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