Lanchester damper effect for quenching the self-excited vibrations of mechanical systems

'roceedings of the NationỉU Centre for SdentiSc Research of Vietnam, Vol. 4, No 1, (1992) (3-24) L A N C H E S T E R D A M P 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 IT E D V IB R A T IO N S O F M E C H A N IC A L S Y ST E M S N g u y e n V a n D ao Institute of Mechanics NCSR of Vietnam Su m m ary . The Lanchester damper effect for quenching both free and forced self-excited vibrations of the mechanical systems with one, two and many degrees of freedom is investi gated by means of the asymptotic method of averaging. Many quantitative estimations for the stationary amplitudes of vibrations and theừ stability are given. In recent years a lot of papers concerned with the dynamic absorber effect for quench ing self-exciied vibrations of mechanical systems with finite degrees .of freedom and also system s w ith distributed param eters have been published [1-20]. In paper [19] certain characteristics of the behaviour of the Lanchester damper for the Q ue nch ing of self-excited vibrations were established. In the present paper the analysis will be extended system at ically for mechanical system s with one, two and many degrees of freedom. Fig. 1 show s a self-excited system which consists of the main mass M and spring with rigidity c and acted on by a “negative” damping force R. The self-excited vibration of the m ass M is to be suppressed by means of a Lanchester damper with mass m and d a m p in g m e c h a n ism (A). 1. IN T R O D U C T IO N ✓ \ R M _ J X S e/ f ~ e xc ite d system Fig. 1. Schematic diagram of a self-excitcd vibrating man M and a Lancheiter damper 4 NGUYEN VAN DAO Usually, the “negative” friction force is of the form = hi>0, 1 = 1,3, (1) he re a n d .s u b se q u e n tly e is a sm a ll p o s itiv e p a ra m e te rs c h a ra c te riz in g th e s m a lln e ss o f th e te rm s . T h e c oe fficie nt hi is a c o m b in a tio n o f lin e a r fr ic t io n a c tin g on m a ss M a nd th e lin e a r p a rt o f th e e x citin g “ n e g a tiv e ” f ric tio n fo rce . The g o ve rn in g e q u a tio n s o f the s yste m u n de r c o n sid e ra tio n are M i + cx = —A( X — t>) + e[h\% — A3Ì 3), ủ = V, (2) mi) -f À (v — z ) = 0. O b v io u sly , i f A is fin ite th en th e re w ill be no s elf-e x cite d v ib ra tio n o f th e m a ss M b ec au se th e zero s o lu tio n X = u = 0 o f e q u at io n s (2 ) w ill be s ta b le . T h is c ase is n o t o f in t ere st fo r this theoretical study. So, it will be supposed that the damping coefficient of Lanchester d am p e r A is a s m a ll q u a n tit y o f o rd e r £. It is n ec ess ary to e x am in e s e pa ra te ly tw o c a s e s : a st ro n g ab s o rb e r w h en th e r a tio o f m asse s -77 is fin it e a nd w e ak o ne w hen th is r a tio is sm a ll. M 2. S T R O N G L A N C H E S T E R D A M P E R The motion equations in this case take the form MX + ci = f [ - A(i - t>) -f /iii - /13X3] , u = vt mi) = e\[i — v) When £ = 0 equations (3) have a partial periodic solution of type X = a c o s f u f - f \p), J1 = — M X = - a t ư sin[vt T v = t) = 0, u = fc, (3) (4) where a, 6 are constants. For £ 7^ 0 but sm all, the formulae (4 ) are considered as the transformation to new variables a(i), 1/>(*), 6(i) w ith'the additional condition â(í) cos(u>i -f _a(t)0(i) sin(u;i + = ,0. (5 ) Substituting the expressions (4) in equations (3) and solving with respect to the derivatives ả, V' gives LAN CHESTER DAMPER EFFECT FOR QUENCHING c _ . u a = “ — / s i ll <p: M w h ere tocnp = — — F COS M tp u t + \p, F = u (A - A i)a s in V? 4- /i3U 3 a3 sin 3 (p. (6) (7) T h e d e riv a t iv e s g ive n b y e q u a tio n s (6 ) a re s m a ll a nd , c on se q ue n tly , a, rp a re e ss e n tia lly c o n sta n t o v er an in t e rv a l w h ere <p ch an ge s by 2tt. T h e a p p ro x im a tio n w h ic h is no w to be in t ro d u c e d consists of a ve ra g in g the rig h t s id es of e q u atio n s (6 ) o ve r 27T in <p. S o, in th e fir st a p p ro x im a tio n w e h av e f ollo w in g av era ge d e qu a tio n s ea 2M a = - /li + - / i 3a;2a 2'), = 0. (8) T o d e te rm in e th e s te a dy s ta te re sp on se o f e q u atio n (8 ) w e p u t à = 0. T h e re is a t riv ia l s o lu tio n , b u t a n o n t riv ia l so lu tio n a lso e x ist s, w it h a 0 g ive n by A 0 = -/ỉ 31»;2 a~ = hi — A, 4 (9) w h ere th e ze ro s u b s c rip t d en o te s the s ta tio n a ry s o lu tio n . S t a b ilit y w ill be in v es tig a te d by s tu d y in g th e b e h a v io u r fo r s m a ll p e r tu r b a tio n o f the ste ad y sta te re sp on se . It is easy to p ro ve t h a t th e z ero s o lu tio n a = 0 is stab le if A - hi > 0, (10) an d the n o n tr iv ia ] s o lu tio n d ete rm in e d by fo rm u la (9 ) is s ta b le if A - hỵ < 0 . T h e d epe n  de nce o f th e s t a tio n a r y a m p lit u d e a<) on th e d am p in g c oe fficie nt A is re p re se n te d in fig. 2. So fo r a st ro n g L a n c h e st e r d a m p e r in cr ea sin g th e d a m p in g fo rce (A ) le ad s to de cre a sin g th e a m p lit u d e o f th e s e lf- ex cite d v ib ra t io n o f m ass M . Fig. 2. The dcpcndcncc of amplitude of vibrating mass hi from the damping coefficient X in the case of a strong Lanchcstcr damper 3. W E AK LAN CH ESTE R D A M P E R Let us consider now the -case when the damper mass m is small of order c. The motion equations then become 6 NGUYEN VAN DAO (11) (12) M x + cx = e[ — A(x — t>) + hịX — /13Ì 3], Ú = V, mi) = \(x — v). The solution of the system of equations (11) will be found in the form I 2 ^ x = aco s V?, (p = ut + vs w = - 7 7, M X = —aa> sin V?, u = au;(i? cos <p + G simp), V = cujj2(—E sin V? + G cos £>), where _ A2 mA . . = "T v™ ” TZ 0 0 ’ ( ) w(A- *+■ m*u/**J A- -f m-uj~ an d ơ, yị> a re fu n c tio n s o f tim e s a tis fy in g th e c o n d itio n (5). S u b s tit u t in g th e e xp re s sio n s (12) in to e q u at io n s (11) a n d s o lv in g th e m fo r th e d e r iv a  tiv es g ive s itJCL — [a(x — v) — hịX 4- /13Ì 3] sin <Py • e 3 (14) wcl\Ịj = ^ [A(i — u) — M 4- /13 X ] cos V,. These equations belong to the standard form to which the averaging technique is applied [21]. So in the first approximation the right sides of equations (14) can be replaced by their averaged values over one cycle of vibration: à = — z [a(uE — 1) + hi — -h^uj2 a2] , x (15) ' ị . - ệ L v C - a . The equations (15) have the following stationary solutions: 1) T h e ze ro s o lu tio n a = 0, w h ich is s ta b le if X(uE - 1) + hx = h1 - 7T~— < 0- (16) À -r In the o p p o site case th e z ero s o lu tio n is u n st a b le an d th e m a ss M v ib ra t e s w it h a s t a tio n a r y V a m p litu d e d e te rm in e d b y : 2 ) 3 , *1 1 , m2(jj2X . . An = jA sw OỈ - fci - m aw r + A3 > °- ( 1 7) LANCHESTER da m per e f fe ct for quenching 7 Fig- s. Some typical curvcs given the dependence of amplitude of vibration on the rn Ị c damping coefficient A in the cast of a weak Lanchestcr damper. Curve 1: hi > y W curve 2: m r r k,< 2 v i r This relation is plotted in fig. 3 giving the dependence of the am plitude of stationary self-excited vibration on the dam ping coefficient A. The m inim um of the am plitudes cor responds to the value A = A. = mu/, (18) and - hi - ịrnM. (19) If the parameters of the system under consideration are chosen so that X = m' J ỉ ' 1201 then the self-excited vibration of the main mass M is com pletely suppressed. So, for a weak Lanchester damper the suppressing effect is achieved only with some intermediate values (A.) of the dam ping force (see Fig. 3). 4. S T R O N G L A N C H E S T E R D A M P E R E F F E C T F O R A S E L F - E X C I T E D v i b r a t i o n a l s y s t e m w i t h t w o d e g r e e s o f f r e e d o m We consider now the self-excited vibrational system w ith two degrees of freedom (fig. 4) which consists of two vibrating masses mi and m 2 w ith elastic elem ents Cl, c 12 and viscous dam ping (/i12) between them. The exciting “negative” dam ping force R is 8 NGUYEN VAN DAO su p p o se d to be o f fo rm (1). To s up p re ss th e s elf-e xc ite d v ib r a tio n o f th ese m a ss es tl L a n c h e ste r d a m p e r (m , A) is used . Fig. 4 ■ Two degrees of freedom self-cxcited system carrying a LanchtsUr damper D e n o tin g th e d is p la ce m e n ts o f m as ses m L, m 2 a nd d a m p e r m fro m th e ir p o sitio n s o f e q u ilib r iu m by xi} 12 a nd u re sp e ct iv e ly , the m o tio n e q ua tio n s o f th e sy ste m c on sid ere d fo r th e ca se o f str on g d a m p e r ( — is fin ite ) ca n be w r itte n in the fo rm m, m l*l + C1X1 + C i2(J i - x 2 ) = - th i2{xi - i 2) + m 2X2 + Ci2(xo — Xl) = —£^12(^2 “ *l) ~ sA(io — v), ủ = V, m ũ = —c A (u — Jo), w here all kinds of friction forces are su p p o sed to be sm a ll qu an titie s of order 5. B e fo re s tu d y in g the v ib ra tio n s g ove rn ed by e q u a tio n s (21) le t us e xa m in e th e se lf e xc ite d v ib r a tio n s o f a tw o d egree s o f fre ed om sy ste m w ith o u t a L a nc h e ste r d a m p e r, whose m o tio n is d e sc rib e d b y e q u at io n s : m Lx L + (cL + cI2)ji - ci2x2 = -fffci3(ii - is ) + j mnXn + 012(12 ~ Xi ) = -*£^12(^2 ~ i i ) • U s in g th e t ra n s fo r m a tio n in to th e n o rm al m o des £ i, £2 • *1 — %1 + s2» x2 = <?lĩi -r &2s2ỉ (23) w here n o J % i = 1; 2, c X c X 2 ^12 m i ĨÌĨ2 \j ( Ci’2 ~ 2 " — ) + y V mi mọ / m l m 2 4c?, (24) w e h ave LàNCHESTER damper effect for quenching 9 •ii + n ỉíi = r r t ^ i + cri ^ ) l Ml s2 *+■ = 7 7 ” (^1 + ^ 2^ 2)1 Mọ (25) ^here Fa = -/i12(ii - x2) + /iiii - hzil, F2 = - h i 2[Ì2 - ii), AÍ,- = mi + ơt2mọ, X = 1 , 2 . Introducing the new variables a, 6, 0 , <f) connected with the co-ordinates {2 by relations (26) £1 = a cos 0, (i = —aO 1 sin 0, Ớ = H it -t- t/>, £0 = 6 cos ry, £0 = —6 0 2 sin »7, rj = fiot -r with the additional conditions ả cos 0 — dip sin 0 = 0, 6 cos rj — bộ sin 77 = 0, equations (25) are replaced by O i ủ = “ T 7 - ( i \ + ơi/ọỊsin tf, Ml Cliarp = — ~ r ( F i 1- ơ i Ạ ) COS 0, M l 0 26 = +ơ2F2)sinr), Mo Qnbặ) = - *77“ (^*1 •+■ f**) cos ĨỊ. A/~ :7) Since a, t/j, ò, ự) are slowly varying variables, then in the first approximation the right sides of equations (27) can be averaged over the time, considering those variables as constants. T he averaged equations will be a — - o 0 “■ hi *+- [ơI — 1)"/112 + — /13 Oj a ~ ■+• - h z fo b -] , o Af [ “ ^ + (Ơ2 1)2^12 + -hztiib2 + - / ^ n ^ a 2] , z Ai 1 4 L ^ = Ậ = 0. The zero solution a = 6 = 0 of the equations Í28) corresponds to the equilibrium of masses rnA and m2. This solution is stable if Ml = hỵ — (ơi — l)2 /ii2 < 0 ) . . ỵ2 = hi — (ơn — 1)2/ijl2 < 0. If these conditions are not satisfied simultaneously then the equilibrium of the masses will be unstable and vibrations m ay occur. The vibration of the first mode with frequency ill and amplitude Oo is determined 10 NGUYEN VAN DAO b y b = 0, A = — hi — (ơi — 1)2A12, 4 (30 ) which is stable if Ả> - [hi - (ơ2 - l ) 2/iia ] — -B- (3 1) T h e v ib r a t io n o f th e se co nd m o de w it h fre q ue n cy n 2 a nd a m p lit u d e 6() is d e te rm in e d b y a = 0, B = \hzĩì\bị = hx - (ơ2 - l) 2/i!2, ^ ^ 4 w h ic h is s ta b le i f f l> Ì [/n -( ơl- l ) 3h13]= ÌA . (33) It is e as y to p ro v e th a t sim u lta n e o u s v ib r a tio n in tw o m o de s is u n sta b le . W e r e tu r n n ow to th e e q u atio n s (21) for a s tr on g d a m p e r. W h e n 5 = 0 fir st tw o e q u a tio n s o f th e s ys te m (21) are c ou p led a nd th e la s t tw o e q u at io n s b eco m e ủ = Vị V = 0 a n d th e y h a ve a s o lu tio n o f th e fo rm X j = a c o s (n xt + íp) -Ị- bcos(ilot + <£), X\ = — aCli sin (n ii + ip) — tfiosinfOoi -f ^), X2 = ơia cos(Qii + 0) + ÒƠ2 cos(ÍÌ2t + <£)» (34) x 2 = —a H iơ i s in (Q it + 0 ) — bíÌ2ơ2 sin (n 2Í ■+■ <£), v = v = 0, Ui = u«> = c on st, h ere th e n o t a tio n s (2 4 ) are u sed . C o n s id e rin g fo rm u la e (3 4 ) as th e tr an s fo rm a tio n in t o th e new v a ria b le s a, t/>, b, w e h av e a sy s te m o f e q u at io n s w h ic h is s im ila r to th e e q u a tio n s (2 7 ) a nd th e c o rre s p o n d in g a ve ra g ed e q u a tio n s w ill be à = — — [ — h i + Ớ ị \ + (ơi - 1)2Aì2 -f ~ h z ĩ í \ a 2 + - ^ n ^ b 2 ], 1 4 2 (3 5) 6 = — [ — /li + Ơ2 A + (cT2 — 1)2/*12 ■+■ -hrfinb2 + r^ nỉa 2]. 2AÍ2 4 z F ro m th e e q u a tio n s (3 5 ) one c an see th e fo llo w in g s te a d y sta te s o f m o tio n : LANCHESTER* DAMPER EFFECT POR QUENCHING 11 1 ) T h e e q u ilib r iu m Xi = i ị = i 2 = X2 = 0 (a = b = o) is s ta b le if (36 ) hi - Xi < 0, hi - < 0, where ^1 = (ơi - 1)2^12 + ~ [ơ2 ~~ l ) 3^12 ơ2^‘ (37) 2 ) S e lf- ex c ite d v ib r a tio n o f m a ss es mi an d m? w it h fre q u en cy n ! a n d a m p lit u d e ao determined by 6 = 0, 3 o n (38) ^0 — —/13OfaQ = hi — (ơ i — 1)**^12 ~ Ơ^À, 4 which is stable if Ao > ị[hx - h ) = jB o . (3 3) 3 ) S e lf-e x cite d v ib ra tio n o f m a sse s m i an d m 2 w it h fre q ue n cy n 2 an d a m p litu d e 60 determined by a = 0 , 3 (40) So = - h 9nĩbỊ = hỵ - (ơ2 - 1) /lÍ2 — Ơ-ỒÀ, 4 which is stable if Bo > 2 ^°' (^) 4) T h e s im u lta n e o u s v ib r a tio n o f tw o m o de s w it h fre q ue n cie s n : , is u n sta b le . F ro m th e fo r m u la e (3 8 ) and (4 0 ) it is e vid e n t th a t fo r a s tr o n g L a n c n e st e r d a m p e r, in creasing its da m p in g coefficien t (A) lead s to th e d ecrease o f sta tion ary am plitu de o f th e se lf-e xc ite d v ib r a tio n o f masses m i an d mo. 5 . W E A K L A N C H E S T E R D A M P E R E F F E C T F O R A S Y S T E M W I T H T W O D E G R E E S O F F R E E D O M In this section the case when the mass m of Lanchester damper is small quantity of order e is investigated. In this case the differential equations of motion (2 1 ) become miXi + (cI + c12)z 1 - c12x2 = sFit + c12( i 2 - Ix ) = s\F2 - X[x2 - v)Ị, " I 42) u = V, m i) -f Xv = A Ì2, w h ere fu n c tio n s Fi a n d F3 a re o f fo rm ( 2 5 ). T h e d iffe re nc e be tw e en th e e q u a tio n (2 1 ) a n d (4 2 ) is t h a t in th e la s t e q u a tio n o f th e s ys te m (4 2 ) th ere is no s m a ll p a ra m e te r c . U s in g t he n o r m a l c o -o rd in a te s (2 3 ) w e ca n tr an s fo rm firs t tw o e q u a tio n s o f (4 2 ) in to th e fo rm 12 NQUYEK VAN DAO (43) £i + nỈÉi = ^ “ 1^1 + ^i[F2 - A(io - v)] j, Ỉ2 + ^2^2 = + Ơ2 [f2 - - v )] 1» an d th e la s t tw o e q u a tio n s o f (4 2 ) b eco m e m ũ + Ati = A(<7a£ i 4- 02s2)> (44) h ere th e n o t a tio n s (2 4 ) are used . In t ro d u c in g th e ne w v a ria b le s ax, ao, £>1, ^2 co n ne cte d w ith th e co -o rd in a te s £1, $2 by th e fo rm u la e £1 = ai cos V?1 , Í 1 = -<*1 ^ 1 sin <Pi, <iPi == H it + \pit £2 = a2 COS <P2, Ỉ2 — — ^2 ^2 sin <p2j v^2 = ^2^ ■+* ^2» (45) à i COS <PI — aiự>i sin ^>1 = 0, 02 COS <P2 — ao02 sin <P2 = Oj we o b t a in m ũ -r Ail = “ AỊơiaxHi sin v^i “H ơoơoOo sin ^>2)» (46) a nd , s o lv in g fo r th e d e riv a tiv e s, w e ha ve A iá i = + ơ i[F ọ - À (i2 - u)] I sin P i , nxa + c7i[F2 - À(i2 - w)]Ị cosv?i, n 2ả 2 = - - ^ ị - Ị F ỵ + ơ 2 [/2 - À (i 2 - 1> ) ]| sin ũ->a2'Ộ2 — ~ T T \ Fí + ơ-JF ọ - A(iọ - v )] Ị COS <p2 , ÀZ2 V * w h e re th e fu n c tio n u is th e s olu tio n of e q u atio n (46): \ ơ ì ai u = 00" xõ(mfli sinv? 1 + AcospJ-i- m*\ìị -+ A- 2fl2 (m flo sin v?2 ■+■ A cos £>2). (48) (47) F o llo w in g th e a s y m p t o tic m e th o d o f a v era g in g [2 1] in th e fir st a p p ro x im a tio n th e r ig h t h a n d s id es o f th e e q u a tio n s ( 47) m a y be re p la ce d b y t h e ir ave ra ge d v alu e s o ve r o ne c y cle o f v i b r a t i o n : [...]... cubic parabolas Y2 = z(z - A.)2, n = *{* - B.)2 The intersection of Y\ and Ỉ 2 and of Yi and y3 gives the root of equations (90) and (91) The greatest value of these roots determines the maximum of the amplitudes of corresponding vibration (see fig 7) The abscissas of the Tightest points of intersection of the straight line Yi with the curves y2 and y3 are the values Amo* and Bmax- W can take e approximately... compare the maxima of amplitudes of vibrations of mass M in the case (a) with ajid (b) without the Lanchester damper it is necessary to find these maxima (am and a* bmax) f r o m t h e f o r m u la e ( 8 4 ) -a n d ( 8 9 ) T h e y a re th e g r e a t e s t v a lu e s a a n d b w h ic h vanish the expressions under the square root in the formulae(84) and(89), i satisfy the following equations m ake e they... t a b le 1, A = * 2 1 3 3 , A a n d s e m i- a x e s - A , LANCHESTER damper effect for quenching 21 It is worth mentioning that in the corresponding system (fig 1) without the Lanche s te r d a m p e r , t h e d e p e n d e n c e o f th e a m p lit u d e b o f m a s s M o n t h e r a t i o f r e q u e n c y V — — U ) is given by the formula 1/2 = ~ ^ 2' (8 9 ) T h e d if f e r e n c e b e t w e... ( 7 1 ) o n e c a n se e t h a t a w eak L a n c h e s t e r d a m p e r is h i g h ly effective on ly w i t h s o m e i n t e r m e d ia t e v a lu e s o f t h e v isc o u s d a m p in g c o e ffic ie n t A : A = À* = m a x { m il.} »=: 1-r JV ; (73) 18 NGUYEN van Da o 8 LANCHESTER DAM PER EFFECT FOR A FORCED SELF-EXCITED VIBRATING SYSTEM In th is s e c t io n it is a s s u m e d t h a t t h e m... ,2 " M -i, N with N decrees of freedom t p 1 fifi /77/ - J - 2/ Fig 5 L a n c k a t t r da mp e r attacked to a s e l f - c z c i u d s y s t e m with N degrees o f f reedom + ( c x + c 12) x i - c i 2 X 2 = e f i , m2x2 + (ci2 + 003)^2 C X1 —C 12 23^3 —£/ 2» m N Ỉ N + Ctf-i.Ar(x/sr - Z A T -i) = ủ = V, m v = e \ ( x # — v), (56) eftfy is LANCHESTER damper effect for 15 quenching w h ere /1 = ^1 *1... s LANCHESTER e 70 is g iv e n by /3 , ( 4^37 2-2 , ag - hx + m 3 7 2A - ^ 2 ^ 3 ^ n j^ o E lim in a tin g t h e p h a s e V'o a n d s o lv in g following a p p r o x i m a t e e x p r e s s io n 2 ^ emuj2\ 2 ^ 71 o/ o 0 I/ A 00, A a o + n r r 2OoA2 ^ f ^ = PsinV-o for. .. a . ao Institute of Mechanics NCSR of Vietnam Su m m ary . The Lanchester damper effect for quenching both free and forced self-excited vibrations of the mechanical systems with one, two and many degrees of. characteristics of the behaviour of the Lanchester damper for the Q ue nch ing of self-excited vibrations were established. In the present paper the analysis will be extended system at ically for mechanical. here is to estim ate the effectiveness of the Lanchester dam per in suppressing the vibration of the mass M under the simultaneous excitation of the mentioned above forces. The governing equations