A new approach to the forced vibration of thin plates

Journal of Sotmd and Vibration (1973) 30(4), 397---417

A N E W A P P R O A C H TO THE FORC17 '-" VIBRATION OF THIN PLATES B. K. DONALDSON

Department of Aerospace Engineering, University of Maryland, College Park, Maryland 20742, U.S.A. (Received22 December 1972) This paper presents a new, approximate method of analysis in the classical style for the response of a thin plate to a single frequency, forced harmonic vibration. The plate may be quadrilateral or triangular, and it may have any boundary conditions. The distinctive feature of the method is the manipulation of certain tractable boundary conditions in order to obtain a solution which satisfies the given boundary conditions when the latter do not allow the direct calculation of a series solution. The tractable boundary conditions are described by finite Fourier series. This method may be applied to other boundary value problems for which there is a complete series solution to the governing differential equation.

1. INTRODUCTION Exact, series solutions for thin plate bending deflections are limited to a relatively few combinations ofplate geometry and boundary conditions. Thus there is a need for broadly applicable approximate methods o f solution. In reference [I] nine such methods o f solution are compared, which either exactly satisfy the governing differential equation or exactly satisfy the essential, or both the essential and the natural, boundary conditions. The nine methods considered in reference [1] are those of (1) boundary point matching, (2) boundary point least squares, (3) Trefftz-Morly, (4) interior point collocation, (5) interior least squares, (6) subdomain collocation, (7) Galerkin, (8) Ritz, and (9) Kantrovich. Other notable approximate methods of solution are the force and deflection finite element methods, the finite d~fference method, and collocation over both the plate interior and the plate boundary. While many of the above approximate methods produce excellent results in many cases, none o f these methods is without limitations. Thus, it is worthwhile to consider new methods of analysis. This article t cesents a new, approximate method of analysis for single plates with straight boundaries whea the applied lateral pressure has an arbitrary spatial distribution, but is harmonic in time. This new method of analysis has much in common with previous classical solutions such as those of reference [2] where Fourier series are used to describe the deflection solutions, the lateral pressures, and the deflection and moment boundary conditions of rectangular plates. However, this new method differs from all previous analyses in the manner of satisfying the boundary conditions. In brief, the derivation o f the new met.hod of analysis begins with a thin, rectangular plate of constant thickness and the corresponding, dynamic form of the fourth-order differential equation for plate bending. A L6vy series solution [3] is obtained for the rectangular plate when there is no lateral load, and the boundary conditions are those where one side of the plate is forced to undergo harmonic vibration while the other three sides are simply supported. At the forced edge, it is the lateral deflection and slope 397

398

13. K. DONALDSON

deflection amplitudes which are specified in a general manner by means of Fourier sine series. Three other L6vy series solutions are obtained for the plate when the boundary conditions are altered so that each of the other three sides in turn is the one side undergoing forced harmonic motion. Then a Navier series solution [3] is obtained when all four sides are simply supported and a lateral pressure acts over the plate surface. These five solutions are superimposed to obtain a single solution for a plate where all four edges are undergoing forced harmonic deflections, and where the plate surface supports a harmonic lateral pressure. All oscillations are at the same frequency, and they are either in or out of phase. Note that because Fourier sine series are used to describe the lateral deflection boundary conditions, the corners of the plate do not deflect vertically. This plate, with its simple boundary conditions, is called the extended plate. The plate of the given problem is called the actual plate. An analysis of an actual plate begins by locating the actual plate within the boundaries of the rectangular extended plate. If the actual plate is located away from the non-deflecting corners of the extended plate, its corners will undergo deflections. The lateral pressure on the extended plate is made to coincide with the given lateral pressure on the actual plate within the domain of the actual plate. The Fourier series coefficients of the extended plate boundary conditions are at the disposal of the analyst, and they are then adjusted to provide the required boundary conditions for the actual plate. This procedure, which involves the solution of only one set of simultaneous algebraic equations, determines the magnitude of the extended plate lateral deflection function. When that function is restricted to the area of the actual plate, it is also the solution for the actual plate lateral deflections. In a second, companion article the application of this method to fiat semimonocoque structures subjected to the same dynamic loading is discussed [4], and in a third, companion article the results of numerical calculations are presented [5]. This method ofanalysis has advantages and limitations. The method is limited to boundary value problems where complete series solutions exist for the governing differential equation. In the case of plate analyses, this requirement precludes general plate in-plane loadings as well as large lateral def{ection amplitudes. As will be fully explained in the subsequent article on numerical results [5], a further limitation is that numerical accuracy is not uniformly good for all types of boundary conditions. On the other hand, the method has the advantages that it can be applied to any quadrilateral or triangular plate with any boundary conditions, and, as will be discussed in the second article [4], the method is easily adapted to fiat skinstringer-frame structures, including those with cutouts. The material that follows begins with the derivation of the equations necessary for employing the extended plate. Then the procedure for actual plate analyses is demonstrated by four example problems. The first three examples concern rectangular plates. They are included because they are instructive, and because later numerical solutions for these problems will show some of the advantages and disadvantages of the present approach. The last example is that of a trapezoidal plate. This example is meant to show the broad applicability of the present method, and to suggest one situation where the present method may compare favorably to all other methods. 2. THE EXTENDED PLATE ANALYSIS As stated in the Introduction, the extended plate is a plate with certain tractable boundary conditions which are used to bring about the required boundary conditions for an actual plate. As such, the extended plate is the basis of the present method of analysis. In this paper, the rectangular extended plate, and therefore the actual plate, will be limited to being homogeneous, isotropic, linearly elastic, and uniformly thin. Neither in-plane forces or shear deformations will be considered. Under these conditions, it is well-known that the plate governing

fORCED VmRATIONOF "rHn~PLATES

399

differential equation of motion can be written as a "~w

04 w

O~ w

pd 02 w

ax, +2ax--P-~&+~p+ 1) at"

q

(1)

z)

The sign conventions of reference [3] will be used throughout, and a list of symbols is given in Appendix I. Side two A

~,X

Side

S~de one

three

S~de four ~y

Figure 1. The extended plate, top view.

.To develop the expression for the deflection amplitude of the extended plate pictured in Figure 1, first consider boundary conditions where sides two, three, and four are simply supported and side one is forced to undergo harmonic motion of frequency to, with vertical deflection amplitude Hi(y), and with slope-in-the-x-direction amplitude KI(y). The amplitudes H~(y) and KI(y) are small, and they are arbitrary except that they must be continuous, have continuous first, second, and third derivatives, and be such that Hi(0) = K~(0) = H~(b) = Kx(b) = 0. Let there be no loads acting on the plate other than the plate inertia loads and those loads at the plate edges which are required to produce these boundary conditions. Designate the deflection function corresponding to the above boundary conditions as w~(x,y,t). An approximate solution for wl can be obtained by writing w~ in the form of a finite L6vy series, i.e., tlT~.}'

wl(x,y, t) = Wl(x,y)sin~ot = sincot ~ Z,(x) s i n - - , .

(2)

b

where all indices on all summation signs vary from one to N unless otherwise noted. The value of N is chosen so as to achieve satisfactory convergence for all quantities of interest. Substitution of equation (2) into equation (1), where in this first case q(x,y, t ) = 0, yields, after cancellation of the common factor sin~t, .~ X"~'(x)sin b -

~

X.(x)sin---~-- +

X.(x)sin--ff-

_ p(o z d ~ X.(x) sin nny D -b '

(3)

n

where each prime indicates one differentiation with respect to the argument. Multiplication of both sides of equation (3) by sin(tony/b) and integration over y from 0 to b produces XJ.'(x) - 2

XT,(x) +

D

X.(x) = 0,

(4)

400

B. K. DONALDSON

where m has been changed to n and use has been made of the orthogonality relation b

2 b

sin ---~- sm --~-- , y =

if n = m.

(5)

0

In this simple case the same result, equation (4), can be obtained by simply noting the linear independence of the various sine functions of the identity equation (3). However, the above weighting procedure for minimizing the error associated with a finite series approximation, will be used in many similar situations where an appeal to linear independence is not possible. Equation (4) is a fourth-order linear, homogeneous, ordinary differential equation. Its characteristic equation for an exponential function solution is the biquadratie r'*-- 2 ( 7 ) 2 r2 + [ ( ~ ) 4

PcoDd] = 0.

(6)

The four roots of equation (6) are +G, - r . , +~., and -~., where

J

"=+

~:. = +

- to

(7)

Note that the quantity ?. is imaginary for any given co whenever fi > n, and it is real whenever n > fi, where = _ co.2

(8)

To avoid any uncertainty on this point the latter two roots of equation (6) are renamed +is. and -is. for n < fi, and +s. and - s . for i1 > !~, where s. is always a real, positive quantity and s.=[

[Pd\ ''2

forn<~, for n > ~.

(9)

Furthermore, instead ofcomposing the solution to equation (4) for n < fi from the exponential functions exp (r. x), exp (-rn x), exp (i&x), and exp (-is. x), which are in part complex valued, the solution for n < fi will be composed of the real, independent functions sinh r. x, cosh r. x, sins.x, and coss.x which as linear combinations of the exponential functions are also solutions to equation (4). Thus, for ii < l],

X~(x) = Cl..sinhr.x + C2.~coshr~x + C a , . s i n s . x + C4,.cossnx. Similarly, for n ;~ fi, the solution to equation (4) can be taken to be

X.(x) = C l . . s i n h G x + C2.~coshr.x + Ca..sinhs~x + C4..coshs.x. By defining the functions

ns.x sin(h)s.x = [sinhs~x

[si

COS Sn X

cos(h)s.x = [coshs. x

ifn < t/, ifn/> ~, ifn < ~ ,

0o)

FORCED VIBRATION OF THIN PLATES

401

the above two solutions may be written as the one equation

X.(x) = C l . . s i n h r . x + C 2 , . c o s h r . x + C a , . s i n ( h ) s . x + C4,. cos(h) s . x .

(11)

The next step is to make the deflection function satisfy the boundary conditions and thereby calculate the constants of integration. The boundary conditions at y = 0 and those at y = b are identically satisfied. The boundary conditions at x = 0 are

W~(O,y) = Ha(y), ti/"1,x(O,y) = Ka(y),

(12)

where each subscripted independent variable following a comma means one partial differentiation with respect to that variable. To obtain a solution, the boundary condition amplitude functions are approximated by the first N terms of their respective Fourier series expansions. Let

Hx(y) = ~ hL.sin.nb y

(13a)

Ka(y) = ~. . k l . . sin nny b

(13b)

and

Thus equations (12) become

nny n~y Z X.(O)sin--ff-= ~ ha,.sin b '

-

X'(O) sin nny nny b = .~kx'"sin ~ ,

(14)

or, for all n, after application of the orthogonality relation, equation (5), equations (12) become

x.(o) =

c2.. + c,..

= Ill..,

X'(O) = r, Ca:, + s, C3., = k l . , .

(15)

The boundary conditions at x = a of zero deflection and zero moment reduce to Wa(a,y) = W,. =(a,y) = 0,

(16)

or

nny X.(a)sin--b-- =

..... nny . .tajs,n-- U = 0.

(17)

As above, these equations lead to

X,(a) = CI., sinh r, a + C2., cosh r, a + Ca., sin(h) s, a + C4., cos(h) s, a = 0 XF,(a) = r2, CI., sinh r, a + r,2 C2., cosh r, a + s,2 C3., sgn (n - h) sin(h) s, a + + s,2 C4.. sgn (n - h) cos(h)s,a = 0. where sgn (X) =

--1

ifg<0.

+1

i f X > 0.

(18)

402

B.K.

DONALDSON

The simultaneous solution of equations (I 5) and (18) for the constants of integration results in the following expression for the deflection amplitude function: Wx(x, y) = ~ J.{hx..[s. cos(h) s. a sinh r.(a - x) - r. cosh r. a sin(h) s.(a - x)] + /1

nrcy

+ k l . . [sin(h)s.asinhr.(a -- x) - s i n h r . a s i n ( h ) s . ( a - x)]}sin ---if-,

(19)

where d. = ( s . s i n h r . a c o s ( h ) s . a - r . c o s h r . a s i n ( h ) s . a ) -x. The function (j.)-I and the similar function (L.) -I which is defined below are often small differences o f two large numbers. A loss of three significant figures with respect to s . s i n h r . a cos(h)s.a or r.coshr.asin(h)s.a is not unusual. Therefore care must be taken to accurately calculate these values. A different formTor W l ( x , y ) could be obtained by choosing zero deflection and zero slope boundary conditions at x = a. The subsequent superposition process would still work, but this choice does not appear to offer any advantage. As a second step towards the deflection amplitude function of the extended plate, again consider the plate of Figure 1, but now with boundary conditions of simple support at sides one, three, and four while side two is being forced by moment and vertical shear distributions sufficient to produce at edge two the vertical deflections H2(x)sino~t and the slopes-in-the-ydirection K2(x)sinoot. The amplitude functions H2(x) and K2(x) are, as before, arbitrary continuous functions except for the requirement of possessing continuous first, second, and third derivatives and having H2(0)= K2(0)= H2(a)= K2(a)---0. The deflection amplitude w2(x,y, t) = W2(x,y)sinoot associated with these boundary conditions can be obtained in the same manner as above when the boundary conditions at y = 0 are written as II I'CX

W2(x, O) = H2(x) = Z h2.. sin ~ , n

a

/ll~X

IV2. r(x, 0) = K2(x) -- ~ k2.. sin ~ . .

a

The solution is then W2(x,y) = ~. L.{h2..[v. cos(h) v. b sinh u.(b - y) -- u. cosh u. b sin(h) v.(b - y)] + n

+k2..[sin(h)o.bsinhu.(b-y)-

sinhu~bsin(h)o.(b-y)]}sin

nltx a

, . (20)

where L. = (v. sinh u. b cos(h) v. b - u. cosh u. b sin(h ) v. b) -x

(21)

and [

[pd~ '/2

=to j

1/2 Ii

(7)']

1/2

for n <~,

for n > ~,

(22)

403

F O R C E D V I B R A T I O N O F T l t l N PLATES

where (23) Repetition of the above procedure for the same type of arbitrary boundary conditions H30,) sin ~ot and K3(y) sinogt at edge three with simple support elsewhere, and then again with arbitrary deflections H4(x)sincot and K,(x)sincot at edge four with simple support elsewhere, leads to the following plate deflection amplitude functions:

W3(x, y) = ~ J.{h3,. [s. cos(h) s. a sinh r. x -

r. cosh r. a sin(h) s. x] -

n

-

-

k3.. [sin(h)s.asinhr.x -

nny r.asin(h)s.x]} sin---~-,

sinh

IV4(x,y) = ~. L.{h4..[v. cos(h) v. b sinh u. y -

(24)

u. cosh u. b sin(h) v..I,] -

n

-

k4.. [sin(h) o. b sinh u.y

IINX

-

sinh u. b sin(h) o.y]} sin - -a

(25)

In order to obtain the fifth and final component of the extended plate deflection amplitt, de function, III5,consider the response ofthe plate shown in Figure 1 when it is simply supported at all four edges, and a pressure which is sinusoidal in time and arbitrary in space acts over its surface. In cases where the pressure amplitude function does not contain significant Fourier terms with index numbers greater than M, the pressure amplitude distribution may be approximated by a finite double Fourier series: i.e.,

q = P(x, y) sin cot =

~f

mTTx

m

sinmt ~ ~. pra., sin

II/~.},

sin--

ran

b'

a

(26)

where a

b

n=y.. Pm..='~4 f f P(x,y)sin m=x. a sm--~-axoy. o 0

After elimination of the factor sino~t, the plate equation becomes tV~, . . . . + 2W5 . xxry + IV5. , m - ~P~

= ~I

~ ~ m

pra,. sin mrCXasin nnyl._ .

(27)

/I

The expression for Ws(x,y) which satisfies both the governing differential equation, equation (27), and the boundary conditions is easily shown to be M

M

IHTTX

tVs = Y Y//ra,. sin ran

.

I1~)'

sm a

,

(28)

b

where pra, II

//ra..

=

.

(29)

It is not difficult to avoid values of co which cause the denominators of the quantities/I . . . . J, and L, to be zero, or very small quantities. More complicated expressions for IVs, which eventually result in single term expressions rather than series summation expressions for the loading factors of the final simultaneous equations can be readily obtained when P(x,y) has a simple form, such as being a constant

404

B.K. DONALDSON

over the plate area. In brief, the procedure is to use a Fourier series expansion for the deflection amplitude function which identically satisfies simple support boundary conditions on two opposite sides in order to satisfy the non-homogeneous governing differential eciuation. This particular solution is then combined with a Ldvy series complementary solution which also identically satisfies the same simple support boundary conditions. Finally, the constants of integration of the L6vy series are adjusted so that the combination of the two series also satisfies the simple support boundary conditions at the other two opposite sides. For the sake of brevity, only the Navier approach will be discussed in this paper. Superposition of the foregoing five deflection amplitude solutions, t1"1 through Ws, provides the deflection amplitude solution for the extended plate. Since this extended plate is restricted to a rectangular shape and zero vertical deflections at the plate corners, its deflection amplitude solution is only of limited usefulness in its present form. For example, the extended plate deflection amplitude solution cannot be directly fitted to the requirements of a rectangular FFSS plate (F = free edge, C = clampled or fixed edge, and S = simply supported edge boundary conditions) because of the general necessity for a deflected corner at the intersection of the two free edges. These restrictions with respect to rectangular shape and zero corner deflections are circumvented in the next section. Other component solutions for the extended plate may be added to certain combinations of the first five component solutions. For example, an additional solution which neutralized the slopes at side four produced by tVx + lV2 + W3 + ;Vs would result in an extended plate with a fixed edge at side four. This approach has little value for single plates; discussion of it may be found in reference [6].

3. THE ACTUAL PLATE ANALYSIS The solution of a given plate problem by use of the above extended plate solution begins by locating the boundaries of the actual plate within the boundaries of an extended plate. The next step is to write the boundary condition equations for the actual plate in terms of the extended plate deflection amplitude function. These boundary condition equations require the arbitrary edge deflections of the extended plate to be such as to produce the required boundary conditions for the actual plate. The last step is to solve simultaneously for the extended plate edge deflection Fourier series coefficients ha,,, ka . . . . . . . which are essential to the extended plate solution. The actual plate solution is then the extended plate solution when the latter is confined to the domain of the actual plate. The details of the above procedure will now be clarified by several example problems. Consider as an example actual plate a rectangular CCCC plate loaded by a given pressure. To obtain a solution for the deflection amplitude function of this actual plate, place the actual plate within the boundaries of a rectangular extended plate as shown in Figure 2. The distances c, e, ~, and ~ are essentially arbitrary, but as discussed later, different ranges of these variables produce different rates of convergence of the series solution for the deflection amplitude function. The extended plate pressure distribution is chosen so as to coincide with the actual plate pressure distribution over the domain of the actual plate. The pressure distribution over the remainder of the extended plate is arbitrary, but it should be a very smooth extension of the pressure distribution over the actual plate in order to avoid poor series convergence. When the actual plate is positioned within the extended plate and the actual pressure distribution is extended, the actual plate becomes just an enclosed portion of the extended plate and therefore the extended plate deflection amplitude function applies also to the actual plate. Then the boundary condition equations of the actual plate may be written in terms of the

FORCED VIBRATION OF THIN PLATES

405

deflection amplittide function of the extended plate, IV= IV~ + W2 + W3 + tV4 + Ws, as follows 9 W(c, y) = O, w ( a - e, y) = O, rv, x(c,y)

= o;

W(x, e) = 0, IV, .(x, e) = 0;

w , , , ( a - e,y)

= o;

W ( x , b - P.) = O,

IV, .(x, b - E) = 0.

(30)

The first of the above equations may be written as (GI. , h i . , + G2. ,kl. ,) sin y + Fs..k2..sinhu.(b

+ ~ [ F 3 . , h 2 . . s i n h u . ( b - - y ) - F 4 . , h 2 . , sin(h) v.(b - y ) +

9 /nzy ---.y) - F 6 . . k 2 , .sin(h) v.(b - y)] + ~ (FI..113.. - F 2 . . k 3 . . ) s i n - - f f - +

+ ~_ IF3,. h4.. sinh u. y - F 4 . . h4.. sin(h) v. y - F s . . k 4 . . sinh u. y + F6,. k4.. sin(h) v. y] + nr~y

+ ~ P s , . sin---~- = 0.

(31)

The values of the constants F~,., G~.., etc., may be found in Appendix II. )x

1I ....

e

!

T

(b-e-~)

(o-c-~) ....

i

I I

b

t

Figure 2. The actual plate and extended plate relation for a fully clamped plate. (The actual plate boundaries are indicated by the inner solid line, while the extended plate boundaries are indicated by the outer solid line.)

Multiplying both sides of equation (31) by the factor (2/b)sin(mrcy/b) and integrating over 0 < y < b yields G l . . , h l . , " + G2. mkl.m + ~ [(Ux . . . . F 3 . , - U2 . . . . F4. n)h2., +

+ (UL... , F s . , --/-/2 . . . . F6.,) k2.,] + F,.,"h3. ," -- F2...k3. m + + ~ [(Us.,".,F3.. -- U 6 . , . , . , r , . , ) h . = . , - (Us . . . . r s . , - 0"6. . . . r 6 . , ) k , . , ] = -Ps.,".

(32)

The remaining seven equations of equations (30) become -GT. ,.It,., - as. ,.k,. ," + ~. [(U, . . . . Fg., - U2 ..... F , o . . ) h 2 . . +

+ (Ux . . . . F n . . ,

U2 . . . . F 1 2 . . ) k 2 . . 1 + FT.,"ha. ," - Fs.,"k3. m -1-

+ ~ [(Us . . . . Fg.. - U6. . . . F, o..) h4., - (0"5. . . . Vl 1.. - [/6 . . . . /;'12. ,) k4., ] = -P6. ,.. (33)

406

B.K. DONALDSON

~. [(U3.,,..F,3.. - U,, . . . . F t , . . ) l h . . + /I

(U3. . . .

F,5. ,, -- U4 . . . . F , 6 . . ) k , . . ] + G,.t.,,,h2.,,, +

+ Gts.,,k2.,. + X [(UT.,... F,3.. -- Us.,,..F,.,.,,)h,.. -n

-- (0"7. . . . F a 5 . , - Ue . . . . Fl6. n)k3..] + F,7.,,h4.,,,- F , s . , , k , . ,, = - P T , , , , n

[(s . . . . r 2 , . . -- G25. mh2. m -

(34)

U,, . . . . Fn.,,)lq.,, + (U3 . . . . F 2 3 . . - U, . . . . F2,..) k t . . ] G26, mka. rn + E [(U7 . . . . F 2 , . . -

U8 . . . . F22.n)h3.. -

It

(35)

-- (07 . . . . F23,. -- Us . . . . F24., .) k3..] + F25. mh4. m -- F26. = k4.,. - - P s . , , , , _r,.., h,.. + ;,... k,... + 7 [(U, .... G~.. - Us .... G,..) h~.. + n

+ (u,.,..,,r

Ui,...C,..)k2..]

+ O,.,,,i,~.,. - G=..,k~.,. +

-+ E [(U, . . . . C ~ . . - U, . . . . G,,.)h,.. - ( U , . . . . d , . . n

- F,.,.h,..,

- F,.,.k,.,.

+ E [(U, ....

09.. -

n

Jr ( U , . . . . O l l . . -- Us . . . .

= -P,.

(37) Jr

(U3 . . . .

+ F,~.,.h=. ,. + Fls.,,,k2.. + 7 [(U~ . . . . r

7 [(u, . . . . C , , . . /I

I

u. . . . . r

F25. m h2. m - - 6 6 .

O,,..)k~..]

+

GIS. m - - U4. m. n G l 6 . n ) k l . n ]

O,~.,. h,.,. - G,,.,,k,.,,

h,.. + (u, . . . . C . , . . -

m k 2 . m "}- ~ [ ( U 7 . . . . n

Jr

- Us.,,.. G , , . . ) h 3 . . -

n

u, . . . .

+

=,

~- [ ( U 3 . m. n C l 3 . n - - U4. m. n C l 4 . n ) h l . n It

-

O,o..)h2..

- (u, .... c,,.. -

n

- (u~..,.. (3,,..

[I2 ....

(36)

(~12..)k2..] Jr GT.,.h3.,.- Gs. mk3.,. Jr

+ 7 [(u, . . . . o , . . - u , . . . . c , , . . ) h , . . -- U,. ,... G,2..) k,..]

U, .... C,,.)k,,,.] =-P,,,.,

G2I.,

-- 08 ....

= -P,..,,

(38)

v, . . . . r (~22. n) h 3 . , - -

- (U~ . . . . G23.. - Us . . . . C2,..)k3..] + G25. mh4.,. - d26.,,k4.,, = - P , . , . .

(39)

Equations (32) through (39) are 8N equations in 8N unknowns. Their solution for the extended plate boundary deflection Fourier series coefficients may proceed in any standard fashion, or, when these equations are cast in matrix form, use may be made of the regular pattern of diagonal matrices in order to eliminate half of the unknown coefficients while only inverting and multiplying diagonal matrices. Once the Fourier series coefficients are determined, the deflection amplitude function for the actual plate is known for all points on the plate. When the series convergence is good, stresses can be determined by differentiation of the deflection amplitude function. The above analyses, as applied to the CCCC plate, is little different from the superposition of standard Lfvy series analyses. In fact, the above analysis reduces to superimposed L6vy analyses when c = t~= e = 6 = 0. No advantage over the superposed L4vy series approach can be claimed in this case from either the point of view of the number of unknowns involved or from the point o f view of convergence o f the series solution. However, since this case involves four partitions of the extended plate, numerical results for this CCCC plate problem will be used to partly demonstrate the validity of cutting the extended plate to obtain an actual plate. There is a small advantage in using an extended plate when one side or two adjacent sides of an actual plate are simply supported. In these cases the number of unknown Fourier series coefficients to be determined is less than that for the superimposed L6vy series approach.

FORCED VIBRATION OF THIN PLATES

407

For example, consider as an actual plate a rectangular CCSS plate which is loaded by a given pressure that is harmonic in time. To obtain an efficient solution for the deflection amplitude function of this actual plate, choose a rectangular extended plate such that II3(y) = Ka(Y) = H4(x) = K4(x) -- 0. Then sides three and four of the extended plate are simply supported because the above boundary deflection functions, rather than being total deflections and slopes, are only increments in deflection and slope beyond those at a simply supported edge. Locate the actual plate within the boundaries of the extended plate as shown in Figure 3 where the respective simply supported sides of the extended plate and the actual plate are coIlinear. Again, the distances c and e are essentially arbitrary. The extended plate pressure distribution is chosen as before, and the actual plate again becomes just a portion of the extended plate. Since the simply supported edges of the actual plate coincide with the simply supported edges of the extended plate, the only boundary condition equations for the actual plate thatneed to be written are W(c,y) = o, W(x, e) = 0, IV, ~(e, y) = 0; IV, ,(x, e) = 0. (40) With W = IV1 + 1II2+ ti"5 in this case, the first one ofthese equations again becomes equation (31) and then equation (32), but now h3., = k3., = ha., = k~., = 0 because sides three and four are simply supported. The remaining three of the above equations are equations (33), (34), and (35) with the same modifications. Equations (36) through (39) simply disappear as zero equals zero identities. Thus there are only 4Nequations to be solved in 4N unknowns in this solution for IV, as opposed to 8N equations in 8N unknowns that would result from superposition of Ldvy series.

~/--

,

H2(x),K2 (x)

V

(~-e)

K~Cy)

Y i

(a-c)

r

>

I

Figure 3. The CCSS actual plate and its associated extended plate. (The actual plate has the dimensions (a-c) by (b-e), while the extended plate has the dimensions a by b.) As an example of a rectangular plate for which c and e cannot be zero, consider an FFSS plate. Non-zero values o f c and e are necessary in order to have a non-zero deflection at the free corner of the plate, and of course, convergence is aided ifc and e are not too close to zero. As before the analysis begins by making the actual plate an included portion of the extended plate by placing the actual plate within the boundaries of the extended plate as shown in Figure 3, and extending the actual pressure loading over the area of the extended plate. Since sides three and four of the two plates are simply supported, the deflection amplitude function is IV= ti"1 + ti"2 + tVs. The boundary conditions that remain to be satisfied are

M:,(c, y) = O, V'~(c,y) = O,

My(x, e) = 0, V,(x, e) = O,

(4 i )

408

B . K . DONALDSON

where M x = --D[W, ~ + t~V,.], Vx = -D[IV, x~x + (2 - it) W, :,j,,], My

=

--D[ ;V,,, + tttV, ,,x],

(42)

V, = -D[IV,,,, + (2 -/t) W,,~].

Thus, for example, the first of equations (41) becomes M,,(c,y) = ~

nny nny A l . . k ~ . . s i n - f f - + A2..hl..sin--b---+ A3..k2..sinh u.(b - y ) +

+ A . . . k2.. sin(h) v.(b - y) + An,. h2.. sinh u.(b - y) +

7}

q- A6.. h2,. sin(h) v.(b - y) + P2t.. sin - -

= 0.

(43)

The constants Aj.. and Bj.. are listed in Appendix II. To make the error orthogonal to the first Nsine functions, multiply both sides of equation (43) by sin (mrcy]b), where m = 1,2, ..., N, and integrate over 0 < y < b. The result is A~.=kl.= + A2...h~.,. + ~ [Ua. . . . A3.. + U2. . . . A4..)k2,. + n

+ (UI . . . . As.. + 0"2..... A6,.)h2,.] + P2t.,. = 0.

(44)

Similarly, for the remaining three of equations (41), B,.,, kl.,, + B~.,~h,.m + ~ [uz.,,.. B3.. + U2. . . . B , . . ) / q . . + n

+ (U, ..... Bs.. + U2. . . . Ba..)h2..] + P22.,. = 0,

(45)

X [(u3 . . . . A,~.. + U, . . . . A,,..)~:I.. + (U~ . . . . A , , . . + U, . . . . A,6. .) /,,. .] + n

+ A4v.., k2.., + A4s. ,. h2. ,. + P25. ,, = 0,

(46)

Y [(u3 . . . . B,3.. + U, . . . . B , , . . ) k , . . + (U3 . . . . B , , . . + U, . . . . B,6..)/,,..] + n

q- B47.mk2, m -F B48. mh2.,, q- P26, m = 0.

(47)

Equations (44) through (47) represent 4N equations in 4N unknowns. Their solution for the Fourier series coefficients determines the deflection amplitude function W. A non-rectangular actual plate provides a more interesting application of the present method with respect to single plates. The CCSS plate pictured in Figure 4 will serve as an example, where the simply supported sides three and four of the extended plate and the actual plate are made collinear, and the actual pressure distribution is extended over the area of the extended plate. The deflection amplitude function is again IV= ti:1 + tt,'2 + ;Is and the boundary conditions to be satisfied are

409

FORCED VIBRATION OF THIN PLATES

9

b

I ,t

+

t

Figure 4. The extended plate and actual plate arrangement for a CCSS right angle trapezoidal plate.

where the geometry of the oblique line is x

a

/'

g--f y = -----7--~ + f .

(49)

The last of these conditions may be rewritten as

W,,

OI,VOx

0W03, +-Ox Oz Oy Oz

f-g

- -

!

t7

W , ~ + 7 tV,, = 0 ,

or

f--g --W,x+ a

W,r = 0.

The boundary conditions at x = c, where c may be zero, are the same as those at x = c for the rectangular CCSS plate, and thus need no further discussion. The boundary conditions along the oblique cut are equally straightforward since they only involve the use of equations (49) in the arguments for IV, tV, x and ll:,r. After multiplying these quantities by (2//)sin Onn~/l) and integrating over ~ from 0 to 1, the oblique edge boundary conditions become Y [(u~ . . . . E , . . -

U,o . . . . E~..)h,.. + ( U ~ . , . . . E ~ . . - U,o . . . . E+..)k,.. +

+ (U, ~, ~ . . E s . . -- U,~ . . . . E~..) h~.. + (U,x . . . . E~.. -- U, ~. . . . E~..) k~..1 = - P ~ . ~,

f-a g ~

(50)

[(Ux.. ,... Ex... -- U~8..,,. Eas. ,) lq.. -- (U,~. ,... E,9.. -- U,s. ,... Eao..) k~.. + 11

+ (U,9 . . . . E2t.. -- U20. . . . E22..)h2.. + (U,9 . . . . E23., - U20. . . . E2+..)k2..] + + Y~ [(u,~ . . . . ~ . .

- u , , . . . . E~o..)/,,.. + (u,3 . . . . E , , . . - V,, . . . . E ~ . . ) k , . . +

n

+ (Uls . . . . Et3.. -- U,6 . . . . Et+. ,) h2,, + (Ua~ . . . . E~s, n - f - - g Pll.,,--Pao.,.. a

U16 ....

Eta, .)/<2..] (51)

410

13. K. DONALDSON

As before the solution of the 4N boundary condition equations for the extended plate Fourier series coefficients determines the extended plate amplitude function. Clearly additional oblique cuts can be made in the extended plate to achieve other geometries, and the boundary conditions associated with these oblique edges can be written in the manner demonstrated. Furthermore, triangular plates can be obtained by simply letting the length of one of the quadrilateral edges be zero. As stated previously, numerical results for the above examples, plus others, will be offered in the third o f this series of three papers [5]. Those results show good accuracy, convergence, and independence o f the dimensions of the extended plate except when there are free edges. Therefore, the present method and the rectangular extended plate recommend themselves when the actual plate is non-rectangular, and the boundary conditions are those of fixed or simply supported edges: This suggests that other boundary value problems where the boundary conditions do not involve many differentiations of the unknown function, would be particularly suitable for the present method. The plane stress problem of the theory of elasticity when the boundary conditions are expressed in terms of the Airy stress function and its derivative normal to the boundary, may be such a problem. ACKNOWLEDGMENTS The author would like to thank Y. K. Lin for inspiration and for reviewing this work. The author would like to thank the Minta Martin Fund at the University of Maryland for various grants that enabled the gathering of numerical results. REFERENCES 1. A. W. LEISSA,W. CLAUSEN,L. HULr~ERTand A. HOPPER1969 American h~stitute of Aeronautics and Astronautics Journal, 7, 920-928. A comparison of approximate methods for the solution of plate bending problems. 2. H. FLE'rCHERand C. THORNE1954 Proceedings of the Second U.S. National Congress of Applied Mechanics, 389--406. Bending of thin rectangular plates. 3. S. TIMOSHENKOand S. WOINOWSKY-KRIEGER1959 Theory of Plates and Shells. New York: McGraw-Hill Book Co. Second edition. 4. B. K. DONALDSON1973 Journal of Sound and Vibration 30, 419--435.A new approach to the forced vibration of fiat skin-stringer-frame structures. 5. B. K. DO.'aALDSO,'qand S. CHANDER1973 .[ourttal of Sotatd and Vibration 30, 437-444. Numerical results for extended field method applications. 6. B. K. DONALDSON1968 Ph.D. Thesis, UniL'ersity oflllinois, Urbana. A forced vibration analysis of flat plates continuous in two directions over open section beams.

APPENDIX I LIST OF SYMBOLS

extended plate length in x-coordinate direction b extended plate length in y-coordinate direction x-coordinates of extended plate partitions C, d plate thickness e, y-coordinates of extended plate partitions ~e y-coordinates of the end points of an oblique cut in an extended plate hj., nth Fourier series coefficient of the function Hj wherej = 1, 2, 3, 4 i square root of minus one kj., nth Fourier series coefficient of the function k t.wherej = I, 2, 3, 4 i length of an oblique cut in an extended plate 111, tl, p summation indices (/

F O R C E D VIBRATION O F T I t l N PLATES

411

double Fourier series coefficient of the pressure amplitude function P(x,y) pressure loading over the plate area rn, Sn quantities defined by equations (7) and (9) t time lln, Vn quantities defined by equation (22) wj(x,y, t) plate lateral deflections. The subscript j refers to a specific set of boundary conditions; j = 1, 2,3,4,5 x, y plate Cartesian coordinates Z coordinate normal to an oblique cut AI, n, Bj.n constant coefficients of plate edge moment and shear equations listed in Appendix II constants of integration in the solution for the elementary Plate deflection amplitude functions D plate stiffness factor, Ed3/12(l - 1~2) D j , m,n constants of the pressure loading; see Appendix II E Young's modulus Pro~ n

q(x,y, t)

Ej .,Fj "1 G t'~, ff_3'.l constant coefficients of plate edge deflection " Gj..)

ttl.n

equations; see Appendix II

independent vertical deflection amplitude function for thejth edge

J~ constant defined in equation (19)

L~ ~L, M,

M,N

2~

P(x,y) e J . nl P J . n UJm m. n

v:,v, ~G X~ ,u P CO 2-~m. it

independent normal slope amplitude function for thejth edge constant defined in equation (21) plate moments in the x and y directions, respectively, per unit length in the y and x directions, respectively maximum values for the indices sgn (n -- ~) sgn ( n - fi) pressure amplitude function for a time varying pressure loading constants of the pressure loading defined in Appendix II constant coefficients defined in Appendix II plate edge Kirchhoff shears per unit length of plate in the y and x directions, respectively plate lateral deflection amplitude; the subscript j refers to a specific set of boundary conditions nth coefficient for the L6vy series solution for the first set of boundary conditions contributing to the extended plate solution (2 -/a) Poisson's ratio coordinate along oblique portion of an extended plate mass density circular frequency of forced vibration double Fourier series coefficient of the deflection amplitude function tVs

APPENDIX II DEFINITIONS

9

b2

]

-- ( ~s~_ Im27r21sinhr~asin(h)sn(a_ c) ] b' ] "

412

B. K. DONALDSON

-~s } cos(h)s.asinhr.(a- c) --

9

- ( ~".s 2 .

/m2n21coshr.asin(h)s.(a-c)] /

[ n 2n 2 \ nnc A3.. = +DL. ( - - ~ - - tad) sin(h)v.bcos -a ' 112~2

)

17TCC

A,., = -DL, ~ a---7- -- Nlw~ s i n h u . b c o s - - . a

.45., = +DL, v, ~ a---7- -- I~u~ cos(h) v. b cos .nnea A6.. = --DL. u. ~ a 2

~llw] cosh u. b cos, a

!12~2 2~ nrce A,3.. = +DJ~ [---~----/lr.) sin(h)s~asin--b--, [112~2 " 2 117[e A44.. = -DJ. ~--~- - N ~ . sinh r,,asin - b'

A4~. . = + DJ. s. {\--~---n" n 2 pr2.) cos(h)s.asin--b tree ' i.12~2

)

lille

A46,. =-DJ.r. ---~- - fillts2, coshr~asin, b '

9

s i n ( h ) v . b s i n h u~(b - e) -

- (iVy]

A48.,=--DL,

Im~-~-2.1sinhu.bsin(h)v.(b-e)] a2 ]

v. u2.

a" ]

--u. (.~v2. /m~-Zlcos2hub.sinJ(h)v(.b-e)al, B , . . = + D J . [ r . ( r ~ - f l t?b2n2 ]~ sin(h)s.acoshr.(a- c) / n2n2\ ] -- S.(l~'s2. - fl"'~)sinhr.acos(h)s.(a-c) ,

413

FORCED VIBRATION OF THIN PLATES

I-[

n~\

B2,.=+DJ~r.sn[~rZ.--fl--ffi--Jcos(h)snacoshr~(a-c) -- ( ~s] -- fl ---~-f-] n2 n2/cosh r. a cos(h) s.( a -- c) B3..=+D

(7) L. (~2~) ---~i----flu2,

-

]

=

sin(h)v, bcos--a

[ n 2 ~Z2 \ n~c L" [---~- -- Nflv2") sinhu"bc~ -'a

(7) L~vn\~a----T---flu z ) cos(h)v.bcos-.a,,~c'

Bs.~= +D --

.~.~_o(n__:)

[112~2 -- 2~ Lntln l - - ' ~ - flNvn) c~176 [ 1121~2

~

rl~C n~e

sinh rn acos

tl~e b '

J,s, ~ n2 n2 - fir2,) cos(h) s, a cos nneb ' ( lbT~) ]k[1121~2 2x~] IlKe B46 n .~---D - - Jnrn [--'-~-- fl]~Sn| coshrnacos , 9

b

(

B47.,=+DL,

n2~2\

u.[u2, - fl---~-) sin(h)v.bcoshu,(b-e) -

- v . (~v2,-fl n2n2/sinhu.bcos(h)v.(b-e)] a2 ]

B4s. , = + DL. u. v, [ [ u, - fl --~ ) cos(h ) v, b cosh u,( b - e) --(Rv2_fln2rt---~2]coshu, bcos(h)v,(b-e)] a2 ]

414

n . K . DONALDSON D3 . . . . = {[(m -- n) 2 x 2 + (g _ f ) 2 u2] [(m + n) z rc2 + (g _ f ) 2 uZ]}-l,

D4 . . . . = {[(m -- n) 2 n 2 + ~l(g _ f ) z 02] [(m + n) 2 n 2 + N ( g _ f ) 2 v2]}-1,

El,. =J.s.cos(h)s.a,

E2,. = J . r . c o s h r . a ,

E4.. = J . s i n h r . a ,

E,.. =J.sin(h)s.a,

Es.. = L.v.cos(h)v.b,

E6,. = L . u . c o s h u . b , nR

ET.. = L . sin(h) v. b,

IlTE

Eg.. =-3-e1..,

E s . . = L . sinh u. b,

E,o..=TE~..,

E,I.,, = TE.,..,

EI~.,, = TE,,.,,,

E13.. = v. E6..,

Ex4.. = u . E s . . ,

E l s . . = v. Es.v,

E16.. = u.E~..,

E17.~=r.Et.n,

Ex8..=s.E2..,

E I g . . = r.E3,.,

E 2 o . . = s.E4..,

E21' n = - - ' a E S " n '

n lz E23.n="~ET.n,

1~17[ E24. n=--~Es.n.

117[

nl~

E22' n=--E6a

. n,

T h e constants F j . . , G j . . , a n d (Tj,. are o b t a i n e d , respectively, f r o m F~.. b y replacing c by 6, a-c, o r a-6, a n d replacing e b y g, b--e o r b-g, as a p p r o p r i a t e . F o r e x a m p l e ,

F I . . = J.[s. cos(h) s. a sinh r. c - r. cosh r. a sin(h) s. c], F t . . = J.[s. cos(h) s. a sinh r. 6 - r. cosh r. a sin(h) s. 6], Gt. . = J.[s. cos(h) s. a sinh r.(a - c) - r. cosh r. a sin(h) s.(a - c)], (71.. = J . [ s . cos(h) s. a sinh r.(a - 6) - r. cosh r. a sin(h) s.(a - ~)],

F2.. = J . [ s i n ( h ) s . a s i n h r . c - s i n h r . a s i n ( h ) s . c ] , F 4 . . = L . u . c o s h u . b s [ n nnc

= L . v. cos(h) v. b sin

Fs .. = L . s i n ( h ) v . b s i n nnc- ,

,

a

r•..=

nRc

s

F6 .

a

a

. = L . s i n h u . b s i n .nnc a

J . r. s.[cos(h) s. a cosh r. c - cosh r. a cos(h) s. c],

F s . . = J . [r. sin(h) s. a cosh r. c - s. sinh r. a cos(h) s. c],

I11"~

Fg. n = - -

a

117[C

L.v.cos(h)v.bcos --, a

Ill'C

n~c

a

a

FIt.. =--L.sin(h)v.bcos

,

nil

nT[c

a

a

Fxo,. = N L . u . c o s h u . b c o s Ill'~

n~c

a

a

Fl2.. = - - L . sinh u . b c o s - - ,

nlze

F13.. = J . s . c o s ( h ) s . a s i n - -b '

Fls,.=J.sin(h)s.asin

,

n~e

F14..=J.r~coshr.asin

nlze

lille

b'

Fa6..=J.sinhr.asin--ff-,

b '

FiT.. = L.[v. cos(h) v. b sinh u. e - u. cosh u. b sin(h) v. el, Illr

Fls""=L"[sin(h)v"bsinhu"e-sinhu"bsin(h)v"e]'

F2t'"='b -J"s"c~176

title

b'

415

FORCED VIBRATION OF TttlN PLATES

nn nne F22..=-~J.r.coshr.acos b '

tin nne F23""="b J"sin(h)s"ac~ " b '

tin nne F~4..= - c J.sinhr.acos "--~-,

F2~..= L.u.v.[cos(h)o.bcoshu.e-coshu.bcos(h)v.e],

F26.. = L . [ u , sin(h) v. b cosh u. e - v. sinh u. b cos(h) v. el. T h e constants P~,. are obtained from Ps,. by replacing c or e by (a-g) or (b-~), respectively: P s . . = Y - / / , ~ . . sin - - , m

P6.. =

//m.. cos --

a

a ill t

P7 .'--- ~ / / . ' ,.

"

,. sin - b

Ps . = '

'

M

DllIf H,. ,, sin - = b Mm m~f + 89 ~. ~ / / p . = sin - j, e 9 . n ~--

m

m

~ b - ) - - H . ,. c o s - " b

ifn=p

andg=f,

if (n -- p) b = + ( g - f ) 171 and g # f ,

b

mm

mrtf

t, m

b

if (11 + p ) b = ( g - f ) m and g # f ,

--89 Y ~ / - / , , . ,,, s i n -

-4

L

~

m n p b a ( g - f ) [ (-l)"§176

M

[ ( n - - p ) 2 b 2 __ m 2 ( g m

_f)2]

0

m, f

-- cos ~ - ]

[(71+

p)2 b 2 _

m2(g

_f)z]

otherwise;

i f n = p and g = f ,

b / L . , . cos ---~m

1 M ~

mr~

m~f

Z - r "'.'c~

m

Pro. n = "

b

~

mrc

-

7o

if (n - p) b = + ( g - f ) m and g # f ,

0

,nrtf n,..

cos --g-

if(n +p)b = (g - f ) m

and g # f ,

m

M M

4YZn..m m 0

otherwise; [(77 p)Z b 2 _ m 2 ( g _ f ) 2 1 [(n + p)2 b z _ m2(g _ f ) 2 1 -

-

ifn=p

"0 1 M 0

Pxl, n =

M M

if (n - p ) b = +(g - f ) n,

m

_~1 M p

mnf

andg=f,

~Pn IIo,,.COS

b

if(n + p)b = ( g - f ) m ,

m

np2(n-P)b4

(-l)"+~

b

.

j

4 ~. ~. 1-1o.= otherwise; p ,. a[b=(n _ p)2 _ m2(g _ f ) 2 ] [ba(n + p)2 _ ma(g _ f ) 2 ]

416

B . K . DONALDSON

q- ~U

P21. n = --]-.D ~m/'-/m. n

~m

sin mn___~Ca

(7) [(7), (m+)'] ,+ [(~)'+ (7)'] m+

P22,.

=

q-.O

/-Ira""

P2s..

=

+ D ~ H ....

+ (2 - It) ~

It

cos - - a

sin

nrc 2 m.e ~+=+O; (_~_)~ .... [(.~)2 +,,,,(;)]co., b

2 f sin tony 2ran sinh u. b UI . . . . = b j b sinh ,t.(b - y) dy = (ran) 2 + (u. b) 2 ' o '-(-1) m U2 . . . .

=

i f n < ~ and m n = v . b ,

2ran sin(h) v. b

otherwise,

(,,m) 2 + R(v. b) ~

2ran sinh r. a U3. m. n = (mrO 2 -t- (r. a) 2'

v,.m.,=

if n < h and mrc = s. a,

-(-1) ~ 2mn sin(h) s. a (mr02 + ~ ( s . a) 2

otherwise,

u,.,,., =-(-1)" u, .....

u , . , .... = - ( - 1 ) m u2. ~,.,.

u7 ....

u8 ....

= -(-1)

~' u 3 . . . . .

v~ . . . . = ~ , . . . .

= -(-1)"

t_z,. m . , ,

'4m,,." I - - - r - ( ~ - : ) r + o k c o - r~ h r ~ o c o + - r - ..

--2m~

U1o 9 ,,,. . = D 2 . s.. [

2 sinhr.asin

-m2~Z-r~a

~

b

(-~)~cos -'7] -~ ~ -

acos

--~-

t-l)

,

cos--b--]

-

-2m~[(n.~)2(g-f)Z-mZn2-Jfls~a2]sin(h)s.asin~}, = / sinh u . ( b - g ) Ull. ,...

if m -- n and g = f ,

[ 93 ..... {4mnn2(g -f)u.

sin(h) v . ( b - g ) U12 . . . .

=

D. .... {4mnn2(g-f)

[cosh u . ( b - f )

-

( - 1 ) m§ cosh u . ( b - g)]},

if m = n and g = f , v . [cos(h) v . ( b - f ) - (-l)m+"cos(h) v . ( b - g)]},

417

FORCED VIBRATIONOF THIN PLATES

.... oim.{4m' 2 rna[ l,'sin' coshr.asin ] 4mnrr2 ~ - f )

V l . . . . . = D2 . ,~. .

- 2mrc

[

~

nng

sna 1(-1) sin --~

nrrf] cos(h) sn a sin ---~-]

(g _ f ) 2 _ m 2 ~2 _ ~ s 2 a 2 sin(h) sn a cos

cos(h) v.(b - f )

-

,

if 112= n and g = f ,

U15 . . . . = [ D . . . . . {4mnn2(g - f ) N v . [sin(h) v.(b - f )

if m = n and f = g, U16 . . . . = [ D3 . . . . {4mnrr2(g - f ) u. [sinh u.(b - f )

- ( - 1 ) =+" sin(h) vn(b - g)]},

cosh u.(b - f )

rna s i n h r n a c o s ~

U17 . . . . = DI . . . .

[ 4mnn2(g-f)

-2mrr

0 U19 . . . . =

- (-1) ~+" sinh u.(b - g)]},

-

fils~a[sin(h)s~acos

(g-f)2-m2n2-~s~a2

cos(h)s.asin

~

(-l)'sin

,

if (m + n) = even and g = f ,

D3 . . . . {2mn[(m 2 _ n 2) n2 + (g _ f ) 2 u]] [sinh u.(b - f )

0 if (m + n) = even and g = f , U2o, ,.. ~ = [ D4, ,., n{2mn[(m 2 - n 2) 7I2 + - N ( g f ) 2

- (-1) "+~ sinh u.(b - g)]},

V2] [sin(h) v~(b - f ) - ( - 1 ) =+" sin(h) vn(b - g)]}.