ynamic absorber effect for self-excited systems

Vdvances in Mechanics 991 • VOL NR y c n e x H M exaH H H H D Y N A M IC ABSORBER E F FE C T FO R SE LF-EX C ITED SYSTEM S Nguyen Van D a o , Nguyen Van Dinh (Vietnam) Contents n t r o d u c t i o n D ynam ic absorber for self-excited systems with single degree of fr e e d o m 1.1 Sclf-excited vibration of a quasi-linear m echanical s y s t e m 1.2 Weak dynam ic absorber [7, ] 1.3 S tr o n g d y n a m i c a b s o r b e r [ , ] 1.4 Som e r e m a r k s 1.5 D am ping the self-excited vibration-caused by the limit energy resource [ ] 4 10 11 1.6 D y n a m i c a b s o r b e r for d r illin g in s t r u m e n t [ 1 ] Dynam ic absorber for self-excited system s with several degrees of fr eed o m 2.1 Self-excited vibration of system with tw o degrees of freedom [ ] 2.2 Strong absorber for the system with two degrees of fr eed o m 2.3 Weak absorber for the system with two degrees of fr e e d o m 2.4 Self-cxcited vibration of system with n degrees of freedom [6, ] 2.5 Weak absorber for system with n degrees of fr e e d o m D ynam ic absorber for system with distributed p aram eters 3.1 D ynam ic absorber for vibrating string [ ] 3.2 D ynam ic absorber for beam [ ] 3.3 Absorber for self-excited vibration o f the plate [ ] C o n c lu s io n R e fe r e n c e s S u m m a r y Pc3fOM e 19 19 20 22 24 26 27 28 33 36 39 39 40 40 Introduction T h e theory of d a m p i n g for a linear system u n d e r the actio n of external h a r m o n ic f o r c e is well k n o w n T h r e e k i n d s o f m e a s u r e s h a v e b e e n u s e d t o d a m p t h e f o r c e d v i b r a t i o n o f th i s s y s t e m : Nguyen Van Dao Nguyen Van Dinh ị C h a n e i n e th e f r e q u e n c y to m o v e a w a y t h e free f r e q u e n c y fr o m th e f o r c e d one I n t r o d u c i n e t h e d a m p e r to d i s s i p a t e t h e v i b r a t i o n a l en er gy B u t t h e d a m p i n g n e c h a n i s m o p e r a t e s o n l y w h e n exists v i b r a t i o n So, by this w a y (e n e rg y d i s s i p a t i o n ) , he v i b r a t i o n is n o t d a m p e d c o m p l e t e l y U s i n g t h e d y n a m i c a b s o r b e r ( m e t h o d of d y n a m i c a l b a la n c e ) a s u p p l e m e n t a r y / i b r a t i o n a l s v s te m — c o n s i s t i n g o f a m a s s fo ll o w e d by a n el astic e l e m e n t U n d e r d e al c o n d i t i o n s w h e n t h e r e a r e n o f r ic ti o n s a n d w h e n th e free f r e q u e n c y of the i b s o r b e r is e q u a l t o t h a t of th e e x t e r n a l force, the v i b r a t i o n of th e m a i n m a s s i i s a p p p e a r s T h e a b s o r b e r c o n t i n u e s to v i b r a t e a n d c r e a t e s th e in e rt fo rce w h i c h j p s e t s t h e b a l a n c e w i t h th e e x c i t a t i o n a l force H e n c e th e d y n a m i c a l b a l a n c e is th e Drinciple of a c t i o n o f t h e d y n a m i c a l a b s o r b e r F o r n o n l i n e a r s y s t e m s t w o first m e a s u r e s ( C h a n g i n g t h e f r e q u e n c y a n d e n e r g y d i s s ip a ti o n ) h a v e b e e n u s ed to limit th e e r o w t h of th e a m p l i t u d e o f v i b r a t i o n o f the m a in m a ss T h e d y n a m i c a b so rb er has b e e n in v e s t i g a t e d firstly for t h e l i n e a r v i b r a t i o n a l s y s t e m t o d a m p t h e f o r c e d v i b r a t i o n N o t l o n g a g o w M M a n s o u r [4 ], w R C l e n d e n i n s R N D u b e v [ ] , A T o n d l [ - ] , p H a e e d o r n [ ] a n d the i u t h o r s o f this m o n o g r a p h p r o p o s e d a n d s t u d i e d th e effect o f th e d y n a m i c a b s o r b e r o r a self-excited s y s t e m It t u r n e d o u t t h a t t h e self-excited v i b r a t i o n m a y be d a m p e d entirely by m e a n s o f t h e d y n a m i c a b s o r b e r w i t h a s u it a b le v a lu e of e x t e r n a l fr ictio n orce In th e p r e s e n t w o r k , to c l e a r u p s o m e q u e s t i o n s a b o u t t h e p r i n c i p l e s o f a c t i o n of the d y n a m i c a b s o r b e r f o r a self-excited s y s t e m we limit o u r s e l v e to a n i n - d e p t h s t u d y :)f a w i d e s p r e a d class o f n o n l i n e a r s y s t e m : a q u a s i - l i n e a r o n e w i t h the h e l p o f a well b a s e d m a t h e m a t i c a l a s y m p t o t i c m e t h o d H e r e it is n e c c e s a r y to d i s t i n g u i s h t w o ^ases: t h e s t r o n g a b s o r b e r w h e n t h e m a s s a n d th e s p r i n g of the a b s o r b e r a r e finite, a n d th e w e a k a b s o r b e r w h e n t h o s e p a r a m e t e r s a re small In b o t h case s t h e fr ic ti on m e c h a n i s m c o n n e c t e d w ith t h e a b s o r b e r h a s a decisive in fl ue nc e o n t h e a m p l i t u d e of self-excited v i b r a t i o n F o r t h e s t r o n g a b s o r b e r , i n c r e a s i n g t h e fri cti on for ce l e a d s to t h e d e c r e a s e o f th e a m p l i t u d e o f self-excited v i b r a t i o n But for th e w e a k a b s o r b e r the effect of d a m p i n g is a c h i e v e d o n l y w ith s o m e m e a n v a lu e s of t h e fr ictio n force In this w o r k m e c h a n i c a l s y s t e m s w i t h on e , t w o a n d seve ral d e g r e e s of f r e e d o m a s well as 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 will he in v e st ig at e d t Ỉ Dynamic absorber for self-excited systems with single degree of freedom 1.1 S e lf - e x c ite d vib tio n o f a q u a si-lin ea r m e c h a n ic a l s y s te m Let us c o n s i d e r a self-ex cited m e c h a n i c a l s y s t e m (Fig 1) w h o s e m o t i o n is d e s c r i b e d by th e diff erential e q u a t i o n (1.1) m x + c x — Í.R (.V, v) Dynamic absorber effect for self-excited systems R m 7777777777777777777777777777, Fig I h e r e t h e f u n c t i o n R { x , x) e xp re ss es t h e a c t i o n o f the self-excited force, th e o t h e r a r a m e t e r s a r e c le a r f r o m t h e figur e; E is a s m al l p o s i t i v e p a r a m e t e r c h a r a c t e r i z e d the m a l l n e s s of t h e c o r r e s p o n d i n g m e m b e r s W i t h o u t loss of g e n e r a l validity, o n e c a n a s s u m e t h a t m = 1, c = T h e s o l u t i o n o f E q (1.1) c a n he w r i t t e n a s 1.2) X = asirup, x = acos(p, (p = t + u'/ in d a a n d ip a r e s lo w ly v a r y i n g f u n c t i o n s o f time S u b t i t u t i n g th e s o l u t i o n (1.2) i n t o 5q (1.1) g iv e s ã c o s (/5 — aiị/sinq) = s R { x , x) a) Taking into account a sin cp + aiị/ COS (/7 = ib) we get f r o m (a), (b): ủ — e R ( x , x)cos (p, (1.3) uijj — — eR ( x , x ) s i n (p S in c e it is a s s u m e d t h a t a a n d |// a r e s lo w ly v a r y i n g , Eqs o v e r o n e cycle as fo ll ow s: R (x , x ) c o s (pdcp a An (1.4) K 2” a ip = — — J R ( x , x )s in (pd(p 2n For (1.5) R ( x , x ) = /jjX — /?3X Eqs (1.4) become (1.6) aệ = 3) m a y be a v e r a g e d Nguyen Van Dao Nguyen Van Dinh m th e r e s u lt o f Eq (1.6) i/ m s a c o n s t a n t T h e s t e a d y s t a t e m o t i o n c o r r e s p o n d s to H e n c e E q s (1.6) give a = 0, A = h J >m h e r e it fo ll ow s t h a t : T h e e q u i l i b r i u m a = is s t a b l e if /jj < a n d u n s t a b l e if hị > T h e r e exist s h a r m o n i c v i b r a t i o n w i t h f r e q u e n c y Ộ = a n d w i t h th e a m p l i t u d e e r m i n e d by A = h !j > T h u s , d e p e n d i n g o n t h e v a l u e /jj o f t h e f r ic ti o n force, t h e m e c h a n i c a l t e m will b e e i t h e r at rest o r v i b r a t e d Weak dynamic absorber [7, 8] e a b s o r b e r is c al l ed a w e a k o n e if its m a s s m a n d stiffness c a r e s m a l l in m p a r i s o n w i t h th e m a i n m a s s m x a n d s p r i n g c J T h e f o l l o w i n g e q u a t i o n s a r e i u m e d for t h e s y s te m in Fig 2: ■ //////////// / / / / / / / / / / / / / / / / , R m, '7777777777777777777777777777777 Fig Xj + c o j x m [ / ỉ 1x - / ? x ? - c ( x - 9) here c m (i>2 — (c2 + c ] ), m2 /ij > , /i3 > x 2) ] , Dynamic absorber effect for self-excited systems T h e first e q u a t i o n o f (1.9) is q u a s i - l i n e a r w hi le t h e s e c o n d o n e is a l i n e a r e q u a t i o n [n t h e first a p p r o x i m a t i o n we shall find th e s o l u t i o n of Eq (1.9) in th e f o r m : Xj = a c os , X, = ( 1 ) a a jjS in X = w J I + lỊ/ , a (M c o sO + N sin ỡ ), X = flC0 j ( — M s i n + N c o s ) where M = c 12(cú2 — (ứỉ m~, [ ( a >2 — coỊ)2 + (ú]Â2y N = c c Oj A m [ ( a >2 — CO?)2 + t O i / ] a n d a \Ị/ satisfy th e a v e r a g e d e q u a t i o n s of ty p e (1.4): ea = 1m c '- 2\2 I ,2-2-1A ^COịŨ > -(X>ÌY tO]/.2] m , [(cư2 — CL>i)2 + t o f / r ] J Cl 2| J f "i< À = \2 - + C O ? /2] li is e a s y to verify t h a t t h e e q u i l i b r i u m a = is u n s t a b l e a n d th e r e exists a s t a t i o n a r y self-excited v i b r a t i o n w i t h th e a m p l i t u d e a d e t e r m i n e d by ( 1.12 ) L ,.,2 h-ịCOịU h J CỈ ^2 \2 _L "2-1 m 1(0)2 — 0JÌ) + W ] / ; J if th e r i g h t h a n d side o f Eq (1.12) is po si tiv e In Fig t h e d e p e n d e n c e o f a m p l i t u d e of v i b r a t i o n o n th e p a r a m e t r Ằ is p r e s e n t e d for th e c ase /?, = 0.05, C ] = 1, c \ 1j m — 0.1, O A = - h 3w \ a T h e c u r v e c o r r e s p o n d s t o a>2 = 1,5 a n d t h e c u r v e to ct>2 = W h e n COJ, co2 a r e in e q u a l i t y , th e e x p r e s s i o n (1.12) is of fo r m : (1.13) h^cúịa2 Cl h i- m 20) Nguyen Van Dao Nguyen Van Dinh n d t h e r e s o n a n t c u r v e is s h o w n in fig by 0" In this case th e effect of d a m p i n g is lichest .3 S tr o n g d y n a m ic a b so rb er [7 , ] ^et us c o n s i d e r th e s o - c a l l e d s t r o n g a b s o r b e r w h e n th e p a r a m e t e r s in, a n d C a re -, inite In this case th e m o t i o n e q u a t i o n s a r e m x + c J X J + fp (.X j X 2) = F ịh x l — h 2x j ) , 1.14) m , X + C2X , + C i ( x — X j ) = — £/ x and t h e y a r e n o t s e p a r a t e , w h e n £ = as Eqs (1.9) U s i n g ' t h e t r a n s f o r m a t i o n in to the n o r m a l m o d e s here M ị = j\ = h lx l - ffl, + d f m h 3x ] , /2 = - /-v2 I n t r o d u c i n u th e n e w v a r i a b l e s a, b \Ị/, Ộ by m e a n s of th e f o r m u l a e Q^acosO, s at + E/ , = £ / ' ? ( ox (3.25) ;i) \ / a4z Ờ z ^v4 ■ 1 ' w i t h th e b o u n d a r y c o n d i t i o n s y (f , x) = z { t , x) = , ^ | ^ ỠX I x = 0, X = / -3 = S ox = I X = , X = / H e r e £ is a s m a l l p o s i t i v e p a r a m e t e r U s i n g th e n o t a t i o n s _ £/ a = (JS ’ e 12 ^1^1 ^ = “ V ’ (?jSj , J l = qS ’ c h° _ ^ "3 = T S ’ r q (3.26) Ỵ c == ■- Advances in Mechanics 1/1991 eS ’ y = Ọị S ị gS’ 34 N guyen Van D a o N guyen Van Dinh we h a v e for th e w e a k a b s o r b e r c ~y c V T T + fl T dr rx (3.27) ct ct ƠÍ W h e n £ = t h e s y s t e m o f e q u a t i o n s (3.27) h a s th e s o l u t i o n * V— 7T 2_ s i n - r c x i ^ c o s a v + /?ws i n c o nf), n= * 71 = - s i n - n x ( a „ c o s a ; j + /)ns i n a j / ) , n= * he re a n a n d /}„ a r e c o n s t a n t s , cư„ = fl n r t t 7 // \\ T T {b — a ) \ —n j + r + / ~ > y/i) «„ + (/> 2- a 2) ■ = (h2 - a 2) ( ^ n ( b - a 2) ị j n ^ / 0i „ a „ + ^ V., +