theoritical method for natural freq

8/12/2019 Theoritical Method for Natural Freq

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M I N I S T R Y O S U P P L Y

R . M . N o . 3 0 3 918,414)A R C Technical Report

A E R O N A U T I C A L R E S E A R C H C O U N C I LR E P O R T S A N D M E M O R A N D A

T h e T h e o r e t ic a l e te r m i n a t io n o fN o r m a l M o d e s a n d F r e q u e n c i e s

o f V i b r a t i o nI. T. MINHINNICK

Crown C opyrig t Reserved

LONDON HER MAJESTY S ST ATIONE RY OFFICEI957

NINE SHILLINGS NET


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T h e T h e o r e t ic a l D e t e r m i n a t io n o f N o r m a l M o d e s a ndF r e q u e n c i e s o f V i b r a t i o n

yI . T . M I N H IN N I C K

COMMUNICATED Y THE DIRECTOR GENERAL OF SCIENTIFIC RESEARCH AIR)MINIST~Y OF SUFPI~Z

Repor ts and Memoranda No. 3o39J a n u a r y z 9 5 6

Summary. In this paper th e various methods that have been devised for the d etermination of the na tural frequenciesand norm al modes of aircraft are discussed and their accuracy and the am ount of work th at they entai l are com pared.An extensive bibliography is given. The discussion is ma inly from th e poin t of view of the flutter analyst, w ho comm onlybases his analyses on the norm al modes, bu t the description an d comparison of the various metho ds should be of generalinterest.1. In t rodu c t i on . By n o r m a l m o d e s a r e m e a n t t h e n a t u r a l m o d e s o f v i b r a t i o n o f t h e s t r u c tu r e .T h e y m u s t s t r ic t ly b e d e f i n e d f o r a n i d e a li z e d s t ru c t u r e , o n e w i t h o u t a n y s t r u c t u r a l d a m p i n gv i b r a t i n g i n s ti ll a ir , t h e a ir b e i n g a s s u m e d t o h a v e o n l y a n i n e r t i a e ff ec t. S u c h a s y s t e m c a nv i b r a t e f r e e ly w i t h c o n s t a n t a m p l i t u d e a t c e r t a i n p a r t ic u l a r f r e q u e n c i e s - - t h e n a t u r a l f r eq u e n c i es .T h e m o d e o f d e f o r m a t i o n o f t h e s y s t e m a t a n y o n e o f t h e s e f re q u e n c i e s is te r m e d a n o r m a l m o d eb e c a u s e t h e s e m o d e s a r e o r t h o g o n a l w i t h r e s p e c t t o b o t h t h e m a s s d i s t r ib u t i o n a n d t h e s t if fn e ssd i s t r i b u t i o n o f t h e s t ru c t u r e . F o r a r e a l s t r u c t u r e , w h i c h p o s s es s e s s t r u c t u r a l d a m p i n g , s o m ee n e r g y m u s t b e s u p p l i e d to i t f o r i t t o v i b r a t e a t c o n s t a n t a m p l i t u d e . T h i s is w h a t is d o n e i n ar e s o n a n c e te s t , in w h i c h a p e r i o d i c e x c i t in g fo r c e is a p p l i e d o v e r a r a n g e o f f r e q u e n c i e s . M a x i m u ma m p l i tu d e s o f v i b r a t i o n w i l l t h e n o c c u r at c e r t a i n fr e q u e n c ie s . I f t h e s t r u c t u r a l d a m p i n g i s n o tv e r y l a r ge , t h e f r e q u en c i e s a t w h i c h t h e s e p e a k a m p l i tu d e s o c c u r w i ll b e e q u a l t o t h e n a t u r a lf r e q u e n c i e s o f t h e d a m p i n g - f r e e s t r u c t u r e , a n d t h e m o d e s o f v i b r a t i o n a t t h e s e f r e q u e n c i e s w i l lb e l i t t l e d i f f e r e n t f r o m t h e n o r m a l m o d e s .T h e c a l c u l a t i o n o f n o r m a l m o d e s , w h i c h w i l l n o r m a l l y b e d o n e i n t h e d e s i g n s t a g e o f t h e a i rc r a f t,is i m p o r t a n t f o r s e v e r a l r e as o n s, F i rs t , b y e x a m i n i n g t h e n o r m a l m o d e s o b t a i n e d a n d p a r t ic u l a r l yt h e p o s i t i o n s o f t h e m o d a l l i n es , it m a y b e p o s s i b le t o t e l l w h e t h e r f l u t t e r i s l i k e l y o r n o t , a n d i na n y c a se su c h a n e x a m i n a t i o n w i ll i n d i c a t e w h a t t y p e s o f f l u t t e r s h o u l d b e in v e s t i g a t e d . I t m a ya l so b e p o s s ib l e t o p r e d i c t w h e t h e r t h e r e i s a n y l i k e l i h o o d o f r e s o n a n t v i b r a t i o n , d u e t o t h ep r o x i m i t y o f t h e n a t u r a l f r e q u e n ci e s t o t h e f o r c in g fr e q u e n c ie s o f t h e p o w e r p l a n t . H o w e v e r ,s u c h f r e q u e n c i e s a r e u s u a l l y h i g h o v e r t o n e f r e q u e n c i e s w h i c h a r e d i f f ic u lt t o c a l c u l a t e a c c u r a t e l y .F u r t h e r , t h e n o r m a l m o d e s a r e c o m m o n l y u s e d f o r t h e a c t u a l f lu t t e r ca lc u l at io n s . T h e o r e t ic a l ly ,a n y s e t of i n d e p e n d e n t d e f o r m a t i o n m o d e s c a n b e u s e d a s d e g r e e s o f f r e e d o m i n s e t ti n g u p t h ef l u t t e r e q u a t i o n s , b u t f r o m p r a c t i c a l c o n s i d e r a t i o n s w e w a n t t o s e l e c t t h o s e m o d e s w h i c h g i v e a na c c u r a t e f l u t t e r s p e e d w h e n t h e n u m b e r o f c h o s e n m o d e s is s m a l l. T h i s is m o r e l ik e l y t o b e ti l ec a se w h e n t h e m o d e s a re r e l at e d t o t h e a c t u a l s tr u c t u r e , a s t h e n o r m a l m o d e s a r e, t h a n w h e nt h e m o d e s c h o s e n a r e q u i t e a rb i t ra r y . I t h a s n o t , h o w e v e r , b e e n c o n c l u si v e ly d e m o n s t r a t e d t h a tt h e n o r m a l m o d e s a r e t h e b e s t f o r t h i s p u r p o s e . I t is p o s s i b le w i t h l es s e f fo r t t o o b t a i n o t h e r

* R.A.E. Rep ort Structures 197, received 2nd May, 1956.689~)


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modes whioh are in some degree related to the particular structure, but an additional reason whyit is desirable to use normal modes is because the y can later be compared wit h the resonance-testmo des.This paper is concerned with the theor etical calculation of norma l modes. It reviews thevarious methods that have been devised and compares them for accuracy and the amount ofwork they entail. The methods fall into types, according to the type of equation used and thetype of semi-rigidity assumed, and this fundamental classification is presented in section 2.The m etho ds themsel ves are described and discussed in sections 3 to 10, in terms, for convenience,

of either purely flexural or purely torsional vibrati ons of a beam. In sections 11 to 13 thediscussion extends to the further complications that occur in practice: coupled flexure-torsionvibrations, the calculation of modes of complex structures, and the calculation of modes of acomple te aircraft. A biblio graphy is given.2 . T y p e s o f M e t h o d s . - - B e f o r e we can consider the calculation of the no rmal modes of a completeaircraft we must first consider the calculation of its component parts (wing, fuselage and tailunit) in isolation. The various metho ds by which these are calculated are based on one of threetypes of funda menta l equation : the basic differential equation, the integral equation incorporatingflexibility coefficients, and the Ra yleig h or Lagrangian equation . The displ acement of thesystem is either specified at a number of finite points, or is expressed as a linear combination ofknown functions; in either case the actual system, which has an infinite number of degrees offreedom, is replaced by one which has a finite num ber of degrees of freedom. In addition, ce rtainapproximations are normally made--the effects of shear deflection, shear lag and rotary inertia

on the flexural vibrati ons and of warpin g of sections in torsion are neglected. The error thusintr oduc ed increases with t he over tone order. Methods which use flexibility coefficients could inprinciple take account of shear effects, but the calculation of flexibility coefficients incorp orati ngth ese effects would not be easy ; experimentally determined values might be preferable, providedthey could be accurately measured. When coupled flexure-torsion vibration is considered, anassump tion has to be made a bout a flexural axis; this is discussed in section 10. First of all,however, we shall consider the si mpler cases of pure flexure and pur e torsion.With the above assumptions, the three types of basic equation are then as follows:

a) Di f f eren t ia l Equa t ion .(i) For pure flexure

I Y ) d y 2 t = oo2[~ y) z y ) . . . . . . . . . . . . . (2.1)This may be written as the set of equations

d Sd y

d M - - S y ) . .d yd~B y ) ~ = M y )

dy

.. (2.2)

.. (2.3)

.. (2.4)

.. (2.5)

with the end conditionsz y ) = ~p(y) -- 0 at a clamp ed end

M y ) = S y ) = 0 at a free en d. I aI

I Q

. . 2 . o ). . 2 . 7 )


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( i i ) F o r p u r e t o r s i o n :e I y ) e o I - o y )~? ~

T h i s m a y b e w r i t t e n a s t h e p a i r o f e q u a t i o n s :d T - ~ y ) o y )a ydOC ( y ) ~ . = T ( y ) , . .cvy

w i t h t h e e n d c o n d i t i o n s :0 ( y ) = 0 a t a c l a m p e d e n d

T ( y ) = 0 a t a f r e e e n d . . .( b ) T h e I n t e g r a l E q u a t i o r a .

(i) F o r a c a n t i l e v e r b e a m i n f l e x u r e :

. . . . . . . . . . . . 2 . S )

2 . 9 )

2 . 1 0 )I

O I

Q

I I

. . 2 . 1 1 )

. . 2 . 1 2 )

z y ) = ~,~ F y , u ) e u ) z u ) d u . . . . . . . . . 2 . 1 3 )0

T h i s e q u a t i o n i n c o r p o r a t e s { h e e n d c o n d i t i o n s . I f s h e a r e f f ec t s a r e n e g l e c t e d , t h e f l e x i b i l it yf u n c t i o n i s g iv e n b y : [ < , , ~ v - y ) v - u )F ( y , u ) = ~ o B ( v ) d r , . . . . . . . . (2.14)w h e r e t h e u p p e r l i m i t is t h e s m a l l e r o f y a n d u .

(ii) F o r a c a n t i l e v e r b e a m i n t o r s i o n :f(y ) = ~,~ e ( y , u ) ,~(u) o (u ) d u , . . . . . . . . (2.15)0

w h e r e a c c o r d i n g t o t h e s i m p l i fi e d t h e o r y t h e f l e x i b i li t y f u n c t i o n i s g i v e n b y :d v . . . . . . . . . . . . . 2 . 1 6 )t h e l e f t - h a n d s i d e s o f e q u a t i o n s ( 2. 13 ) a n d ( 2 .1 5 )z ( y ) - - z (O ) - - y ~ (0 ) . . . . . . (2 .17)o y ) - o o ) , . . . . . . . . 2 . 1 s )

~ ( Y ~ ) = ~ o c ~ )( i i i ) F o r a n u n s u p p o r t e d s t r u c t m - e ,m u s t b e a l t e r e d a s f o l l o w s :

z ( y ) i s r e p l a c e d b y0 ( y ) i s r e p l a c e d b y

w h e r e t h e o r i g i n y = 0 is t a k e n t o b e o n e e n d o f t h e b e a m .F u r t h e r e q u a t i o n s a re t h e n r e q u ir e d . T h e s e ar e o b t a i n e d b y e q u a t i n g t h e t o t a l i n e r t i a fo r ce s

a n d i n e r t ia m o m e n t s t o z er o, i . e . ,j (u) z ( u ) d u = 0 . . . . . . . . . . (2 .1 9)

( u ) u z ( u ) d u = 0 . . . . . . . . . . (2 .20)0

~ ~ ( ~ ) 0 . ) a ~ = 0 . . . . . . . . . . 2 . 2 1 )3

f o r f l e x u r e , a n d

f o r t o r s i o n .68984) A2


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F o r a s y m m e t r i c a l b e a m , t h e o r i g in y = 0 s h o u l d b e t a k e n t o b e t h e c en t r e of t h e b e a m . F o re x a m p l e , f o r a p a i r o f w i n g s w i t h f r e e r o o t , t h e o r i g i n y = 0 i s t a k e n a s t h e w i n g ro o t . T h e c a s eso f s y m m e t r i c a l v i b r a t io n a n d a n t i - s y m m e t r i c a l v i b r a t i o n m a y t h e n b e c o n s i d e r e d s e p a r a te l y ,a n d i t is o n l y n e c e s s a r y t o c o n s i d e r o n e h a l f of t h e b e a m . F o r s y m m e t r i c a l f le x u re w e w r i t eW (0 ) = 0 , a n d e q u a t i o n ( 2. 20 ) i s d i s c a r d e d , b u t e q u a t i o n ( 2 .1 9 ) m u s t s t i l l b e s a t i s f i e d ; f o r a n t i -s y m m e t r i c a l fl e x u r e w e w r i t e z (0 ) = 0 a n d e q u a t i o n ( 2.1 9 ) is d i s c a r d e d b u t e q u a t i o n ( 2.2 0 )s a ti sf ie d . F o r s y m m e t r i c a l t o r s to n 0 (0 ) is r e t a i n e d a s a n u n k n o w n p a r a m e t e r ; f o r a n t i-s y m m e t r i c a l t o r s i o n w e w r i t e 0 ( 0) = 0 a n d d i s c a r d e q u a t i o n ( 2 .2 1 ).

( c ) R a y l e i g h ' s E q u a t i o n .I f co=J = m a x i m u m k i n e t i c e n e r g y a n d V = m a x i m u m p o t e n t ia l e n e r g y w e h a v e :

o~~ ~ Y = ~ V , . . . . . . . . . . . . . . (2 .2 2)w h e r e ~ Y a n d d V a r e t h e c h a n g e s in J a n d V r e s p e c t iv e l y d u e t o a s m a l l c h a n g e i n m o d e s h a p e.

F o r p u r e f l ex u r e, w e m a y w r i t e :Y =

F o r p u r e t o r s i o n :v = - ~J = V = l

f*op y) {~ y)}~gy . .f [ B ( y ) { z ( y )} ~ d y . . .

f o ~ ( y ) { O ( y )} ~ a y . .~ * C ( y ) { o ( y ) } ~ g y .

o

. . 2 . 2 3 )

. . 2 . 2 4 )

(2.25)(2.26)

F u n c t i o n s z ( y ) o r 0 ( y ) c a n b e f o u n d t o s a t i s f y t h e s e e q u a t i o n s ( d i ff e r en t ia l , i n t e g r a l o r R a y l e i g h )o n l y f o r c e r t a i n c h a r a c t e r i s t i c v a l u e s o f o), w h i c h a r e t h e n a t u r a l f r e q u e n c i e s o f t h e p h y s i c a lp r o b l e m ; t h e f u n c t i o n s z ( y ) o r 0 ( y ), w h i c h a r e t h e n f o u n d , a r e t h e n o r m a l m o d e s .S u p p o s e t h a t z , ( y ) , z s ( y ) a re t h e n o r m a l m o d e s i n p u r e f l ex u r e c o r r e s p o n d i n g to t h e d i s t i n c tn a t u r a l f r e q u e n c i e s c o,, co, r e s p e c t i v e l y . T h e n i t c a n b e d e d u c e d f r o m e q u a t i o n (2 .1 ) t h a t z , ( y ) a n dZ s ( y ) a r e o r t h o g o n a l w i t h r e s p e c t to b o t h t h e m a s s d i s t r i b u t i o n a n d t h e s t if f n es s d i s t ri b u t i o n ,

t h a t i s :f l t ~ ( Y ) z , ( y ) & ( y ) d y = 0 . . . . . . (2 .27)[ B (y )~ : (y)~ : (s) ey = 0 . . . . . . . 2 . 2 5 )

S i m i l a r l y i f O ~ ( y ) a n d O / y ) a r e tw o d i f fe r e n t n o r m a l m o d e s i n t o rs i on , t h e y m u s t s a t is f y t h eo r t h o g o n a l i t y r e l a t i o n s :[ * ~ . y ) o , y ) < y ) g y = o . . . . . . 2 . 2 9 )

o

. [ c y ) o / (y ) o / (y ) ey = 0 . . . . . . . 2 . 3 0 )A n o r t h o g o n a l i t y r e la t i o n c o n s i s te n t w i t h t h e s e c a n b e d e r i v e d f r o m t h e i n t e g r a l eq u a t i o n .

F o r f l e x u r e , t h i s i s : [ [ F ( y , u ) ~ ( u ) ~ ( u ) ~ ( y ) ~ , ( y ) d ~ e y = o . . . . . . 2 . 3 1 )J 0 d Oa n d f o r t o r s i o n z

o. . (2 .32)


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The various meth ods will now be described and discussed, in terms, for convenience, of purelyflexural or pure ly torsi onal vibrations. (Later, in sections 11 to 13, the app lica tion of thesemet hods to more complic ated cases will be considered.) The classification of each meth od isindi cate d with t he heading, as follows: for the t ype of semi-rigidi ty assumed, PD denotes theuse of point displacements, DF denotes the use of displacement functions : for the typ e of equati onon which the method is based, DE denotes differential equation, IE denotes integral equation,and EM denotes energy method.

3 . T h e H o l z e r - M y k l e s t a d M e t h o d (PD /DE) .--A step-by step metho d of solving the differentialequations for pure torsion was intro duce d by Holzer in 1921, and was ext end ed by Myklesta d tothe case of flexural vibration.

The beam is divided into a number of segments, and for pure flexure the distributed mass ofeach segment is replaced by a discrete mass at the centre of gravi ty of the segment. The pointswhere these masses are situa ted are num ber ed consecutively from a free end of the bea m, whi chfor definiteness we take to be t he r ight -han d end of the beam.We let m~ =

z , =~ r

rr z

mass at point rvertical dispIacement at point rslope of bend ing curve a t point r (posi tive when z,,_~ > z~ > z,+~)shearing force to immediate right of point rbending mome nt at point rdistance betw een point r and point r + 1.

We also introd uce flexibility coefficients, and for th e purpose of defining them, the segme ntIr is supposed clampe d at the point (r + 1). Then

f~ = displacem ent of point r due to unit load at point rg~ = slope of bendin g curve at poi nt r due to unit loa d at th at point

= also displacement of point r due to unit rot ary couple at th at pointh,, = slope of bendi ng curve at p oint r due to uni t rot ary couple at t hat point.

The differential equations (2.2) to (2.5) for flexural vibration may now be replaced by thefollowing set of difference equations

S~+1 = Sr + m ~ % . . . . . . . . . . . . (3.1)M ~ I = M S ~ J ~ . . . . . . . . . . . . (3.2)

~ ~ = ~ - - M ~ h ~ - - S ~ l g ~ . . . . . . . . . . 3 . 3 )z~ ~ = z~ - - ~o , J~ - - M r g r - - S ~ i f ~ . . . . . . . . . (3.4)

At r = 1, the free end of the beam, we have S~ = M~ = 0. A part icular value is assumed forthe frequency ~o, we write z~ = 1 and l eave ~o~ as an unk no wn paramete r. By usi ng the differenceequations for r = 1, 2, 3, etc., in turn, we can find the values of S , M , . , ~ and z~ at each pointin turn in terms of the unk now n ~1 until we arrive at the far end of the beam. At this far end,two end conditions have to be satisfied. ~ For example, if it is a cl amped end we must havez,, = ~o,~ = 0. The value of ~01 can no w be de te rmin ed by sat isfying one of the se end condit ions .The process is now repeated with varying values of o) until both end conditions are satisfied.The final value of co is then a natural frequency of the structure, and the corresponding valuesof z, determine the modal shape.

5


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The process can be given a physical meaning. For example, for a symme tric beam performingsymmetrical flexural oscillations, the conditions to be satisfied at the centre of the beam areS,~ = ~o,~ = 0. If we have chosen wl so that ~p,~ = 0, th en for an arbi tr ary value of co we ha veS,~ not zero, and we may wri te 2S,, = F cos cot, where F cos cot is an ext ernally appli ed forceat the be am centre requ ired to maint ain t he oscillations of frequency co. This external forcebecomes zero at the resonance frequency or natural frequency of the structure.For pure torsion, each segment of the b eam is replaced by a discrete polar mo men t of inertia J,.If T, = torque to imm edia te right of point r

0r --= angle of twist at point r~ twist at point r due to unit torsional couple at that point with segment clampedat point (r -b 1),

the difference equations are:T, +I = T , + co ] ,.O0~+~ : O~ - - ~ T,+ 1 .

. . . . . . . . . . . . 3 . 5 )3 . 6 )

The pr ocedure is similar in principle to th at .for pure flexure, but it is simpler since there areonly two equa tions in each set of difference equations, and ther e is no paramet er such as ~o~which must be kept unknown during the step-by-step process.The m etho d will determine overtone modes and frequencies as well as the fundamen tal withoutthe necessity of deter mini ng the fun dame nta l mode and frequency first. The accuracy, however,becomes worse as the over tone order increases ; for reasonable accuracy there sho uld be at leastfour times as many segments as there are frequencies required.4 . T h e S to d o l a M e t h o d (PD /DE ).- -In the Stodola metho d the differential equations are solvedby iterative integration.For a cantileve r beam performing flexural oscillations, the following equations m ay be derivedfrom the differenti al equat ions (2.2) to (2.5) an d consti tut e one stage of the it era tive process.

l

M , ( y ) S , u ) d u

f, /y - o(4.1)

4 . 2 )(4.3)

f,. (y ) - - ~ ( u ) d u . . . . . . . . . . . . (4.4)co2 z ,( l ) = z ,_ l ( l ) , . . . . . . . . . . . . . . (4.5)

where S , ( y ) , etc., are the approximations to S ( y ) , etc., obtained during the rth iteration; thecantilever is clampe d at the point y = 0 and y = l is the free end.The f actor co2 has be en dro pped in equa tion (4.1); this is permiss ible since the mo de shapesz ( y ) can be dete rmi ned only to within a constant factor. The quanti ties Sd y ), M / y ) , ~ ( y ) arenow consistent with z ~ (y ) as regards absolute magnitude but not with z~_~(y) . The dropping ofthe factor co in equat ion (4.1) is compen sat ed by eq uat ion (4.5), where z~(l) is chosen as a measureof the absolute magnitude of z d y ) .


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A s h a p e i s a s s u m e d f o r z 0(y ) ; t h e i n t e g r a t i o n s i n e q u a t i o n s ( 4.1 ) t o ( 4.4 ) a r e t h e n p e r f o r m e d ,a n d m , d e t e r m i n e d f r o m e q u a t i o n ( 4. 5) , f o r r = 1 , 2 , 3 , e t c ., i n t u r n . T h e p r o c e s s i s r e p e a t e du n t i l t w o c o n s e c u t i v e s h a p e s zr y) ( s a y z N ( y ) a n d zN + d y) ) a g re e t o t h e r e q u i r e d a c c u r a c y . T h i si s t h e n t h e f u n d a m e n t a l m o d e a n d con i s t h e f u n d a m e n t a l f r e q u e n c y .S i n c e t h e i n t e g r a t i o n s w i l l h a v e t o b e p e r f o r m e d n u m e r i c a l l y , t h e m o d e s h a p e z ,. (y ) w i l l h a v e

t o b e s p e c if i e d a t a f i n i t e n u m b e r o f p o i n t s , a n d t h e q u a n t i t i e s S,, y), M,,(y ) , ~ , , (y ) de te rmineda t a f i n it e n u m b e r o f p o i n t s . T h e s i m p l e s t m e t h o d i s t o s p e c i f y z / y ) , M , ,( y) a t e q u i d i s t a n t p o i n t s~-a 3a/2, . . ( - - )a , wh ere na l , an d to spec i fy S , y ) an d ~jJ,.(y ) a t th e in te r m ed ia t ep o i n t s y = a , 2 a , . . . ~ a . T h e i n t e g ra t i o n s m a y t h e n b e r e p la c e d b y s im p l e s u m m a t i o n s , g i v i ng :

w h e r e e i t h e r

or

IS , k a ) = Z m ~z,_~[ s q- -)a ] . . . . . . . . . . . 4 .6 )

s + l ) ~

m , = a [ ( s + a]~ - -M , [ k - - { ) a ] ---- a Z k a ) . . . . . . . . . . . . 4 .7 )

s k

i M ,[ ( s - - )a] . . . . . . (4.8)= a , : , B [ s - ) a ]kz,E k + )a] = a ~ ~ , sa) . . . . . . . . . . . . . (4.9)

s = l

O t h e r m e t h o d s o f n u m e r i c a l i n t e g r a t i o n , s u c h a s S i m p s o n ' s r u l e, m a y a ls o b e u s ed .F o r b e a m s w i t h o t h e r e n d c o n d i ti o n s a n u n k n o w n p a r a m e t e r m u s t b e a d d e d t o o n e o r m o r eo f e q u a t i o n s (4 .1 ) t o ( 4. 4) . T h i s p a r a m e t e r i s k e p t a s a n u n k n o w n d u r i n g e a c h i t e r a t i o n c y c l ea n d i t s v a l u e is d e t e r m i n e d f r o m o n e o f th e e n d c o n d i t i o n s a t t h e e n d o f th e c y cl e .T h e o v e r t o n e m o d e s m a y b e f o u n d i n t u r n b y t h e a b o v e p r o c e ss if a t t h e e n d o f e a c h i t e r a t i o nc y c l e t h e d e r i v e d m o d e s h a p e z~ y) i s m o d i f i e d b y a d d i n g t o i t m u l t i p l e s o f e a c h o f th e l o w e r m o d e si n su c h a w a y t h a t t h e m o d i f i e d m o d e s h a p e i s o r t h o g o n a l t o a ll t h e l o w e r m o d e s . T h e p r o c e s sb e c o m e s i n c r e a s i n g l y l a b o r io u s a s t h e o r d e r o f t h e o v e r t o n e i n c r e a s e s ; a li m i t t o t h e n u m b e r o fo v e r t o n e m o d e s a n d f r e q u e n c i e s w h i c h c a n b e a c c u r a t e l y d e t e r m i n e d i s s e t b y t h e n u m b e r o fi n t e g r a t i o n s t a t i o n s u s e d .5 . So lu t ion o f In tegra l E qua t io ns by Nu m er ica l In teg ra t ion a nd Co l loca tion ( P D / I E ) . - - F o r ac a n t i l e v e r b e a m i n f l e x u r e :

f y ) z u ) d u . . . . . . . . . s . 1 )0W e c a n d e r i v e f r o m t h i s e q u a t i o n a n i t e r a t i o n s e q u e n c e i n t h e s a m e w a y a s f o r t h e S t o d o l am e t h o d . H o w e v e r , s i n c e i t w il l b e im p o s s i b l e i n p r a c t i c e t o p e r f o r m t h e i t e r a t i o n s e x a c t l y , t h em e t h o d r e d u c e s t o o ne u s i n g n u m e r i c a l i n t e g r a t i o n a n d c o l l oc a t io n . T h i s r e p l a ce s t h e i n t e g r a le q u a t i o n s b y a m a t r i x e q u a t io n , w h i c h n e e d n o t n e c e s s a r i ly b e s o lv e d b y i t e r a t io n , a l t h o u g h t h ei t e r a t i o n m e t h o d i s u s u a l l y t h e m o s t c o n v e n ie n t .T h e c o l l o ca t i o n p o i n t s w i l l c o i n c id e w i t h t h e p o i n t s a t w h i c h t h e i n t e g r a n d s a r e e v a l u a t e dfo r u se in th e in teg ra t io n fo rm ulae . L e t the se po i n t s be t i l e p o in t s y ---- y~ an d le t ff (y~) ---- ~ ,e t c . , a n d F y r, y ,) = F , .

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I f t h e n u m e r i c a l i n t e g r a t i o n m u l t i p l i e r s a r e d e n o t e d b y c ~ , e q u a t i o n ( 5 . 1 ) r e d u c e s t o :z~ = o ; ~ F,,,OCs~,Z~. . . . . . . . . . . . . (5.2)

s l

W e n o w i n t r o d u c e t h e f o l l o w i n g m a t r i c e s :m : d i a g o n a l m a t r i x w h o s e s t h d i a g o n a l t e r m i s e ,~ ,F = s q u a r e s y m m e t r i c m a t r i x I F , ,]z = c o l u m n m a t r i x [z,,] .

T h e n e q u a t i o n (5 .2 ) m a y b e w r i t t e n i n t h e m a t r i x f o r m :~ F ~ z = z . . . . . . . . . . . . . . . . . (5.3)

O w i n g t o t h e d i s c o n t i n u i t y i n th e t h i r d d e r i v a t i v e o f t h e f l e x i b i l i t y f u n c t i o n F ( y , u ) a t t h ep o i n t y = u , a n d p o s s i b l e d i s c o n t i n u i t ie s i n t h e m a s s a n d s t if f n e ss d i s t r ib u t i o n s , t h e r e i s n oa d v a n t a g e i n u s i n g i n t e g r a t i o n f o r m u l a e of h i g h e r o rd e r t h a n S i m p s o n ' s ru l e. F o r t o r s i o n a lv i b r a t i o n s t h e f l e x i b i l i ty f u n c t i o n h a s a d i s c o n t i n u i t y i n it s f i r st d e r i v a t i v e , a n d o n l y t h et r a p e z o i d a l o r s u m m a t i o n r u l e s c a n c o n v e n i e n t l y b e u se d .

6 . T h e M o r r i s D i s c r e t e M a s s M e t h o d ( P D / I E ) . - - A p a r t i c u l a r c a se of t h e l a s t m e t h o d d e s c ri b e di s t h e d i s c r et e m a s s m e t h o d , a s d e v i se d i n d e p e n d e n t l y b y a n u m b e r o f w r i t e rs a n d a s d e v e l o p e d .i n E n g l a n d b y J . M o r ri s , a n d i s t h e s i m p l e s t m e t h o d t o a p p l y i n p r a c t ic e . F o r p u r e f l ex u r e , t h es y s t e m i s d i v i d e d i n t o a n u m b e r o f s e g m e n t s a n d t h e m a s s o f e a c h s e g m e n t c o n c e n t r a t e d a t i tsc e n t r e o f g r a v i t y , s o t h a t t h e s y s t e m i s r e p l a c e d b y a w e i g h t l e s s b e a m c a r r y i n g d i s cr e t e m a s s e srn~ a t a d i s t a n c e y ~ f r o m t h e r o o t . I f t h e i n t e g r a l e q u a t i o n ( 5.1 ) i s a p p l i e d t o t h i s s y s t e m o fd i s c r e t e m a s s e s , w e a t o n c e o b t a i n a n e q u a t i o n s i m i l a r t o ( 5 . 2 ) , n a m e l y :z , = ~ ~ F , m s Z ~ . . . . . . . . . . . . . (6.1)

s l

T h i s e q u a t i o n c a n b e o b t a i n e d b y p h y s i c a l re a s o n i n g . T h e f l e x i b i l it y c o e ff ic i en t i s d e f i n e d a st h e d i s p l a c e m e n t a t p o i n t y~ d u e t o u n i t l o a d a t t h e p o i n t y , , t h e r o o t b e i n g p r e s u m e d e n c a s t r 6 .W h e n t h e b e a m i s v i b r a t i n g , t h e i n e r t i a l o a d a t p o i n t Ys i s m s o ) y,, s o t h a t b y e q u a t i n g t h e p r o d u c to f f l e x i b i l i t y t i m e s l o a d , s u m m e d o v e r t h e b e a m , t o d i s p l a c e m e n t , e q u a t i o n (6 .1 ) i s o b t a i n e d a to n ce . T h i s e q u a t i o n le a d s t o th e m a t r i x e q u a t i o n :z = c o ~ F m z , . . . . . . . . . . . . . . (6.2)

i d e n t i c a l i n fo r m t o e q u a t i o n (5 .3 ) in t h e c o l l o c a t i o n m e t h o d , w h e r e m i s n o w a d i a g o n a l m a t r i xw i t h s t h d i a g o n a l e l e m e n t m , .T h e m o s t c o n v e n i e n t m e t h o d o f s o l v in g t h i s m a t r i x e q u a t i o n i s b y i t e r a ti o n . U s e m u s t b em a d e o f t h e o r t h o g o n a l p r o p e r t y o f t h e m o d e s i l l d e t e r m i n i n g t h e o v e r t o n e m o d e s a n d f r e q u e n c i e s ;t h i s is t h e m a t r i x e q u i v a l e n t o f e q u a t i o n ( 2.2 7) . A l t h o u g h t h e w o r k t h i s i n v o l v e s i n c r e as e s w i t ho v e r t o n e o r d e r , t h e r e i s t h e c o m p e n s a t i n g a d v a n t a g e t h a t t h e o r d e r o f t h e m a t r i x e q u a t i o n i sr e d u c e d a t e a c h s t a g e. I n g o i n g f r o m e a c h m o d e t o t h e o n e n e x t h i g h e r , a p p r o x i m a t e l y o n es i g n i f i c an t f i g ur e i s l o st . T h e l o w e r m o d e s m u s t t h e r e f o r e b e e v a l u a t e d t o a l ar g e n u m b e r o fs i g n i f ic a n t f ig u r es . T h i s l os s of a c c u r a c y i s a fe a t u r e o f m o s t m e t h o d s o f s o l u t i o n o f t h e m a t r i xe q u a t i o n .T h e m e t h o d s t h a t h a v e s o fa r b e e n m e n t i o n e d a l l s u f fe r f r o m t h e d e f e ct s i n h e r e n t i n r e p l a c i n gt h e s t r u c t u r e b y a d i s cr e t e m a s s s y s t e m . D u n c a n h a s s h o w n b y c o n s i d e r i n g t h e t o r s i o n o f au n i f o r m c a n t i l e v e r t h a t t h e e r r o r in t h e f r e q u e n c y o b t a i n e d b y m e a n s o f t h e d i s c re t e m a s s m e t h o di s i n v e r s e l y p r o p o r t i o n a l t o t h e s q u a r e o f t h e n u m b e r o f d e g r ee s o f f r e e d o m u s e d. T o o b t a i na n a c c u r a c y o f 1 p e r c e n t w e m u s t u s e s ix t im e s a s m a n y d e g r ee s o f f r e e d o m a s t h e r e a r e d e s i r e d


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modes. The accuracy for flexure ma y be a little better. If we decide tha t an accuracy to withi n2 per cent should be aimed at, then at least four times as many degrees of freedom as there aredesired modes should be used for either flexure or torsion. In addit ion to ensuring the accuracyof the frequency, this should give a sufficiently large numbe r of ordinates to dete rmin e the modeshapes with reasonable accuracy.7 . T h e L a g r a n g i a n o r R a y l e i g h - R i t z M e t h o d (DF/EM).--We consider the case of flexure of abea m to illustrate t he meth od, and let th e di splaceme nt be expressed as a linear combinatio n ofassumed functions of the form:

= z y ) q , , . . . . . . . . . . . . ( 7 . 1 )r = l

where Z r ( y ) are the assumed functions and q, are the generalized co-ordinates.We can now use the simplified Langrangian equations (or the Rayleigh equations):

j g m~: aVaq, - - aq~' . . . . . . . . . . . . . . (7.2)where co~3 = ma xi mu m kinetic energy, and V = ma xi mu m potent ial energy.

If the expression (7.1) for z ( y ) is substituted in the equations (2.23) and (2.24) forY and VIand the differentiations with respect to q,, then performed, we obtain the matrix equation:[ c o M - - E J q = O , . . . . . . . . . . . . . . . . (7.3)

where A is an inertia matrix with elements :A ~ , = ( y ) Z , ( y ) Z , ( y ) d y , . . . . . . . . . . (7.4)

and E is an elastic stiffness matr ix with elements := B ( y ) Z / ' ( y ) z , ' / y ) d y ( 7 . 5 )

0

Since the mass distribution (y) and stiffness distri bution B ( y ) will not in general be expressedas mathematical functions, the coefficients (7.4) and (7.5) will have to be evaluated by usingsome me th od of numeric al inte grat ion; Simpson's rule is the best for this purpose. To keep theerrors in this process sufficiently small, the number of integration stations must be at least fourtimes th e num ber of desired modes, and if possible six times as great. This rule is similar to th atgiven for the discrete mass metho ds of the preceding paragraphs. The order of the mat rixequations to give results of a prescribed accuracy will, however, be much less in the Lagrangianmet hod than in the Morris discrete mass method. This reduction in the order of the matr ixequat ion is obtaine d at the expense of more complica ted expressions for evaluati ng the coefficientsof the matrices.

If the order of the matri x equati on is not greater than four, the deter minantl o : A - - E [ . . . . . . . . . . . . (7.6)

can be expand ed and the resulti ng equation solved for the fr equency co. If the order of thematrix equation is greater than four, either the iteration method or the escalator method ( s e eAppendix II) can convenient ly be used. If the iteration met hod is used and a cantilever beamis being considered, the first step is to find the reciprocal of the matrix E, and then to form theequation :c o 2 E - 1 A q = q . . . . . . . . . . . . . . . (7.7)

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Alternatively, the matrix E m ay be expressed as the product of a lower triangular matri x andits transpose, that is:E L L . . . . . . . . . . . . . . . (7.8)

If we now write ~ = L q , equation (7.7) becomes:co~ L -~ A (L-l) c~ = c] . . . . . . . . . . . . . . . (7.9)

We thus have to find the reciprocal of a triangular matrix, which is easier than finding thereciprocal of E.

The accuracy of the m eth od will depend on the choice of displacement functions Z / y ) . It ismost desirable that these displacement functions should satisfy not only the bound ary conditionsat the fixed end of the beam, but also those at the free end. The bound ary condit ions at the freeend are given in the first place as dynamical conditions, but from them can be deduced geometriccondit ions: there are three different cases, depe nding on the ratio of flexural rigidit y to massper unit l ength at the free end.Duncan has determi ned sets of displacement functions of polynomial form which satisfy thesegeometric end conditions s e e Appe ndix I). The first four of these for a blu nt- end ed bea m areshown in Fig. 1. For a uniform beam, accurate results for modes and frequencies can be foundby using one more functio n than t here are desired modes. For less ideal bodies, however, itappears that rather more are necessary. Even for a uniform beam, the full accuracy can beobta ined only by keeping a large numbe r of figures in the calculations. Figures are lost for the

overt one modes ; there is some loss of figures for the generalized co-ordinates q, during the solutionof the matrix equation and a further loss for the displacements z when the numerical values ofthe q~ are subst itut ed in equa tion (7.1). This loss of figures would be even greater if the bo und arycondition s at t he free end were not satisfied. The reason for the loss is tha t the chan ge in shapeas we pass from one displacement function to the n ext is only a slow one. A way of Considerablyreducing it is to start the calculations not vdth the Duncan functions themselves but with linearcombina tions of them . In this ne w set of functions the first is identic al with the first of theDun can functions. For the second we combine the first two Dunca n functions to give a nodalpoint in roughly the expected place. The third function is a combination of the first threeDun can funct ions to give two nodes where we expect the m to be, and so on. The first thre efunctions derived in this way are shown in Fig. 2.For a freely supported beam, one or possibly two of the displacement functions must representthe rigid mode displacements of vertical translation or roll or both. The remaining displacementfunctions may convenientl y be take n to be Duncan s displacement functions, or combinations

of them ; these satisfy the conditions Z / y ) = Z / ( y ) = 0 at the point y = 0, which may be tak ento be the end of an asymmetric beam or the centre of a symmetric beam.In this case the ma tri x E will have a null row and null column for each rigid mode. Beforethe matrix equation can be expressed in the form (7.7), as required in the iteration meth od, therigid mode co-ordinates will therefore have to be eliminated from the equation. This eliminationwill not, however, be necessary if the escalator method is used.Displacement functions of a different kind have been introduced by Rauscher of America.Whereas Duncan s functions do not depend on the particular structure, apart from the en dconditions, Rauscher s functions depend on t he elastic properties of the structure, and mi ghttherefore be expected to give more accurate results.Rauscher starts b y dividing the beam into a number of segments, equal to the n umber offunctions he wants to find, and imagines the loading curve which is typical of any one of thenormal modes to be given approximat ely by a polygon, linear in each segment. This polygonalloading is in turn replaced by the sum of a numb er of triangular loadings, each triangle extendingover two segments, except the tip triangle, which extends over one only (see Fig. 3). Eac htriangular loading is then considered in turn, and by integratin g four times, introducing thestiffness distribution in th e process, the displacement produced by the triangular load is obtained.

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S i n c e w e r e q u i r e o n l y t h e s h a p e o f t h e d e f l e c t i o n c u r v e , t h e h e i g h t o f t h e t r i a n g l e i s n o t i n f a c ts p ec i fi e d. T h e d i s p l a c e m e n t s o b t a i n e d i n t h i s w a y f or al l t h e t r i a n g u l a r l o a d s t h e n g i v e t h ed i s p l a c e m e n t f u n c t i o n s . T h e f u n c t i o n s f o u n d b y t h i s p r o c e ss fo r a u n i f o r m b e a m , w i t h f o u r d e g r e eso f f r e e d o m c o n s i d e re d , a r e s h o w n in F ig . 4. T h e s e f u n c t i o n s ar e m o r e a l ik e t h a n D u n c a n ' sf u n c t i o n s a n d t h e r i s k o f c o m p u t a t i o n a l i n a c c u r a c i e s i s t h e r e f o r e g r e a t e r , s o t h a t i t i s e v e n m o r en e c e s s a r y t o c o m b i n e t h e m l i n e a r l y i n th e w a y t h a t h a s b e e n d e sc r ib e d a b o ve . T h e s e c o n dd e r i v a t i v e s o f t h e f u n c t i o n s w i l l h a v e b e e n f o u n d h a l f - w a y t h r o u g h t h e i n t e g r a t i o n p r o c es s , so n od i s a d v a n t a g e a r i s e s i n n o t h a v i n g t h e f u n c t i o n s e x p r e s s e d i n al g e b r a i c f o r m . I f t h e r e a r e r a p i dc h a n g e s i n t h e m a s s d i s t r i b u t i o n , a r e f i n e m e n t o f t h e f u n c t i o n s c a n b e m a d e - - i n s t e a d o f t r i a n g u l a rl o a d s w e t a k e l o a d s w h i c h a r e t h e p r o d u c t o f a t r i a n g l e a n d t h e m a s s d i s t r i b u t i o n .

W h a t i s r e a l l y d o n e in t h e c o n s t r u c t i o n o f t h e R a u s c h e r f u n c t i o n s i s t o s t a r t w i t h a s e t o fa r b i t r a r y f u n c t i o n s a n d o b t a i n a s e t o f i m p r o v e d f u n c t i o n s b y o n e r o u n d o f i t e r a t i o n . A na l t e r n a t i v e w a y o f o b t a i n i n g f u n c t i o n s a p p r o p r i a t e t o t h e p a r t i c u l a r s t r u c t u r e c o n s i d e r e d i s t os t a r t w i t h a s e t o f s i m p l e r a l g e b r a i c fu n c t i o n s , s a y f u n c t i o n s ~ +~, a n d i n t e g r a t e t o o b t a i n i m p r o v e df u n c t i o n s .A m a t r i x e q u a t i o n i d e n t i c a l w i t h t h a t o f t h e L a g r a n g i a n m e t h o d is o b t a i n e d i f t h e G a l e r k i nm e t h o d i s a p p l i e d t o t h e d i f f e r e n t ia l e q u a t i o n . T h i s is a m e t h o d o f m a k i n g t h e m e a n e r r o r i n t h e

d i f f e r e n ti a l e q u a t i o n a m i n i m u m .8. The Complementary Ene rgy Method ( D F / E M ) . - - O n e d i s a d v a n t a g e o f t h e L a g r a n g i a n m e t h o di s t h a t t h e e l a s t i c c o e ff i ci e nt s d e p e n d o n t h e s e c o n d d e r i v a t i v e o f t h e d i s p l a c e m e n t . T h e d i s p la c e -m e n t f u n c t i o n s w h e n c o m b i n e d t o g e t h e r c a n r e p r es e n t t h e t r u e m o d e o n l y t o w i t h i n a c e r t a ine r ro r . D i f f e r e n t i a t i o n a l w a y s in c r e a s es s u c h e r r or s so t h a t t h e p o t e n t i a l e n e r g y i n th e m o d e si s g i v e n l e s s a c c u r a t e l y t h a n t h e k i n e t i c e n e r g y .T h i s d i f f e r e n t i a t i o n i s a v o i d e d i n t h e c o m p l e m e n t a r y e n e r g y m e t h o d , w h i c h w a s f ir s t d e v e l o p e db y G r a m m e l a n d r e i n t r o d u c e d b y W e s t e r g a a r d a n d R e i s sn e r. T h e p o t e n t i a l e n e rg y f or f le x u rei s e x p r e s s e d i n t h e f o r m

v = o M 2 y ) d y . . . . . . . . . . . 8 . 1 )w h e re t h e b e n d i n g m o m e n t M y) i s g i v e n b y t h e d o u b l e i n t e g r a l :

M y ) = d u d v . . . . . . . . . S .2 )y

I f t h i s f o rm o f t h e p o t e n t i a l e n e r g y i s u s e d i n a p p l y i n g L a g r a n g e ' s e q u a t i o n s , w e o b t a i n am a t r i x e q u a t i o n o f t h e f o r m :EA - - ~ K ~ q = 0 , . . . . . . . . . . . . . . . . (8.3)

w h e r e t h e i n e r t i a c o ef f ic i en t s A ~ a r e t h e s a m e a s i n t h e L a g r a n g i a n m e t h o d ( e q u a t i o n (7 .4 )) a n dt h e e l a s t i c c o e f f i c i e n t s a r e g i v e n b y

w h e r ez 1K ,.s = M ~ y ) M ~ y ) dyo~ y) . . . . . . . . (8.4)

M / y ) ---- d v . . . . . . . . . . . 8 .5 )y

I n t h i s m a t r i x e q u a t i o n t h e f r e q u e n c y o c c u r s i n c o n j u n c t i o n w i t h t h e e l a s t i c m a t r i x , a n d n o tw i t h t h e i n e r t i a m a t r i x a s i n t h e o r d i n a r y L a g r a n g i a n m e t h o d .11


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I f t h e e l a s t i c p r o p e r t i e s o f t h e b e a m a r e e x p r e s s e d i n t e r m s o f f l e x i b i l i t y f u n c t i o n s , t h ee x p r e s s io n f o r t h e p o t e n t i a l e n e r g y m a y b e w r i t t e n i n a d i f f er e n t f o r m b y e q u a t i n g i t t o t h e w o r kd o n e b y t h e i n e r t i a f o rc e s . I f w e e x p r e s s t h e d i s p l a c e m e n t i n t h e f o r m ( 2.1 3 ), w e t h a n h a v e :i t y ) I t u ) z u ) du dy 8 . 6 )1 0 42 JO . . . . .

I t f o ll o w s b y d i f f e r e n t i a t i o n t h a t t h e c o e f fi c i en t s K ~, a r e g i v e n b y :K , = 0 i t y ) z , y ) F y , I t u ) d y . . . . . 8 . 7 )

T h i s f o r m o f t h e e x p r e s s io n f o r K , , i s p a r t i c u l a r l y c o n v e n i e n t w h e n f l e x i b il i ty c o e ff ic ie n ts h a v eb e e n d e t e r m i n e d e x p e r i m e n t a l l y .T h i s m e t h o d g i v e s m o r e a c c u r a t e f r e q u e n c i e s t h a n t h e L a g r a n g i a n m e t h o d , b u t i t h a s t h ed i s a d v a n t a g e t h a t m o r e f ig u re s m u s t b e k e p t i n t h e c a l c u l a t io n s t o o b t a i n t h i s a c c u r a c y .T h e m o d e s a r e n o t n o t i c e a b l y d i ff e r en t , b u t i m p r o v e d m o d e s h a p e s c a n b e o b t a i n e d b y u s i n gt h e f o r m u l a :

( ( y ) = F ( y , u ) I t ( u ) z ( u ) d u . . . . . . . . . . (8.8)i fs = 1 qs o F ( y u ) I * ( U ) Z , ( U ) d u , . . . . . . . . (8.9)w h i c h i s d e r i v e d f r o m e q u a t i o n s ( 2 .1 3) a n d (7 .1 ). T h e i n t e g r a l s u n d e r t h e s u m m a t i o n s i g n w i l lh a v e a l r e a d y b e e n o b t a i n e d i f t h e e l a st i c c o e ff ic ie n ts K , , h a v e b e e n d e t e r m i n e d f r o m e q u a t i o n( 8.7 ), a n d l i t t l e a d d i t i o n a l w o r k i s t h e r e f o r e e n t a i l e d .

T h e a b o v e t r e a t m e n t a p p l ie s s p ec i fi c a ll y t o a c a n t i l e v e r b e a m , a n d a l l t h e d i s p l a c e m e n t f u n c t i o n sm u s t s a t i s f y t h e c o n d i t io n Z ( y ) = Z ( y ) - - 0 a t t h e c la m p e d e n d. F o r a n u n s u p p o r t e d b e a ma d d i t i o n a l f u n c t i o n s r e p r e s e n t i n g t h e r i g id m o d e s a r e r e q u i re d , b u t a d i f f i c u lt y n o w a r is e s in t h ea p p l i c a t i o n o f t h e m e t h o d .W h e n t h e o r d i n a r y L a g r a n g i a n m e t h o d is u se d , t h e e q u a t i o n s o b t a i n e d b y d i f fe r e n ti a t io n w i t hr e s p e c t t o t h e r i g i d m o d e c o - o r d i n a t e s a r e i d e n t i c a l w i t h e q u a t i o n s (2 .1 9) a n d ( 2.2 0 ), t h a t i s , t h e y

a r e m a t h e m a t i c a l e x p r e ss i o ns o f t h e v a n i s h i n g o f t h e t o t a l v e r t i c a l i n e r t i a f o rc e a n d t o t a l i n e r t i am o m e n t . T h i s w il l n o t b e t h e c a se i f t h e c o m p l e m e n t a r y e n e r g y m e t h o d i n t h e a b o v e f o r m isu s e d ; t h e r e s u l ts o b t a i n e d w i l l b e i n c o r r e c t a n d t h e f o ll o w i n g m o d i f i c a t i o n i s m a d e .T h e e l a st i c t e r m s i n t h e e q u a t i o n s o f m o t i o n a r e n o w o b t a i n e d b y e v a l u a t i n g t h e d e r i v a t i v e s~ V / a q ~ , w h i c h a p p e a r i n L a g r a n g e s e q u a t i o n s , s u b j e c t t o t h e c o n d i t i o n t h a t t h e t o t a l f o rc e a n dm o m e n t e q u a t i o n s ( 2.1 9) a n d (2 .2 0) b e sa t i sf i e d . T h i s is e q u i v a l e n t t o e x p r e s s in g t h e r i g i d m o d ec o - o r d i n a t e s i n t e r m s o f t h e r e m a i n i n g g e n e r a l i z e d c o - o r d i n a t e s b y u s i n g e q u a t i o n s (2 .1 9) a n d( 2 .2 0 ), a n d s u b s t i t u t i n g t h e s e e x p r e s s i o n s i n t h e p o t e n t i a l e n e r g y (8 .1 ) o r ( 8.6 ) b e fo r e p e r f o r m i n gt h e p a r t i a l d i f f e r e n t i a t i o n .A m a t r i x e q u a t i o n i d e n t ic a l w K h t h a t o f t h e c o m p l e m e n t a r y e n e r g y m e t h o d is o b t a i n e d if t h eG a l e r k i n m e t h o d is a p p l ie d to t h e i n t e g r a l e q u a t i o n , m a k i n g t h e m e a n e r r o r i n t h e e q u a t i o n am i n i m u m .9 . C o ll oc a ti o~ w i t h A s s u m e d M o d e s ( D F / I E ) . ~ F o r f le x ur e, t h e d i s p la c e m e n t z ( y ) i s a g a i ne x p r e s s e d i n t h e f o r m :

z ( y ) = E Z , ( y ) q , , . . . . . . . . . . . . (9. .1)~ =1w h e r e t h e Z / y ) a r e t h e a s s u m e d f u n c t i o n s .

F o r a c a n t i l e v e r b e a m c l a m p e d a t y = 0 , t h e f u n c t i o n s Z , ( y ) m u s t s a t i s f y t h e c o n d i t i o nZ , (0 ) = Z , (0 ) = 0 . F o r t h i s c a se , t h e e x p re s s io n (9 .1 ) fo r z ( y ) i s s u b s t i t u t e d i n t h e i n t e g r a le q u a t i o n ( 2 . 1 3 ) ; i f t h e o r d e r o f i n t e g r a t i o n a n d s u m m a t i o n i s i n t e r c h a n g e d , w e o b t a i n :i i~ ( y ) q~ = a~ q , F ( y , u ) i t ( u ) Z ~ (u ) d u . . . . . . . (9.2)s = l . s = l 0

2


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T h i s e q u a t i o n c a n n o t n o w b e s a t i sf i ed a t a l l p o i n t s o f th e b e a m , b u t a n a p p r o x i m a t e s o l u ti o nm a y b e o b t a i n e d b y s a t i s f y i n g i t a t n s e l e c te d c o l l o c a t i o n p o i n t s y l , y ~, . . . y ~ .I f w e w r i t e :

a n d l e t

G~s = F ( y , u ) t ~ ( u )Z , ( u ) d u . . . . . . . . (9.3)Z ~ , = Z , y , ) . . . . . . . . . . . . . . (9.4)

G = n t h o r d e r s q u a r e m a t r i x w i t h t y p i c a l e l e m e n t G~sZ = n t h o r d e r s q u a r e m a t r i x w i t h t y p i c a l e l e m e n t Z , ,q = m a t r i x c o l u m n { q l, q2, q , ,} ,

w e o b t a i n t h e m a t r i x e q u a t i o n :[co~G - - Z7~q = 0 . . . . . . . . . . . . . . . . . (9.5)

I n t h e e n e r g y m e t h o d s d e s c ri b e d i n t h e l a s t tw o s e c t i on s t h e m a t r i c e s i n t h e m a t r i x e q u a t i o na r e s y m m e t r i c . I n t h e c o ll o c a t i o n m e t h o d , h o w e v e r , th e m a t r i c e s G a n d Z a r e n o t s y m m e t r i c ,a n d i f m a t r i x i t e r a t i o n i s u s e d t h e c a l c u l a t i o n o f t h e o v e r t o n e m o d e s a n d f r e q u e n c i e s i s a l i tt l em o r e t r o u b l es o m e T h e a c c u r a c y i s i n g e n e r a l n o t q u i t e a s g o o d a s i n t h e e n e r g y m e t h o d s , b u tm a y b e b e t t e r i f t h e c o l lo c a ti o n p o in t s h a p p e n t o h a v e b e e n c h o s en f o r t u n a t e l y .

F o r a n u n s u p p o r t e d b e a m , t w o o f t h e d i s p l a c e m e n t f u n c t i o n s m u s t r e p r e s e n t t h e r i g i d m o d e so f v e r t i c a l t r a n s l a t i o n a n d r o ll . T h e m o d i f i c a t i o n ( 2.1 7) i s n o w m a d e t o t h e l e f t - h a n d s i d e of t h ei n t e g r a l e q u a t i o n ( 2. 13 ) a n d e q u a t i o n ( 9.2 ) i s a l t e r e d a c c o r d i n g l y . T h i s e q u a t i o n i s n o w s a t i s f i e da t o n l y ( n - - 2 ) c o l l o c a ti o n p o in t s , a n d t h e d e f i c ie n c y i n t h e n u m b e r o f e q u a t i o n s i s m a d e u p b ys a t i s f y i n g e q u a t i o n s (2 .1 9) a n d (2 .2 0) w h i c h e q u a t e t h e t o t a l i n e r t i a f o rc e a n d t o t a l i n e r t i am o m e n t t o ze ro . A m a t r i x e q u a t i o n o f s i m i l a r f o r m to e q u a t i o n (9 .5 ) i s a g a i n o b t a i n e d , b u t t h em a t r i x c o r r e s p o n d i n g t o Z h a s t w o n u l l r ow s a n d t w o n u l l c o lu m n s .10. N u m e r i c a l E x a m p l e C o m p a r i n g t h e V a r i o u s M e t h o d s . - - T h e f o l l o w i n g r e s u l t s f o r t h e f l e x u r a lm o d e s o f a t a p e r e d b e a m o f c o n s t a n t w i d t h , w i t h d e p t h a t t h e t i p o n e f if t h o f t h e d e p t h a t t h er o o t , a r e t a k e n f r o m a r e p o r t b y S i d d a l l a n d I s a k s o n 1.

M e t h o d

L a g r a n g e . . . . . . . . . . . .

C o m p l e m e n t a r y e n e r g y . . . . . . . .

C o m p l e m e n t a r y e n e r g y . . . . . . . .D i s c r e t e m a s s . . . . . . . . . . . .M y k l e s t a d . . . . . . . . . . . .

D e g r e e s o fM o d e s f r e e d o mP e r c e n t e r r o r in f r e q u e n c y

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. . D u n c a n (

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+ 2 4+ 0 4 5- - 0 0 5- - 0 0 9- - 0 0 9- - 0 0 9

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+ 4 2 . 0+ 9 . 3 + 6 6 - 2+ 4 . 3- - 0 . 6 5 - - 1 2 . 2

1 . 8

+ 2 . 8 2+ 0 . 6 9

I f a n a c c u r a c y t o w i t h i n 2 p e r c e n t i s o u r o b j e c t iv e , t h e n t h e f i r st l i n e s h o w s t h a t t h e L a g r a n g i a nm e t h o d w i t h t w o o f D u n c a n s f u n c t i o n s a l m o s t g i ve s t h i s a c c u r a c y fo r th e f u n d a m e n t a l f r e q u en c y .F o r t h e f i r s t o v e r t o n e f r e q u e n c y , h o w e v e r , a t l e a s t f o u r of D u n c a n s f u n c t i o n s w o u l d b e r e q u i r e d ;i f w e m a k e t h e r e a s o n a b le a s s u m p t i o n t h a t t h e e r r or s i n t h i s f r e q u e n c y w i t h i n c re a s i n g n u m b e r13


15/32

o f f u n c t i o n s f o r m a g e o m e t r i c s e q u e n c e * , t h e n f o u r f u n c t i o n s w o u l d b e s u f f i c i e n t t o g i v e a n e r r o ro f n o t m o r e t h a n 2 p e r c en t . T h i s s u g g es t s t h a t t h e L a g r a n g i a n m e t h o d i n c o n j u n c t io n w i t hD u n c a n s f u n c t i o n s w i l l g i v e s u f f i c ie n t l y a c c u r a t e r e s u l t s i f t h e n u m b e r o f a s s u m e d f u n c t i o n s i st w o g r e a t e r t h a n t h e n u m b e r o f d e s ir e d m o d e s. I f D u n c a n s m o d e s a re u se d in t h e c o m p l e m e n t a r ye n e r g y m e t h o d w e s ee t h a t t h e n u m b e r o f f r eq u e n ci e s g i v e n a c c u r a t e ly i s on e le ss t h a n t h e n u m b e ro f a s s u m e d m o d e s . I f t h e m o d e s a r e r e p l a c e d b y R a u s c h e r s m o d e s , t h e r e is a f u r t h e r i m p r o v e -m e r i t - - t w o f r eq u e n ci e s a r e g i v e n t o w i t h i n a n a c c u r a c y o f 2 pe r c e n t w h e n o n l y t w o d i s p l a c em e n tf u n c t i o n s a r e u s e d .

F o r t h e M o r ri s d i s c re t e m a s s m e t h o d , h o w e v e r , e v e n w i t h f i v e m a s s e s t h e s e c o n d f r e q u e n c y i sn e a r l y 3 p e r c e n t i n e r ro r . T h e M y k l e s t a d m e t h o d w i l l g i v e e r r or s w h i c h a r e v e r y l i t t le d i f fe r e n tf r o m t h o s e o f t h e M o r r i s d is c r e t e m a s s m e t h o d ; w i t h 1 1 m a s s e s t h e f i r s t t w o f r e q u e n c i e s h a v ee r ro r s a l m o s t t h e s a m e a s t h e c o m p l e m e n t a r y e n e r g y m e t h o d u s i n g th r e e o f D u n c a n s f u n c t i o n s.O n e p o i n t w h i c h m a y b e n o t e d i s t h a t w h i l e t h e e r r o r i n t h e L a g r a n g i a n m e t h o d i s a l w a y sp o s i t i v e ( t h a t i s, t h e c a l c u l a t e d f r e q u e n ci e s a r e to o h i g h ) , t h e e r r o r i n t h e c o m p l e m e n t a r y e n e r g ym e t h o d m a y b e p o s i ti v e or n e g a ti v e .T h e m o d e s h a p e s f or t h e f u n d a m e n t a l m o d e a r e q u i t e g o o d i n e v e r y c as e . F i g . 5 s h o w s t h e

s h a p e o f th e f i r st o v e r t o n e m o d e w h e n t w o d e g r ee s of f r e e d o m a r e u s e d, c o m p a r e d w i t h t h e e x a c tm o d e s h a p e. T h e R a u s c h e r f u n c t io n s g iv e a b e t t e r m o d e s h a p e t h a n d o D u n c a n s f u n c t i o n s ;t h e r e i s l i t tl e d if fe re n ce b e tw e e n t h e L a g r a n g i a n a n d c o m p l e m e n t a r y e n e r g y m e t h od s , b u t t h em o d e s h a p e o b t a i n e d b y t h e l a t t e r m e t h o d c a n be i m p r o v e d b y i n t e g r a ti o n , a s h a s b e e n i n d i c a te da b o v e . T h e r e i s t h e n l i t t l e d i ff e re n c e b e t w e e n t h e m o d e o b t a i n e d a n d t h e t r u e m o d e i n t h i sp a r t i c u l a r c a s e .T h e b e s t m e t h o d t o b e u s e d fo r a p a r t i c u l a r p r o b l e m w i l l d e p e n d o n t h e c a l c u l a t i n g a i d sa v a i l a b l e . I f d es k m a c h i n e s a r e u s e d , i t i s r e c o m m e n d e d b y t h e a u t h o r t h a t i n g e n e r a l t h eL a g r a n g i a n m e t h o d s h o u l d b e u s e d i n c o n ju n c t i o n w i t h d i s p l ac e m e n t f u n c t io n s o f t h e R a u s c h e rt y p e , b u t i f th e f l e x ib i l it y f u n c t i o n is k n o w n f or t h e s y s t e m t h e c o m p l e m e n t a r y e n e r g y m e t h o dw i t h D u n c a n s m o d e s s h o u l d b e u se d , a s i n t h i s ca s e t h e m o d e s c a n b e i m p r o v e d w i t h o u t d i f fi c u lt y .T h i s w o u l d e n a b l e a g i v e n a c c u r a c y t o b e o b t a i n e d w i t h t h e m i n i m u m n u m b e r o f d e g r e e s o ff r e e do m . T h i s s a v i n g o f w o r k i s o f fs e t b y t h e i n c r e a s e d w o r k i n v o l v e d i n s e t t i n g u p t h e m a t r i xe q u a t i o n s a s c o m p a r e d w i t h t h e L a g r a n g i a n m e t h o d i ll t e r m s o f D u n c a n s d i s p l a c e m e n t f u n c t io n s ,a n d e s p e c i a l l y c o m p a r e d w i t h t h e d i s c re t e m a s s m e t h o d s . I n a d d i t i o n , g r e a t e r c a r e i s n e e d e d t oa v o i d c o m p u t a t i o n a l e rr o rs . H o w e v e r , t h e r e d u c t i o n i n t h e o r d e r o f t h e m a t r i x e q u a t i o n is t h em o s t i m p o r t a n t c o n s i d e r a t i o n , a n d i n a l l p r o b a b i l i t y t h e r e i s a s a v i n g o f e f fo r t.I f a d ig i t a l c o m p u t e r w e r e a v a i la b l e , h o w e v e r , i t w o u l d b e i m p o r t a n t t o k e e p t h e p r o g r a m m i n ga s s i m p l e a s p o s s i b le e v e n t h o u g h t h e n u m b e r o f d e g r ee s o f f r e e d o m w a s r e l a t i v e l y l a r ge . S i n c es t a n d a r d s u b - p r o g r a m m e s e x i s t f o r t h e m a n i p u l a t i o n o f m a t r ic e s , t h e M o r ri s di s cr e te m a s s m e t h o dw o u l d b e t h e b e s t m e t h o d t o u s e .O n e m e t h o d w h i c h h a s n o t b e e n m e n t i o n e d i s a m e t h o d d e v i se d b y D u n c a n b a s e d o n L a g r a n g e se q u a t i o n s w h i c h m a k e s u s e o f t h e o r t h o g o n a l p r o p e r t i e s o f t h e m o d e s i n s u c h a w a y t h a t o n l yt w o d e g re e s o f f r e e d o m n e e d b e c o n s i d e r e d a t a n y s t a g e o f th e c a l c u l at i o n , w h i c h e v e r m o d e i s

b e i n g e v a l u a t e d . T h e s o l u t i o n o f t h e m a t r i x e q u a t i o n i s t h e r e f o r e q u i t e si m p l e , a n d t h e m e t h o di s w e l l s u i t e d f o r d e s k - m a c h i n e c a l c u l a t io n . T h e r e i s a n a c c u m u l a t i o n o f e r r o r as t h e o v e r t o n eo r d e r in c re a se s , a d d i t i o n a l t o t h a t i n t h e o r d i n a r y L a g r a n g i a n m e t h o d , b u t w h e n a p p l ie d t o au n i f o r m b e a m g o o d a c c u r a c y i s o b t a i n e d f o r t h e f i r s t t h r e e f r e q u e n c ie s .11. Coupled Flexure Torsiora. When t h e c o u p l i n g b e t w e e n f l e x u r e a n d t o r s i o n h a s t o b e t a k e ni n t o a c c o u n t , s o m e a s s u m p t i o n h a s t o b e m a d e a b o u t a f l e x u r a l a x i s. S u p p o s e t h a t s o m e l i n ec a n b e fo u n d i n t h e s t r u c t u r e s u c h t h a t a c ou p le a p p l i e d a t a n y p o i n t a b o u t t h e t a n g e n t t o t h eT h e e r r o r g i v e n b y t i le L a g r a n g i a n m e t h o d d e c r e a s e s m u c h m o r e r a p i d l y w i t h t h e n u m b e r o f d e g r ee s o f f r e e d o mt h a n d o e s t h e e r r o r g i v e n b y t h e d i s c r e te m a s s m e t h o d .

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l i n e p r o d u c e s n o f l e x u r a l c u r v a t u r e o f t h e l i n e a t t h a t p o i n t . I f t h i s l i n e (t h e lo c u s of f l e x u r a lc e n tr e s ) i s s t r a i g h t a n d n o r m a l t o t h e l i n e o f c l a m p i n g a t t h e r o o t ( n o r m a l t o t h e c e n t r e - li n e f o ra n a i r c r a f t w i n g ) , t h e l i n e i s t e r m e d a f l e x u r a l a xi s .T h e b a s i c e q u a t i o n s o f s e c t i o n 2 n e e d s o m e m o d i f i c a t i o n f o r t h i s c a s e .a ) D i f fe r e n t i a l E q u a t i o n s . - - E q u a t i o n s ( 2.1 ) a n d ( 2.8 ) a r e r e p l a c e d b y t h e e q u a t i o n s :

a ~ a ~ ty ~ B y ) a y 2 t = ~ 2 { ~ y ) ~ y ) + ~ y ) o y ) } , . . . . . . . . 1 1 . 1 )d c y ) , , y = , . . . . . . . . (11.2)

w h e r e ~ ( y ) = m a s s m o m e n t a b o u t f l e x u r a l a x i s p e r u n i t l e n g t h .T h e s e e q u a t i o n s m u s t n o w b e c o n s i d e r e d a s a p a i r i n s t e a d o f s e p a r a t e l y .S i m i l a r l y , e q u a t i o n s ( 2 . 2 ) a n d ( 2 . 9 ) a r e r e p l a c e d b y t h e e q u a t i o n s :

d Sa y - ~ { , . y ) z y ) + ~ y ) o y ) } , . . . . . . . . 1 1 . 3 )a Td y - - m = { ~ ( y ) y ) + ~ y ) o(y )} . . . . . . . . . (11.4)

E q u a t i o n s ( 2 . 8 ) t o ( 2 . 5 ) a n d ( 2 . 1 0 ) r e m a i n u n a l t e r e d .b) T h e I n t e g ra l E q u a t i o n . - - E q u a t i o n s (2 .1 3) a n d ( 2.1 5) a r e r e p l a c e d b y t h e p a i r o f e q u a t i o n s :

~ y ) = ~ ~ y , ~ ) { ~ ~) ~ ~ ) + ~ ~ ) o ~ ) } a ~ . . . . . 1 1 . 5 )o

o y ) = ~ ~ y , ~ ) { ~ ~ ) ~ ~ ) + ~ ~ ) o ~ ) } a ~ . . . . . 1 1 .6 )I f a s t r a i g h t f l e x u r a l a x i s i n t h e a b o v e s e n s e d o e s n o t e x i s t , a m o r e g e n e r a l f o r m o f t h e s ee q u a t i o n s m a y b e u s e d, p r o v i d e d t h a t s e c ti o n s n o r m a l t o t h e b e a m m a y s t il l b e a s s u m e d n o t t od i s to r t . W e c a n t h e n w r i t e :

z y ) = ~o~ F y , ~ ) { ~ u ) ~ ~ ) + ~ u ) 0 ~ )} a uco~ T y , u ) {$(u) z u ) + z u ) 0(u)} d u , . . . . (11.7)

y ) = ~ ~ u , y ) { ~ ~ ) z u ) + ~ u ) 0 u ) } a u

+ ~ ~ y , u ) { ~ u ) ~ u ) + ~ ~ ) 0 u ) } a u , . . . . 1 1 . 8 )w h e r e t h e a x i s of y i s n o w t a k e n a s a n y c o n v e n i e n t a x i s a l o n g t h e l e n g t h o f t h e s t r u c t u r e , a n d t h en e w i n f l u e n c e f u n c t i o n i s d e f i n e d a s :

g ~( y, u ) = d i s p l a c e m e n t a t p o i n t y o n a x i s d u e t o u n i t t o r s i o n a l c o u p l e a t p o i n t uN ( u , y ) = a n g l e o f t w i s t a t s e c t i o n y d u e t o u n i t l o a d a t p o i n t u o n a x i s .

T h e f u n c t i o n s F y , u ) a n d ( y, u ) h a v e t h e s a m e p h y s i c al m e a n i n g a s b e f or e ex c e p t t h a t t h e ya r e re f e r r e d t o t h e r e f er e n c e a x i s i n s t e a d o f t o t h e f l e x u r a l ax i s . T h e y w i l l n o t n o w , h o w e v e r ,b e d e fi n e d b y e q u a t i o n s (2 .1 4) a n d ( 2. 16 ). T h e t h r e e f u n c t i o n s F , ~ a n d N c o u ld b e m e a s u r e de x p e r i m e n t a l l y , a n d c a n i n p r i n c i p l e t a k e a c c o u n t o f t h e s h e a r e f f e c t s .15


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E q u a t i o n s ( 2 . 1 9 ) t o ( 2 . 2 1 ) , w h i c h e x p r e s s t h e f a c t t h a t t h e t o t a l i n e r t i a f o r c e a n d m o m e n t sa r e z e r o , a r e r e p l a c e d b y t h e e q u a t i o n sf l = 0 . . . . . . . . . (11.9)f ~ { ~ ~ ) ~ ~ ) + ~ ~ ) o ~ ) } ~ a ~ = o , . . . . . . . . 1 1 . 1 o )

0 { ~ u ) z u ) + ~ u ) 0 u ) } d u = 0 . . . . . . . . . 1 1 . 1 1 )T h e s e e q u a t i o n s a p p l y w h e t h e r t i l e a x i s o f y i s t h e f l e x u r a l a x is o r n o t .(c ) R a y l e i g h ' s E q u a t i o n s . - - T h e e x p re s s io n s (2 .2 3) a n d (2.2 5 ) fo r Y a re n o w re p la c e d b y th es i n g le e q u a t i o n :

9 = o{tt(y) z 2 ( y ) + 2~(y) z ( y ) O ( y ) + n ( y ) O~(y)}d y , . . . . (11.12)a n d t h e e x p r e s s i o n s ( 2.2 4) a n d (2 .2 6) f o r V a r e r e p l a c e d b y t h e s i n g l e e q u a t i o n

; I= ~ o B ( y ) \ d y U + C ( y ) \dv]~ d y . . . . . . . . . (11.13)T h e m e t h o d s d e s c r ib e d i n t h e p r e v i o u s s ec t io n s c a n n o w r e a d i l y b e g e n e r al i z ed b y u s i n g t h ea b o v e e q u a t i o n s . I n t h e d i s c re t e m a s s m e t h o d s t h e s y s t e m i s r e p l a c e d b y a w e ig h t le s s b e a mc a r r y i n g d i s c re t e m a ss e s a n d m o m e n t s o f i n e r t i a a t t h e e n d o f a f in i t e n u m b e r o f t r a n s v e r s e r i g i da r m s . I n t h e M y k l e s t a d m e t h o d s i x di ff e re n c e e q u a t i o n s h a v e n o w t o b e c o n s i d er e d t o g e t h e r ,e q u a t io n s (3.1 ) a n d (3.5 ) b e in g su i t a b ly m o d i f i e d . T w o u n k n o w n s 01 a n d ~/J1, i n s t e a d o f o n ly o n e ,h a v e t o b e c a r ri e d t h r o u g h e a c h s ta g e o f t h e w o r k a n d d e t e r m i n e d b y t h e b o u n d a r y c o n d i t i o n sa t t h e f a r e n d . S i m i l a r l y i n t h e S t o d o l a m e t h o d a se t o f s i x i n t e g r a l s h a s t o b e c o n s i d e r e d a te ac h. i t e r a t i o n s te p . I n t h e c o l l o c a t i o n a n d M o r r is d i sc r e t e m a s s m e t h o d s , e q u a t i o n s (5 .2 ) a n d(6 .1 ) a re r e p l a c e d b y a p a i r o f s u m m a t i o n e q u a t i o n s , a n d t h e m a t r i x e q u a t i o n c o r r e s p o n d i n gto (5.3 ) o r (6 .2 ) c a n b e r e a d i ly d e r iv e d .I n t h e m e t h o d s u s i n g d i s p l a c e m e n t f u n c t i o n s w e e xp r e ss t h e f l ex u r a l a n d t o r s i o n a l d i s p la c e m e n t sb y t h e p a i r o f e q u a t i o n s

z ( y ) = ~ Z , ( y ) q ~ , . . . . . . . . . . . . . . (11.14)s= l

o y ) - - ~ o s y ) x . . . . . . . . . . . . . . . . . 1 1 . 1 5 )S= Iw h e r e Z s ( y ) , O , (y ) a r e t h e f l e x u r a l a n d t o r s i o n a l d i s p l a c e m e n t f u n c t i o n s r e s p e c t i v e l y , a n d qs,x s t h e c o r r e s p o n d i n g g e n e r a l i z e d c o - o r d i n a t e s . T h e n u m b e r o f f l e x u r a l f r e e d o m s n a n d t o r s i o n a lf r e e d o m s v m u s t b e t h e s a m e i n t h e c o l l o c a ti o n m e t h o d , b u t c a n b e d i f f e re n t i n t h e e n e r g y m e t h o d s .T h e r e s u l t i n g m a t r i x e q u a t i o n c a n b e w r i t t e n i n p a r t i t i o n e d f o r m , t h e p a r t i t i o n l i n e s d i v i d i n ge a c h m a t r i x i n t o f o u r s u b - m a t r ic e s o f w h i c h t h e t w o d i a g o n a l su b - m a t r i c e s a r e p u r e fl e x u re a n dp u r e t o r s i o n m a t r i c e s a n d t h e t w o r e m a i n i n g s u b - m a t r i c e s c o n t a i n t h e c o u p li n g t e r m s . I n t h e

m a t r i c e s c o r r e s p o n d i n g to Z o f t h e c o l l o c a ti o n m e t h o d a n d E o f t h e L a g r a n g l a n m e t h o d , t h ec o u p l i n g s u b - m a t r i c e s a r e n u ll , b u t i n t h e c o m p l e m e n t a r y e n e r g y m e t h o d t h e r e a r e n o z e r oe l e m e n t s .I t m a y s o m e t i m e s be c o n v e n i e n t t o t a k e a s s u m e d m o d e s e a c h o f w h i c h a r e p a r t l y f l ex u r a l a n dp a r t l y t o r s io n a l . T h e n in s t e a d o f (11 .1 4) a n d (11 .1 5) w e w r i t e

nz y ) = Z z : y ) q s, . . . . . . . . . . . . . . 1 1 . 1 6 )s = l

o ( y ) = ~ , O ~ (y ) q , , . . . . . . . . . . . . . . (11.17)S=I

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s o th a { :t h e m o t i o n i n e a c h d e g r e e o f f r e e d o n qs i s r e p r e s e n t e d b y t h e p a i r o f d i s p l a c e m e n t f u n c t i o n gZs ( y ) a n d O / y ) . W h e n t h e s e e q u a t i o n s a r e u s ed , t h e e x p r e s s i o n s f o r t h e e l e m e n t s o f t h e m a t r i c e si n t h e m a t r i x e q u a t i o n a r e m o r e c o m p l i c a t e d t h a n w h e n t h e s i m p l e r e q u a t i o n s (1 1.1 4) a n d ( 11 .1 5)a r e u s e d . A p a r t i c u l a r c a s e w h e n e q u a t i o n s ( 11 .1 6) a n d ( 1 1.1 7) w o u l d b e a p p r o p r i a t e i s w h e nt h e n o r m a l m o d e s o f a s t r u c t u r e h a v e b e e n f o u n d a n d t h e n o r m a l m o d e s o f t h e s t r u c t u r e a f t e rs o m e m o d i f i c a t i o n of i t ar e r e q u i r e d . T h e n o r m a l m o d e s o f t h e o r i g i n a l s t r u c t u r e c o u l d t h e nc o n v e n i e n t l y b e u s e d a s a s s u m e d m o d e s i n t h e c a l c u l a t io n s .

1 2 . C o m ] ) l e x S t r u c t u r e s . - - E x c e p t f o r t h e i n t e g r a l e q u a t i o n i n t h e f o r m (1 1.7 ) a n d ( 11 .8 ), t h ed i s c u ss i o n of t h e l a s t s e c t i o n a s s u m e d t h a t t h e r e w i l l b e a s t r a i g h t f l e x u r a l a x is , b u t i n g e n e r a lt h i s w i l l n o t b e t h e c a se . I f w e c o n s i d e r a s w e p t - b a c k w i n g , t h e l oc u s o f f l e x u r al c e n tr e s m a y b es t r a i g h t f o r m o s t o f t h e s p a n , b u t i t w i l l c e r t a i n l y b e c u r v e d n e a r t i l e w i n g r o o t a n d w i l l m e e tt h e c e n t r e - li n e a t r i g h t a n g l es (see F i g . 6 ). I n d e a l i n g W i t h s u c h a w i n g t h e s i m p l e s t a p p r o x i m a t i o ni s t o a s s u m e t h a t t h e r o o t t r i a n g l e A B C is r i g id . A b e t t e r a p p r o x i m a t i o n i s t o a s s u m e a s h a p e fo r-t h e d e f l e ct i o n o f t h e c u r v e d p a r t o f t h e l i n e o f f l e x u r a l c e n t re s , s a y p a r a b o l i c . I f t h e l i n e A Ca n d a l l o t h e r l i n e s r a d i a t i n g f r o m A i n t h e r o o t t r i a n g l e a r e a s s u m e d t o r e m a i n r i g i d , t h e m o t i o no f a l l p o i n t s i n t h e t r i a n g u l a r a r e a i s p r e s c ri b e d . F l e x u r a l a n d t o r s i o n a l d i s p l a c e m e n t f u n c t i o n sm a y t h e n b e a s s u m e d r e l a ti v e t o th e r o o t t r i a n g l e f o r t h e r e m a i n d e r o f t h e s t r u c t u r e a n d t h ee q u a t i o n s o f m o t i o n s e t u p .I n t h e m o r e g e n e r a l c a s e , w h e r e t h e l o c u s o f f l e x u r a l c e n t r e s i s e v e r y w h e r e c u r v e d , i t m a y b ep o ss ib l e t o a p p r o x i m a t e t o i t b y a p o l y g o n a l c u rv e . E a c h s t r a i g h t p a r t o f t h e p o l y g o n is t r e a t e ds e p a r a t e l y a n d c e r t a i n c o n t i n u i t y r e l a t i o n s s a t i s f i e d a t t h e j u n c t i o n o f c o n s e c u t i v e p a r t s .H o w e v e r , i t i s p o s s ib l e t o t r e a t t h e p r o b l e m p r o p e r l y i n t h i s c a s e o n l y if f l e x i b i l i t y c o e f f ic i e n t sa r e k n o w n a t v a r i o u s p o i n t s o v e r t h e s u r fa c e , a n d t h e s a m e a p p l i es if t h e d i s t o r t i o n o f c h o r d w i s es e c t i o n s .is n o t s u f f i c i e n t l y s m a l l , a s m a y b e t h e c a s e f o r w i n g s of S m a l l a s p e c t r a t i o . I n t h i s c a s et h e m o t i o n o f t h e w i n g is d e s cr i be d n o t b y a c o m b i n a t i o n o f fl ex u r e a n d t o r s io n b u t b y t h ed i s p l a ce m e n t o f p o i n t s o v er th e w h o l e w i n g ar ea . T h e i n t e g r a l e q u a t i o n m a y t h e n b e w r i t t e n :

z , , y ) = f f F x , y . . v ) v ) v ) e v . . . . . . . r e . l ) ,w h e r e p(x , y ) = m a s s p e r u n i t a r e aF(x,y , u , v ) = d e f l e c t i o n a t p o i n t ( x, y ) d u e t o u n i t l o a d a t p o i n t (u, v)

= F u , v ; x , y ) .T h i s e q u a t i o n c o u l d be s o l v e d b y t h e M o r ri s d i sc r e t e m a s s m e t h o d , c o l l o c a ti o n w i t h a s s u m e d/m o d e s o r t h e c o m p l e m e n t a r y e n e r g y m e t h o d . T h e d i s cr e te m a s s m e t h o d w o u l d b e t h e s im p l e s ti n p r ac t i c e . T h e w i n g s u r f ac e is d i v i d e d i n t o a n u m b e r o f s m a l l a r e a s a n d t h e m a s s i n e a c h a r e ai s c o n c e n t r a t e d a t i t s c en t r e o f g r a v i t y . T h e m a t r i x e q u a t i o n f o r f r e q u e n c y a n d d i s p l a c e m e n tc a n t h e n b e s e t u p i n a w a y s i m i l a r t o t h a t f o r a s i m p l e b e a m . F l e x i b i l i t y c o e ff i ci e nt s a r e r e q u i r e df o r e a c h o f t h e s e c e n t r e s o f g r a v i t y , g i v i n g t h e d e f l ec t io n a t o n e c e n t r e o f g r a v i t y d u e t o u n i tl o a d a t e ac h of t h e o t h e r s i n t u r n . D . W i l l i a m s 9~ h a s d e v e l o p e d a m e t h o d s u i t a b l e f o r a d i g i t a Ic o m p u t e r b y m e a n s o f w h i c h t h e s e f l e x i b i l i t y c o e ff ic i en t s c a n b e c a l c u l a t e d .

D i s p l a c e m e n t f u n c t i o n s f o r w i n g s o f t h i s g e n e r a l t y p e f o r u s e i n e i t h e r t h e c o l l o c a t i o n o rc o m p l e m e n t a r y e n e r g y m e t h o d s c a n b e o b t a i n e d b y a g e n e r a l i z a ti o n o f R a u s c h e r s m e t h o d .F i g . 7 s h o w s a d e l t a w i n g d i v i d e d i n t o s q u a r e s ( t h e n u m b e r o f c h o r d w i s e a r e a s s h o w n a r e p r o b a b l ym o r e t h a n a r e n e c e s s a r y ) . I n s t e a d o f o b t a i n i n g t h e d e fl e c ti o n s d u e to a t r i a n g u l a r l oa d , w e m u s tn o w c o n s id e r a p y r a m i d l o a d c o v er i n g f o u r a d j a c e n t s q u a r e s w i t h a p e x a t th e c o m m o n p o i n to f t h e s q u a r e s , a n d d e t e r m i n e t h e d e f l e c t e d s h a p e o f t h e w i n g u n d e r s u c h a l o a d .13. C o m p l e t e A i r c r a f t . - - W h e n t h e f r e q u e n c i e s a n d m o d e s o f a c o m p l e t e a i r c r a f t a r e r e q u i r e d ,a c o n v e n i e n t m e t h o d i s f ir s t t o o b t a i n s e p a r a t e l y b y a m a t r i x m e t h o d t h e f r eq u e n c ie s a n d m o d es .o f t h e w i n g s e n c a s t r 6 a t t h e r o o t a r i ~ d - o f - t h e f u s e l a g e - t a i l c o m b i n a t i o n e n c a s t r e a t i t s j u n c t i o n

1768984) B-


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w i t h t h e w i n g ._ m a t r ix f o r m T h e e q u a t io n f o r t h e c o m p le t e s y s t e m m a y t h e n b e w r i t t e n in t h e p a r t i t i o n e di ] . . . . . 1 3 1w h e r e t h e m a t r i c e s W , F a n d R r e p r e s e n t t h e w i n g , f u s e la g e - t a il c o m b i n a t i o n a n d r i g i d m o d et e r m s r e s p e c ti v e l y . T h e r i g id f r e ed o m s w i ll b e v e r t i c a l t r a n s l a t i o n a n d p i t c h fo r s y m m e t r i cm o t i o n , a n d ro l l, l a t e r a l d i s p la c e m e n t a n d y a w f o r a n t i s y m m e t r i c m o t i o n . T h e m a t r i c e s ~. a n dr e p r e s e n t t h e c o u p l i n g t e r m s a r i s i n g f r o m t h e r i g i d f r e e d o m s . W e fi r s t c o n s i d e r o n e r i g i d f r e e d o mo n l y w h i c h is r e p r e s e n t e d b y o n e r o w a n d o n e c o l u m n o f t h e m a t r i x i n ( 13 .1 ). T h e n f r o m t h ek n o w n s o l u ti o n s o f t h e f i x e d ro o t e q u a t i o n s :

W q ~ = O , F q ~ = 0 . . . . . . . . . . . . . . . (13.2)w e c a n b y u s i n g t h e e s c a l a t o r m e t h o d s ee A p p e n d i x I I o b t a i n a n e q u a t i o n f o r f r e q u e n c y i nt h e f o r m

c , - - d , 2 - - e - - f Z , . . . . . . . . . . 1 3 . 3 )

w h e r e 2 = 1 /~ o~. T h i s e q u a t i o n c a n b e s o lv e d fo r 2 b y a m e t h o d o f s u c c e s si v e a p p r o x i m a t i o n s ,a n d t h e e l e m e n t s o f t h e c o l u m n m a t r ic e s q~, ql a n d q~ a r e t h e n o b t a i n e d f r o m e x p r e s s i o n s o fs i m i l a r f o rm t o t h e l e f t - h a n d s id e of t h e e q u a t i o n . A s e c o n d r i g id fr e e d o m i s n o w a d d e d a n d a ne q u a t i o n o f t h e f o r m (1 3.3 ) i s a g a i n u s e d , w h e r e t h e q u a n t i t i e s a t o f n o w d e p e n d o n t h e s o l u t i o nw i t h o n e r i g i d fr e e d o m . I f t h e r e a r e t h r e e r i g i d f r e e d o m s , t h e p ro c e s s i s a g a i n r e p e a t e d .T h e e s c a l a to r m e t h o d i s e q u i v a l e n t t o s e t t i n g u p a n e w m a t r i x e q u a t i o n i n t e rm s o f t h e f i x e dr o o t m o d e s o f t h e w i n g a n d t h e f u s e l a g e - ta i l c o m b i n a t i o n w h i c h h a v e a l r e a d y b e e n d e t e r m i n e d ,b u t t h e m e t h o d d o e s t h i s s y s t e m a t i c a l l y a n d o n e d o e s n o t h a v e t o t h i n k o f i t c o n sc i o us ly .T h e e s c a l a t o r m e t h o d c a n n o t , h o w e v e r , b e u s e d i f t h e c o m p l e m e n t a r y e n e r g y m e t h o d i s u s e dt o s e t u p t h e m a t r i x e q u a t i o n . T h i s is b e c a u s e o f t h e d f f f ic u l ty i n t r o d u c e d b y t h e r i g i d f r e e d o m sm e n t i o n e d a t t h e e n d o f s e c t io n 8. W h e n t h e m o d i f i ed p ro c e d u r e d e s c r ib e d t h e r e is a d o p t e d ,i t i s f o u n d t h a t t h e m a t r i c e s W a n d F o f e q u a t i o n (1 3.1 ) a r e d i f fe r e n t f r o m t h e m a t r i c e s W a n dF i n t h e f i x e d r o o t e q u a t i o n s ( 1 3 .2 ), a n d i n a d d i t i o n e q u a t i o n (1 3.1 ) d o e s n o t c o n t a i n a n y n u l ls u b - m a t r i c e s . T h e b e s t p r o c e d u r e i s t h e r e fo r e t o u se t h e f ix e d r o o t m o d e s a s n e w a s s u m e dm o d e s i n s e t t i n g u p t h e e q u a t i o n s f o r t h e c o m p l e t e a i r cr a f t, u s i n g t h e o r d i n a r y L a g r a n g i a n m e t h o dt o d o so . T h e i n e r t i a c o e f f ic i e n ts a r e r e a d i l y o b t a i n e d f r o m d i f f e r e n t i a t i o n of t h e k i n e t i c e n e r g y ,a n d t h e e l a st i c t e r m s t h e n f o ll o w a t o n c e s in c e th e f ix e d r oo t n a t u r a l f r e q u e n c i e s a r e k n o w n .W i t h o n e ri g i d m o d e w e t h e n o b t a i n a t o n c e a n e q u a t i o n o f t h e f o r m ( 13 .3 ); w h e n f u r t h e r r i g idm o d e s a r e a d d e d w e c a n a g a in u se th e e s c a l a t o r m e t h o d . T h i s m e t h o d c o u l d al so b e u se d if t h ef i x e d r o o t m o d e s o f t h e w i n g a n d f u s e l a g e - ta i l c o m b i n a t i o n h a v e b e e n o b t a i n e d b y e i t h e r t h eM y k l e s t a d o r t h e S t o d o l a m e t h o d .T h e p r o c e d u r e d e s c r ib e d a b o v e o f c a l c u l a t i n g t h e f i x e d r o o t f r e q u e n ci e s a n d m o d e s f i r st a n du s i n g t h e s e r e s u l t s i n o b t a i n i n g t h e f r e q u e n c i e s a n d m o d e s o f t h e c o m p l e t e a i r c r a f t i s c e r t a i n l y

a d v i s a b l e i f d e s k m a c h i n e s a r e u s e d fo r t h e c a l c u l a ti o n s . W h e n a n e l e c tr o n ic c o m p u t e r i s u s e d,: on ~ h e o t h e r h a n d , t h e s a v i n g i n c a l c u la t i o n t im e c o m p a r e d W i t h t h a t r e q u i r e d t o s o l v e e q u a t i o n( 13 .1 ) d i r e . c t l y w o u l d p r o b a b l y b e i n s u f f i c i e n t t o c o m p e n s a t e f o r t h e c o m p l i c a t i o n i n t h ep r o g r a m m i n g . E v e n i n t h i s c a se , h o w e v e r , i t m i g h t s t il l b e a d v i s a b l e t o a d o p t t h e a b o v e p ro c e d u r e ,f o r t h e r e a r e g r o u n d s f o r b e l i e v i n g t h a t t h e f i x e d r o o t m o d e s a r e b e t t e r s u i t e d f o r u s e in f l u t t e r. c a l c u l a t io n s t h a n a r e t h o s e o f t h e c o m p l e t e a i r c r a ft . T h e m o d e s o f t h e c o m p l e t e a i r c r a f t w o u l ds t il l h a v e t o b e c a l c u l a te d , f or it is o n l y b y a s u b s e q u e n t c o m p a r i s o n o f t h e m w i t h t h e r e s o n a n c e -t e s t m o d e s t h a t t h e c a l c u l a ti o n s c o u ld b e ch e c k e d .

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F l e x u r a l r i g i d i ty , o r f i e x u r a l s t if f n es s p e r u n i t l e n g t h . Y o u n g s m o d u l u st i m e s s e c o n d m o m e n t o f a r e a o f s e c t io n a b o u t n e u t r a l a x i sT o r s i o n a l r i g i d i t y , o r t o r s i o n a l s t i f f n e s s p e r u n i t l e n g t h

----- [ E , , ] S q u a r e m a t r i x o f e l a s t i c s ti f f n e s s c o e ff i c i en t sF l e x i b i l i t y o r i n f lu e n c e f u n c t i o n . D i s p l a c e m e n t a t p o i n t y d u e t o u n i t l o a da t p o i n t u

= F y , , y , ) F l e x i b i l i t y c o e f f i c i e n t= [ F,s ~ S q u a r e m a t r i x o f f l e x i b i l i t y c o e f f ic i e n t s

D i s p l a c e m e n t a t p o i n t ( x, y ) d u e t o u n i t l o a d a t p o i n t ( u , v )F l e x i b i l i t y c o e f f ic i en t f o r f l e x u r e o f r t h s e g m e n t o f b e a m ( d i s p l a c e m e n t d u et o u n i t l o a d )

= [G ,,I S q u a r e m a t r i x o f c o e ff i ci e n ts fo r m e t h o d o f c o l l o c a t io n w i t h a s s u m e dm o d e sF l e x i b i l i t y c o e f fi c ie n t f o r f l e x u r e o f r t h s e g m e n t o f b e a m ( s lo p e d u e t o u n i tl o a d )F l e x i b i l i t y c o e ff i ci e n t fo r fl e x u r e o f r t h s e g m e n t o f b e a m ( s lo p e d u e t o u n i tc oup l e )U n i t m a t r i xP o l a r i n e r t i a o f r t h s e g m e n t o f b e a m

= [ K , ~ S q u a r e m a t r i x o f e l as t ic c o ef fi ci en t s i n t h e c o m p l e m e n t a r y e n e r g ym e t h o d ( d i m e n s i o n s s t i f f n e s s d i v i d e d b y f r e q u e n c y ~)L e n g t h o f b e a mL o w e r t r i a n g u l a r m a t r i xB e n d i n g m o m e n t a t p o i n t y

= M y , )E i t h e r r t h i t e r a te t o b e n d i n g m o m e n t o r b en d i n g m o m e n t d u e to lo a d i n g~ ( y ) Z r ( y )D i s c r e t e m a s s a t p o i n t Y rD i a g o n a l m a t r i x w i t h e l e m e n t s m , o r , t * Y,)N u m b e r o f d e g r ee s o f f r e e d o mC o l u m n m a t r i x w i t h e l e m e n t s q ,G e n e r a l i z ed c o - o r d i n a t e a s s o c i a te d w i t h r t h d e g r e e of f r e e d o mS h e a f i n g f o r c e a t p o i n t yr t h i t e r a t e t o s h e a r i n g f o r c eT o r q u e a t p o i n t y(1 /~ o 2) t i m e s m a x i m u m k i n e t i c e n e r g yI n t e g r a t i o n v a r i a b l e s

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S q u a r e m a t r i x [ Z , , JN u m e r ic a l i n t e g r a t i o n c o n s t a n ty / ZT o r s i o n a l d i s p l a c e m e n tO y~)T o r s i o n a l d i s p l a c e m e n t i n r t h n o r m a l m o d eD i s p l a c e m e n t f u n c t i o n f o r to r s i o nM o m

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