form will be considered for R(0.

RANDOM VIBRATIONOF A CIRCULARPLATE

447

With the use of equations (12) through (18) and the ergodic assumption, one obtains [6] oo

co 2 1 2 x a

a

1

(A"(t)A'a(t)):p 2hz Nm.Nt.. f f f f f fh,..(r,)ht,(z2)(p(r,,ODt-z,)p(r2,0z, t-x2)) -~-oa

0

0

0

0

• rl r2 Rm.(rDRt~(r2) sin not sinqO2drt dr2 dO1 dO2dTl d~2 - r h~ N,.. '

h,..(~) ht,(~2) R ( ~ - T D d q d~,

p2h2N,..s 2n _

5..z6,,~

.0 2 h 2 N,..,

1 2n _

_

h,..(~Del'zdq

htq(r2)e-lO'2dr2 A(~)dc0

oo

_- p26,.z6.r 1 ~f H.~(-a 0 Hz,(co) A(co) dco. h z N...~ 2-'~ where A(to) is the Fourier transform of

R(z) and

(I 9)

is given by

A(co) = ( R ( x ) e -'~" d~.

(20)

6,,, and 6,q are Kronecker's delta functions. Similarly, one has 6=I 6.r 1 (B,~,(t) B,q(/)> = pC h' Nm.c 2n d H,..(-co) H,q(to) A(co) do,

(21)


(22)

From (A,..(t)Alq(t)>,(B,.,(t)Bl~(t)) and (Am.(t)Btq(t)).the various mean square quantities characterizing the response of the plate can be computed. For example, the mean square displacement of the plate is {A,,~(t) sin nO+ a_(t) cos nO}R,..(r)


m-0 n-0 N

= Y~ X {
+


(23)

m-0n-0

It is evident from equations (12), (13), (14), (19) and (21) that =

for

n # 0,

(24)


for

n = 0.

(25)

Hence, one obtains from equation (23) M

U

(w2(r, 0, t)> =

(26) mmO,q-O

448

s. CHONAN

It is evident from equation (26) that for an applied force represented by equation (18) the mean square displacement of the plate becomes independent of the circumferential co-ordinate 0. The calculation of the mean square moments follows the same line as that given in the above procedure. Here, one computes only (Mr2>. The bending moment Mr is given by [a ~w

vow

v a27~

Mr=-Dt--~-rZ+r~r +~-i--~).

(27)

and one has = D'

+

+

0-~/{A...(t) sin nO + B,..(t) cos nO} R..(r)

=Dl~ ~ (B~.(t))[(~+v a nZv\., ..o._o

|]2

(28)

,o,

where n2v\

[~,..\2

-~ r Or +<,-,,)H

[

r~

J.(ct~..)

zi.

etrr"("-i)(_~):)(z) =.. ( ~)] .'. ,<,..

-(1-v)I.--~.)[[

~-

+-,.+, at

<<-

+\a ] J"

R(T) in equation (18) is required to be such that the total power of the input is finite. A form having this feature is R(z) = Pwo e-I'~~ (29) Substituting equation (29) into equation (20) gives A(oa) = 2Pwlol(o~2 + wlo).

(30)

Substituting equation (30) into equation (21) and calculating the integral with the use of the Jordan lemma and the residue theorem, one finally obtains, omitting the details, Pa 4 Do(t2o + ll) (B~.(t)> = (ph)./2 D3/Z (N,..c/a2)ll?,..(D~ + i#7o + ~,..)'

(3 I)

where q = (a4/phD) 112c,

Do = (pha41D) II2 o~o.

Here, ~/is the non-dimensional damping coefficient, and Do the non-dimensional correlation parameter of the applied load. By substituting equation (31) into equations (26) and (28), the mean square displacement and moment of the plate are obtained in the following non-dimensional forms: (w2>* = - - ~ m=O n~O

(N,..c/a 2) #l?m.(D~ + 1112o+ ~,..) . . . . . . .

RANDOM VIBRATION OF A CIRCULAR PLATE

(N~.d a2) tlY,.~(O~ +

<3Ir2> = ra.O

~If2o + 7~,.)

449

~-~]R,~,(r*)

r* Or*

, (33)

n-O

where (w2). =

(Ph)tl2 D312 pa + ,

2 (Ph)ttz

(Mr2>

~

* =

#.i

,

t.,D,I,

/, ~__. a

Here,
r/= 0.001,

v = 0.3.

The results were obtained by taking M = 9 and N = 5 and including the first 60 terms o f the series solutions. For each value o f m and n the eigenvalues a~, and fl,,, are determined from equations (6). Some variations of the mean square displacement and moment with increasing I.I

I

I

I

1,0 O - - - O - - O = . . ~ O - .o ~ -

"

lg

I

t

I

O--

O--

O--O-~

I

I

e,.---e

0.9

-

AA .J~ It

~t

V IV

0.8

0.7

A I

0.6,

I

I

I

I

9

o M

mean square Figure 2. Convergence o f the series solutions. - - - - , Mean square displacement; ~ , m o m e n t . A, N = 0; O, N = 1 ; El, N = 2; It, N = 5; Ta2ID = O, ka+/D = O, k,a/D = 0, v = 0-3,12o = 1, ~7= 0"001, r* =0"5.

number o f terms are shown in Figure 2 as functions of M and N. In the figure, (w2)*o and (Mr2)'~o imply the results obtained by taking the first 60 terms of the solutions. It is evident that while the displacement can be obtained with the use of only the fundamental mode, one-term model, higher mode functions have to be considered for accurate determination of the moment. For various values of the edge constraint parameter (v + k,a[D), radial tension parameter Ta2/D and foundation parameter ka+/D, the mean square displacement (MSD) and the mean square moment (MSM) were evaluated from equations (32) and (33) and are plotted in Figures 3 through I0.

450

s. CHONAN 2 I I

J

I

I

_

I

j

I

I

t

I

z

v

0

0"5

Figure 3. Variation o[ profile of m e a n square displacement with increasing edge rotational constraint. , Multimode solution; . . . . , fundamental m o d e solution. Yu2/D = O, ka4JD = 0,/20 ~ 1, r/= 0"00l.

i

I0

i

i

,.

.

I

,,

OD

i

i

i

J

,, 9

0

|

S

0.5 111,

Figure 4. Variation of profile o[ m e a n square m o m e n t with increasing edge rotational constraint. Multimode solution; . . . . , fundamental m o d e solution. Tu21D -- O, ka41D = 0, v = 0.3,/20 = I, r/-- 0.00].

z- 4

0.2--

I 0

I

T

I

! 0.5 /,0

Figure 5.Variation of profile of mean square displacement with increasing initial radial tension, ka41D = O, v + k,a/D

= ~,/20

= l , Vl =

0"1301.

RANDOM VIBRATION OF A CIRCULAR

PLATE

451

Some variations o f the profiles of MSD and MSM with increasing edge rotational constraint are shown in Figures 3 and 4. The circular plate with a simply supported edge is realized by taking k, = O. (v + k,a[D) = oo implies a plate with a clamped edge. It is evident that the response of the plate is quite sensitive to the edge constraint parameter. MSD is maximum at the center of the plate regardless of the value of (v + k;a/D), while the position o f the maximum MSM is transferred from the center to the edge of the plate as (v + k,a/D) 20

i

~A

I

I

I

I

I

I

I

I

I

[ 0.5

I

~

I

I

Ta~/D=-4

V

-2

I

I

[

0

T*

Figure 6. Variation of profile of mean square moment with increasing initial radial tension, ka4/D = 0, k,a/D=Qo, v = 0"3, Oo= 1, ~/= 0'001. 0,15

i

i

i

!

I

i

i

I

I

i

0.10

50 0.05

--

75

0.5 f*

Figure 7. Variation of profile of mean square displacement with increasing foundation stiffness.Ta2/D = 0, v + k,a]D = co, -(2o= 1, q = 0-001. increases. For a comparison, the results from the fundamental mode, one-term solution were obtained and are plotted with dashed lines. The effect of initial radial tension on the profiles of MSD and MSM is shown in Figures 5 and 6. MSD increases monotonously at every position of the plate as the radial compressive force increases, while an increasing tensile force has an opposite effect. It is noted that MSM in the vicinity of r* = 0.6 is not sensitive to the variation of the initial radial force as compared with that of other positions. Variations o f the profiles of MSD and MSM with increasing stiffness of the foundation are presented in Figures 7 and 8. MSD decreases as the stiffness of the foundation increases.

s . CIiONAN

452

N

.,o~/o=o V

5

-

25

75/

[ I I I

0

I 11

]

0.5 r9

Figure 8. Variation of profile of mean square moment with increasing foundation stiffness. Ta'/D = O, k,a/D = = , v = 0"3, I2o = 1, r / = 0"001. It is also noted that M S M in the vicinity o f r* = 0.6 is not sensitive to the variation o f the foundation stiffness. In general, it can be said from Figures 3 through 8 that the mean square displacement takes o n a maximum value at the center o f the plate regardless o f the magnitudes o f the t

~

i

[

i

i

L

I~ 4- k t e l O "0"3

I0_1ii

~'~ k,o/D ,O.3 I

v

I0-i

iO-Z-4(b)I ~ ; I 0

I

i

4

-4

0

4

Ta ZlD

Figure 9. Variation of mean square displacement at the center of the plate ka'/D = 0 ; (b) ka*/D = 5 0 ; (e) ka'/D = 100.

vs.

initial radial tension. (a)

453

RANDOM VIBRATION OF A CIRCULAR PLATE

elastic edge constraint, the initial radial tension and the stiffness of the foundation, while the mean square moment takes on a maximum value at the center of the plate or along the edge of the plate depending upon the relative values of these factors. For several values of (v + k,a/D) and ka4/D, the mean square displacement at the center of the plate was evaluated and is plotted as a function of Ta2/D in Figures 9(a) through (c). Clearly from the figures, MSD decreases with an increase of any one of (v + k,a[D), Ta2/D and ka*/D. Furthermore, it is evident that an increase ofka4/D decreases remarkably the effects of (v + k,a/D) and TaZ[D on MSD. v +kt O/D,O.3

I0 z

eJ

~

~o

V

\%.

-

I

i'~,,,,,,.,

-4 I0

~...2

'

~3-. .

-~

'

I

0 i

l

l

. . . . . . . .

"

, ,"

.

4 8

,

.._ t

~

~

i

,

- . . . . . . . . . . . . . .

"

I0 v 4-J,a/O

~ ,o~

V

i0_ I -4

( b T)

r

r

T 0

~

T

I0-'

r 4

-4

0

To 2/D

Figure 10. Variation o f m e a n s q u a r e m o m e n t vs. initial radial tension. (a) ka4/D = 0; (b) ka41D = 50; (c) ,r*=0; .... ,r*=l.

ka41D=lOO, v=O.3, Oo=l, Fl=O.O01.

With (v + k, alD) and ka4/D taken as parameters, the mean square moments in the plate are presented in Figures 10(a) through (c). The solid lines are the results at the center of the plate, and the dashed lines are the results along the edge of the plate. For v + k,a/D = 0.3 the plate is simply supported and MSM along the edge of the plate is zero regardless of the values of Ta2/D and ka4/D. It is observed that MSM at the center of the plate is greater than that along the edge of the plate for v + k,a/D = 0"3, 1 and 3 and for all the values of Ta2/D and ka4/D under consideration. On the other hand, MSM along the edge ofthe plate is greater than that at the center ofthe plate for v + k,a/D = oo. It is noted that the position of the maximum MSM for v + k,a/D = 10 is under the control of the relative values of Ta2/D and ka4/D. In this case, MSM at the center of the plate is greater than that along the edge of the

454

s. CHONAN

plate for ka4[D = 0 and Ta~[D < 2 and for ka4[D = 50 and Ta2[D < - 2 . For other values o f ka4[D and Ta2/D, M S M takes the maximum along the edge o f the plate, 5. CONCLUSIONS A theory has been developed for the random vibration o f an elastically supported circular plate with an elastically restrained edge and an initial radial tension. The main result is that the mean square displacement of the plate, in general, takes on a maximum value at the center of the plate, while the mean square moment takes on a maximum value at the center of the plate or along the edge of the plate depending upon the relative values of the elastic edge constraint, the initial radial tension and the stiffness o f the foundation. ACKNOWLEDGMENT The author wishes to acknowledge the consistent guidance and encouragement of Professor H. Saito of T o h o k u University. REFERENCES 1. R. S. Wr~r~rR 1965 Transactions of the American Society of t~Iechculical Engineers, Series E32, 893-898. Forced axisymmetric motions of circular elastic plates. 2. A. L. SCHLACKJR., P. G. Kr.SSELand W. N. Do,~o 1972 American Institute of Aeronautics attd Astronautics Journal 10, 733-738. Dynamic response of elastically supported circular plates to a general surface load. 3. P. A. A. LAURA, J. L. Po.),mo and L. E. LuISONX1976 Journal of Soundand Vibration45, 225-235. Forced vibrations of a circular plate elastically restrained against rotation. 4. S. CHONAN 1976 Journal of Sound and Vibration 49, 129-136. The free vibrations of elastically connected circular plate systems with elastically restrained edges and radial tensions. 5. A. C. ERINGEN 1957 Journal of Applied Mechanics 24, 46-52. Response of beams and plates to random loads. 6. S. H. CRA~DALL and W. D. MARK 1963 Random Vibration in 3[echanical Systems. New York, London: Academic Press.