c 1 * I 

NASA CONTRACTOR 

REPORT 

NASA CR-178964 

THE EFFECT OF A SMALL INITIAL CURVATURE ON THE 

FREE VIBRATION OF CLAMPED, RECTANGULAR PLATES 

By A. A. Adeniji-Fashola 

Resident Research Associate 

National Research Council 

Science and Engineering Directorate 

Systems Dynamics Laboratory 

and 

A. A. Oyediran 

Department of Mechanical Engineering 

Lagos State University 

Ojo, Lagos, Nigeria 

Interim Report 

October 1986 

Prepared for 

NASA-Marshall Space Flight Center 

Marshall Space Flight Center, Alabama 35812 

1bXTJ.IAL CURPATUBE ON TBE FREE VIEAiATJSOrJ CP 

(EASA-CE-178964) XbK EEEECX CE A SHALL Nel- 1E6€4 

Ibaticnal Academy of Sciences - bational Unclas 

CLAHPEC, 3ECPANGUlAB ELAIES Interim BeFort 

Besearch) 24 p CSCL 2OK G3/39 43798 
TECHNICAL REPORT STANDARD TITLE PAGE' 

1. REPORT NO. 12. GOVERNMNT ACCESSION NO. 13. RECIPIENT'S CATALOG NO. 

NASA CR- 178964 I 

1. TITLE AND SUBTITLE 

The Effect of a Small Initial Curvature on the Free 

Vibration of Clamped, Rectangular Plates 

A. A. Adeniii-Fashola* and A. A. Oyediran** 

7. AUTHOR(S) 

9. PERFORMING ORGANIZATION NAME AND ADDRESS 

5. REPORT DATE 

October 1986 

6. PERFORMlNG ORGANIZATION CODE 

8. PERFORMING ORGANIZATION REPOR r R 

10. WORK UNIT NO. 

George C. Marshall Space Flight Center 

Marshall Space Flight Center, Alabama 35812 

11 1. CONTRACT OR GRANT NO. 

NASW- 3458 

13. TYPE OF REPORS. & PERIOD COVEREC 

12. SPONSORING AGENCY NAME AND ADDRESS 

Contractor Report 

National Aeronautics and Space Administration 

Washington, D .C . 20546 1.1. SPONSORING AGENCY CODE 

15. SUPPLEMENTARY NOTES 

Prepared by Systems Dynamics Laboratory, Science and Engineering Directorate. 

*National Research Council. 

16, ABSTRACT 

An analytical method of obtaining the natural frequencies and mode shapes of 

clamped, rectangular plates having a small initial curvature is presented. 

the singular perturbation technique is used to reduce the fourth-order plate vibration 

problem to the simpler membrane problem with modified bcundary conditions that 

account for the bending effects. 

ratios varying between 0.1 and 1.0 and for the dimensionless normal prestress between 

0.1 and 1.0 have been presented for values of E, the normalized bending rigidity, 

rangkg between Os 0010 md 0.2500. 

Specifically, 

The eigenfrequencies for plates with inverse aspect 

It is established that a small initial curvature has no effect on the frequency of 

vibration of the plate. However, its effect is manifested in the eigenmodes. 

** Department of Mechanical Engineering, Lagos State University, Ojo, Lagos, Nigeria 

17. KEY WORDS 1 18. DlSTRlBUTlON STATEMENT 

Small initial curvature ; clamped, 

rectangular plates ; normal prestress ; 

singular perturbation technique ; 

eigenfrequencies and eigenmodes 

I Unclassif ied-Unlimited 

I 

19. SECURITY CLASSIF. (d thlm repart) 20. SECURITY CLASSIF. (or thim wee) 21. NO. OF PAGES 22. PRICE 

Unclassified Unclassified 23 NTI S 
TABLE OF CONTENTS 

Page 

SUMMARY 1 

I. INTRODUCTION ......................................................... 1 

11. PROBLEM FORMULATION ................................................ 2 

111. METHOD OF SOLUTION .................................................. 4 

IV. CONCLUSION ............................................................ 13 

REFERENCES.. ................................................................ 14 

..................................................................... 

iii 
CONTRACTOR REPORT 

THE EFFECT OF A SMALL INITIAL CURVATURE ON THE FREE VIBRATION 

OF CLAMPED, RECTANGULAR PLATES 1 

SUMMARY 

An analytical method of obtaining the natural frequencies and mode shapes of 

clamped , rectangular plates having a small initial curvature is presented. 

the perturbation technique is used to reduce the fourth-order plate vibration problem 

to the simpler membrane problem with modified boundary conditions that account for 

the bending effects. Excellent results are obtained when the bending rigidities are 

small compared with the in-plane forces. 

Specifically, 

I. INTRODUCTION 

The problem of a plate having a small initial curvature has long been recognized 

From small deflection theory, the total deflection is the sum of the initial deflecTimoshenko and Woinowski-Krieger 

[ 11. 

tion and the deflection resulting from the applied load on a flat plate. 

particular solutions are available in the literature. 

[l] presented an exact analysis for the static deflection of simply-supported plates 

having a small initial curvature. 

plates are, however, not available in the open literature. 

However, few 

The solutions to the vibraticn problems of such 

On the other hand, much work has been done on cylindrically-curved panels. 

The work of Sewall [2] revealed that natural frequencies of vibration of cylindricallycurved paneis are obtainable in ciosed form for only two boundary conditions. Interestingly, it was found that the panel natural frequency deviates from the flat plate 

frequency only by a curvature term. This curvature term vanishes for large radii of 

curvature. However, Sewall's formulation, which is complex and involves integrals, 

does not lend itself to easy application. Blevins [ 31 presented a simplified analysis 

based entirely on Sewall's work. The Donne1 equations for a cylindrical shell were 

used and mode shapes of a single-span beam satisfying pertinent boundary conditions 

were obtained. 

other boundary conditions. 

The Rayleigh-Ritz method was used to generalize Sewall's results to 

The problem presented here is different from those of Sewall and Blevins. We 

consider the free vibrations of a clamped, normally-prestressed plate with a small 

initial curvature using the method of singular perturbations. 

are those for curved plates. 

vibration of clamped, initially-flat plates even in the absence of normal prestress 

forces. Explicit expressions for eigenvalues and mode shapes for the free vibration 

of initially-curved , normally-prestressed plates have , however, been obtained in the 

present study, in the limit when the normalized bending rigidity is small. 

The governing equations 

There are no known analytical expressions for the free 

In the absence of shearing prestress, the asymptotic eigenvalue results for 

initially-curved , normally-prestressed plates are found to correspond to those of 

initially-flat , normally-prestressed plates. 

modified. 

However, the eigenmodes are highly 

It is shown here that the eigenmodes can be decomposed into the following 
three components: 

deflection resulting from the static equilibrium condition. 

(i) the initial deflection, (ii) the flat plate deflection and (iii) the 

11. PROBLEM FORMULATION 

We consider a thin rectangular plate having an initial deflection wo'(x',y'). If 

the plate is further deflected by an amount wl'(x',y',d, then the total deflection, 

w'(x' ,y' , T) , is 

W'(X',Y',T) = w '(x',y') + W1'(X',Y'yT) 0 

The equilibrium equation given by Timoshenko and Woinowski-Krieger [ 13 is 

for the case in which there is no shearing prestress. 

tion ( 2.2) becomes 

In non-dimensional form, equawhere 

No is the characteristic normal prestress value while L is the characteristic length 

used in normalizing the deflection w' and the co-ordinates x' and y'. 

malized using the characteristic frequency, a, such that 

Time is nort = aT and N~/~~~L~ = 1 (2.5) 

where v is the mass per unit area of the thin plate. 

plate, we require that w = aw/an = 0 on all edges. 

For a fully-clamped, rectangular 

If we let the dimensionless initial deflection wo(x,y) be given by a double 

Fourier series of the form 

2 
W W 

W0(X,Y> = c c Pmn Sin(nm) Sin (mi-ry/b) 

m=l n=l 

where b is the dimensionless length of the rectangular pate in the y direction, the 

non-dimensional governing differential equation ( 2.3) becomes 

2 2 2 cow 

Sin(n.rrx) Sin(m.rry/b) (2.7) - C C qmn 

a w1 +-- 2 a w1 2 a w1 -8, -- 2 82 - ay2 at2 

E v w1 

ax 

24 

m=l n=l 

where 

It should be noted that the plate length in the x direction has been taken as the 

characteristic length, L. 

If we now make the substitution 

in equation ( 2.7) , we obtain the following dynamic equation : 

2 2 2au au 

ax 2 ay2 at2 

2 

EVU-P1 -- B2 -+-=o 24 2au 

with boundary conditions 

u(O,y,t) = u(l,y,t) = u(x,O,t) =u(x,b,t) = 0 

and initial conditions 

(2.10) 

(2.11) 

and the static equilibrium equation : 
with boundary conditions 

(2.12) 

(2.13) 

Equation (2.10) is the governing differential equation for the free vibration of an 

initially- flat , anisotropically-prestressed rectangular plate, while equation (2.12) corresponds to the static problem of a similar plate subjected to some generalized loading. 

Thus, the initial curvature effect is equivalent to that of some fictitious loading. 

111. METHOD OF SOLUTION 

The governing differential equations (2.10) and (2.12), subject to the condition 

The procedure is to seek an approximate soluequations (2.11) and (2.13), respectively, are solved by the method of singular perturbations, in the limit when E -+ 0'. 

tion valid in the core region of the domain. Thus, w2 write 

u 0 =uo+Eul O + E2 u20 + 0 (E 3 1 

0 

The resulting membrane-type governing differential equations, valid in the core 

region, are 

V 

uo 

40 

uo + Ao2 uvo = v uv-2 - c lo el2 u: xx +@2 v v-o 

YY o =1 

and 

4 

2 

aw 20 

xx YY 

61 +v 

(3.2) 

v = 0, 1, 2, ... 
Here, the subscript v denotes the order in E, while the superscript o denotes outer 

solution. 

carded, 

opposed to equations (2.10) and (2.12) which are fourth order. 

Terms in which the dependent variable has a negative subscript are dis0 0 and We $v note that the equations (3.2) for uv are of second order as 

It is easy to see that the expansions given in equation (3.1) cannot satisfy all 

the boundary conditions. Thus, the resulting solutions for uv and $v , which are 

equivalent to states of membrane deformation, will not be valid near the plate edges 

where bending deformations exist. 

the deformations change rapidly to satisfy the boundary conditions while the fourth 

order governing differential equations (2.10) and (2.12) are preserved. The solutions of these equations in the bending regions will then serve as modified boundary 

conditions for the membrane-type equations ( 3.2). 

0 0 

Across these bending regions (boundary layers), 

For example, near x = 0, a stretching transformation takes the form 

x = XI& . 

In this case, we seek expansions of the form 

(3.3) 

The governing differential equations become 

v- 2 

i 

-u 

i i 2 ui 

+ B2 v-2 v- 4 U 

vxxxx XXYY YY 

and 

i 2i i 

- +v-4 

YY YYYY + B2 +v-2 

i 2i - 

- -2 qv-2 

XXYY c"xxxx - % +\Ixx 

Equations similar to (3.5) can be written for the regions near x = 1, y = 0, and y = 

b. 

curvature. 

The dynamic and static problems will now be considered for arbitrary initial 

5 
3.1 The Dynamic Problem 

The dynamic problem for a fully-clamped , rectangular plate undergoing sinusoidal 

vibration can be described by the equation 

2 2 2 

2au- zau xu=o EVU-B1 - 62 - - 

aY2 

24 

2 

ax 

subject to 

au 

U(0YY) = ax (0,y) = 0 ; 

It is easily verified from the literature [4,5,6] or otherwise that an approximate solution to equation (3.6), subject to the boundary conditions (3.7), can be written as 

2 22 2 + (m.rr/B2b )(2y-b) sin (nTx) cos (mry/b)} + 2Ao {(n T /B1 )x(l-x) 

2 

cos (n-rrx) cos (mny/b) + (n.rr/2B1 62)(2f32b-61)(2~-1) cos (n.rrx> sin (mny/b) 

+ (m1~/26,B,~b~)(2B,- B2b)(2y-b) sin (nnx) cos (m.rry/b) 
where 

z1 = A. nT sin (mny/b) 

n+l 

1 z2 = (-1) 

z3 = (Ao mT/b) sin (nTx) 

m+l 

z3 z4 = (-1) 

n+ 1 

z5 Z6 = (-1) 

z7 = (Ao mr/b 2 BlB2) {n?rbBZ(2x-l) cos (nTx) + 2B1 sin (nTx)) 

m+l 

z7 "*= (-1) 

while the eigenvalues A2 are given as 

2 m2 r2/b2 + E 4 {B1 2 n2 T 2 + B2 m2 x2/b2) A = B12 n2 T2 + B2 

+ E' {(T 44 /b )(n2b2 + m 22 ) + (12 r 24 /b (n 24 b + m 2 >I + 0 (E~) (3.10) 

The eigenvalues for plates with inverse aspect ratios, b, varying between 0.1 

and 1.0 and for the dimensionless normal prestress values B1, B2 between 0.1 and 

1.0 are presented in Figures 3.1 through 3.4 for values of E ranging between 0.0010 

and 0.2500. 

It is seen from Figure 3.l(a) that the zeroth and second order solutions for 

b = 1 (square plate) are essentially indistinguishable for values of E of 0.0025 and 

0.0010. 

two curves, indicating the solution to be more sensitive to the value of the normalized 

bending rigidity, E. 

decreasing values of the normalized prestress B1 and B2 as shown in Figure 3.l(a)iii 

and Figure 3.l(b). 

However, as b decreases, a considerable difference develops between these 

A similar trend, albeit more pronounced, is observed for 

It is observable fram Figure 3.l(b)iii that the difference between 

7 
2 the second order and zeroth order eigenfrequency (A ) can be as much as four orders 

of magnitude at mode numbers m = n = 15 for values of E = 0.0250 and higher. 

Figure 3.2 shows the second order eigenfrequencies for E = 0.0010 plotted 

against the mode numbers with the inverse aspect ratio b as parameter. 

trend of decreasing values of the eigenfrequency with increasing b is easily inferred 

from the eigenfrequency expression equation (3. lo). 

The observed 

The same trend of decreasing eigenfrequency with increasing b is also evident 

in Figures 3.3(a) and (b) in which the additional effects of combinations of m and n, 

the mode numbers in the x and y directions respectively, Figure 3.3(a), and combinations of P1 and b2, the normalized prestress in the x and y directions respectively, Figure 3.3( b) , have been highlighted. 

Finally, Figures 3.4(a) and (b) show the eigenfrequencies plotted against the 

normalized prestress for various combinations of values of m, n, and b. At higher 

values of m and n, Figure 3.4(b), the eigenfrequency is essentially independent of 

P1 and P2 at the higher E values. 

Next, we consider the static problem for an arbitrary initial curvature, 

3.2 The Static Problem 

The governing differential equation for a fully-clamped , rectangular plate takes 

the form 

subject to 

(3.11) 

(3.12) 

$(x,b) = - ad) (x,b) = 0 

aY 

Using the particular form of wo(x,y) given in equation (2.6), and by comparison with equation (2.12), the right hand side of equation (3.11) is seen to take the 

form 

8 
Sin (nmx) Sin (m71y/b) qmn 

where q was defined in equation (2.8) in terms of pmn, the Fourier coefficients 

obtained by expressing any function f(x,y), describing the initial curvature, as a 

double Fourier series. The procedure is to solve equation (3.2b) for v = 0, 1, 2,. . . subject to appropriate boundary conditions. It is easy to see that the boundary value 

problem for $o takes the form 

mn 

2 2 

2 a $0 2a$o - Sin (nmx) Sin (mny/b) qmn 2 +f32 --- 61 - 

ax aY2 

(3.13) 

subject to 

Clearly, the solution to equation (3.13) satisfying all the conditions of equation 

(3.14) is 

Sin (nmx) Sin (mny/b) (3.15) $0 = Ymn 

with 

(3.16) 

To the next order of approximation, q1 must satisfy the differential equation 

+ 82 2 - a2+1 =o 2 a2$l 

aY2 

81 - 2 

ax 

subject to 

(3.17) 

(3.18) 

9 
In previous studies [ 7,8], equation (3.17) subject to equations (3.18) was solved 

numerically. In general, any of the standard numerical methods of solving parabolic 

partial differential equations such as the Gauss- Seidel, Jacobi and the under- and 

over-relaxation (SOR) methods can be used. However, using the singular perturbation analysis technique , analytical expressions can be obtained to leading and higher 

orders with increasing accuracy as E -F 0'. It can be shown that 

+ {A3 cosh (a2x) + A4 sinh (a2x)) sin (mv/b) 

where 

and the constants AI, . . . , A4 are observed to take the form 

A = - mn y /B2b 1 mn 

(3.19) 

(3.20) 

1 A 2 = - A1{(-l)m + cosh (cllb))/sinh (alb) 

A =-nny 

Aq = - A3{(-l)" + cosh (a,))/sinh (a,) 

3 mn' '1 

(3.21) 

The method of solution is further extended to $2. The boundary value problem to be solved for $2 is 

2 a2q2 

@1 - 2 

ax 

2 a2+2 

aY2 

B2 - 

4 

$0 =v (3.22) 

subject to 
$2(x,o) = - (A2a2/B2) sin (n.rrx) - (m.rr/bB2) (A3 cosh (a2x) + A4 sinh (a,x)) 

$2(~,b) = (A3 cosh (a2x) + A4-sin (a2x)) (-1) m m.rr/Bjb 

+ (a1/B2) (A1 sinh (alb) + A2 cosh (alb)) sin (n.rrx) . (3.23) 

We seek the solution to equation (3.22), subject to equations (3.23), in the 

form 

$2a = [g, cosh (sly> + g2 sinh (clly)l sin (nm) 

+ [g3 cosh (a2x) + g4 sinh (a,x)] sin (mv/b) 

+ 7 sin (n.rrx) sin (m.rry/b) 

with 

22 2 2 22 2 22 2 7 = - (n2.rr2 + m .rr /b )/(B1 n IT + 6, m IT /b ) L 

and 

g1=-A2 a 12 /B 

g2 = al [A1 sinh (alb) + 2A2 cosh ( ulb)] /B2 sinh (alb) 

g3 = - Aq a2/B1 

g4 = a2 [A3 sinh (a,) + 2A4 cosh (a,)] /PI sinh (a,) 

Also, the expression for IJJ~~ take6 the form 

(3.24) 

(3.25) 

(3.26) 

11 
where 

m El = - E2[(-1) + cosh (alb)] /sinh (alb) - E3b - E4b coth (alb) 

E2 = - mn [A K + A4 K21/B2b 31 

22 2 E =A B n 1~113~ a1 3 11 

E4 = E3 A2/A1 

(3.27) 

n E = - E6 [ (-1) + cosh (a,)] /sinh (a2) - E 7 - E8 coth ( a2) 5 (3.28) 

E6 = - nn [A1 K3 + A2 K41/B1 

1 

E8 = E7 A4/A3 

while 

n+ 1 K1 = nn[1 + (-1) 

K2 = (-l)n+l nn sinh (a2)/(a2 + n2 n2) 

K3 = mn [ 1 + (-1) 

K4 = (-1) 

cosh (~~)]/(a~~ + n2 IT^) 

m+ 1 222 cosh (alb)] /b(a12 + m IT /b ) 

m+l 222 

m.rr sinh (alb)/b(a12 + m IT /b ) 

(3.29) 

2 This completes the solution of the problem to 0 (E ). 

12 
IV. CONCLUSION 

The small initial curvature effect on the free vibration of clamped, rectangular 

plates has been presented ., 

varying between 0.1 and 1.'0 and for the dimensionless normal prestress between 0.1 

and 1.0 have been presented for values of E ranging between 0.0010 and 0.2500 in 

Figures 3.1 through 3.4. 

The eigenfrequencies for plates with inverse aspect ratios 

It is established here that a small initial curvature has no effect on the frequency of vibration of the plate. 

The eigenmodes obtained are of increasing accuracy as E -t O', similar to the trend 

for the eigenfrequencies presented in Figures 3.1 through 3.4. 

However, its effect is manifested in the eigenmodes. 

The solutions obtained from above provide a guide for researchers that impleThis utility of the singular perturbation 

ment the finite difference, finite element or other numerical schemes on the computer 

for the solution of curved plate problems. 

technique in providing simplified expressions which yield solutions valid in the 

asymptotic limit and which can serve as useful guides for the pure numerical analysis 

approach has been attested to by Nayfeh and Kamat [7] and Ramkumar et al. [8]. 

13 
REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

14 

Timoshenko, S. P. and Woinowski-Krieger, S. : Theory of Plates and Shells. 

2nd Edition, McGraw-Hill Kogakusha, 1959, pp. 393-395. 

Sewall, J. L. : 

Supported or Clamped Edges and Comparison with Some Experiments. 

TN-D-3791, 1967. 

Vibration Analysis of Cylindrically-Curved Panels with SimplyNASABlevins, R. D. : Natural Frequencies of Shallow, Cylindrically-Curved Panels. 

J. Sound and Vibration, Vol. 75, 1981, pp. 145-149. 

Schneider , W. : 

Einer Eingespannten Rechteckigen Membran. Acta Mechanica, Vol. 13, 1972, 

Einflu 6 Einer Kleinen Biegesteifigkeit auf die Querschwingungen 

pp. 293-302. 

Hutter, K., Olunloyo, V. 0. S. : Vibration of an Anisotropically Prestressed 

Thick Rectangular Membrane with Small Bending Rigidity. Acta Mechanica, Vol. 

20, 1974, pp. 1-22. 

Oyediran, A. A. and Gbadeyan, J. A. : Vibration of a Prestressed Orthotropic 

Rectangular Thin Plate Via Singular Perturbation Technique. 

accepted for publication, 1986. 

Acta Mechanica , 

Nayfeh, A. H. and Kamat , M. P. : A Numerical Perturbation Technique for the 

Transverse Vibrations of Highly Prestressed Plates. 

Vol. 61(1), 1977, pp. 95-100. 

J. Acoust . SOC. Am. , 

Ramkumar, R. L., Kamat, M. P., and Nayfeh, A. H.: Vibrations of Highly 

Prestressed Anisotropic Plates via a Numerical-Perturbation Technique. - Jnt . J. Solids Structures, Vol. 13, 1977, pp. 1037-1044. 
0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) 

(i) b = 1.00 (SQUARE PLATE) ; p1 = 1.00 p2 = 1.00 

5000 

> 4000 

0-0500 2 3000 

0.0025 E 

1 

N 0.2500 X, 

0.1000 0 3 

0.0250 w 

0.0010 5 2000 

(3 

0 2 4 6 8 10 12 14 16 

MQDE NUMBER (M = N! 

(ii) b=0.50;p1 =1.00 ;p2=1.00 

12500 

N 

0.2500 6 

0.0500 5 

0.0250s 7500 

0.0025 

0.0010 LL 

z 5000 

(3 

0.1000 >. 0 10000 

w 

2500 

0.2500 

0.1000 

0.0500 

0.0250 

0.0025 

0.0010 

0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) 

0 2 4 6 8 10 12 14 16 

MODE NUMBER fM = N) 

0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) MODE NUMBER (M = N) 

(iii) b = 0.10 ; p1 = 1.00 ; p2 = 1.00 

FIGURE 3.1(a) SECOND ORDER ElGEN FREQUENCIES SHOWING THE EFFECT OF b. 

ZEROTH ORDER VALUES ARE SHOWN DOTTED 

15 
0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) 

x1o5 

4 

- el 

6- 

>3- 

u 

zw22 w I 

K. 

LLz- / 

w1 2. 

w. 

0 / 

MODE NUMBER (M = N) 

(i) b = 0.10 ; p1 = 0.10 ; p2 = 1.00 

E = 0.0025 

/ 

/ 

0.3010 /’ 

0.0025 

0.1000 

0.0500 

0.0250 

0.0025 

0.0010 

> 

0 

z 

w 

3 

K 

LL 

z 

w 

w 

El 

(3 

0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) MODE NUMBER (M = N) 

(11) b = 0.10 ; p1 = 1.00 ; 02 = 0.10 

x105 

= 0.0025 

0.1000 

0.0500 

0.0250 

0.0025 

0.001 0 

0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) 

(iii) b = 0.10 ; p1 = 0.10 ; p2= 0.10 

FIGURE 3.l(b) SECOND ORDER EIGEN FREQUENCIES SHOWING THE EFFECT OF p1 AND p2 

ZEROTH ORDER VALUES ARE SHOWN DOTTED. 

16 
0.2 

0.4 

0.6 

0.8 

1 .o 

-- . 

0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) 

(i) fi1 = 1.00 ;p2= 1.00 

: 0.2 

0.4 

0.6 

0.8 

1 .o 

0 2 4 6 8 10 12 14 1'6 

MODE NUMBER (M = N) 

(ii) O1 = 0.10 ; o2 = 1.00 

1 o4 

(v 

x, 103 

10-1 

0 2 4 6 8 10 12 14 16 

MODE NUMBER (M = N) 

(iii) p1 = 0.10 ; 02 = 0.10 

0.2 

0.4 

0.6 

0.8 

1 .o 

FIGURE 3.2 SECOND ORDER VALUES (E = 0.0010) OF THE EIGEN FREQUENCY FOR COMBINATION 

OF VALUES OF AND 02 

17 
1 

hl 

x, 

> 

u 

z 

W 

3 

U 

LL 

z 

W 

W 

El 

I? 

108 -1 I I I I 30000 

0.0 0.2 0.4 0.6 0.8 1 .o 

PLATE INVERSE ASPECT RATIO (b) 

(i)M=l; N=l; ~l=l.00;~2=1.00 

1 

hl 

x, 

> 

u 

3 

z 

0.2500 

LL 

0.1000 z 

0.0500 

0.0250 W 

0.0025 

0.0010 

- 

hl 

6 

> 

0 

z 

W 

3 

K 

LL 

z 

W 

W 

z1 

I? 

1 06 

1 o5 

0.2500 

0.1000 

1 o3 0.0500 

0.0250 

0.0025 

107 , , I , [ , f , ~0.0010 

0.0 0.2 0.4 0.6 0.8 1 .o 

PLATE INVERSE ASPECT RATIO (b) 

(ii) M = 1 ; N = 5 ;PI = 1.00 ; p2 = 1.00 

0.0 0.2 0.4 0.6 0.8 1.0 

PLATE INVERSE ASPECT RATIO (b) 

(iii) M = 5 ; N = 1 ; p1 = 1.00 ; p2= 1.00 

20000 

10000 

0 

1500 

A 

hl 6 1250 

> 

0 

3 

5 500 

250 

750 

z1 

z 

W 

0 

0.0 0.2 0.4 0.6 0.8 1 .o 

PLATE INVERSE ASPECT RATIO (b) 

0.0 0.2 0.4 0.6 0.8 1 .o 

PLATE INVERSE ASPECT RATIO (b) 

0.0 0.2 0.4 0.6 0.8 1.0 

PLATE INVERSE ASPECT RATIO (b) 

FIGURE 3.3(a) SECOND ORDER EIGEN FREQUENCIES SHOWING THE EFFECT OF M AND N 

ZEROTH ORDER VALUES ARE SHOWN DOTTED 

18 
\ 

0.2500 

1 

0.1000 

0.0500 

0.0250 

0.0025 

0.0010 

0.2 0.4 0.6 0.8 1.0 

PLATE INVERSE ASPECT RATIO (b) 

(i) M=5 ; N=5 ; p151.00 ;02=1.00 

106 

x, lo5 

5 lo4 

103 

- 

(v 

> 

u 

3 

LL 

5 

2 102 w 

10 

0.0 0.2 0.4 0.6 0.8 1.0 

PLATE INVERSE ASPECT RATIO (b) 

(ii) M = 5 ; N = 5 ; p = 0.10 b2 = 1.00 1 

108 I I I I 1 1 

z * 102 2 

W 

1 

> " 20000 5 

0, 

: 10000 

3 

K 

w 

w 

E 

0 

I 

(v 

5 

> 

u 

z 

w 

3 

0, 

a 

z ,2500 w 

.loo0 12 

.E300 w .0250 

.0025 

LL 

0:O 0.2 0:4 0:s 018 1.0 

PLATE INVERSE ASPECT RATIO (b) 

(iii) M = 5 ; N = 5 ; p1 = 0.10 p2 = 0.10 

1250 

1000 

750 

500 

250 

0 

0.0 0.2 0.4 0.6 0.8 1.0 

PLATE INVERSE ASPECT RATIO (b) 

0.0 0.2 0.4 0.6 0.8 1.0 

PLATE INVERSE ASPECT RATIO (b) 

0.0 0:2 0:4 0.'6 0:8 1:O 

PLATE INVERSE ASPECT RATIO (b) 

FIGURE 3.3(b) SECOND ORDER EIGEN FREQUENCIES SHOWING THE EFFECT OF 01 AND 2 

ZEROTH ORDER VALUES ARE SHOWN DOTTED 

19 
I I I I 

I 

0.2500 

0.1 000 

0.0500 

0.0250 

0.0025 

0.0010 

0.0 0.2 0.4 0.6 0.8 1.0 

NORMALIZED PRESTRESS 

(ii) M = 1 ; N = 5 ; b = 1.00 

I 

N 

-P ~ .- 0.1000 

. 0.0500 /- /- 0.0250 

_- 4 0.0025 

6 > lo3 :--- 

2 102 . 0.0010 

0 - z 

w 

w 

a 

U 

2 __ 10 : 2 

w 

1 

0.0 0.2 0.4 0.6 0.8 1.0 

NORMALIZED PRESTRESS 

(iii) M = 5 ; N = 1 ; b = 1.00 

FIGURE 3.4(a) SECOND ORDER EIGEN 

FREQUENCIES SHOWING THE EFFECT OF 

M AND N (ZEROTH ORDER VALUES ARE 

SHOWN DOTTED) 

20 

0.0 0.2 0.4 0.6 0.8 1.0 

NORMAL1 ZE D PRESTRESS 

(i)M=5 ; N=5 ; bz1.00 

lo6 ii0.2500 

1 o5 

1 o4 

1 o3 

102 

10 

0.1 000 

0.0500 

0.0250 

0.0025 

0.001 0 

0.0 0.2 0.4 0.6 0.8 1 .o 

NORMALIZED PRESTRESS 

(ii) M = 5 ; N = 5 ; b = 0.50 

I E =' 0.2500 I I I 1 

__ 0.0500 

- 0.0250 ~- 

. 0.0025 

a _- __-----: 0.0010 

_- w 

w 

102 

010 0:2 0:4 0:s 018 1 :o 

NORMALIZED PRESTRESS 

(iii) M = 5 ; N = 5 ; b = 0.10 

FIGURE 3.4(b) SECOND ORDER EIGEN 

FREQUENCIES SHOWING THE EFFECT OF b. 

ZEROTH ORDER VALUES ARE SHOWN 

DOTTED 
APPROVAL 

THE EFFECT OF A SMALL INITIAL CURVATURE ON THE FREE 

VIBRATION OF CLAMPED, RECTANGULAR PLATES 

By A. A. Adeniji-Fashola and A. A. Oyediran 

The information in this report has been reviewed for technical content. 

of any information concerning Department of Defense or nuclear energy activities or 

programs has been made by the MSFC Security Classification Officer. This report, 

in its entirety, has been determined to be unclassified. 

Review 

AMV &wl+ 

G. F. McDONOUGH 

Director, Systems Dynamics Laboratory 

L 

*U.S. GOVERNMENT PR INTI NG 0 F F ICE 1986-730-067/40036 21