V o lu m e : H A N O I 1991 Proceedings of the National C entre f o r Scientific Research of Vietnam, Vol (1991) (3-18) Mechanics M U L T I- D Y N A M IC A D S O R B E R E F F E C T F O R Q U E N C H IN G T H E S E L F - E X C I T E D V IB R A T IO N S O F M E C H A N IC A L S Y S T E M S W IT H D S T R H U T F J ) P A R A M E T E R S N c l v f n V I ) ao an Institute of \l* rh'imcf S C S R of Vietnam S u m m e r y M u l n - d y n:\mu: al >•■irl.tr r cllW ti f r quenching fh» * if-rxcit.-H vil rM ns of string*, be am s unci plaUrf M e investig ated T h e a sy m p to tic m efliod uf avet*:\git.g $ applied to a weakly nonlin ear s y s t e m in order to find Its solutions The series solutions, "st'piiTiite vciriCiblos" techniques are used T h e l a b i l i t y of the sta t i o n a r y vibr ation s IS studied by nu'iu’ j of variational e q u a tio n s and the R o u th - Hurwitz criterion I N T R O D U C T I O N In r e c e n t y e a r s m u c h i n t e r e s t h a s b e e n t a k e n in t h e e f f e c t o f d y n a m i c a b s o r b e r f o r q u e n c h i n g se lf - exc i t ed s y s t e m s 11-181 Most i n ve s t i ga t i on s wer e f ocused on self-excited system s w ith discrete paramet er: ? T h e p r o b l e m o f v i b r a t i o n o f d i s t r i b u t e d p a r a m e t e r s e l f - e x c i t e d system s having one d y n a m i c a b s o r b e r h a s been s tu d ie d in 112, 171 In the p re se n t p a p e r the a n a ly s is w ill be e x te n d e d to su ch s t r in g s , b e a m s a n d p la te s w it h m u lt i-d y n a m ic a b s o rb e rs F ig s h o w s a s t r i n g w i t h t w o fi xed e n d s a t I = and X — t an d carrying N d y n a m i c a b s o r b e r s at l o c a t i o n s X i , I , , T K • L e t t h e i r s t r u c t u r a l c h a r a c t e r i s t i c s : m a s s e s ( m , ) , d a m p in g s (Aj) a n d s t i f f n e s s e s ( k ị ) b e g i v e n T h e s t r i n g is a c t e d OI1 b y “ n e g a t i v e ” f r i c t i o n forces d irec ted along t h e y - a x i s a n d d is t r ib u t e d along its len gt h w i t h t h e intensity (1 ) where hi and /13 are positive constants Let /i be the mass per unit length of the string, S() the te n s io n i n t h e s t r i n g , y ( i , i ) t h e v e r t i c a l d e f l e c t i o n a t t i m e t o f a p o i n t l y i n g o n the s t r i n g a x is a t a d is ta n c e X from the o r ig in , Zi t h e d is p la c e m e n t o f i - t h a b s o rb e r fro m its p o s itio n o f e q u ilib r iu m , e a sm all dim ensionless positive p a m e te r characterizing the sm allness of th e corresponding te rm s a n d l et a n o v e r d o t d e n o t e t h e d e r i v a t i v e w i t h r e s p e c t t o t T h e X c o - o r d i n a t e is t a k e n a l o n g t h e le n g th o f th e s t r in g w it h the o rig in at the le ft end an d the y c o -o rd in a te is p e r p e n d ic u la r to the -axis T h e governing equations of the string and absorbers are m 2, + k,[z, - y ( x , , t ) ] = - e A , [ i , - ỳ ( x , , t ) ] i * = ,2 , , (2) NQUYBN VAN DAO where Ft = k % - t /( x t , i ) ] + e \ i [ i i - ỹ(x«-91)] = -rriiZi, [zi (3 ) Ố is the D a c fu n c t io n T h e b o u n d a r y c o n d it io n s fo r th e s trin g u n d e r c o n s id e tio n are t / ( , i) = y[l,t) = 0, F ig = a t X = a n d X = I (4 ) Scheme of Vibrating tiring carrying N dynamic absorber» T h e s t a t io n a r y s o lu t io n s of q u a s ilin e a r e q u a tio n s (2 ) w ith b o u n d a ry c o n d it io n s ( ) a n d th e s p e c ia l rô le o f th e d a m p in g m e c h a n is m s (A ,) in q u e n c h in g the se lf-e x c ite d v ib r a t io n s o f th e s t r in g w ill be in v e s tig a te d T w o ty p e s oi d y n a m ic a b s o rb e rs w ill be c o n s id e re d : S tro n g a b s o r b e rs w h e n ỴỴlị the tio s o f t h e ir m a sses — /i s m a ll q u a n titie s o f o rd e r € Ịq■ a n d stiffn e sse s -zr a re fin ite an d w e ak ones w h e n these r a t io s are Sị) MULTI- DYNAMIC A BS ORB E R E F F E C T a FR E E UNDAM PED VIBRATIONS OP A STRING W ITH ABSORBERS; NATURAL FREQUENCIES Let us c o n s i d e r the c a s e when e = Then the equations (2) become - 5° ^ k ' í z,)’ “ V"(x í' * ) ] tf( * - x d i f f e r e n t T h e s o l u t i o n o f rhosc e q u a t i o n s w i t h h o u m l a r y c o n d i t i o n s ( )c o u l d from be s o u g h t ill the form wlurh is formally sim ilar to (C) with functions Y „ ( i ) } t„ of type (12), (10): y ( x t ) f ; Y„{x)Q„(l), rl (0 = ( 20 ) 2> " Q » (0 > 1= where the unknown functions of t : Qn(t) are to be determined i n t o e q u a t i o n s ( 2) , m u l t i p l y i n g t h e first e q u a t i o n o f ( ) b y Substituting ihtise expressions Ytn(z)dx and in t eg ti ng ov er th e d o m a i n | , £ | , t h e n m u l t i p l y i n g t h e s e c o n d e q u a t i o n o f (2) b y z ịflị a n d a d d i n g t h e m u t i l i z i n g tl ie o r t h o g o n a l i t y relation (8) we arrive at Q, „(t ) + Vut Qm(t ) ~ irr~Gn,> tn (21) where i d G ,„= 00 X / R { ^ ) Y „ , [ z ) d x + M * « » - y „ ( x t) ] [ y „ , ( x 1) - i « = !,= ! m = 1,2, U s in g the “ v a ria tio n o f p a r a m e t e r s ” te c h n iq u e , assum e the so lu tio n o f e q u a tio n s ( ) to be of the f o r m : NGUYEN VAN DAO Qm — &mCOSỚm ,Q tn — Gmcưm 81ĩlớm j (2 ) where a m , 0m are new functions of tim e, which satisfy the relation ãm COS dm -h a m (u 'm 9m ) sin ỡm — S ubstituting the expressions (22) into equations (21) and solving for the derivatives ả,nt ỡrn gives (23) em — ^ tn ~~ TT— M ttị U rt Cft ị )T l G tn COS $ m Since a m and 6tn — cutnt are slowly varying functions of tim e, the change in th e values during a tim e period (2n/(jjf„) is very small Hence, one may replace equations (23) by their tim e-averages, assum ing a rn and $rn - c J,n t to be c onstant If this is done, the resulting equations are very sim ple For exam ple, if only the first sp atia l mode (m = 1) with lowest frequency (J is retained then we j have ea 2A / (2 ) where (25) #3 = h i f Y*{x)dx (I According to the m a th e m a tic a l basis of the method of averaging |20| solutions o f e quations (24) correspond a sy m p to tic a lly to the solutions of equations (23) for sufficiently sm a ll param eter e N oting the expressions (22) one can see th at constant solution of equations (24) corresponds to periodic solution of equations (21) T he asym ptotic validity of the procedure is for all tim e t > if equations (24) have stable constant solutions In this section the periodic solutions of equations (21) are of interest and only the ste a d y sta te responses are investigated The stability of the solu tio n s will be studied by e xam ining the behaviour for small perturbations and the characteristic ro ots of the linear variational system for th a t solution From equations (24) it is easy to verify that 1) The trivial solution dị = which corresponds to the equilibrium of the sy s te m under consideration is stable if MULTI-DYNAMIC ABSORBER E F F E C T R ecalling the form ula ( l ) for the “negative” friction force one can see th a t the inequality (26) is valid w h e n the d a m p in g force* (Aj) of the absorbers are sufficiently great 2) T he n o n te r o constant amplitude of vibrations it determ ined by ffjjw’a? = Hu- (27) R e a l a m p lit u d e d i e x is ts if H n > (2 ) This in equality is also the condition under which the constant am plitude d] is stable If t h e r s t t w o m odes ( m = 1, m = 2) w ith f r e q u e n c i e s UJI, LJ2 are r e t a i n e d th e n by u s in g t h e averaging p r o c e d u re fo r e q u a tio n s (2 ) the f o l l o w i n g avera g ed e q u a tio n s are o b ta in e d : àị - Y j ỹ ~ { H u ” ~ / / 31^1 èị = u/1, - -H 32vịaị)ì è2 = W, J a7 = (29) - - / / 32^ '? « ? - Ị ^ 3 ^ q 2)» where / //12 = /11 I - Y j ( x ) d x - J A,[*l2 - ya(x,)]3f u •'*» / = /13 Ị Y?(x)YỈ(x)ji, ' (30) 1 # 33 ^3 j I) yf(l)dx From e q u a tio n s (29) one can prove that ) T h e solution dị = ao = is stable if //11 l a \ = H l2t (32) is St able if A* W ỈT 3) T h e so lution Ũ2 = 0, Qi Ỷ determined by Ax = = //,„ (33) NGUYEN VAN DAO 10 is stable if H i2Hzi H 32 • 4) T h e s o l u t i o n a-L Ỷ 0» a Ỷ is s t a b l e if H 31H 33 “ //3 > W E A K D Y N A M I C A D S O R B E R S W “ s t u d y now t h e se l f- e x c i te d v ib t io n of the string carrying N wea k ab sorbers , w h e n the T k ration of t heiir m a s s e s — a n d s t i f f n e s s e s ~ r are s m a l l q u a n t i t i e s o f or d e r e T h e m o t i o n e q u a t i o n s hei u So then become ( 34) i= m, z, + Ắ, [ 2, - ụ ( x , , t ) ] - - Ằ , [i - ỹ(Xi,t)]> c p, = Ắ [í , - p ( x , , t ) ] c, (3 ) H À, [ 2, - ỹ ( i , , t ) ] , * 1,2, V , wi()i r.he s a m e b o u n d a r y c o n d i t i o n s ( ) Here in c o n t r a s t to e q u a t i o n s (2) t h e e q u a t i o n ( 34 ) b e c o m e s indepeinlent of equations ( ) when £ = The solution of e q u a t i o n s ( 34 ) a n d (35) with bounrlar) condi t i on:- (4) w h e n c — c a n he a s s u m e d in t h e f o r m oc T(X, V) = ^ V sin ~ x ( C 9ịcosuỉut + D n shi(jJnt ) t »= oc V— > 2, = 2_ (36) n /7 -ỊTx fi Yn(x,)]s(x X,) , = _ pY,t(x) -w , w ri > (45) and (40) n = 1,2, T h e rem aining formulae of the previous sections are formally transfered to the case of vibra­ tion of th e b e a m , and th e analogous conclusions of those section s will also be obtained For the sake of brevity the corresponding formulae and conclusions are not given here M U L T I-D Y N A M IC A B S O R B E R S E F F E C T F O R S E L F -E X C IT E D V I B R A T I O N S OF A P L A T E T h e p ro b le m u n d e r c o n s id e tio n in th is p a r a g r a p h is th e n a t u r a l c o n tin u a tio n o f the p re v io u s ones for a more c om plicated system - a plate carrying N absorbers This problem is of interest in a irc ft a n d s t r u c t u r a l in d u s trie s W it h th is in m in d , a t h e o r e t ic a l s o lu tio n w it h the lim it a t io n s o f Small deflection th e o ry is presen ted Fig s h o w s a r e c t a n g u l í p l a t e w i t h s i d e s a a n d b, s i m p l y s u p p o r t e d a l o n g all f o u r e d g e s and earring N dy n a m ic absorbers at locations X = z t , y = y, (i = 1,2, % ) with masses m ,, N stiffn e sse s k% a n d d a m p in g s Aj T h e p la te is s u b je c te d in th e v e r t ic a l d e c tio n O z to a “ n e g a tiv e ” friction force - f l ( f f ) distributed uniformly over all its area T he m otion equations of the system p la t e -a b s o r b e r s re d u c e to a7 + D V 4Ỉ - N < Ft6 ( x - x , ) ( y - y.) = s R { g t )> ,= l m,ũ, + kt[u, - *(*,-,y,-,e)] = -cA,[ti, - i(i,,y,,i)] t = 1, 2, — (47) AT, where Ft = k, [u, - z ( z , , y n ] + « - / 33 97 ( a * + d y ) 2* - i(x,-, y ,,t)], (48) NGUYEN VAN DAO 14 z = z ( x , y , is the tra n s v e rs e d e fle c tio n o f the p la te , ịị is th e m ass d e n sity o f the p la te m a t e r ia l, D is th e f l e x u l rig id it y o f th e p la te , is the D a c d e lta function, Ui is v e r t ic a l d is p la c e m e n t , m, k i t Xi are the mass, stiffness and d a m p in g respectively of i-th absorber Let us consider the s im p ly su p p o r t e d p late w it h th e b o u n d a r y co n d itio n s Fig Schr "'ế? jJ a ploX carrying N dynamic absorbers e = g = at x = 0» x = a » V = 0, y = y, - (4 ) For d e g e n e r a t e s y s t e m o f e q u a t i o n s (4 ) w h en e = we assum e a s o lu tio n in th e f o r m 2( x , y , t ) = ^ Z m n(z ,y ) T m n (t), m.n = (5 ) U| = ^ mf*Tffin ( t ) , rn.fi = i w h ere U jmn are c o n s ta n ts , Z mM( x , y ) a re fu n c tio n s o f X an d y satisfyin g the b o u n d a r y c o n d it io n s (4 ), so th a t y / \ _ z m „ ( x , y) - Zinti ~ d~ Z inn —~ = a t X = 0, I = a, y = 0, y = 6, (51) 15 MULTI-DYNAMIC ABSORBER E F F E C T and also satisfying the orthogonality relations f f 7 , Al, f = / 0, * / ( m - j ) + ( n - j ) / , *-1 Substituting the expressions (50) into equations (47) and u s i n g the separation of variables technique ve obtain T mn -* Ttnn D V AZ,nn(x, y) - £ M “«mn = < /.)]< 5(l - x , ) ( y - y, ) _ nZtnn[x J y) kị [ut’mP| — rrifi ( x , , y, )Ị (53) uimn lince x , y and Í are independent, the equalities (53) m ust be a consta nt Let this constant be Thus we have Trnti + wJtlftTtnn = 0, (54) N h'^tntt ^rntị (^) y) — Ì2V Zfnti (^J y) ■■^ ^ + ^lun (~t >y» )] ^(2? —X| )ỏ ( y — t/j ) — 0, 1=1 (^ i ^ w m n ) u im n ~ (55) ii)1 (5 ) T he follow ing expression satisfies equation (55) and boundary condition (51) z,nn{x,y) = -^ Y z%'tl s i n — I s in y y (57) ;».»/= i Jy su b stitu tin g (57) into equation (55) then multiplying the r e s u lt by sin “ “ X sin ^ y d x d y and CL b I i t e g r a t i i i g o v e r X, V f r o m t o a and t o r e s p e c t i v e l y , w e o b t a i n 7*"/ _ mn 2J miẨCịu;;vinZvflvl(Zil yv -)8in — I , sin -=—y« J 1- _ a _ / _ n \ ( ĩ l \ 2í VL\2^ \ í l + ( y ) ] ) ( * - - m ,C l/* * ) „ ( eo\ \ assuming th a t the den o m in a to r of the expression (58) es not vanish we have z>nn(x,y) i U2 s in: “ p * sin: ~ i-* )* mn “I y { — j?L — J - I _u r p£ ■ * «= W - D i< T >’ + < T > ’ n m,fc,zmn(ii, y,) sin —I, sin £ frỉ -r ~ — ki - r — i } • J (59) ' 16 NGUYEN VAN DAO I f we lot = 1» an d y = y, ( i = , , ) th e n ~ o ( a* ủ \^ l „ l\ ,.E , p* qn \ m ịujịìnkị 6Ìn— x ì s in ^ r yi ềiiì— XjBin^r yj - D[{^) +( y ) j i j(*i -"V-'mJ , J*-< *•.*>- ° (60) ( j = , , , yv) A nontriviid solution of the hom og en ou s algebraic equations (GO) exists if and only if the d e t e r m i­ nant of the coefficient* vanishes i e d e t | ^ l A| = 0, (61) v» here * - V ^ L : 9* sin x I sin : p* : 9* L V* s i n — X j s i n y, — ^ _ - a £ _ _ _ J — A ’ r02 Ì - u [ ( ? ) a + (? )? }(* - « « J E q u a l i o n ((>1) J v e s the* e i g e n v a l u e s ^ nu ( m »r* = , , ) 51 Tin* solution of qiiasilinoar e quations (47) will be found ill the form oc *(*iy»y) ^ ^ ( Gi ) £ m » i(Z i y)Q »nr,(0> m.n = i oo = ^ ^ ^im »»Q w n(0» HI n = w h ere u lt„ „ , z ,„ „ ( x , y ) are d e te rm in e d fro m (5 ), (5 ) an d Qmn [ are fu n c t io n s o f t im e to be determined S u b stitu tin g the ex pressio ns (63) into the 6rst equation of (47), m u ltip ly in g by Zfij ( z , y)dxdy an d in te g r a tin g o v e r th e p la te re g io n , th e n a d d in g the re s u lt w it h th e r e m a in in g equations of (47) alter m ultiplying them by Uifj and sum m ing up on i from Ỉ to N, we have M Q j+»ĩ,Q *i = jị-L 'H 3) J — , , , w h o le th e o r th o g o n a lit > re la tio n s (5 ) are also ta k e n in to accoun t and • •J L'J = I> ỊI n H {yt )Z,j(x,y)Jxdy(I (6 ) oo “ V ^ ^ ^ ^ ^i[u*»HM * lnfl (x, , y, )j [uj,y — " (l| I J/| )] tH.ti I: n L e t U5 c o n s id e r ilie t r a n s fo r m1a t io i o f v a r ia b le s Q,J = A , j C o s 0JÍ Q tJ S u b s t it u t in g (6 ) in t o (6 ) an d s o lv in g f o r Ả ặj t ỏỆ t ị = - A , j U tj * i n tj one obtains the syste m (c c ) 17 MULTI-DYNAMIC ABSORBER E F F E C T Atj — L'j sini,;, (67) J = tJ*, — “TJ j co ® J =: 1, 2, F ollow ing the m e t h o d of averaging, in the first approxim ation one can replace the right hand sides o f (67) by t h e t im e averages If we keep only the first m ode corresponding to = j = th e n the averaged e q u a tio n s are Ả“ = n r S G“ - \ G^ A^ ' (6 ) 011 — W| 1, w here tẫ b G n = hl / N Z ị l ( x , y ) d x d y - Ỵ ' A ị u i i - Z u ( x , , y , ) ] 0 a Ơ 01 = h 1=1 (69) b ỊỊ , z^(x,y)dxdy 0 It sh o u ld be no ted that, w ith the assum ptions mentioned above we have in the first a p p r oxim ation Q n — A l l cos dị lf • (70) ( x , y , t ) c- Z u ( x , t / ) / i u CO$0U From (68) it id readily verified that the zero solution A 1 = is asym ptotically stable if < (71) T h is inequality is satisfied for example with -sufficientlygreat values of the dam ping coefficient5 A, The nontrivial sta tio n a r y solution ( Al l Ỷ 0) of (68) is determ ined by - ^ ì uj i Á \ { = c u , i ơn>0 w h ic h (72) (73) is stable if T h e r e la t io n s ( ) a n d (6 ) sh o w th a t in c re a s in g the d a m p in g fo rc e s ( A i) le a d s to d e c r e a s in g th e am p litu d e o f s e lf-e x c it e d v ib r a t io n o f the p la te (All) I t is w o r t h m e n t io n in g th a t the w e ak a b s o rb e r effect fo r th e b e am s is s im ila r the o n e g iv e n in se c tio n for t h e strings CONCLUSIONS The results obtained are qualitatively the same as with discrete systems 11], but a system 10, with multi-dynamic absorbers is more effective than that having only one absorber, because the resulting effect becomes stronger (see formulae (25), (30), (41), (69)) in quenching self-excited vibrations of the strings, beams and plated 18 NGUYEN VAN D A O T he dam ping m echanism s of dynamic absorbers play an important role in that quenching For strong absorber, increasing the dam ping forces leads to a decrease of the am plitude of self­ excited vibrations But for weak absorbers, the suppressing effect is achieved only w ith som e intermediate values of dam ping forces ACKNOW LEDGEM ENT This work was performed at the Institute of Mechanics of TH Darm stadt, FRG, where author held a study visit supported by the D A A D from September to November 1990 The author wishes to express his deep appreciation to Prof Dr Peter Hagedorn for helpful discussions during the preparation of this paper and for enabling him to make use of all facilities of the Institute of Mechanics of TH D arm stadt The support of D A A D is gratefully acknowledged REFERENCES w M M an sou r, Q uenching of limit cycles of a Van der Pol oscillator, J Sound and V ibration , (197 2) 395 w R G le n d en in g and R N D u bey, An analysis of control m e th ods for galloping system , Trans A S M E , B , No (1973) A TonHI, Q u e n c h in g '•'f 5eIf-ex cifed vibrations, J Sound and Vibration 42 (1975) 252 A Tondl, Q u e n c h in g of self-excited vibrations Effect of dry friction, J Sound and V ibration 45 ( ) 285 A Tondl, A p plicatio n of tu n ed absorbers to self-excited sy s tem s with several masses, Proc of X l- t h Con feren ce on D y n a m ic s of M achin es, P g u e 1977 p Hagedorn, U b er die T ilg u n g selbsterregender Schwingungen, ZAMP (197 8) 815 p Hagedorn, On the c o m p u t a t i o n of d a mp e d wind-excited vibrations of overhead transmission lines, J Sound and Vi brat ion ( ) (1982) 253 Nguyen Van D in h , T h e tu n e d absorber in self-excited system , J Mechanics, Hanoi, No 3-4 (1979) 21 Nguyen Van Dno, A note on the dyn am ic absorber for self-excited system , J M echan ics, Hanoi, No (19 ) 10 Nguyen Van D a o and N gu y en Van D in h , D y n a m ic absorber for self-excited sy stem s, Proc of X IV -th C on feren ce on D y n a m ic of M achin es, P rague, Septem ber 1983 Nguyen Van Drto, Q u e n c h in g the self-excited oscillations o f mechanical sy s tem s , International C o n feren ce on Nonlinear Oscillation* (X ), B ulgaria , 19Ồ4 N guyen Van D.IO, D y n a m ic absorber for self-excited sy s tem with distributed parameters, J M ec h a n ic s, H an oi, No (198 5) 15 Nguyen Van Di\o, D y n a m ic absorber for drilling instrum ent, Proc of IC N O -X I, 693-698, B u d a p e st 1987 Nguyen Van D a o , D y n a m ic absorber for self-excited sy s te m with limit energy resource, J M ec n ic s, H3IÌOĨ, 14 11 12 13 14 No (1989) 15 Ngu yen Van Dno, Som e rem arks on the effect of dyn am ic absorber, J Mechanics, H anoi, No ( 9 ) 10 16 Nguyen Van D a o and N gu y en Van Dinh, D y n a m ic absorber for sy stem s with distribu ted param eters, F ro c of NCSR V i e tn a m , (1 990 ) 17 Nguyen Van D a o and N g u y e n Van Dinh, Dyn am ic absorber effect for self-excited sy s tem s , A d v a n c e s in M e ­ chanics, Warsaw, 12, No (1 991 ) 18 T I K u tn e tso v n ti