Dynamic absorbers for systems with distributed parameters

PROCEEDINGS OF THE NATIONAL CENTRE FOR SCIENTIFIC RESEARCH OF VIETNAM V o lu m e HANOI 1990 P r o c e e d in g s o f the N a tio n a l C e n tre fo r Scie ntific Reaeă rch o f V ie tn a m , Vol (1990) ( $ - ) M echanics DYNAM IC ABSO RBERS F O R S Y S T E M S W IT H D I S T R I B U T E D P A R A M E T E R S N guyen V an D a o and N g u y e n V an D in h Institute 0Ĩ Mechanics HANOI S u m m a r y In thi* p a p e r the dynam ic absorber is used to quench th e Belf-excited v ib ratio n of th e strin g , beam a n d p la te H ere, as in the c&it of th e Bystem w ith several degrees of freedom [7-11], th e friction m ech an ism h u th e decisive role in the quenching v ib tio n s For th e stro n g absorber increaaing th e friction force (À) leads to d e creaain g 'th e am plitude of »elf-excited v ib ratio n For th e weak ab»orber (m , c are sm all) th e effect of quenching will be achieved only with Borne m ean values of th e friction force D Y N A M I C A B S O R B E R F O R A V I B R A T I N G S T R I N G Let US c o n sid e r th e v ib ration of the string w ith two fixed ends at = and X = t (Fig ) It is a s s u m e d t h a t the string is acted on by the ex te rn a l force along its w h o le length w it h the ( 1) w h ic h e x c it e s th e v ib r a t io n of the string in the y-direction To q uench this v ib tio n one can U9e th e d y n a m ic a b so rb er h a n g in g in parallels w ith In d ir ec tio n y R Ữ gA ck JTi b Fig Let ụ b e th e m a ss d e n sity of the string; To is the in itial strain , we have the follow in g e q u a tio n s of m o t io n o f t h e s y s t e m p resen ted in Fig : NGUYEN van Dao and NGUYEN Van DINH Ẽ1 - j + clu - y(M)]6(z - b) -T t \ = iR ,, , \ m ũ 4- c|ti — y(è, t)Ị = —£A dy{b,t at S(x - 6), 1.2) d y ( b , t) dt where € is sm all p a m eter characterized the sm allness of the corresp on d in g terms, is the Dirac f u n ction T h e b o u n d a r y co n d itio n s are y ( , t) = y ( £ ,i) = 0, B^_y di dx2 dx2 1=0 = 3) T h e s o lu tio n of e q u a tio n s (1.2) w ith b ou n d ary d itions (1 ) is found in the form oo y{x,t) = ^ w „ { t ) Y „ ( x ) t n.= V' / \ _ • 1.4 ?l 7r yT 1(x) = sin — I , here ỈVn (t) are u n k n o w n fu n ction s of tim e which are to be d eter m in ed B y m u ltip ly in g two s i d e s ' o f (1.2) with s i n — x d z and in tegratin g on X from G t ) £ the follow in g eq u a tio n s for Wn and u w ill be obtained w n + Ô ị w n - c n u = i P i% u ~ u u —uJ7Ytl(b)Wtl = eP-2, (1.5) where Pi = - Ịy V „ ( f c ) u - jY?(k)}W u - p = _ A | U _ y„(6)Vi/„|, m cn = ) - y n (fc), w2 = - m In these eq u a tio n s the fu n ctio n s Wj (y ỹí n) are n egle cted In practice, the most im p o r ta n t case is n = U sin g the n o rm a l c o o r d in a te s £2 * (1.7) u)2 Y n (b) , u;2 r n (6) “ ““2 — vị » “2 —“ -o» U/ u>2 — i / j = ^ + u)2 - n / ( ^ - a J ) + c n i*>2 y n (6 ), 2*1 = a ’ + w2 + N ( ^ _ a )2 + 4CnU,2y ri(i)i / DYNAMIC ABSORBERS FOR SYSTEMS WITH DISTRIBUTED PARAMETERS w e have •— j d i P + ), £1 + fc'i £1 = £2 + £2 = €C (-P H Ẩ2 i -^2-^2)» v\ ( 1.8 ) ịll /i£ •+• m ỉ ị B y m e a n s o f transform i t i o n in to th e new variables < , ,(12 , 11 Í = a i c o s f li, = - a ^ i s in i, fli = i t -f , / (1.9) Ỉ2 = £ fl2 COS $ > 2~ — Qo s in $2) $2 = ^2^ v*2I w e get 2Tl T ^ di = -eM ^ i - — r d l P ) sin i , 2m ^1^101 + —r đ í P2) = C O S * !, ( 10 ) 2m t/2 a = —€/c2 (/^1 — —-d'>P2) sin # , t / 2a V;2 = — £ fc o ( P i + — - d o P ) COS ớ2 In the first ap ;ro x im a tio n one can replace (1 ) b y their averaged e q u ation s as follows: o ^2^2 = o o o 0\1 - (?jA - 77 /13(^11a iA +'2^2a2)]» lfi * *'1*1 = 2/i - + ^ a ? ) ] , 10 v a l ipi = u2a t/-2 = 0, ( 11) 9, = ^(d, - V'n(fi))2 F ro m (1.11) w e o b ta in the sta tio n a r y s o lu tio n s : ) a = , a i 7< d e t e r m in e d by £ ( 12 ) w h i c h is sta b le if (1 ) ) Ữ! = , Ữ # d e t e r m in e d b y ^2 = ~ ^ ^ a2 = - < ?2^ (1 ) w h i c h is sta b le if Ả2 > - (/ỉ 3) a i 7^ 0, Ữ2 ^ 0; b u t th is so lu tio n is u n sta b le A) — - A\ (1 ) NGUYEN VAN DAO and NGUYEN Van DINH T h u s , in the first ap p r o x im a tio n the string vib rates C IS f \ í I \ • y ( x , i ) = a, cos( t -I- i/'t) sin — I , /i Uj = a xd x cos( u t t + t ), I1-16) i = 1, From th e form u lae ( ), ( 1 ) it follows t h a t the vib tion of the str in g d ecr eases w hen the friction (A) is increased So in the case considered increasing the friction is an effective m e t h o d of q u e n c h in g th e v ib tio n of the string ( F ig 2) A Fig.2 We co n sid er now the weak d y n a m ic absorber It is a ssu m e d th a t th e stiffness c and the mass m of th e a b so rb er are sm all T h e n , instead of (1.2) w e have th e follow ing e q u a tio n s o f m o tio n "r s =£ { R (§7 )+ | ~J'MW* -f +A u — ° cu c ) - > r mu 4- c |u — y ( , i ) j = —A u - dyj b t dt dy(b,t)' dt 6[x-b) , 1.17) T h e e q u a tio n s for Wn and u are w n + f i w n = Ớ = 1^2, here a ; = hy [ Y ỉ ( x ) d x - X ( - r (fc))2, */o hi = h3 [ Y?( x)YỈ( x) dx, (2 2 ) J2' = /í3 [ Y2(x)dx Jo T h e triviđ.1 sola': ŨIÌ [ a = of (2.21) is sta b le if ft] < and /i2 < or ii A is sufficient big T h e solu tio n a\ = 0, a Ỷ d eter m in ed by /4 _ (2) 2 _ Lã /12 * 'ô 0^2^2 -3 is sta b le ii (2 ) „, n ^1-^3 A2 > - 2hl T h e so lu tio n a — a ^ d eterm in ed by ^ — -JIA 1),a-'jaj rlj ,.3 ,2 „2 — • ' is s t able if 1) ^2‘^3 2/iị T h e n o n trivial so lu tio n a Ầ 7* 0, Ữ2 7* is alw a y s u n sta b le ^ ^ T h u s , w e o b t a in e d here the sa m e reốult as in th e p rev io u s paragraph : in creasing th e friction force (A) leads to d e c r e a s in g the a m p litu d e o f v ib r a t io n o f the string 12 N GUYEN VAN DAO and NGUYEN VAN DINH D Y N A M IC A B S O R B E R F O R B E A M In th is p a r a g p h a r e c t a n g u la r c a n tile v er b e a m of le n g t h ty cross s e c t io n , Y o u n g ’s m o d u lu s E , m o m e n t of in er tia / , m a ss d e n s it y p is sid er ed u nd er th e a ctio n o f a force d istrib u ted alon g its le n g t h as s h o w n in F ig A n o t h e r b e a m (ty i , Ẽiị J i , Pi) c o n n e c t e d w i t h th e first th rou gh an e la s tic layer (c ,A o ) is used as a d y n a m ic a b sorb er for q u e n c h in g th e se lf- e x c it e d v ib r a tio n o f th e first b e a m • Y - ' V : ’' ^1 % ■'•'A-'-' -.'7 k f 7Ỉ7^ y Fig W it h the a d o p t io n of th e e l e m e n ta r y b e a m theory, th e e q u a tio n s o f m o tio n o f the b e a m s are: d 2\j - > • ( ! ? - £ ) ■ (3.1) (é-ẳ) w ith b o u n d a r y co n d itio n s: y ( t i x) = z ( t i i ) = 0, d2y _ dz2 here € is a s m a ll p o s it iv e p aram eter U sin g the n o ta t io n s : _ dx2 at = 0, X —t (3.2) at X = 0, X= t 13 DYNAMIC ABSORBERS FOR SYSTEMS WITH DISTRIBUTED PARAMETERS w e have for t h e w ea k absorber (c,Ao are small): I dy ( dy\3 1ã í _ d 2z dA z dz V y f d 2z ^< z \ ~ * \ d t * + l dĩ*) (3.3) dy W h e n € = th e s y s t e m of eq u a tio n s (3.3) has the solution: sin —nx(an cosc ưnt + ịn s i n cưní) , y = n (3.4) = ^ s i n ^ n x ( a n cose Jn t -f p n sina;n i), here an an d bn are c o n s t a n t s , — (t ) ■ (e ? - e2) ( ™ ) + T + A2wn ] a " + («1 - «2 ) ( ^ ) " r~2 M - ">(=) - ( ei - e ") ( ~ p ) (=)*»■ + Ằwt l a „ + l ( eỉ - e ) ( ^ ) /3» = (3.5) + T + A2wn 6„ («?-«*) ( t ■f ) ’ + ■ > ( = ) * » > We u?f n ow (3.4) as the form ulae of tran sfo rm ation in to the new variables a n , bn w it h s u p p le m e n ­ tary re la t io i.s d n cosunt -f n sin (jjnt = (3.6) r S u b s t i t u t i n g e x p r e ssio n s (3.4) into (3.3) and com p arin g the coefficients o f h a rm o n ic s sin -~nx w e o b ta in e q u a t io n s for ân, n T h en c om b in in g th em w ith (3.6) w e get th e eq u a tio n s for ả n , fl in the s t a n d a r d form s For ã ị and bị w e have u>iài = t /?■ a i - — ! h ^ a ỵ ( ị + b \ ) -f ei ( ^ ) ” ^ sin2 -h ị + (3.7) —€ I / 11^1 — —O'Ih^biịâị + èỊ) — J Z\ cos2 A v e r a g in g t h e right hand sides of (3.7) over the tim e we get “ lài = ị I uiỵhi - $d ịe] [j'j e* ( ) ~ - U>ỶJ _ + ^ l)| > (3.8) -idi -f Wj/l! - t\ ( | ) -w ? a x+ ~ u ỉ^j + ? )| I 14 NGUYEN VAN DAO and NGUYEN VAN DINH whre (e? - e7) di = (3.9) d2 = A The coordin ates O 61 corresponding to the low est frequency O play an im portant role in x, J th e vib ration o f the b ea m If the vibrations w ith high frequencies &re neglected, the eq u atio n s ( ) w ill characterixe the vib ration of the beam T h e stationary solutions o f th ese equations are 1) dị =s bị = (3.10) 2) 16 uỉịhsA = ị h ị - fda Ị e ị or = /li - f A7 = a Ị + ^ (?-«) ( ) A ô a3 ? Jâ ( ) a+ i ' ĩ - »)1 H ’ (3.11) TVivial solu tion (3.10) 1ô stable if (ô?- *)ã ( ? ) • * hi< ( (3.12) * ( f ) * A ’ + («? - «a) ( £ ) + and the solu tio n (3.11) is always stable w ith positive values o f A We have m et w ith the form ula of type (3.11) and we have seeji w h at the m inim um o f A achieves w ith an averaged value o f A (see Fig 3) It is worth m entioning th at when tw o lowest frequencies of tw o b eam s are equal: Wi = V u lYn + c( Yn - ( ) The solu tion of the equations (3.19) w ith boundary conditions (3.16) w ill be found in the form Y*(x ) = \ 7"z' j = l ( ) Zn(z) = yX >=1 j7T By m u ltip lyin g equations (3.19) w ith sin - ỵ x d x and integrating on X from to t we o b tain the follow ing expressions for unknown coefficients Z njy Y nj ElịĩlỴ — — 0, + c - p iS iw * = 0, + c-pSu,l -cY n j + Eịli (3.21) j = 1,2 16 NGUYEN VAN DAO and NGUYEN VAN DINH N ontrivial solu tio n of (3 ) corresponds to the values u>„ which satisfy the frequency equations: — c + c - pSuil El = (3 2 ) c - piSiuị — c j = , , , oo T h u s w e have a set of values of frequencies u ị , ,cô ị > d epending on j For quasilm ear eq uation s (3.13) we put oo y(t1 ~ 2Mi ~ 4^ lu,‘ v i ) A l ’ V> - w ) (3 ) here h\ = h < l j Y?(x)dx - À* I ( Y f c " ’ = f c ? / n (x )á x 0 l - z j ’ d *, (3 ) DYNAMIC ABSORBERS FOR SYSTEMS WITH DISTRIBUTED PARAMETERS 17 From (3.29) it follows: 1) The equilibrium Ai = is stable if /lỊ < This condition is satisfied, for exam ple, w ith sufficiently great valu es o f A 2) There ex ists the v ib tio n a l regim e w ith the frequency U and am plitude A I determ ined by ỈI = h\ > If two v ib ration s w ith tw o first frequencies (3.31) u>2 are m aintained then w e have: 11 \ Ả3 = ĩ k (h‘ ~ h°w' rị> = w2, where I I h i = fc? I Y f d x - \ ' j ( Y - z 2)2dx, 0 i t h[2) = hị j Y ' d x , h; = h Ị °3 (3.33) Y?Y*dx From these it follows: ) T he equilibrium A ị = Á = is stable if h\ < 0, h \ < , for instance, if A is sufficiently great In th is case, follow ing the formulae (3.23), (3.27) the vibration of the beam does not occur 2) There ex ists the v ib tio n a l regim e w ith frequency Ui ( Ả = 0) determ ined by = hm> L if o L* L( ) •-J J A ^ ấ l ),U I A ^ J 1 (3 ) 2h; 3) There ex ists the v ib tio n a l regime w ith frequency C (A i = ) determ ined by Ư \ h ^ w ị Á ị =h-2 > if (3.35) 4) T h e v ib tio n a l regim e w ith two frequencies U>1 and u>2 is alw ays unstable It is w orth m en tionin g th a t the effect of absorber w ill disappear if the coefficient o f A in the expressions of /lỊ, h>2 vanishes, for exam ple, if 18 NGUYEN VAN DAO and NGUYEN VAN DINH A B S O R B E R F O R S E L F -E X C I T E D V I B R A T I O N OF T H E P L A T E The problem presented in the previous paragraph m ay be considered for a more com plicated system , nam ely, for th e plate Self-excited vibration of a plate (D , /Ắ, u) under the action o f the sm all force •* * - * (ir)1 s can be quench w ith the aid of a dam per which consists of another plate ( D , Pi , z) connected w ith th e m ain plate through an viscoelastic layer (c , Ao) The m otion equations for th is system arc: d 2u / du — c(u — z) — X dz\ \ dt ~ d t ) ' D iV 2+ d 2Z = (4.1) a -e(z dt where _ \7 11 w e h a v e (4 13) À in the previous paragraphs the formulae (4.11) and (4.13) show the effectiven ess o f a d ynam ic absorber w ith mean values of À In general, the effectiveness of the absorber for quenching the vib ration w ith frequency 0j nm iafound in the case when tw o natural frequencies unm and wnm o f the p la t e s co in c id e N ow , let us consider the case of a strong absorber when c, Di and fẤi are not sm a ll In this case the m o tio n o f the plates is governed by the equations: —4 ầ d 2z D ịV z + V z) = É d2u DV u + + c(u - e/i, (4.14) + c(* - u) = e/a- here (4.15) h “ ~ x ° (ãỉ ■