Non-linear dynamic response of circular plates subjected to transient loads

Non-linear Dynamic Response of CircularPlates Subjected to TransientLoads by

R. s. ALWAR

and

YOGENDRA

NATH

Department of Applied Mechanics Indian Institute of Technology, Madras, India The Chebyshev polynomials have been applied to the large amplitude motions of circular plates under transient loads, with and without damping. The nonlinear differential equations are linearized by using Taylor series expansion for one of the terms. It is shown that there is good agreement between the results obtained by the present technique and the available results. The advantage of this technique is essentially due to the fact that the Chebyshev polynomials are rapidly converging polynomials. It is shown that very accurate results can be obtained with only four terms of the Chebyshev series which may not be possible with conventional methods.

ABSTRACT:

Notation deflection and stress function non-dimensional deflection and stress function thickness of plate radius of plate non-dimensional radius

WV? *, 4 h r, a L

Eh3/12( 1 - v’) Young’s modulus, Poisson’s ratio and the density of the plate material intensity of load non-dimensional load time non-dimensional time damping parameter non-dimensional damping parameter rth Chebyshev polynomials in the range -15 p 5 1 and 0 I p 5 1, respectively number of terms in the series arbitrary functions

JZ v, Y

7

&.J,

5.J

%J,

d,J

coefficients of

Chcbyshev

series

i

subscript J step of marching variable superscript (N) order of derivative Superscript’ First term of the expansion (a, P, 6, 7 and 5) Houbolt coefficients (A, 8, 4, =$and CL)

to be halved

R. S. Alwar and Y. Nath I. Introduction

There are many situations, such as seismic tests, nuclear explosions, sonic booms analysis, etc., in which plate-like structures are subjected to transient loads and large amplitude motions occur. If the amplitude of motion is of the same order as the thickness of the plate then for the mathematical description of motions, the classical linear plate theory is inadequate and the use of non-linear plate theory is inevitable which takes into account the interaction between the bending and stretching of the mid-plane. In the case of circular plates subjected to axisymmetric dynamic loads, the governing equations reduce to two coupled non-linear partial differential equations. These equations because of their complexity are mostly solved by using approximate methods of various types, such as rate-form linearization (11, finite difference (2), Kantrowitch averaging (3). In the present analysis, an attempt has been made to give an analytical solution for large amplitude response of circular plates using Chebyshev polynomials. It is well known that the Chebyshev polynomials approximation to a function F(p) is nearly minimax. In other words, if the unknown constants in Chebyshev series are determined by collocation with the zeros of the Chebyshev polynomials as the collocation points, then the resulting series will give a minimum of the maximum absolute error. In addition, if a given function is expressed as a sum of Chebyshev polynomials, by using the property of orthogonal functions, it is seen that of all the ultraspherical polynomials, the Chebyshev polynomials have the fastest rate of convergence. The application of Chebyshev polynomials to the solution of ordinary differential equations has been discussed by Fox (4), and the solution of simple first-order ordinary and non-linear differential equations has been discussed by Clenshaw (5) and Norton (6). Recently, the non-linear differential equations occurring in fluid flow problems connected with laminar boundary layer theory have been solved using Chebyshev polynomials after linearizing the equation using a quasi linear technique (7). To the authors’ knowledge there has been no attempt to apply this rapidly converging series to solve non-linear problems in structural mechanics. In this paper, we wish to apply this technique to large amplitude response of circular plates with and without damping subjected to step function loads. The results are presented for both the simply supported as well as clamped edge boundary conditions. The method can be extended to elastically restrained conditions without additional complexities. II. Field Equations The governing differential equations of motion used in the present analysis are of the form given by Huang (3). This particular form is chosen as one of the equations of second order which facilitates the analysis to a great extent. The

528

Journal

of the Franklin

Institute

Circular Plates Subjected to Transient Loads equations

are as follows:

.

a4w 2a3w D $T+;g-T-+~~

1 a*w

1 aw

(la)

+ yhag+

r ar*

ychary= q(r, t). (lb)

By introducing

the following dimensionless

parameters,

we have

p=rla,*=w/a;P=

c

Equations

P=qa3/D

(1) become 2-

P*y+P-$-$+-

p3~+2p*~_p$+~-~2

all;

-

I-V* 2

f 0

2

ak2 pap (

. p2 aqaii apitp++-T (

=p3

The boundary

conditions

(a) clamped immovable

1 =

0

(24

-a25 ap

1

>

P(P,+g+~

a2*

_aii, I

*

t2bj

are: edge p=l,

@=O

as ap-

-0

(34

ai 4=0; --

Pap

529

R. S. Alwar and Y. Nath (b) simply supported

immovable edge iQ=o

p= I, a2f0 -+v-=o p ap2

(c) symmetry

aii

(W

ap

condition at the centre a* p = 0,

apa3G ap3-

-0

-0

(3c)

&=o.

IZZ. Linearization The non-linear

Technique terms in Eqs. (2) are linearized in the following way.

Let X, be any function of space and time at any time step J, i.e. (X);=(X),

* (X),.

(W

Using Taylor series, (_%), can be expressed as

(R),=(x),_,+~J_l .AT+($g+. ..

(4b)

Using the backward difference scheme, C&/a7 and a2J%/aT2can be expressed as ,_1

=

{<%-I

-

(&2VA7

=

{(%-,

-

~(X)J_~ + (5&_3}/67’.

Substituting Eqs. (4b), (4~) and (4d) in (4a) the following relation is obtained: (X): = (X),(2.5(X),_,

-2(_%),_,+0.5(X),_,+.

. .}.

(W

Assuming that the values are known at time steps J- 1, J- 2 qtc., any nonlinear term at time step J can be linearized in a similar way. In the present analysis only three terms in the Taylor’s series expansion are retained, and in such a case Eq. (4e) can be written as (*): = (ff),M%,

+ EC%,

where J=l A=B=C=O.O

530

+

@%,I

(4f)

Circular Plates Subjected to Transient Loads J=2 A=B=O c= 2.0 J=3 A = 0.0 B = 2.5 c = -2.0 J>3 A = 2.5 B = -2.0 c = 0.5.

The above procedure (2b)

leads to the following linearized version of Eqs. (2a) and

+ B(!!)J_2

+

Cr$),_,) =0 @a)

P3($)J+2P2@,-P($),+(~),-12(;)7[(~),

* (A~),_,+B~),_*+c~),-,)

+ (&J(

A($),_,

+B($),-,

+

C@),_,}]

IV. Properties of Chebysheu Series The rth order Chebyshev

polynomial

T,(p)=cosrfl,

From the Trigonometric

is expressed as

cos8=p,

-1lpl.

Identity

cos(r+1)8+cos(r-1)8=2cos8cosrf3 the recurrence

(6) (7)

relation can be written as T,+*(P) = 2~7%~) - T,r-I,(p) TO(P) = I, T1(p) = p; etc.

Vol. 303, No. 6, June 1977

(8) 531

R. S. Alwar and Y. Nath

We now find it convenient to use the Chebyshev polynomials for the range 0 5 p 5 1. The Chebyshev polynomials for this range can be derived from the polynomials of the range -11 p 5 1 as T?(p)= 2nd the I.._ .I._

mrwmnnrlino w”‘“‘y”‘““‘~

T,(2p-l),

rm-lwrenoe I”V...IVIIVV C+dP)

The orthogonality

=

(9)

relatbn 1.9” he ~vnr~ccm4 I”1%..1”11 .,-a. “V “^y”“Y.7””

= WP

T:(p)

OSpzSl

1,

-

m-T(P)

-

OP c&u

q-,,(P);

(10)

T?(p) = 2p- 1; etc.

conditions for the range 0 I p 5 1 are as follows: m=ln

0, 1 F *il

?“x,(p)c(p)p-p*)-+p

=

m = n#O I 42, 7r, m=n=O.

(11)

If a continuous function g(p) in the range 0 5~5 1 is expressed as g(p) = (a0/2) +C”lzl a,TT(p), then the coefficients of the above series are determined by using orthogonality relations (6) as follows: 7=+

c’g(p)(p-p’)-:dp I 2 Pl a,=- ~ o g(p)T:(p)(pJ

(12a) P*)-~

dp.

KW

The recurrence relation between the derivatives of Chebyshev polynomials can be expressed as $:;l’(p)

= 4Tyk’(p),

01 p 5 1

(13a)

where superscript denotes the order of derivative. Expressing the derivative of g(p), that is gck’(p) as a Chebyshev polynomial with coefficients a!k), it can be shown from Eq. (13a) that following relationship exists between the coefficient of gtk+‘j(p) and g’“‘(p) (k+l) = 4ra(k) a&k-:f’ - aI+ r *

The expressions for products like p’TT(p) and TT(p)Tf(p)

(1W

can be expressed as (14)

where 2r

0 i

T:(P)

532

. T:(P)

2r = (2r=i{c+r(~)

l)!i! + C-,,(P)).

(1%

Journal of The Franklin Institute

Circular Plates Subjected to Transient Loads

If g(p) and f(p) are two Chebyshev

series

g(p)= :+

8

(164

a,TT(p)

r-1

,zlbrTT(p)

f(p) = ++

Mb)

then the product of g(p) and f(p) can be expressed

074

g(p)f(p) = 2 + $ c,TT(p) r-1 the coefficients c, of the Eq. (lla) Co=

are given by

t’aibi

(17b)

i=O n+V?l

C, ~1

C

r= 1,2,3..

ai(bli-,(+b,+i),

i=O

.n

(174

where br=O ai=O bi=O c,=O

r>m i >n i>m r>n

for for for for

Ii- rl indicates the absolute value of (i-r) halved.

and the coefficient of a0 must be

V. Analysis The integration of Eqs. (5a) and (5b) are carried out space-wise using Chebyshev polynomials, and time-wise using the implicit integration scheme, viz. Houbolt scheme. The deflection W(p, T), stress function I&J, r) and their derivative with respect to p, in Chebyshev series are expressed as

as

-(l) --;+;$I w.

@y’TT

ap=” a29

-(2)_

';

; Nz2

(18b)

#'TT

(184

@:“‘T:

(184

ap2=w r=l a3ii,

ap3-

_

w(3)

-

-(3) w. 2 +

N-3 c r=l

(18e) (W Vol.303.No.6,June 1977

533

R. S. Alwar and Y. Nath

Wb) a2tj ___/2)-

6;

; Nz2

(194

9;29'7.

r=l

The load term P(p, 7) can be expressed as

(20) The product terms in Eqs. 5(a) and 5(b) can be expressed in Chebyshev series as follows. A typical term p3(a4K#p4) can be given as

r=O +B~~~~2,,~+~~~~~3,,}~

Another

(W

typical product term can be written as p2(3f;),,

= p2[ (y$;

@,?T?}(;!;’

(21b)

e%J:]]

N-l, =

p2

c

&,.J:

N-l =

c:

{;&+,,+tL+1,,

+$&,J

r=O

+$d,l-l,.r (21c)

+&d;-,,JD’T where hi-1

;i,J

=$

(NE*)’

tj~,:‘{G~~Lil,J-l+

I;i::)i,J-1)

CW

is0

r>(N-1).

&=Ofor

We)

Using Eqs. (18) to (20) and expressing the product terms as above Eqs. (5a) and (5b) can be written as follows: N-2, c

EGm2,,

-+

ai!'?l,J+~~rT:+$1~~~l,~+~l(I~~~2,~}

r=O +w%.J

+~i!,:'+a~~~~,,,,}-{II;,,>

+14 2

534

1 {zE;+~,~+$G,,

+%~r-l~,JWf = 0 (224 Journal

of The Franklin Institute

Circular Plates Subjected to Transient Loads N-4, c

[{I

-(4) 64w,+3+~?i~~*,J+~~‘lrl!l,J+~~~~+~~~~~~,,.,+~~~~~~,,,+~~~~~3,,,

>

r=O

+2{~~,l’!2,J+f~j3!1,J+~~‘I~+aittfr”l,,..r

+7%{?*,.J

- {W%,J +~~)!T:+a~~~~,‘L,,,,}+{~,!~} - 12 t 0

2{&6,+2.J+% 4

- 12 ; 0 N-4,

2{i%r+2,J

-

+tiir+l,J

+t6,,-,,,J+~6,~-2,,J}

+%,J

+&%,J +&,.J

+ii6,r-2,,J)17-?

c [{~~+,+BTT+,+~‘T”c”+l+~TT+a~T;f_,,+~T~-,, r=O

+$‘77-3,)

+

r+l,J

(

5) -&

p(d,+

* (Ai+,,J

(a@r,J +

+

th’&,J-1+

p*r,J-l+

$+,,J-2+

a%,J-2

5@,,J-3

+

+

77%,J-3

/.‘)

II I=

0.

Note that the inertia terms and damping terms are expressed in the Houbolt form in the above equations. In Eqs. (22a) and (22b)

(224 (224 For r = 0 they take special values given as N-l, Co,J =

c

i=O

+;:‘{Aw& ’

+B&-,+ I,

Cti,I,~_3}

cm

N-l

Wg) N-2, ho,J

=

C

i=O

Jli,J{AG$-i + BKJ~,Y-~+ CG$-3}

Wh)

((u, p, 6, r) and {) and (A, 0, 4, 5 and CL)are the constants of the Houbolt scheme in the evaluation of acceleration and velocity function respectively. These constants depend upon the type of loading and the initial conditions see Appendix). Vol. 303, No. 6, June 1977

535

R. S. Alwar and Y. Nath

Similarly Eqs. (3a), (3b) and (3~) can be expressed. (a) P = 1

N-l, c

r=O

G,‘,“T:=O

CM

N-l, c

{&jl:‘l

+#)+$j,~I,)T~-

r=O

y

2’

&TT

=

0;

r=O

(b) P = 1

f’ i&T:=0

r=O N-2,

N-l

c

r=O

(4

{$?:‘,‘I

+&‘,2’+&$?4)T’+

Y c ’ C$‘TT = 0 r=O

(23b)

P=0 N-l,

til”T:=O

,F6 N-2 1

‘,$3’T+()

r=O

(234

Superscript ’ denotes that coefficient of TT for r = 0 must be halved. Equations (22a) and (22b) are the generating equations for the evaluation of the unknown coefficients of the Chebyshev series. Using Eq. (13b), these equations can be expressed in terms of G,, KJ~+~, . . . I&, &+1,. . . etc. Now by equating the coefficients of weight functions for r = 0, 1, 2. . . k etc., a set of simultaneous algebraic equations in terms of W1, W2. . . Wk, &, IcT2,. . . I& are obtained, which are solved by matrix inversion. Instead of three terms of the Taylor series expansion for the linearization, one term of the expansion can be taken and the accuracy can be improved by iteration. For this particular example, it has been shown that the computer time taken for the solution of algebraic equations linearized with three terms of Taylor series expansion needs approximately one third of the time with the iteration technique, using one term of Taylor series expansion. VI. Numerical

Results and Discussion

Results obtained by the present analysis corresponding to a non-dimensional load qa4/Eh4 = 10 and damping parameter C = 16 are plotted and compared 536

Journal of The Franklin Institute

Circular Plates Subjected to Transient Loads I 11

11

’

11

11 Static

11 solution\

1 -

-------t -

-

-

T

Fmtte difference

-

Authors’

immovable clomped edges

&S.O T =I00

FIG. 1. Comparison of the damped response. with the results obtained by the finite difference technique (2). It can be seen from Fig. 1 that there is reasonably good agreement between the two results. The influence of the damping factor for the non-dimensional loads P(T) = 2.184 and P(T) =2.730 on the dynamic response of circular plates for the clamped and simply supported edge conditions are shown in Figs. 2, 3,4 and 5, respectively. From these note that the magnitude of the critical damping parameter is not sensitive to change in the boundary conditions and small 18

I

I

I

I

I

I

I

Immovable clomped edges 16-

FIG. 2. Effect

Vol. 303,No.6,June

1977

of damping

on the dynamic

response.

537

R. S. Alwar and Y. Nath

16

12 f 08

FIG. 3. Effect of damping on the dynamic response.

changes in load intensity. By introducing artificial damping, i.e. increasing the time step size in the Houbolt scheme, the static solutions are obtained as shown in Fig. 6. It can be seen that there is excellent agreement with the results given by Way (7). This can be taken as the check for the present analysis. In order to get an idea of the fast rate of convergence of the Chebyshev series, the

:_ % O* 06-

-02

0

FIG.

538

I 01

I 02

I 03

I 04 l-

I 0.5

I 06

I 07

06

4. Effect of damping on the dynamic response.

Journal

of The Franklin

Institute

Circular Plates Subjected to Transient Loads

FIG. 5. Effect of damping on the dynamic response.

values of the coefficient

in the series w”, I?,, tiij2. . . are plotted in Fig. 7. 2 Therefore, we can see from this that accurate results cah’be obtained even with four terms in the series, which is not-attainable with conventional methods. The present technique can be easily extended to the transient analysis of composite plates, plates with variable profiles, axisymmetric shell problems, etc.

14

I

I

I

I

I

I

Clamped immovable edges IZlF=o to-

Ar=O. I

085 f

-

Authors’

---

woy

qo4/Eh4

FIG. 6. Static deflection by introducing artificial damping. 539

R. S. Alwar and Y. Nath 2.4

0

01

02

FIG. 7. Convergence

03

04

of Chebyshev

05

06

series.

References (1)J. R. Lehner

“Nonlinear static and dynamic deformations of and S. C. Batterman, shells of revolution”, Int. J. Nonlinear Me&. Vol. 9, pp. 501-519, 1974. “Finite amplitude response of circular plates (2) W. A. Nash and H. Kenematsu, subjected to dynamic loading”, IUTAM Symp. Herrenalb, p. 311-316, 1969. (3) C. L. Huang and B. E. Sendaman, “Large amplitude vibrations of a rigidly clamped circular plates”, Int. J. Nonlinear Mech. Vol. 6, pp. 451-468, 1971. (4) L. Fox, “Chebyshev methods for ordinary differential equations”, Computer J., Vol. 4, pp. 318-331, 1962. (5) C. W. Clenshaw and H. J. Morton, “The solution of nonlinear ordinary differential equations in Chebyshev series”, Computer J., Vol. 6, pp. 88-92, 1963. (6) H. J. Norton, “The iterative solution of nonlinear ordinary differential equations in Chebyshev series”, Computer J., Vol. 7, pp. 76-85, 1964. (7) M. A. Jaffe and J. Thomas, “Application of quasilinearization and Chebyshev series to the numerical analysis of the laminar boundary layer equations”, J. AIAA, Vol. 8, pp. 483-490, 1970. (8) L. Fox and I. B. Parker, “Chebyshev Polynomials in Numerical Analysis”, Oxford University Press, London, 1968. (9) J. C. Houbolt, “A recurrence matrix solution for the dynamic response of elastic aircraft”, J. Aero. Sci., Vol. 17, pp. 540-550, 1950. (10)S.Way, “Bending of circular plates with large deflections”, J. Appl. Mech., ASME Vol. 56, pp. 627-636, 1934. “Theory of Plates and Shells”, McGraw(11)S.Timoshenko and S. Woinsky-Krieger, Hill, New York, 1959.

540

Journal of The Franklin Institute

Circular Plates Subjected to Transient Loads

Appendix The following relations process,

as suggested by Houbolt

=

are used to start the recurrence

_I{%,J+l- 2W,J+ WJ-1) AT’

(Al) 642)

+3w,,,-6~,,,-1+~,,,--2}. The initial conditions

are as follows:

.I=0

- =0 wr,o aw

() a,

a% 0 a72

Substituting Eq. (A3) in Eqs. (Al) intervals of time are obtained as

= 0

,,o

,,o

=

(A3)

P(T).

and (A2), the fictitious

@,,_I = P(T)A?-

coefficients

at negative

a,,1

*,,-z = SPAT’-

(A4)

81?,.~.

(AS)

These values of KJ,,_~and CJ_~ as expressed in Eqs. (A4) and (AS) are substituted in the following equations to yield the coefficients (a, p, 6, n and 5) and (A, 8, 4, ,$ and (.L) during the initial increment of time, i.e., a%

( 1 a72 ,,I

(

=$

{a*,.J + @%-I

aw a, ,,, = & )

w,,,

+ w,,-,+

w,,,-3

+ 51

+ tw,J-1 + dJ*,,,J-2 - &L-3

(A7)

+ CL)

J=l a==,

6 AT

p=o,

h=18/6A~,

s=o,

0=0,

n=o,

5=-2P(T)

c#J=O, [=O,

/.L=-

SPAT’ 6A

7

J=2, a=72,

*=$

Vol. 303, No. 6, June 1977

2

q=o, 8=$,

7

,#J=o,

t=o,

{=-P(T)

CL=-

SPAT

6AT

541

R. S. Alwar and Y. Nath J=3 a=g

2

P =$,

6=$,

q=o,

5=0,

5=0

/.b=o

J>3

a=& B=$, a=$, v=-$, A==

542

11

-18 @=6L\7’

#d-,

pz&

i=o /L=o.

Journal

of The Franklin

Institute