A new approach to the forced vibration of flat skin-stringer-frame structures

Journal of Sound and Vibration (1973) 30(4), 419--435

A N E W A P P R O A C H TO THE F O R C E D VIBRATION OF FLAT SKIN-STRINGER-FRAME STRUCTURES B. K. DONALDSON

Department of Aerospace Engineering, Unirersity of ~Iaryland, College Park, Maryland 20742, U.S.A. (Receired 22 December 1972) This paper presents a new, approximate method of analysis for the response to forced harmonic vibration of a finite structure composed of a fiat plate continuous over, and rigidly attached to, a grid of open section beams. The beam grid may have any form, the fiat plate may have polygonal cutouts, and the structure as a whole may have any combination of boundary conditions. The method of analysis is one where finite series solutions are used to satisfy the boundary conditions and equations of continuity as well as the plate and beam governing differential equations.

I. INTRODUCTION This paper presents a new approximate method of analysis for the response of a finite, flat skin-stringer-frame structure undergoing forced harmonic vibration. A fiat skin-stringerframe structure is one which is composed of a thin flat plate, the skin, continuous over a grid of beams, the stringers and frames. The skin is assumed to be rigidly attached to the beam grid and it is considered to be divided into a series of smaller plates each of which is of uniform thickness. The boundaries of an individual plate are defined by either the beam grid, the boundaries of the structure as a whole, or the requirement for uniform thickness. The beam grid is similarly divided into beams of constant cross-sectional properties whose end points are defined by either the joints or end points of the beam grid or the requirement for constant cross-sectional properties. All the beam and plate elements of the structure are assumed to be linearly elastic, homogeneous, and isotropic. There are many approximate methods of analysis available for use with such a structure. Finite element methods, Rayleigh-Ritz, and finite differences are perhaps currently the most popular, and each can generally be expected to produce good results. However, none Of the currently available methods works equally well with all skin-stringer-frame combinations. Therefore, a new approach to the skin-stringer-frame response problem is worth considering. The new approach offered in this article is in the classical style of matching plate edge lateral and slope deflections and equilibrating plate edge moments and Kirchhoff shears. Typical of previous work in this vein is that described in reference [1] where the subject of the analysis is a continuous plate also considered to be composed of individual rectangular plates. In that analysis, the dynamic form of the fourth-order plate equation and the lateral deflection and moment boundary conditions of each plate are satisfied by the superposition of Ldvy series. Since the boundary conditions are expressed in terms of Fourier sine series, the analysis of reference [1] is limited to continuous plates where the individual rectangular plates do not deflect laterally at their corners. The method described in reference [2] typifies another approach to satisfying the edge conditions between the component plates of a 419

420

13. K. DONALDSON

larger, continuous plate. In that reference the subject of the analysis is a single row of rectangular plates which are simply supported along the two edges which are common to all the plates of the row, and which are continuous over open section, simply sut~ported beams at the interior edges of the individual plates. In this case the L6vy series solution is cast into transfer matrix form thereby efficiently condensing the various boundary condition equations at the interior plate edges and beams. While that approach is compact, it is limited to a single row of rectangular plates, and it is at its best when dealing with free vibrations. The method of analysis offered in this article overcomes the limitations that are a result of plate components which are restricted to a single row, or which are constrained to zero lateral deflections at their corners. This is accomplished by using the plate analysis method presented in reference [3]. When that method is applied to the plate elements of a flat skinstringer-frame structure, a separate, fictitious, rectangular, "extended" plate is considered to enclose each actual plate of the structure. In brief, the extended plate has boundary conditions which consist of arbitrary lateral and normal slope deflections, except at the corners where, because the arbitrary edge deflections are represented by Fourier sine series, the lateral deflections are zero. By considering an actual plate element to be just an enclosed portion or partitioned segment of its associated extended plate, the actual plate boundary conditions are generally removed from the constraints at the extended plate corners, as they must be free from such constraints in the actual structure. This relationship between an actual plate and its extended plate also means that the actual plate deflection amplitude function can be obtained from that of the extended plate by simply restricting the domain of the extended plate amplitude function to the area of the actual plate. The deflection amplitude function of the extended plate is obtained in a straightforward manner by superposition of L6vy series which satisfy the deflection boundary conditions at the extended plate edges, and a series solution for the extended plate loading which coincides with the actual plate loading over the area of the actual plate. Differentiation of the extended plate deflection amplitude function provides the actual plate edge moments, Kirchhoff shears, and concentrated corner reactions which load the adjacent portion of the beam grid. Then with the beam loading so determined, the non-symmetrical open section beam bending and non-uniform torsion equations of motion can be integrated to obtain beam deflection amplitude functions. The beam constants of integration and the extended plate edge Fourier series coefficients, and hence the beam and plate amplitude functions, are determined by writing and solving simultaneously the eight beam boundary condition equations per beam and the equations of continuity between the plates and the beams. This is the only point in the analysis where a simultaneous solution involving more than two unknowns is required. The necessary plate equations are taken from reference [3], but the beam equations and the solution process are detailed in this paper. The present method has several limitations. Shear deformations are omitted, and it does not seem possible to extend the method to account for general in-plane forces. More serious limitations involve the differentiation of the series solution for the plate lateral deflections to obtain the plate edge loads which slows convergence, and the rapid linear increa?se in the number of unknown quantities with the increase in the number of plate elements. The latter difficultyrequires a decrease in the accuracy of the solution with an increase in the complexity of the structure, and vice versa, when the structure goes beyond just a few plate elements. The advantage of the present method with respect to other differential equation series solutions is, again, the greater geometric variety possible with this method. In addition to the lack of constraints concerning the arrangement of plate elements and plate element corner deflections, there is greater geometric flexibility because non-rectangular plate elements can also be part of the skin-stringer-frame analysis [3]. Note that cutouts in the structure are just the absence of plate elements. The advantages of the present method with

FORCED VIBRATIONOF BEAMS AND PLATES

421

respect to such methods as finite elements and R a y l e i g h - R i t z , if any, are not clear except for the fact there is no need to c h o o s e elements or functions.

2. THE PLATE ELEMENT Consider a skin-stringer-frame structure that is loaded by a pressure which has an arbitrary spatial distribution, but which is harmonic in time. The pressure loading acts directly u p o n the plate elements forcing them to deflect from their unloaded equilibrium position. Consider any rectangular plate element o f that structure: say, plate element alpha (see Figure I).

.

r .I

--

I,

Y

>

e

I i i i I

--x--

I I(b-e-~) i

e

L '

I

u

Figure 1. Plate element alpha geometry. (l'he inner dashed line is the actual plate alpha boundary while the outer dashed line is the extended plate alpha boundary. The distances c, e, ~',and ~ are essentially arbitrary.) F r o m reference [3], the lateral deflection amplitude function o f the actual plate alpha, using the terminology, symbols (see Appendix I) and sign convention o f that reference, is, in general, N

W(x,y)

= ~. J . { l h . . [ s .

cos(h) s. a sinh r . i a - x ) - r . cosh r. a sin(h) s . ( a - x)] + nTry

+ k l . . [ sin(h) s. a sinh r . ( a - x ) - sinh r. a sin(h) s . ( a - x)]} sin ---if-- + N

+ ~. L,{h2. ,[v, cos(h) v, b sinh u.(b - y ) - u. cosh u, b sin(h) v~(b - y)] + n

tlltX

+k~..[sin(h)v.bsinhu.(b-y)-sinhu.bsin(h)o.(b-y)]}sin

a

+

N

+ ~ J.{ha..[s.cos(h)s.asinhr.x-

r.coshr.asin(h)

s.x] -

n

nny

-- k~,. [sin(h) s. a sinh r. x - sinh r. a sin(h) s. x]} sin T

+

H

+ ~. L , { h 4 . . [ v . cos(h) v. b sinh u n y - u, cosh u, b sin(h) v. y] Ill~X

-k4,.[sin(h)v.bsinhu.y-sinhu.bsin(h)v.y]}sin a ~,r m

+ ~. ~ H., . s i n

m~x

-Iv

n~y

sin

,

(1)

where c < x < a - c and e < y < b - e, and all s u m m a t i o n s range from either one to N or one to M, where M and N a r e chosen so as to achieve satisfactory convergence in all quantities

422

m.K. DONALDSON'

of interest. Since there is no possibility of confusion at this point, the subscript alpha has been omitted from the symbols W=, h, .... u,=. etc. The combination circular and hyperbolic functions used in equation (1) are defined as follows

sinsnx

if n < h, if n ;~ h,

sin(h)s.x = ~sinhs.x /coss~x cos(h)s, x = [coshs.x

ifn < h,

sin v.y sin(h) v.y = ~sinh v.y

ifn<~,

COSVny cos(h) v.y = [cosh v.y

if n < ,~, ifn ~ ii.

if n ~ h,

ifn ~ ,q,

In equation (1), the 8N extended plate edge Fourier series coefficients ha. a through k4. u are the unknown quantities to be determined. The actual plate edge distributed moments and Kirchhoff shears can be obtained by differentiating equation (1). For the purpose of explanation it is sufficient to consider only the side x = c of the actual plate. The distributed moment M~,(c,y) and the Kirchhoff shear V~,(c,y) at that edge are

nrcy nr~y M~(c,y) = ~ A~..ka..sin T + A2,.ha..sin ---~- + A3..k2,.sinhu.(b - y ) + R

+ A4,. k2,. sin(h) v.(b - y) + As,. h2.. sinh u.(b - y) +

9

+ A 6 . , h 2 . , sin(h)

nr~y

.

nr~y

v,(b - y ) - A * . , •3., sin - - ~ + A 2., 113., sin T

- A*.. k4,. sinh u.y - A*,. k4.. sin(h) v.y

-

+ A*.. h4.. sinh u.y +

+A*..h4,.sin(h)v.y+P2,..sin~-~--],

V~,(c,y)=

~[

nny nny Bl..kl,.sin--ff--+ B2..hl..sin--~-+ B3..k2..sinhu.(b-y)+

+ B4., k2., sin(h) v.(b - y) + Bs., h2., sinh u, (b - y ) +

nr~y + B6..h2..sin(h)v.(b-y) + B * . k 3 . . s i n ~

nrcy

B~* .h3..sin ---~- +

+ B*.. k4.. sin h u. y + B ~ . k4.. sin (h) v. y - B*,. h4, n sinh u. y -- Bg'./7... sin(h) v.y

9 nny]

+ P22.. s,n ---~-],

0)

where the quantities Aj,., Bj.., and Pj.. can be found listed in reference [3]. A short catalogue of such moments and shears can be found in reference [4]. The concentrated forces at the

FORCED VIBRATIONOF BEAMSAND PLATES

423

corners of the plate element will be accounted for when the beam boundary conditions are examined. 3. THE BEAM ELEMENT Let the beam element at the side x = c of actual plate alpha be beam element epsilon. Then beam epsilon is loaded along its length by the plate alpha distributed moment and shear described by equations (2) and (3), plus a similar distributed moment and shear produced by the plate element on the other side of beam epsilon which will be called plate element beta. F o r the sake of some generality, let beam epsilon be a non-symmetrical open section beam (see Figure 2). The moment-deflection relations [5] for such a beam element are M e = - E I ~ w g - E I c . vo. M, = -EIr

wg - - E I , v~.

(4)

T , = - E C ~ , 0 ~' + a J e 0',

where each prime on the beam deflection symbols w, v, and 0 indicates one partial differentiation with respect to the beam length coordinate (in this case, y) and each dot indicates one partial differentiation with respect to time. The sign conventions for the beam loading and the beam stress resultants are shown in Figure 3, with the exception of the torsional loads and deflections which are positive according to the right-hand rule. By considering the equilibrium of the infinitesimal segment of beam epsilon shown in Figure 3, the differential equations for beam epsilon can be found to be

EIr w~" +

E I c , vg' = pn.

EIc. w~,"+ Et. v'~' = &, E C w 0 " - - GJe 0" = -cr

(5)

The distributed loads and torques acting on beam epsilon due to the beam motion and the plate alpha reactions are p . = V~ - pAii'~, pc = N x - - p A i ~ . "cr = M x - c,T N:, + e~ Vx - p J c O + (6r - - tr pAfi'c - - (fin - ~,7) pAiJc .

(6)

When the distributed loads are eliminated between equations (5) and (6), the easily visualized result is the beam equations of motion. Those equations involve both the deflections at the A

--~

Ic

-'~

Figure 2. Beam epsilon cross-section. (The)-axis is positive out of the paper. The letter C identifies the centroid, O the shear center, and ..4 the point of attachment. The point of attachment will generally be at the rivet line. It defines the horizontal dimensions of the plates, and it is assumed to be the point on the crosssection at which the continuous edge loads of the plates act on the beam. Plate alpha lies to the right of point A while plate beta lies to the left.)

424

B. K . D O N A L D S O N

p,(r)

Sr t- dSr (o)

(>')

y> ..~+ dS~

(b) Figure 3. An infinitesimal segment of beam epsilon. (Consistent with the plate analysis conventions, the beam deflectionsand distributed loads are positive down and]or positive in the direction of the corresponding axis.) (a) Side view of beam epsilon facing towards plate alpha; (b) bottom view of beam epsilon. shear center and the deflections at the centroid. In order to facilitate later comparisons between the beam and plate deflections, the beam equations will be expressed only in terms of the deflections at the point of attachment which are, of course, also the plate edge deflections. The linear relations between the displacements at the point of attachment, the centroid, and the shear center are Wo = ) v . -

er O,

),'~

t5c O,

=

w~ -

Vo= v= + t, 0, vc = vo + ,~. 0.

(7)

The high degree o f relative rigidity of the structure in the plane of the skin allows only negligible horizontal motion of the beam at the point of attachment. Hence it is assumed that vo = 0. This constraint is, of course, in keeping with the assumption that the plates only undergo vertical motion. The general in-plane force N~,(y) which was completely omitted from the plate analysis is useful here. The N~, of the second of the three beam equations of motion can be used to eliminate the N~ in the third of those equations. When this is done, and equations (7) and Vo= 0 are substituted into the first and third equations of motion, V~ is eliminated from the third equation by use of the first equation, and harmonic motion is assumed such that )% - W ( y ) sin tot,

0 = O(y) sin tot, the result is, after cancelling sin tot, EI~ W " - p t o 2

AW-

E A O " + 6~pto2 A O

=

V~,,

- E A W " + 6~pto~ A W + ECw~ 0 " - G J, O" -- pto2 jo 0 = M~,,

(8)

425

FORCED VIBRATION OF BEAMS AND PLATES

where A = c c l c - e.I.~,

J, = Jo + ( ~ + ,5~.)A, and Cwa = Cw + e~l~ - 2ece. lc. + ~. I.,

which is always greater than C,~. The second of the original three equations of motion now simply defines the quantity Nx in terms ofthc quantities Wand O, and is o f n o further interest. For the sake of simplicity the force resultants contributed by plate beta have bccn omitted from the above formulation. These resultants do not interfere with or complicate the above procedure sincc the above M~, V,, and Nx can be taken as the net resultants of the two plate elements. Now. it is necessary to obtain a solution for the simultaneous equations, equations (8), which are termed the beam amplitude equations. First, however, note that these two equations decouplc if 6~ = A = 0, which would be the case, for cxamplc, if the beam cross-scction possessed an axis of symmetry, and its point of attachment to the continuous skin lay on that axis. The solution to the beam amplitude equations is found by combining a complementary solution and a particular solution. The complementary solution is found by sccking a solution to the homogeneous equations in the form of exponential functions, namely, W ( y ) = r exp ay,

O(y) = Yexpay.

(9)

Substitution of equations (9) into the homogeneous form of equations (8) yields + [ E l e a 4 - pco2 A] cp -- [ E A a 4 -- 6r

A] y = O,

- [ E A a 4 - 6 ; p m 2 A] q~ + [EC,,.o a 4 -- G J , a 2 - pm2 Jo] Y= 0.

(I0)

In order to have a non-trivial solution for the constants (b and Y, the determinant of their coefficients must be zero. The expanded form of the determinant equation is E2(Ir Cwo - A 2) a s - EIr G J e a 6 + poj2 E ( 2 A A 6 r - IeJ,, - C w o A ) a 4 + + a J e A p c o 2 a 2 + p 2 c o ' A ( J o - 6~A) =- 0.

(11)

This is a fourth-order equation in a 2. (Since this analysis is one involving forced vibration, the frequency eo is a given quantity just as the other constants that appear in the above coefficients.) An exact solution to equation (11) is not worthwhile since numerical solutions are a routine matter with present-day computing machines. Since complex roots as well as real roots are possible, to illustrate all possible solutions for a 2, it will be assumed that one of the quartic roots is positive, another negative, and the third and fourth are complex conjugates. These roots can be designated as y2 - C , (r + i~) 2, and (r - ir 2, respectively. Then the eight roots of the biquartic are al = y, a2 = - y , a3 = iv, a4 = - i v , as = r + i~, a6 = - r i~,, a7 = r ir and as = - r + i~. By redefining the constants of integration associated with the exponential functions, the solutions for W and O can be written entirely in terms of real functions as follows: W = cP~coshyy + q'2 sinh~,y + 4)3cosvy + 'b4sin vy + cPscosh eY c o s e y + + 4~6sinh r cos ~,y + q'~ sinh e y sin e y + cP8cosh r sin ey, O = ]q cosh),y + Y2sinh~,y + Y3cosvy+ Y4sinvy+ Yscoshr

+ Y6sinhr

+ YTsinh r

Yscosheysiney.

ey+ (12)

426

n.K. DONALDSON

If the roots of equation (11) are distributed in a fashion other than that assumed here, the above complementary solutions are, of course, adjusted accordingly. The relation betwcen the two sets of constants of integration can be determined from either of equations (10). Let ~Fj= Fj 4',.

(13)

F j = E A a ~ - 6~pOJ 2 A "

(14)

Then, from the first of equations (10),

Since )3 and q~j must be real, the Fj must also be real even i f a j is complex. Thus the writing of beam complementary solutions is simply a matter of evaluating the roots of equation (I I) and selecting the appropriate functions for equations (12). Particular solutions for the non-homogeneous portions of equations (8) can be obtained systematically by use of the method of undetermined coefficients. Recall that the nonhomogeneous terms supplied by plate alpha are specified by equations (2) and (3), and those terms that are supplied by plate beta are of the same form and thus need not be considered in this explanation. Consider any two matching terms of the first N terms in each of the expressions for the distributed moment and shear: say, A l . , , k l . , . s i n O n n y / b ) and B l . , , k ~ . , , s i n Q n n y / b ) . The particular solutions for these two terms may be written as tony

W = tPl.,.kl,,.sin ---~--, O = 12x.,.kl.,.sin mz~y b

(15)

Employing these two solutions in equations (8) with the above moment and shear terms leads to RI.,. ~1.,. - $1.,. f21.,. = BI.,., - - S l . , . ~'l.,. + T I . " I 2 t . , . = A I . " , (16) after cancellation of k~. ,. sin (mrcy/b), where RI. " =

EIc - pco 2 ,4,

$1. ,,I =

E A - 6cp~o ~ A ,

7"1. " =

ECwa +

G J,, -- p~oZ J,.

The solution of this pair of equations for ~ L " and f2L,. for all m from one to N is a straightforward matter. Exactly the same procedure can be used with the remainder of the nonhomogeneous terms. For example, the form of the particular solutions for the third group of non-homogeneous terms is IV = 7/3. ,. k2. ,, sinh u,.(b - y ) , O = f23. ,, k2. ,, sinh u,,(b - y). The constants ~3," and 03.,. are determined from the simultaneous equations R 3 , " ~J3." -- $ 3 , " 0 3 . " = B 3 . " , - $ 3 . , . Itt3,,. if" T3. m - Q 3 . " = A3. m.

FORCED V I B R A T I O N OF BEAMS A N D PLATES

427

The expressions for the constants R3.,~, S3.m, and Ta.,,, plus those for all other non-homogeneous terms, are listed in Appendix II. Thus the entire particular solution may be written as

nny

IV(y)= ~. ~ l . . k l . . s i n J ~ - ~ + 7"2..h1..sin--~-+

7"~..k2..sinhu.(b-y)+

+ 7"4.. k2.. sin(h) v.(b - y ) + 7"5.. h2.. sinh u.(b - y ) + 7'6.. h2.. sin(h) v.(b - y ) -

tiny

nny

- q'*..k3..sin--~---+ 7"*,.h3..sin--~----- 7tZ.k4..sin(h)v.y + 7"* 5,.

7"*..k4,.sinhu.y-

nny] h4..sinhu.y + ~P6*..h4..sin(h)v.Y + 7"2~.. sin'--b-- /

O(y)=~[f2,.k,.sinn-~-+.+t22,.sin~-],.

(17)

where, for the sake of easy bookkeeping, the 7"j,. and I2j.. directly correspond to the Aj.. and Bj... Finally, the combination of equations (12) and (17) provides the total beam amplitude solution. 4. AN EXAMPLE PROBLEM Now that general plate and beam amplitude expressions are available, all that remains is to write the individual beam and plate boundary condition equations for any given structure. These equations lead directly to solutions for the beam constants of integration and the

b

Figure 4. A simple example of a skin-stringer-frame structure. (The plate and the beams are simply supported at edges x = a and y = b. The plate and the beams are rigidly connected in the interior of the structure. The plate is called plate alpha while the beam at x = c is beam epsilon and the other is beam nu.) plate edge Fourier series coefficients, and hence to a solution for the amplitudes of the total structure. This process will be illustrated by means of a simple example. Consider a structure which is composed of a single rectangular plate simply supported at two adjacent edges and elastically supported by open section beams along the other two edges. Let the beams also be simply supported where they abut the foundation (see Figure 4). Let the pressure distribution over the plate surface be P(x,y)sintot. Before proceeding with the solution for the amplitudes of this simple structure, it is convenient to make explicit the amplitude expression for beam nu. The distributed loads transmitted to beam nu from plate alpha are

p, = + V,(x, e), "rx= -M,(x, e)

428

B.K. DONALDSON

and

M,(x. e)

~, [A43.. k l . . sinh r.(a - x) + A44., sin(h) s.(a -- x) + A4s.. h~..sinh r,(a '- x) + + A46.,h~.,sin(h)s,(a-x)+A,7.,k2.,sin-

?/1IX

/'/7[X

a

a

' +A48.,h2.,sin

+

+ P 2 5 . sin nnx.~__] 9

a

N[

V~(x, e) = ~

(18)

J'

B43., k l . , sinh r,(a - x) + . . . + B48., h2., sin .mrx + P 2 6 . , s i n ' a

a

. (19)

With this information, the technique of the previous section allows the beam nu amplitude solutions to be written immediately as

W,(x) = ~b~,cosh Y~x + r

sinh ya x + r

cosh)'2 x + q~4~sinh)'2 x + ff'5, cos v, x +

§

+ 'b6vsin v, x + q'7,cosv2x + q'8,sin v2x + ~ [7'43.,k,.,sinhr,(a /Z t + ku,4., k~.. sin(h) s,(a - x) + W4s,, lb., sinh r,(a - x) +

+ 7~46..h~.,sin(h)s.(a-x)+ 7J47.,k2.,sin + ~25,.sin

ngx

llTZx

+ 7~,s..sin -

a

x ,

a

+ (20)

where, for the sake of variety, the apparently more common situation of two positive roots and two negative roots for the beam nu quartic equation has been assumed. N o t e that the corresponding equation for plate alpha, equation (1), and the corresponding equations for beam epsilon, equations (12) and (17), can be simplified by setting k3.,,=h3.,,=k4.,,= h4., = 0, because plate alpha is simply supported at its third and fourth sides. N o w it is convenient to proceed to the" final equations. The beam boundary condition equations at y = b and x = a are

W,(b) = W;(b) = 63~(b) = Off(b)= O. W~(a) = W((a) = O.(a) = O;(a) --- 0.

(22)

The fourth and eighth o f the above equations state that the beam warping stresses at y = b a n d x = a are zero [5]. Since these equations are rather simple, it suffices to detail only the first and the last:

W,(b) = 0 = q'~, cosh)', b + ~2~ sinh)', b + ~3~ cos v, b + ~4, sin v, b + q~5,cosh ~bbcos ~,b + + ~6, sinh ~bbcos ~b + r O,~(a) = 0 -- 72 F , r + y2 F , r

-- v2 F7 r

sinh ~bbsin ~bb + ~8, cosh q~bsin ~bb,

cosh 71 a + 72 F2 r

sinh ~, a + y2 F3 ~b3vcosh)'2 a +

sinh Y2a - v2 Fs '/'5, cos v, a -- v2 F6 r v2a -- v~ Fs cb8 sinv2a.

sin vl a -

FORCED VIBRATION OF BEAMS AND PLATES

429

T h e beam juncture equations are W,(e) = W,(c),

O,(e) = - W , ; ( c ) ,

w;(o = +o.(0, o;'(c) = o, Mr

-- T~,,(c) = O,

o;(e) = o,

s . # ) + &.(c) - R(c, e) = O,

Ty,(e) + ~,tr

= O,

(23)

where again the warping stresses are taken to be zero. This would be approximately true if there were an air gap and a bracket connection between the two beams. The first five of the above equations need no amplification. The sixth of the above equations is the juncture shear equilibrium equation where the concentrated plate corner force, R(c,e) is equal to 2D(I -IOtV~.~, r This is the longest o f the eight equations, and it will be detailed as representative o f the three equilibrium equations. First expand the sixth equation as

E,I~, WT(e) + E, A, O~'(e) + E,Ir WT(c) + E, A, OT(c) + 2D.(I -- F) W,. ~,(c, e) = 0.

(24)

F o r the sake of simplicity, at this point, assume that beam epsilon and beam nu have the same cross-sectional properties so that E, = E, - E, I~, = lc, = / , and At =A, =A. However, despite the obvious contradiction, the different distribution o f roots from equation (11) for beams epsilon and nu will be maintained in order to continue to demonstrate the case when there is more than one root in a single classification. Let 2(D/E)(I - 1 0 = A; then equation (24) becomes 7~(Ic + AFt) ~1, sinh 7, e + 7~(Ir + A F2) ~2, cosh 7. e + v~(I~ + A F3) q~3,sin v, e --

-- v~(lr + AF4) ~04,cos v, e + (Ic + AFs) ~s,(~b 3 sinh q~ecos We -- 3q52~kcosh q~esin We -- 3q~k 2 sinh ~be cos We + 03 cosh ~besin We) + (It + A/'6) ~,(~b 3 cosh qbe cos We - 3~b2 ~, sinh q~e sin ~9e - 3q~b 2 cosh q~ecos ~ke + ~3 sinh q~e sin ~,e) + (I c + AFT) x x q~7~(~3 cosh ~e sin ~,e + 3q52 ~, sinh q~ecos Ipe - 3qbtp2 cosh q~e sin 0e _ ~,3 sinh (be cos 0e) + (Ir + A Fs) @8,(~b3 sinh q~esin We + 3~b2 ~, sin q~ecos ~,e --3qb~b2 sinh qbe sin O e - if3 c~

+ l ( ' ~ ) (I, ~2.. + AI22.,,)cos

~{[[nn'

~e c ~

3

[ ( " i f } (1r ~u' c* +~ A 12'" ")

nne].

--if-/k, .. +

hL. + [u](I, ~3.. + Af2a..)coshu.(b-e) +

+ vanN(Ir ~ , . . + Z 124..) cos(h) v.(b - e)] k2,. + [u3.(IcIPs,. + Z 125, .) cosh u.(b - e) + + v~./V(4 ~P6.. + Af26..)cos(h)v.(b-e)]h2..+

(I~ 7~2,.. + A O 2 L . ) c o s b ) +

/

+ ~,~(I~ + A F~) ~ . sinh "~ c + ~,~(Ir + A F2) ~/'2. cosh ~,~c + 7](1r + A F3) q'a. sinh ~'2 c + + ~](Ir + A F,) q~,. cosh ~ c + v](l~ + A F~) ep~.sin va c - v](lr + A F~) ~ . cos v~ c + /

+ v](I~ + A FT) ~7~ sin v2 c - v](l~ + A Fs) ~ . cos v2 c - ~. ~ [r~.(l~ ~,a.. + A O.a..) n

X

(

• cosh r , ( a - c) + s.~ ~(I~ ~ , , . . + A O44..)cos(h)s.(a- c)] k~., +

+ [r~.(Ig7 ~ . . + Af24s ' ,) cosh r.(a -- c) + s3.2q(l~ ~P,6., + Af2,6..) cos(h) s,(a - c)] Iq.. +

[(7;

+ -

(/,Tt~,v..+AfL~,..)cos

7]

k2..+l|aJ(lr

a..)cos

7]

h2..+

430

B.K. DONALDSON

+

(Ir t/'2s' . + At22s ' .) cos--~-J-- A

./.cos--'~" {r~s.[cos(h) x n

x s~ a cosh r~(a - c) - cosh r. a cos(h) s.(a - c)] h i . . + [r. sin(h) s~ a cosh r.(a - c) - s.sinhr.acos(h)s.(a-c)]kl,.}-.4

L.cos a {u.v.[cos(h) x

~

x v. b cosh u.(b - e) - cosh u~ b cos(h) v~(b - e)] h2.. + [u. sin(h) v~b cosh u~(b - e) -ff cos b

-v.sinhu.bcoshv.(b--e)lk2..}+A

~

/1,,.,cos

a

-

=0.

(25)

The final terms of the first two summations and the last summation make the above equation non-homogeneous, In this example problem the remaining two equilibrium equations can be" simplified by noting that the second derivatives of the twists at the juncture are zero. While equation (25) and the other juncture equations are somewhat lengthy, there is a good deal of duplication of terms among all these equations and as a consequence it is not a difficult matter to program these equations for machine computations. It is of greater significance that there are only a relatively few different possible boundary condition equations or components of those equations, and when these are written and programmed once, they need never be derived again. They can simply be patched together as required by the structure under study. At this point there are 16 + 4N unknowns. They are the sixteen constants of integration q~, through q'8, and ,'b~, through q~s~, and the 4N extended plate edge Fourier series coefficients hi,,, kl.,, h2.,, and k2,,, where n = 1, 2, . . . , N. The above beam boundary conditions provide sixteen equations for these unknowns. The remaining 4N equations are obtained by requiring continuity between the beam and actual plate edge deflections. Specifically these equations are W,(y) = W~(c,y), O,(y) = -W.,

lu

~ (c,y),

(26) (27)

= W . ( x , e),

(28)

O , ( x ) = W . , , ( x , e).

(29)

Consider equations (26) and (27). First extend the range of definition of those four deflection functions to be the entire interval 0 < y < b. Then these two identities can be made into 2N equations by using the Galerkin error orthogonalization technique on the interval 0 < y < b. Applying the operator b

2 ~

~.

tony

( )sin ---ff--dy, o

to both sides of equation (26) leads to, for m = 1 , 2 , . . . , N, ZI~. ,~ ~1~ + Z.~. ,~ ~,~ + Z3~. m ~a, + Z4~. ,~r

+ Z~. ,~ q~s~+ Z6~. ,. ~6, + ZT~.,~ q~7~+ N

+ Zs,,m q'8, + 7',.ink,,,. + ~P2.,~h2,m + Y. [(U, . . . . ~3..+/-/2 . . . . t / ' . , . ) k . . . + n

+ (Ul . . . . ~ s , . + /-/2. . . . t/J6,.) h2,.] + t/'zl, m = G l . m h l . m + G2.~,kl.,~ + N

+ Y [(U, . . . . Fa. ,, - U . . . . . F4. ,,) h 2 . . + (U, . . . . Fs,. - U. . . . . F6, .) k2..l + Ps. ,.. n

(30)

FORCED VIBRATION OF BEAMS AND PLATES

431

Applying the same operator to both sides of equation (27) yields, for m = 1, 2 . . . . . N, ZI~. m FI ~xe + Z2e. m F2 q~2~ + Z3~. ,n F3 ~b3~ + Z4,. m F4 q~4~ + Zse. m Fs ~5* +

+ Z6z, mr6 qb6c+ Z T , , m F T q 5 7 , + Z 8 , , r a r s ~ s t + f 2 t , m k l . r a +

~r~2.rahl,m-2V

N

+ Y. [ ( v , . , . . a ~ . . + v ~ . , . , o , . , ) k ~ . . + ( v , . . . . o ~ . . + v2 . . . . o ~ . . ) h , . . ] + o ~ , . , n

N

= aT. , h i , ra + G8, rakl, m -- ~. [(UI . . . . Fg., - U2 . . . . Flo. n)h2, n + n

+ (U, . . . . F H . . -

U 2....

(31)

F12. n)k2. n ] - P6,,."

Equations (28) and (29) can be treated in the same fashion on the interval 0 < x < a. Equation (28) then becomes

"~1'vl, m (~lv "~ '~2vl. m (~2v "[- ~'lv2, m (P3v '+ '~2v2, m (~4'V "[- ~-----'3vl,m (/)5,"~ '--'~4'r m (Z)6'.' "~ IV

+ ~3,~. ,~ r

r, + --%,2. r, a'8, + Y /I

[(v~. . . .

~',3., +

u, . . . .

~',,.,) k,., +

dC (U3 . . . . ~145. n "JC U4 . . . . ~F46. n ) h l . n ] + ~//47. m k2. ra + ~'tas. mh2. m + t/'tz5., N = ~ [(U3. . . . F13. n - Ua, ra,, F1,. n) h,. ,, +,(U3 . . . . F l s . n - / - / 4 . . . . F16,.)kl..] + tl

(32)

+ Gtv.=h2. m+ Gls, mk2. m+PT..,.

Equation (29) becomes 'Earl m FI qbl, + ~'2,x. ,~ F2 q'2v + ~=1,2.,~ F3 ~3v + "~%~2v2,m F4qb4v + E3,1. m F5 qbs* +

+ ~4n. ,./'6 r

+

~'3v2. m F 7 (~Tv "[- ~4v2. m F 8 r

N -t- ~. [ ( U 3 . . . . n

~"~43. n +

+ 04 . . . . O 4 * . n ) k l . n + (U3 . . . . a4s, n + U4. . . . a46. n)itt.n] + 047. tnk2. m + N

+ f24s. m1,2. ,. + f22s. ,. = - - ~ [(/-/3. . . . F z , . . - U4 . . . . F22, .)1,1.. + (U3 . . . . /7.3.. n

- U4. . . . F24..) kx..] + G,s. ,. h2. ,. + G26. ~,k,. ,. - P8, .1.

(33)

These 4N continuity equations may be regrouped with respect to the constants of integration and the Fourier series coefficients, and the constant terms may be placed alone on the righthand sides. When the same is done for the 16 boundary condition equations, the combination of the 16 and 4N equations may be cast in the matrix form [Q] {~} = {~.}

(34)

and solved for {~}, the vector of beam constants of integration and plate Fourier series coefficients. This completes the determination of the beam and plate amplitude functions. Numerical results for the forced vibration of a plate and beam structure of the type discussed above will be presented in a subsequent paper [6]. 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. E. H. DILL and K. S. PISTER 1958 Proceedings o f the Third U.S. National Congress o f Applied Mechanics, 123-132. Vibration of rectangular plates and plate systems. 2. Y. K. L1N and B. K. DONALDSON1969 Journal of Somld and Vibration 10, 103-143. A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels.

432

B.K. DONALDSON

3. B. K. DONALDSON 1973 Journal of Sound and Vibration 30, 397--417. A new approach to the forced vibration of thin plates. 4. B. K. DONALDSON 1968 Ph.D. Thesis, University of Illinois, Urbana, Illinois, U.S.A.'A forced vibration analysis of flat plates continuous in two directions over open section beams. 5. The Collected Papers of Stephen P. Timoshenko 1953. New York: McGraw-Hill Book Co. 6. B. K. DONALDSONand S. CHANDER 1973 Journal of Sound and Vibration 30, 437--444. Numerical results for extended field method applications.

APPENDIX I LIST OF SYMBOLS a b c, 6 d e, ~ /0.,

extended plate length in x coordinate direction extended plate length in y coordinate direction x-coordinates of the extended plate partitions plate thickness y-coordinates of the extended plate partitions nth Fourier series coefficient of the lateral deflection at t h e j t h edge o f the extended plate i square root of minus one k~., nth Fourier series coefficient of the normal slope deflection at t h e j t h edge of the extended plate Ill, II, p summation indices

b [ pm2 d] 1`+ e

n[

D j

t---ff-J Pc, P~ beam lateral force loading per unit length of beam axis, acting in the /~ and t! directions, respectively

s,

[m( '~T -

t

time

v,

09

-

+

-

[ ( - ~ ) : ' - a ~ ( - - ~ - - ) l ' 2 ] ''2

i f n
and

i f n < ii, and ifn~,'/

deflections of the beam point of attachment to the continuous skin in the t! and/~ directions, respectively ~c, Iv,:. deflections of the beam centroid in the tl and ~"directions, respectively Vo, WO deflections of the beam shear center in the r/and ~"directions, respectively .',t",y plate and beam grid Cartesian coordinates A beam cross-sectional area Aj.., Bj.. constants defined in reference [3] .4*J , n~ B~,, see Appendix II [~a~ Wa

FORCED VIBRATIONOF BEAMSAND PLATES C~, C.,. D E G

I~. I,. I~, Jo.J~ .I, L.

M~, M,

M:, M. At, N N

N~,N, Pl, n

[Q] R

R~.,, &.,, rj., &,s, T,,,T, ~..]J, m~

IV Zj,n

433

warping constants of the open section beam cross-section with respect to the shear center and the point of attachment, respectively plate stiffness factor, Ed3/12( l - it 2) Young's modulus constants defined in reference [3] shear modulus of elasticity centroidal moments of inertia and product o f inertia for the beam cross-section polar moments of inertia of the beam cross-section about the point of attachment and the centroid, respectively St. Venant's constant for uniform torsion (s,, sinh r,, a cos(h) s~ a - r,, cosh r,, a sin(h) s. a)(v,, sinh u. b cos(h) v. b - u. cosh u. b sin(h) v./5)plate moments in the x and y directions, respectively, per unit length in the y and x directions, respectively beam moments about the beam cross-sectional axes ~ and ~1, respectively maximum values for the indices m, n, and p sgn (n - h) sgn (n - h) plate in-plane forces in the x and y directions, respectively quantities defined in reference [3] matrix of coefficients for the unknown beam constants of integration and the extended plate edge Fourier series coefficients concentrated plate corner force in the lateral direction quantities defined in Appendix II used to obtain the beam particular solution beam shear forces in the ~ and q directions, respectively beam torque about the x and y axes, respectively quantities defined in reference [3] beam or plate lateral deflection amplitude quantities defined in Appendix II and used in the beam-plate continuity equations names, used as subscripts, to identify plate elements

O, l, h', )., a, tO I e, lt, v, 4, o, rr, ] names, used as subscripts, to identify beam elements p, z, o, 4, X, V )

{a} column matrix of the beam constants of integration and the plate Fourier series

coefficients 7 the square root of a positive quartic root of equation (11) d~r 6~ distances from the point of attachment to the centroid in the ( and r/directions, respectively e~, t~ distances from the point of attachment to the shear center in the ( and q directions, respectively (, tl beam cross-section coordinates parallel and perpendicular to the skin with their origin at the centroid 0 angle of beam twist {2} column matrix of load related terms It Poisson's ratio v square root of the absolute value of a negative quartic root of equation (11) p plate or beam mass density a root ofequation (11) o f a n y nature ry torque loading per unit length acting upon a beam parallel to the y axes 4, V real and imaginary parts of complex conjugate roots of equation (11) considered as an eighth-order equation co circular frequency of forced vibration Fs proportionality factor between beam constants of integration O A

amplitude of beam twist 2 D 0 - U)

E ,E~.. quantities defined in Appendix II, and used in the beam-plate continuity equations /L,,,. plate loading factor defined in reference [3]

434

B. K. DONALDSON

)'J, '~s constants of integration for beam torsional and lateral deflection amplitudes, respectively ~Ps... -Qs.. coefficients of the particular solution for the beam lateral and torsional deflection amplitudes, respectively

A P P E N D I X II DEFINITIONS IN ADDITION TO THOSE OF REFERENCE [3]

when c is replaced by (a - c).

A~., = Aj., B's.. -- B s . ,

w h e n c is r e p l a c e d b y ( a -

c) ;

For use with sin n m y : b 17

RI. m =

~

E I c -- po~ z .4,

finn\" S,..=~T ) Ea-~cp~ T1. ,. =

EC~,o +

GJe - Po~ZJ.;

DII~X

for use with sin - - . " a

Rz ,. =

(7)"

E h - peo2 A,

$2.,, =

T~. m =

E A - 6cpoga A ,

( 'ry' ~

(m--5

ECwo +

for use with sinh u,.(b - y )

GJe -- pogZJ.;

and sinh Umy:

R3. m = U~ EIc -- po~ 2 A , $3.,. = u~ E A -- 6cpo92 A , I"3, ,. = u.,4 EC,.,o - u~ G Jr. -- p w 2 A ;

for use with sin(h) vm(b - y ) and sin(h) Vmy: R4. ., = v~ EIr - pco2 A , $4.,. = v~ E A - 6cpoaZ A , 7"4. ,, = v~ E C w ~ - ~lv~ G J , - po~Z J , ;

for use with sinh r,.(a - x ) Rs.,,=

r~EIr

and sinh r,, x: A.

S s . m = r ~ E A -- 6 c p ~ 2 A . Ts. ,. = r~ E C w ~ - r~ G J e - p w z A ;

FORCED VIBRATION OF BEAMS AND PLATES

for use with sin(h)s,,(a - x) and sin(h)smX:

R6. m=

s ~ EIr - PO) 2 A ,

$6. ,, = s ~ E A - 6~pco 2 A , 7"6... = s ~ E C w o - f l s ~ G J ~ - po~Z J . . b

Z~,m =

cosn ?y sm---~-- ay 0

2ran [1 - (--l)m cosh ~b], - (mn)Z + (Tb) z -2ran Z2. m

(mn)2+(~b)2(-l)msinh?b,

Z

(;

.2n_~m .. [ l - - ( - 1 ) m c ~

a. ,, : ~ t m n ) z - (vo) 2 l

(0

if m n =

vb,

--2tort m 9 Z 4 . m : { (mn'~"~-- (vb) 2 (--l) sm vb,

[,1 Z$,m ~

if ran .= vb,

(mn - ~,b) [1 - ( - 1 ) ~ cosh Cb cos g,b] - ( - 1 ) " Cb sinh Cb sin ~,b + (r 2 + (ran - ~,b) 2 +

(mr: + Ipb) [l -- ( - 1 ) " cosh Cb cos ~pb] + ( - l ) " Cb sinh Cb sin Ipb (r

(-1)"[r

cosh Cb sin ~,b - (mn + ~,b) sinh Cb cos ~pb]

Z6, m

(r

(-1)m[r

ZT. m

z + ( m n + Ob) 2

2 + ( m n + ~,b) 2

cosh Cb sin Ipb + (ran -- ~,b) sinh Cb cos ~,b] (r 2 + (rim : ~b) 2

Cb[l - ( - 1 ) " ] cosh Cb cos ~b - (nm + d,b) ( - 1 ) " sinh Cb sin d,b (r 2 + (ran + Ob) 2 Cb[l - ( - 1 ) m] cosh Cb cos d,b - (ran - r (r

Z s , rtI

( - 1 ) msinh Cb sin ~,b

+ ( . m - ~b) ~

(-1)"[(ibb sinh Cb cos ~bb - (ran - ~bb)cosh Cb sin r (r 2 + (mn - ~b) ~ (-1)"[r

sinh (ibbcos ~,b + (nm + Ob) cosh Cb sin r (r 2 + (,nn + r z

(note that 7-.3. ,, and Z+. ,, are continuous); ~1. ,, = Zj. = when b is replaced by a.

435