FEM Eigenvalueanalysisoftaperedtwistedtimeshenkobeam

7/28/2019 FEM Eigenvalueanalysisoftaperedtwistedtimeshenkobeam

1/14

Journa l o f Soun d and Vibra tion (I 978) 56(2), 187-200

FINITE ELEMENT EIGENVALUE ANALYSIS OF TAPEREDAND TWISTED TIMOSHENKO BEAMS

R. S. GUPTADepartment o f Mechanical E ngineering, Punjab Engineering College,Chandigarh-11, India

ANDS. S . RAO

Department o f M echanical Engineering, Indian Institute o f Technology,Kanpur-208016, India(Received 13January 1977,and in revised o rm 14September 1977)

The stiffness and mass matrices of a twisted beam element with linearly varying breadthand depth are derived. The angle of twist is assumed to vary linearly along the length of thebeam. The effects of shear deformation and (otary inertia are considered in deriving theelemental matrices. The first four natural frequencies and mode shapes are calculated forcantilever beams of various depth and breadth taper ratios at different angles of twist. Theresults are compared with those available in the literature.1. INTRODUCTION

The analysis of tapered and twisted beams has wide application in many industrial problems.The vibration and deflection analysis of compressor blades, turbine blades, aircraft pro-peller 131ades, helicopter rotor blades, gear teeth, springs of electromechanical devices,electrical contact switches, etc., all can be made by using such beam elements.Tapered beams have been analyzed by many investigators using different techniques.Rao [1] used the Galerkin method to calculate the fundamental natural frequencies ofbeams tapered in depth. Housner and Keightley [2] applied the Myklestad [3] procedure todetermine the first three modes of a tapered beam. Mart in [4] obtained the frequencies of atapered beam using the assumption that the eigenvalues and eigenvectors of a tapered beamcan be expa.nded about those of a beam with zero taper in terms of the taper parameters. Raoand Carnegie [5] used the finite difference method to obtain the frequencies and mode shapesof tapered blading. MaNe and Rogers [6, 7] solved the differential equation of vibration oftapered beams with different boundary conditions using Bessel functions and tabulated theresults of the first five vibrational frequencies for different breadth and depth taper ratios.In analyzing pretwisted beams, different approaches have been used by various investiga-tors. Mendelson and Gendler [8] used station functions while Rosard [9] applied the Mykle-stad method. The Rayleigh-Ritz method was used by Di Prima and Handelman [10], Carnegie[11] and Dawson [12]. Rao [13] analyzed pretwisted beams using the Galerkin method.Carnegie and Thomas [14] used a finite difference procedure for the analysis of pretwistedbeams.

The finite element technique has also been applied by many investigators, mostly for thevibrat ion analysis of beams of uniform cross-section. All these investigations differ onefrom the other in the nodal degrees o f freedom taken for deriving the elemental stiffness andmass matrices. McCalley [15] derived consistent mass and stiffness matrices by selecting the187


2/14

188 R . S . G U P T A A N D S . S. R A Ot o t a l d e f l e c ti o n a n d t h e t o t a l s l o p e a s n o d a l c o - o r d i n a t e s . A r c h e r [1 6] a n a l y z e d v a r i o u s b e a m sw i t h s p e c i f i c b o u n d a r y c o n d i t i o n s . K a p u r [ 1 7 ] t o o k b e n d i n g d e f l e c t i o n , s h e a r d e f l e c t i o n ,b e n d i n g s l op e a n d s h e a r s l o p e a s n o d a l d e g r e e s o f f r e e d o m a n d d e r iv e d t h e e l e m e n t a l m a t r i c eso f b e a m s w i t h l i n e a r l y v a r y i n g i n e r ti a . C a r n e g i e e t a L [1 8] a n a l y z e d u n i f o r m b e a m s b y c o n s i-d e r i n g f e w i n t e r n a l n o d e s i n i t. N i c k e l a n d S e c o r [ 1 9 ] u s e d t o t a l d e f l e c t i o n , t o t a l s l o p e a n db e n d i n g s l o p e o f t h e tw o n o d e s a n d t h e b e n d i n g s l o p e a t th e m i d - p o i n t o f t h e b e a m a s t h ed e g r e e s o f f r e e d o m t o d e r i v e t h e e l e m e n t a l s t if f n es s a n d t h e m a s s m a t r i c e s o f o r d e r s e v e n .T h o m a s a n d A b b a s [2 0] a n a l y z e d u n i f o r m T i m o s h e n k o b e a m s b y t a k i n g to t a l d ef l ec t io n , t o t a ls l op e , b e n d i n g s l o p e a n d t h e d e r i v at i v e o f th e b e n d i n g s l o p e a s n o d a l d e g re e s o f f re e d o m .

I n t h i s w o r k t h e f i n i t e e l e me n t me t h o d i s a p p l ie d f o r f i n d i n g t h e f r e q u e n c i e s o f n a t u r a lv i b r a t i o n o f d o u b l y t a p e r e d a n d t w i s te d b e a m s . T h e s t if fn e s s a n d m a s s m a t r ic e s o f th e b e a me l e me n t a r e d e v e l o p e d b y t a k i n g b e n d i n g d e f l e c t i o n , b e n d i n g s l o p e , s h e a r d e f l e c t i o n a n ds h e a r s l o p e i n t w o p l a n e s a s n o d a l d e g r e e s o f fr e e d o m . T h e e f fe c ts o f s h e a r d e f o r m a t i o n a n dr o t a r y i n e r ti a , w h i c h a r e o f s i g n if i ca n t i m p o r t a n c e a t h i g h e r m o d e s o f v i b r a t i o n , a r e c o n s i d -e r e d i n ~ he d e r i v a t io n . T h e e l e m e n t a l m a t r ic e s o f a d o u b l y t a p e r e d b e a m w i t h o u t p r e tw i s ta n d a p r e t w i s t e d b e a m w i t h o u t ta p e r c a n b e d e r i v e d a s s p e ci a l c a s e s o f t h e p r e s e n t ma t r i c e s .T h e s ti ff n es s a n d m a s s m a t r ic e s o f a t a p e r e d b e a m a n d t h a t o f a u n i f o r m b e a m w i t h o u ts h e a r d e f o r m a t i o n a r e a l s o s p e c ia l ca s e s o f th e g e n e r a l m a t r ic e s . T h e f i rs t f o u r n a t u r a l f r e -q u e n c i es o f v ib r a t i o n h a v e b e e n c a l c u l a t e d f o r u n t w i s t e d a n d a p r e t w i s te d d o u b l y t a p e r e dc a n t i l e v e r b e a m s b y u s i n g t h e f in i te e l e m e n t t h u s d e v e l o p e d . T h e e f fe c t s o f v a r i a t i o n o fd e p t h a n d b r e a d t h t a p e r r a t i o s o f t h e b e a m h a v e a l so b e e n s t u d ie d . T h e r e s u l ts c o m p a r e w e l lw i t h t h o s e r e p o r t e d i n t h e li t e r a tu r e .

2 . E L E M E N T S T I F F N E S S A N D M A SS M A T R I C E S2.1. D I S P L A C E M E N T M O D E L

F i g u r e l ( a ) s h o w s a d o u b l y t a p e r e d , t w i st e d b e a m e l e m e n t o f l e n g t h / w i t h t h e n o d e s a s 1a n d 2 . T h e b r e a d t h , d e p t h a n d t h e t w i st o f th e e l e m e n t a r e a s s u m e d t o b e l i n ea r l y v a r y i n ga l o n g i ts l en g t h . T h e b r e a d t h a n d d e p t h a t t h e t w o n o d a l p o i n t s a r e s h o w n a s b t, l h a n d b 2, h 2,r e s p e c ti v e l y . T h e p r e t w i s t a n g l e s a t t h e t w o n o d e s a r e d e n o t e d b y 0 1 a n d 0 2, r e s p e c t iv e l y ( al is t o f n o t a t i o n i s g i v e n i n A p p e n d i x B ). F i g u r e l ( b ) s h o w s t h e n o d a l d e g r e e s o f f r e e d o m o f t h ee l e me n t , w i t h b e n d i n g d e f l e c t io n , b e n d i n g s l o p e , s h e a r d e f l e c ti o n a n d s h e a r s l o p e i n t h e tw op l a n e s t a k e n a s t h e n o d a l d e g r e es o f f r e e d o m .

T h e t o t a l d e f l e c t i o n s o f t h e e l e me n t i n t h e y a n d x d i r e c t i o n s a t a d i s t a n c e z f r o m n o d e 1 ,n a m e l y , w (z ) a n d r ( z ) , a r e t a k e n a s

w ( z ) = w ~ ( z ) + w e ( z ) , v ( z ) - - v ~ ( z ) + v ~ ( z ) , ( l )w h e r e w b ( z ) a n d v b ( z ) a r e t h e d e f l e c t io n s d u e t o b e n d i n g i n t h e y z a n d x z p lanes , r e spec t ive ly ,a n d w ~ ( z ) a n d v ~ ( z ) a r e t h e d e f l e c t io n s d u e t o s h e a r i n t h e c o r r e s p o n d i n g p l a n e s .

T h e d i s p la c e m e n t m o d e l s f o r w b ( z ) , w ~ ( z ) , v b ( z ) a n d v ~ ( z ) a r e a s s u m e d t o b e p o l y n o m i a l s o ft h i r d d e g r e e . T h e y a r e s i m i l a r i n n a t u r e e x c e p t f o r t h e n o d a l c o n s t a n t s . T h e s e e x p r e s s io n s a r eg i v e n b y w b ( z ) = ( u l / I 3 ) (2z a - 3 1 z 2 + / 3) + ( u 2 / l a ) (31z 2 - 2z 3) -

- ( u 3 / p ) ( z 3 _ 2 1 z 2 + t 2 z ) _ ( u 4 / l 2 ) ( z 3 _ l z 2 ) ,W s ( z ) = (us /13)1"2z 3 _ 3 l z 2 - 1 3 ) + ( u 6 / l 3 ) ( 3 1 z 2 - 2 z 3 ) -

- ( u T / t ~ ) ( z 3 - 2 1 z " + 1 2 z ) - ( u s / t ~ ) ( ~ - t z 2 ) ,v ~ ( z ) = ( , 1 9 / l 3 ) ( 2 z 3 - 3 1 z 2 - 1 3 ) + ( U l o / . 1 3 i ( 3 / z 2 _ 2 z 3 ) _

- ( u l J l Z ) ( z 3 - 2 z 2 + 1 2 z ) - ( u 1 2 / 1 2 ) ( z 3 - l z 2 ) ,v , ( z ) = ( u 1 3 / 3) (2z3 - 3 z 2 - 13) + ( u , , / 1 3 ) ( 3 1 z 2 - 2z 3) -

- ( u x s / l 2 ) (z 3 - 2 l z 2 + l ' z ) - ( u z , / l 2) ( z 3 - 1 z 2 ) , (2 )


3/14

TAPERED AND TW ISTED TIMOSHENKO BEAMS 189

Ul U5 uZ U6

L , / " u / l ~ u BIuII / u /

/ : Y'~ IY / /u~ UI3 ~ 1o x r u14

u \ y '( c )

F i g u r e 1 . ( a ) A n e l e m e n t o f ta p e r e d a n d t w i s t e d b e a m ; ( b ) d e g r e e s o f f r e e d o m o f ar t e l e m e n t ; (c ) a n g l e o ftwist 0.w h e r e u l , u 2 , u a a n d tt4 r e p r e s e n t t h e b e n d i n g d e g r e e s o f f r e e d o m a n d u s , u6, u7 and U s a r e t h es h e a r d e g r e es o f f r e e d o m i n t h e y z p l a n e , it9, trio , ti ll a n d u~2 r e p r e s e n t t h e b e n d i n g d e g r e e s o ff r e e d o m a n d u~3, t tz4 , i t15 a n d u16 t h e s h e a r d e g r e e s o f f r e e d o m i n t h e x z p l a n e .2 .2 . ELEMENT STIFFNESS MATRIX

T h e t o t a l s t r a in e n e r g y U o f a b e a m o f l en g t h 1, d u e t o b e n d i n g a n d s h e a r d e f o r m a t i o n , i sg i v e n b y

U = o LI. 2 k O z Z ] + E l , , , O z z O z2 ~- 2 k O z z / j + - - 2 - t k W ] + k ~ z ] j l d z . ( 3 )A s t h e c r o s s - s e c t i o n o f t h e e l e m e n t c h a n g e s w i t h z a n d a s t h e e l e m e n t is t w i s te d , th e c r o s s -

s e c t i o n a l a r e a A , a n d t h e m o m e n t s o f i n e r t i a I,`,` , I . a n d I,` y w i ll b e f u n c t i o n s o f z :A ( z ) = b ( z ) h ( z ) = {b z + ( b , - b , ) z / l } { hx + (h 2 - h z ) z / l } = ( l / 1 2 ) ( c , z * + cz l z + c3 1 2) , ( 4 )

w h e r ecl = (b2 - b ,) (h2 - h i) , c2 = b ~ ( h 2 - h x ) + h t ( b 2 - b ~ ) ,

I ~ ( z ) = I ,` , ,` , cos z 0 + I r y . s in z 0 ,I . ( z ) = Iy .y , co s 2 0 + 1 ,`, ,` , s in 2 0 ,I,`r(z) = ( I x , , ` , - I y , y , ) 89 in 20 ,

c3=blhl , (5 )

(6 )


4/14

1 9 0 R . S . G U PT A A N D S . S . RA Ow h e r e x ' x ' a n d y ' y ' a r e t h e a x e s i n c li n e d a t a n a n g l e O, t h e a n g l e o f tw i s t, a t a n y p o i n t i n t h ee l e m e n t , t o t h e o r i g i n a l a x e s x x a n d y y a s s h o w n i n F i g u r e l ( c ) . T h e v a l u e o f / = , , , = 0 a n d t h ev a l u e s o f l~ , ~ , a n d I , ,y , c a n b e c o m p u t e d a s

b ( ~ ) h ' ( ~ ) II x , ~ , ( z ) - - l ' - - - - - ~ - 1 2 Y [ a ' z 4 + a 2 I z 3 + a 3 1 2 z 2 + a 4 1 3 z + a s ] ' (7 )w h e r e

w h e r e

a , = ( b 2 - b , ) ( h 2 - h , ) 3 , a z = b , ( h 2 - h , ) 3 + 3 ( b 2 - b , ) ( h 2 - h , ) 2 h , ,a 3 = 3 { b , h ~ ( h 2 - h , ) 2 + ( 6 2 - b , ) ( h 2 - h ~ ) h ~ } ,a , = 3 b , h~(h2 - h , ) + (b 2 - - 6 , ) h ~ , a s = b x h ~ ,

~ , . , . ( z ) = h ( z ) . b 3 ( z ) = ~ [ a , z" + a 2 t : + a ~ " z 2 + d , t 3 + a , t ' ] ,12 1214

( 8 )( 9 )

t h e e l e m e n t s t if fn e s s m a t r i x c a n b e e x p r e s s e d a s[ A K ] [0 ] [ D K ] [0 ]

[ o ] [ C K ] [ 0 ] [ 01 1 ,[ K ] = [ D K ] [0 ] [ B K ] [0 ] /1 6 x 1 6

[o 1 [o 1 [o 1 [ C K l ]

a n di l a 2 w b \ [ a " vb

w h e r e [ A K ] , [ B K ] , [ C K ] a n d [ D K ] a r e s y m m e t r i c m a t r i c e s o f o r d e r 4 a n d [ 0] i s a n u l l m a t r i xo f o r d e r 4 . T h e e l em e n t s o f m a t r i c es [ A K ] , [ B K ] , [ C K ] a n d [ D K ] a r e f o r m u l a t e d i n A p p e n d i xA .

0 5 )

( 1 6 )

d , = ( h2 - h , ) ( 6 2 - b , ) 3 , d 2 = h , ( b 2 - b , ) 3 + 3 ( h 2 - h , ) ( 6 2 - b , ) 2 b , ,d 3 = 3 {h , b , ( 6 2 - 6 , ) 2 + ( h 2 - h , ) ( b 2 - 6 , ) 6 ~} ,d 4 = 3 h, b ~ ( b 2 - b , ) + ( h 2 - h , ) b ~ , d s - - h , b ] . (113)

B y s u b s t i t u t i n g t h e e x p r e s s i o n s f o r w b , w : , v b , v s, A , I x x , I xy a n d I yy f r o m e q u a t i o n s ( 2 ) , ( 4 )a n d ( 6) in e q u a t i o n ( 3) , t h e s t r a in e n e r g y U c a n b e e x p r e s s e d a s

v = 8 9 ( l l )w h e r e u i s t h e v e c t o r o f n o d a l d i s p l a c e m e n t s u , , u2 . . . . . u , 6 , a n d [ K ] is t h e e l e m e n t a l s t if f n e ssm a t r i x o f o r d e r 1 6. I n t e r m s o f th e i n t e g ra l s d e f i n e d a s

f [ a 2 w ~ \ 'E I , , , ,~ - - ~z~J d z = [u~u2u3u4l r [AK] [u~u2u3u4] , ( 1 2 )z / 0 2 . \ 2

i . l a w , \ 2~ A G \ / - ~ - z d z = [ u , u 6 u 7 u s ] [ C K ] [ u , u 6 u 7 u s ] , ( 1 4 )0


5/14

TAPERED AND TWISTED TIMOSHENKO BEAMS 1912. 3. ELEMENT MASS MATRIX

T h e k i n e t i c e n e r g y o f t h e e l e m e n t T , i n c l u d in g t h e e ff e ct s o f s h e a r d e f o r m a t i o n a n d r o t a r yin e r t i a , i s g iv e n b y

o ; L w +p /a 'w~\/azv~'~ p[= [a~w ,\q+ g i , , , t a - - - ~ t ) t - ~ . ~ ] + . ~ g l O - - ~ ) l d z . ( 1 7 )

B y d e f i n i n g

a n d

! 2

J - ~ - \ ~ - / d z = [ a , a ~ a 3 a , ]" [ a M ] [ a , a , a 3 a , ],0l 2

(3 T \ 0 - - f f ~ / d z = [ a , u z a , ~ , ] r [ B M ] [ a , u z u 3 t i , ] ,0

!f T \ a - ~ - ~ / d z = [a 9 a ,o a , , a , = ] " [ C M ] [ a g . , o . , , a , . ] .o

(18)

(19)

(20 )

I t is t o b e n o t e d h e r e t h a t a l l t h e f o r c e d b o u n d a r y c o n d i t i o n s c o u l d b e s a ti sf ie d b y t h e p r e s e n tm o d e l . A m o n g t h e n a tu r a l b o u n d a r y c o n d i t i o n s , i f t h e c o n d i t i o n o f z e r o b e n d in g m o m e n t i st o b e e n f o r c e d a t a f r e e e n d , t h e e l e m e n t d u e t o T h o m a s a n d A b b a s [2 0] is e x p e c t e d t o b eb e t t e r t h a n t h e p r e s e n t o n e .

f r e e e n d : O wd O z =O a n d Ov,/Oz=O; (24)c l a m p e d e n d : w , = 0, wb = 0, v, = 0, vb = 0, Owb/Oz 0 a n d OvdOz= 0 ; (25 )

h in ge d en d : w, = 0 , wb = 0 , v , = 0 an d vb = 0 . (26 )

c o n d i t i o n s :

1

J o g \ O z Ot] ~Oz Ot] d z = [ a , a z z~ 39 , ] r [ D M ] [ 9 9 9 , o u , , ~ , = 1 , ( 2 1 )w h e r e ~ d e n o t e s t h e t im e d e r i v a t iv e o f th e n o d a l d i s p l a c e m e n t tq , i = 1 , 2 . . . . . 1 6, t h e k in e t i ce n e r g y o f t h e e l e m e n t c a n b e e x p r e ss e d a s

T = 8 9 f i, ( 2 2 )w h e r e [ M ] i s t h e m a s s m a t r i x g i v e n b y

[ [ A M ] + [ a M ] [ A M ] [ D M ] [0 1 ]/ /= [ [ A M ] [ A M ] [ A M ] [0 1 1[M] ( 2 3 )/ [DM] tAM ] tAM] + [ C M ] t A M ] / '~6x16/ /L [0] [0] [ A M ] [AMlJ

a n d [ A M ] , [ B M ] , [CM ] a n d [ D M ] a r e s y m m e t r i c m a t r i ce s o f o r d e r 4 w h o s e e l em e n t s a r ed e f i ne d in A p p e n d i x A .2.4. BOUNDARYCONDmONS

T h e f o l l o w i n g b o u n d a r y c o n d i t i o n s a r e t o b e a p p l i ed d e p e n d i n g o n t h e t y p e o f e n d


6/14

192 R .S. GLIPTAAND S. S. RAO3 . N U M E R I C A L R E S U L T S

T h e e l e m e n t s ti ff ne s s a n d m a s s m a t r ic e s d e v e l o p e d a r e u s e d f o r t h e d y n a m i c a n a l y s i s o fc a n t i l e v e r b e a m s . B y u s i n g th e s t a n d a r d p r o c e d u r e s o f s t r u c t u r a l a n a l y s i s , t h e e i g e n v a l u ep r o b l e m c a n b e s t a t e d as

( iX] - - o ,2 [M] ) U = 0 , ( 27 )w h e r e [ K ] a n d [ M ] d e n o t e t h e s t i ff n e ss a n d m a s s m a t r i c e s o f th e s t r u c t u r e , r e s p e c t i v e l y , Ui n d ic a t e s n o d a l d i s p l a c e m e n t v e c t o r o f t h e s t ru c t u r e , a n d w i s t h e n a t u r a l f r e q u e n c y o fv i b r a t i o n .

A s t u d y o f t h e c o n v e r g e n c e p r o p e r t i e s o f t h e e l e m e n t i s m a d e b y t a k i n g t h e s p e c ia l c a s e o f au n i f o r m b e a m w i t h a le n g t h o f 0 .2 5 4 0 m , b r e a d t h o f 0 .0 7 62 m , d e p t h o f 0 .0 7 0 4 m , E = 2 .0 7 10 ~ I N /m z , G = 3E/8, m a s s d e n s i t y o f 8 00 k g / m 3 , / a = 2 / 3 a n d 0 = 0 ~ F o r t h i s b e a m , t h ef i r s t , s e c o n d , t h i r d a n d f o u r t h n a t u r a l f r e q u e n c i e s o b t a i n e d b y t h e p r e s e n t m e t h o d ( w i t h 4e l e m e n t s ) h a v e b e e n f o u n d t o h a v e 0 " 0 0 y , 0 . 07 ~ o, 0 . 3 0 ~ a n d 0 . 6 0 ~ e r r o r s , r e s p e c t i v e l y .T h e f i rs t f o u r n a t u r a l f r e q u e n c i e s o b t a i n e d b y u s i n g 8 e l e m e n t s a r e 8 4 5. 8 , 39 8 9 .5 , 8 8 3 6. 8 a n d1 38 27 .1 H z w h i l e t h e e x a c t v a l u e s a r e 8 4 5 .8 , 3 9 8 8 .9 , 8 8 3 4 .2 a n d 1 38 18 .1 H z , r e s p e c t i v e l y[2 0]. T h e c o n v e r g e n c e o f th e n a t u r a l f r e q u e n c i e s o f a p r e t w i s t e d d o u b l y t a p e r e d c a n t i l e v e rb e a m h a s a l s o b e e n s t u d i e d a n d t h e r e s u lt s a r e s h o w n i n T a b l e 1 . I n t h is c a s e t h e n a t u r a lf r e q u e n c i e s g i v e n b y t h e m e t h o d o f re f e r e n c e [ 21 ] h a v e b e e n f o u n d t o b e s l ig h t l y h i g h e r t h a nt h o s e p r e d i c t e d b y t h e p r e s e n t m e t h o d . I t c a n a l s o b e s ee n t h a t r e a s o n a b l y a c c u r a t e r e s u l tsc a n b e o b t a i n e d e v e n b y u s i n g f o u r f in it e e l e m e n t s .

T A ~ L z 1Natural frequencies of a tapered and twisted Tirnoshenko beam ( H z )

N u m b e r o f e le m e n ts F i r s t m o d e S e co n d m o d e T h i r d m o d e F o u r t h m o d e1 304-8 1187.0 2259.3 4519-22 298.7 1146.8 1685-3 4046.53 298-1 1139.2 1652.2 3647.44 297.9 1137.9 1647.3 3593.55 297.8 1137.5 1646.0 3585.66 297.8 1137.4 1645.3 3578.87 297.8 1137.3 1645.1 3578.58 297-8 1137-3 1645-0 3578.3A c c o r d i n g t o t h e m e t h o d o fre ~r en ce [21] 299.1 1142-8 1653"3 3595.7

D ata: length o f beam = 0.1524 m, breadth at ro ot = 0-0254 m, depth tap er ratio = 2.29, breadth taperratio = 2"56, tw ist = 45 ~ shear coefficient = 0.833, mass density = 800 kg]m 3, E = 2.07 x 10 l N]n l2,G = (3/8)E.T a b l e 2 s h o w s t h e f r e q u e n c y r a t io s o f a n u n t w i s te d t a p e r e d b e a m f o r v a r i o u s c o m b i n a t i o n s

o f d e p t h a n d b r e a d t h t a p e r r a ti o s . S ix fi ni te e l e m e n t s a r e u s e d t o m o d e l t h e b e a m . I t i s o b s e r v e dt h a t f o r c o n s t a n t d e p t h t a p e r r a t i o t h e f r e q u e n c y r a t io o f al l t h e f o u r m o d e s i n c re a s e s w i thb r e a d t h t a p e r r a ti o w h i le f o r c o n s t a n t b r e a d t h t a p e r r a t io t h e f r e q u e n c y r a t i o d e c r e a s e s f o rt h e f i r s t m o d e a n d i n c r e a s e s f o r t h e s e c o n d , t h i r d a n d f o u r t h m o d e s w i t h a n i n c r e a s e i n t h ed e p t h t a p e r r a t i os . T h e s h e a r d e f o r m a t i o n e f f ec ts r e d u c e t h e f r e q u e n c y o f m o d a l v i b r a t i o n .T h e p r e s e n t r es u l ts c a n b e s e en t o c o m p a r e w e l l w i t h t h o s e r e p o r t e d b y M a b i e a n d R o g e r s [ 6]w h i c h a r e a l s o i n d i c a te d i n T a b l e 2 . F i g u r e 2 s h o w s c o m p a r i s o n o f th e r e s u lt s g i v e n b y t h ef in i te e l e m e n t m e t h o d w i t h t h o s e r e p o r t e d b y R o s a r d [9 ] f o r a t w i s t e d b e a m o f 0 . 0 25 4 m x0 . 00 6 3 5 m c r o s s - s e c t i o n a n d 0 . 2 79 4 m l e n g t h . I t c a n b e s e e n t h a t t h e t w o s e t s o f re s u l ts a r eq u i t e c o m p a r a b l e .


7/14

77

II0

II

gI I

I1

, + - -

' 0 , . ~

Z00

lJ

II

"0

000II

E~J

",d"o0IIo

~JcJ ~J,o~ o

~j~3

.~0~ 0 ~

0 . . . . . .

~j 0 0 0


8/14

1 9 4 R. S. GUPTA AND S. S. RAO9

8 -

7

6.o

g4 ~

3

2

I

1 I

Third mode

Second mode

First mode

I II0 ZOAngle of twis t (degrees)

30

Figure 2. Comparison of results for an uniform twisted beam. -- -- , Values by Rosard method; - -values by present method.

F i g u r e s 3 a n d 4 s h o w t h e v a r i a ti o n o f m o d a l f r e q u e n c ie s w i t h b r e a d t h t a p e r r a t io f o r b e a m sh a v i n g 0 ~ 3 0 ~ 6 0 ~ a n d 9 0 ~ t w i s t w i t h c o n s t a n t d e p t h t a p e r r a t i o w h i l e F i g u r e s 5 a n d 6 s h o ws i m i l a r v a r i a t i o n s f o r b e a m s w i t h c o n s t a n t b r e a d t h t a p e r r a t i o a n d v a r y i n g d e p t h t a p e rr a t i o . H e r e t h e l e n g t h o f t h e b e a m i s t a k e n a s 0 . 2 5 4 m a n d t h e r o o t c r o s s - s e c t i o n a s 0 .0 7 6 x0 . 0 3 8 t i m e s t h e l e n g t h o f t h e b e a m . A g a i n t h e e ff ec t s o f b r e a d t h a n d d e p t h t a p e r s a re s e e n t o

I i I

4r - Second mode

O~o6O*

F First mod e/ ] / - 9 o ~' / / / // / / s .

/ l l O / / / / l l ~ - . ~ . ~f i l l / 1~ .~ , - ~ - - = - -

I I I2 3 4

Breadth toper rQliOF ig u re 3 . E f f ec t s o f s h e a r d e fo r m a t i o n a n d b r e a d t h t a p e r r a t i o o n t h e f i rs t a n d s e c o nd n a t u r a l f requenc ieso f a t w i s te d b e a m . e , M e t h o d o f r e fe r en c e [ 2 2 ]; , w i t h o u t s h e a r d e f o r m a t i o n ; . . . . , T i m o s h e n k o b e a m ;d e p t h t a p e r r a t i o = = 3.


9/14

T A P E R E D A N D T W I S T E D T I M O S H E N K O B E A M S1 4 I I I

13 \ \\ \ ~. \12 \ \ \

/ f / f - Fourth mode11 - \ \ \ ~ ' ' , -0 I0 ~ . ~

9 - ~"t~

8 / f / f -Th i rd mode~ +

6 - 3 0 "

4 I i I I2 3 4 5Breadth toper ratio, l

195

Figure 4. Effects of shear deformation and breadth taper rat io on the third and fourth natural frequenciesof a twisted beam. , Without shear deformation; . . . . . , Timoshenko beam; depth taper ratio cr = 3.

be pronounced at higher modes of vibration. Here also the effect of shear deformation isseen to reduce the moda l fr equencies at higher rates in higher modes of vibrat ion in all thecases. The results found by the meth od of Carnegie and T hom as [22] for the first two nat ura lfrequencies are also indicated in Figures 3 and 5. It can be seen that the present resultscompa re excellently with those of Carnegie and Tho mas.

. 9

u~

I I I

~ 9 0 ~Firsl mode -~ I~, 60 ~

I I I2 3 4Depth taper ratio,a

Figure 5. Effects of shear deformation and depth taper rat io on the first and second natural frequencies of atwisted beam. , Without shear deformation; . . . . , Timoshenko beam; breadth taper ratio fl = 3. o,Method of reference [22].


10/14

196

a twisted beam.

13

12

II

I 0._o

9

7

6

5

4

R. S. GUP TA AND S. S. RAO

I I I

FcxJrth mode~ ~ ~)o

T h i r d m o d e --~ o

i i I2 3 4D e p t h t a p e r r a t io

Figure 6. Effects of shear deformation and depth taper ratio on the third and fourth natural frequencies of, Without shear deformation; . . . . . , Timoshenko beam; breadth taper ratio/Y = 3.4. CONCLUSION

The finite element procedure developed for the eigenvalue analysis of doubly tapered andtwisted Timoshenko beams has been found to give reasonably accurate results even with fourfinite elements. The effects of breadth and depth taper ratios, twist angle and shear de forma-tion on the natura l frequencies of vibration of cantilever beams have been investigated.The present results are found to compare very well with those report ed in the literature. Theelement developed is expected to be useful for the dynami c analysis of blades of roto-dyna micmachines.

REFERENCES1. J. S. RAo 1965 Aeronau t i ca l Quar ter l y 16, 139-144. The fundamental flexural vibration ofcantilever beam of rectangular cross-section with uniform taper.2. G. W. HOUSNERand W. O. KEIG}{T[.EY1962 Proceedings o f the Ame rican S ocie ty o f Civi l Enghz-eers 88, 95-123. Vibrations of linearly tapered beam.3. N. O. MYKLESrAD1944 Journa l o f Aerospace Sc i ence 2, 153-162. A new method for calculatingnatural modes of uncoupled bending vibrations of aeroplane wings and other types of beams.4. A. I. MARTIN1956Aeronaut ical Quarterly 7,109-124. Some integrals relating to the vibration ofa cantilever beams and approximations for the effect of taper on overtone frequencies.5. J. S. RAO and W. CARNEGIE 1971 Bul le t in o f Mec hanica l Engineerhtg Education 10, 239-245.Determination of the frequencies of lateral vibration of tapered cantilever beams by the use ofRitz-Galerkin process.6. H.H. MAB1E,and C. B. ROGERS1972Journa l o f t he Acous t i ca l Soc ie t y o f Am er ica 51, 1771-1774.Transverse vibrations of double-tapered cantilever beams.7. H. H. MABm and C. B. ROGERS 1974 Journa l o f t he Acous t ica l Soc ie t y o f Amer ica 55, 986--988.Vibration of doubly tapered cantilever beam with end mass and end support.8. m. MENDELSONand S. GENDLER 1949 N A C A TN-2185. Analytical determination of coupled

bending torsion vibrations o f cantilever beams by means of station functions.9. D. D. ROSARD 1953 Jottrt ta l o f App l ied M echan ics 20, 241-244. Natural frequencies of twistedcantilevers.10. R. C. DI PRIMA and G. H. HANDELMAN 1954 Quar ter l y on App l i ed Mathemat i c s 12, 241-259.Vibration of twisted beams.


11/14

T A P E R E D A N D T W I S T E D T I M O S H E N K O B E AM S 19711. W. CARNEGIE1959 Proceedings o f the Inst i tu te o f i t lech anica l Engineers 173, 343-346 . V ib ra t ionof p r e twi s t ed ca n t i l eve r b l ad ing .12. B . DAW SON 1968 Journal of Mechanical Engineerhtg Science 1 0 , 381-388. Cou pled bend ingv i b r a t i o n s o f p r e t w is t e d c a n t il e v e r b la d i n g t r e a t e d b y R a y l e i g h - R i t z m e t h o d .13. J. S. RAG 1971 Journal o f the Aeronautical Socie ty o f lndia 23 , 62 -64 . F l exura l v ib r a t ion o fp r e twi s t ed beam s o f r ect angu lc . r c ross - sec t ion .14. W . CARNEGIE an d 3". THOMAS 19 72 Journal of Engineerhtg for Industry, Transactions o f the .Am erican Socie ty o f Mechanical Enghwers 94 , 255-266 . The coup led bend ing -bend ing v ib r a t ionof p r e twi s t ed t ape r ed b l ad ing .15. R . M cCALLEY 1963 General Electr ic Company, Schenectady, N ew York, Repo rt No. D IG /SA ,6 3 - 7 3 . R o t a r y i n e rt i a c o r r e c ti o n f o r m a s s m a t r i c e s.16. J . S . ARCHER 1965 Am erican Institute o f Aeronautics an d Astronautics Journal 3, 1910-1918.Con s i s t en t m a t r ix f o rm ula t ions fo r s t r uc tu r a l ana lys i s u s ing fin it e e l emen t t echn iques .17. K . K . KAPUR 1966 Journa l o f the A cous t ica l Soc ie ty o f A mer ica 40, 1058-1063. Vib rat ion s of aT i m o s h e n k o b e a m , u s i n g f in i te e l e m e n t a p p r o a c h .18. W. C A R N E G I E , J . THOMAS and E. DOCUMAKI 196 9 Aeronautical Quarterly 2 0 , 321-332. Ani m p r o v e d m e t h o d o f m a t r i x d i s p l a c em e n t a n a l y s is i n v i b r a ti o n p r o b l e m s .19. R. NICKEL an d G . SECOR 1972 bzternational Journal o f Nume rical M ethods in Enghwering 5 ,

243-253 . Co nvergen ce o f cons i s ten t ly de r ived Timo she nko beam f ini te e l emen t s .20. J. THOMASand B. A . H . ABBAS 1975 Journal o f So un da nd Vibration 4 1 , 2 9 1 - 2 9 9 . F i n i te e l e m e n tm o d e l f o r d y n a m i c a n al ys is o f T i m o s h e n k o b e a m .21. 3 . S. RAG 1972 Journal o f Engineerhtg for bMustry , Transactions o f the American Socie ty o fMe chanical E ngineers 94 , 343-346 . F l exu ra l v ib r a t ion o f p r e twi s t ed t ape r ed can t i l eve r b l ades.22. W . CARNrGm an d J. THOMAS1972Journa lo fEng ineer ing for lndus try , Transac tions o f the Amer i -can Socie ty o f Mechan ical Engineers 94 , 367-378 . The e f f ec t s o f shea r de fo rmat ion and ro t a ryine r t i a on the l a t e r a l f requenc ies o f can t i l eve r beams in bend ing .

A P P E N D I X A : E X P R E S S IO N S F O R [ AK] , [ BK] . . . . , [ D M ]T h e f o l lo w i n g n o t a t i o n is u se d f o r c o n v e n i e n c e :

1w , = J z ' - ' d z , i = 1 ,2 . . . . . n , ( A I )0

L l = I t - l , i = 1,2 .. .. . n, (A2)[ z ]Vl = f z ' - ' co s 2 (02 - 0 , ) + 01 dz , i = 1 , 2 . . . . . n , ( 1 3 )0 , [ z ]St = f z l-~ sin 2 (02 - 01) + 01 d z , i = 1 , 2 . . . . . n , ( 1 4 )0

w h e r e O~ a n d 0 2 d e n o t e t h e v a l u e s o f p r e t w i s t a t n o d e s I a n d 2 , re s p e c t i v e ly , o f th e e l e m e n t .As wb, w~, Vb a n d v~ a r e a l l th e s a m e i n n a t u r e e x c e p t f o r t h e i r p o s i t i o n s i n t h e s t if fn e s s a n d

m a s s m a t r i c e s , o n e c a n u s e )~ t o d e n o t e a n y o n e o f t h e q u a n t i t i e s w b, w s, v b o r v s a n d i n as im i l a r m an n er t he se t (111,112,113, a4 ) ca n be us ed to r ep res en t a ny on e o f t he se t s (u~, u2, u3, u4),(us, u6, u7, us) , (ug, U , o , u l , , u12) o r (u13, u14, u ,s , U16 . T h u s

1] 13 _113 -3 or .2 12Z) q_~3(3122_223)_~__~4(23 -i f (z ) = ~ - ~ ( 2 z 2 - 3 1 z 2 + ) ~ ( ~ - , . , ~ + 1 1 2" IZ2), (AS)d ~ 111 ~ ( 3 Z 2 - - 4 1 z " --~z = - f ~ (6 z 2 - 6 1 z) - + 1 2 ) + ~ ( 6 1 z - 6 z 2 ) - ~ ( 3 z 2 - 2 1 z ) , ( A 6 )

d 2 if, l] 1 . a3 t]2 a 4d z 2 = I-S ( 1 2 z 6 1 ) - . ~ ( 6 z - 4 1 ) + i f ( 6 1 - 1 2 z ) - 7 ~ ( 6 z - 2 1 ) . ( t 7 )


12/14

19 8 R. S. GUPTA AND S. S. RAOB y l e t t i n g P ~ , t . k ( i = 1 . . . . 4 ; j = i , . . . , 4 ; k = t . . . . . 7 ) d e n o t e t h e c o e f f ic i e n t o f z k - X l 7 -k f o rt h e a~zT~ t e r m i n t h e e x p r e s s i o n o f ~ 2 , Q~.~ ,k( l = 1 . . . . 4 ; j = i . . . . . 4 ; k = 1 . . . . 5 ) t h ec o e f f i c i e n t o f z k - ~ l 5 - k f o r th e f i ~ t e r m i n th e e x p r e s si o n o f ( d ~ ] d z ) z, R ~ . j , k ( i = 1 . . . . 4 ;

j = i , . . . . 4 ; k = 1 . . . . 3 ) t h e c o e f f i c ie n t o f z k - ~ l 3-k f o r t h e a ~ a j t e r m i n th e e x p r e s s i o n o f( d 2 ~ ' /d z 2 ) 2 , H i . j ( i = I . . . . . 4 ; j = i . . . . . 4 ) t h e i n d e x c o e f f i c i e n t o f l to a c c o u n t f o r t h e d i f f e r -e n c e i n i n d e x o f I d u e t o m u l t i p l i c a t i o n o f r o t a t i o n a l d e g r e e s o f f r e e d o m ~1 a n d a 2 a n d t h ed i s p l a c e m e n t d e g r e e s o f f r e e d o m a 3 a n d t74, t h e v a l u e s o f P ~ . j . k , Q ~ . j .k , R ~ . j , k a n d H ~ ,~ c a n b eo b t a i n e d a s sh o w n in T a b l e s A I a n d A 2 .

T A BL E A 1V a h t e s o f H i . j , R i . j . k , Q l . j. ~

R ~ . ~ ., f o r k = Q t . j . , f o r k =)k Ar "~ r 9i j H~ . j 1 2 3 1 2 3 4 5

1 1 0 1 4 4" 0 - 1 4 4 " 0 3 6 "0 3 6 . 0 - 7 2 - 0 3 6" 0 6 - 0 0 . 01 2 0 - 1 4 4 " 0 1 44 "0 - 3 6 . 0 - 3 6 " 0 7 2" 0 - 3 6 " 0 0 - 0 0 ' 01 3 1 - 7 2 " 0 8 4" 0 - 2 4 . 0 - 1 8 . 0 4 2" 0 - 3 0 " 0 6 ' 0 0 ' 01 4 1 - 7 2 " 0 6 0" 0 - 1 2 " 0 - 1 8 " 0 3 0 ' 0 - 1 2 " 0 0 "0 0 . 02 2 0 144-0 -1 44 "0 36"0 36-0 -7 2" 0 36"0 0 -0 0 -02 3 1 7 2 -0 - 8 4 " 0 2 4 . 0 1 8 :0 - 4 2 " 0 3 0" 0 - 6 - 0 0 ' 02 4 1 7 2 -0 - 6 0 ' 0 1 2" 0 1 8 " 0 - 3 0 ' 0 1 2" 0 0 - 0 0 . 03 3 2 3 6 - 0 - 4 8 " 0 1 6" 0 9 "0 - 2 4 " 0 2 2 . 0 - 8 - 0 1 "03 4 2 3 6" 0 - 3 6 - 0 8 . 0 9 "0 - 1 8 " 0 1 1 .0 - 2 " 0 0 "04 4 2 3 6 -0 - 2 4 " 0 4 - 0 9 "0 - I 2 - 0 4 - 0 0 - 0 0 "0

T A B LE A 2V a h l e s o f P t , j.

P I . j . ~ f o r k =ri ] I 2 3 4 5 6 7

I 1 4 . 0 - 1 2 . 0 9 . 0 4 . 0 - 6 . 0 0 . 0 1 .01 2 - 4 . 0 1 2 .0 - 9 . 0 - 2 , 0 3 .0 0 . 0 0 -01 3 - 2 . 0 7 . 0 - 8 . 0 2 . 0 2 . 0 - 1 - 0 0 -01 4 - 2 . 0 5 . 0 - 3 , 0 - 1 . 0 1 .0 0 . 0 0 -02 2 4 . 0 - 1 2 . 0 9 . 0 0 - 0 0 . 0 0 . 0 0 . 02 3 2 .0 - 7 . 0 8 .0 - 3 . 0 0 . 0 0 . 0 0 . 02 4 2 - 0 - 5 . 0 3 . 0 0 . 0 0 . 0 0 . 0 0 - 03 3 1 -0 - 4 . 0 6 - 0 - 4 - 0 1 -0 0 - 0 0 - 03 4 1 .0 - 3 . 0 3 . 0 - 1 . 0 0 - 0 0 . 0 0 . 04 4 1 . 0 - 2 . 0 1 -0 0 . 0 0 . 0 0 . 0 0 . 0

EVALUATION OF [ B K ]A s t h e p r o c e d u r e f o r th e d e r i v a t i o n o f [ A K ] , [ B K ] . . . . . [ D ~ I ] i s t h e s a m e f o r e a c h , t h e

e x p r e s s i o n f o r [ B K ] i s d e r i v e d h e r e a s a n i l l u s t r a t i o n . O n e h a s

(A8)


13/14

T A P E R E D A N D T W I S T E D T I M O S H E N K O B E A M S 19 9wh e r e ~ = vb a n d

t t 2 = U l o .t 2 3 ) u H9 4 ku12

P u t t i n g t h e v a l u e o f l y r a n d f f i n e q u a t i o n ( A 8 ) g iv e s

i / a 2 r v \ ' E l , % . + ( I , . , . { ( 0 , 0 z + x. . > c o , , , , , 0 , ) ]x g ( 1 2 z - 6 1 ) - ~ ( 6 z - 40 + ~ ( 6 ! - 1 2 z ) - ~7(6z - 20

w i t hd z , ( A9 )

B K x . a = coe f f i c ien t o f a~ ax = coe f f i c ien t o f u9 u9E

1211~+


14/14

2 0 0 R .S . GUP T A AND S . S . RAO3 5

C K , s = I t G ~ ~ [ c , L ( , + j + n , .,, U o - i - j ) O t . j . j ], 1 = 1 , 4 ,9 1 8 ~ 9 . . . ,l = l j = l J = / , . . . , 4 , ( A I 3 )5 3

I=1 1=1I = l . . . . . 4 , J = I , . . . . 4 ,

( A I 4 )3 7

A M , . s = g ~ x ~ [ c , L , . s . m . j i U t u _ , _ s ) P , . , . , ] , I = I . . . . . 4 ,= J=l

5 5B M t s = P ~ ~ [ {d , L c,+ j+ m , ) U o , - , - j ) Q t . s . j } +9 1 2 g P ~ I -1 / = t

+ ( a, - d l ) { L ( t + l . n , . : V o l _ l _ j ) Q t . s . j } ] , I = 1 . . . . 4 ,$ 5

PC M t s =9 12gll----- ~ . ~ [{ a ,L ,+ j+ ,,,.,, U cu _,_ j, Q ,.j.~ } +/ - 1 J = l

+ ( d i - a l ) { L ( i . ~ + n , . ~ ) V ( u - ~ - ~ ) Q 1 . s . j } ] , I = 1 . . . . 4 ,5 5_ P

1=1 1=1

J = I , . . . . 4 , ( A l 5 )

J = l . . . . . 4 , ( A I 6 )

J = I . . . . . 4 , ( A ! 7 )

I = 1 . . . . 4 , J = I , . . . . 4 .

(AI8)A P P E N D I X B : N O M E N C L A T U R E

AbEgGhIx ,,, IT,),, x~,[K ]1L[M]

tUU1)

Wx , yZf r e q u e n c y r a t i oO~f l0Pp

a r e a o f c r o s s - s e c ti o nb r e a d t h o f b e a mY o u n g ' s m o d u l usa c c e l e r a ti o n d u e t o g r a v i t ys h e a r m o d u l u sd e p t h o f b e a mm o m e n t o f i n er ti a o f b e a m c r o ss - se c t io n a b o u t x x , y y a n d x y ax i s , r e sp ec t iv e lye l em en t s t i ff n e ss m a t r ixl e n g t h o f a n e l e m e n tl e n g t h o f t o ta l b e a me l e m e n t m a s s m a t r i xt i m e p a r a m e t e rn o d a l d e g r e e s o f f r e e d o ms t r a i n e n e r g yd i s p l a c e m e n t i n x z p l a n ed i s p l a c e m e n t i n y z p l a n ec o - o r d i n a t e a x e sc o - o r d i n a t e a x is a n d l e n g t h p a r a m e t e rr a t io o f m o d a l f r e q u e n c y t o f re q u e n c y o f f u n d a m e n t a l m o d e o f u n i f o r m b e a mw i t h th e s a m e r o o t c r o s s - s e c t i o n a n d w i t h o u t s h e a r d e f o r m a t i o n e f fe c tsd e p t h t a p e r r a t io , = h d h ab r e a d t h t a p e r r a t i o , = b~[b2a n g l e o f t w i s tm a s s d e n s i t ysh ea r co e f f i c ien tS u b s c r i p t s : b , b e n d i n g ; s , s h e a r .

Date post:	03-Apr-2018
Category:	Documents
Upload:	pavan-kishore
View:	225 times
Download:	0 times

FEM Eigenvalueanalysisoftaperedtwistedtimeshenkobeam

Documents