ynamic absorber effect for self-excited systems

Dynamic absorber effect for self-excited systems 5... Dynamic absorber effect for self-excited systems 93... Dynamic absorber effect for self-excited systems 13... Dynamic absorber effec

Vdvances in Mechanics

D Y N A M I C ABSO R BER E F F E C T

F O R S E L F -E X C IT E D SY ST E M S

Nguyen Van D a o , N guyen Van Dinh (Vietnam)

Contents

n t r o d u c t i o n 3

D ynam ic absorber for self-excited system s with single degree of fr e e d o m 4

1.1 Sclf-excited vibration of a quasi-linear m echanical s y s t e m 4

1.2 Weak dynam ic absorber [7, 8 ] 6

1.3 S t r o n g d y n a m i c a b s o r b e r [ 7 , 8 ] 8

1.4 Som e r e m a r k s 10

1.5 D am ping the self-excited vibration-caused by the limit energy resource [ 1 2 ] 11

1.6 D y n a m i c a b s o r b e r for d r ill in g i n s t r u m e n t [ 1 1 ]

2 D ynam ic absorber for self-excited system s with several degrees of fr e e d o m 19

2.1 Self-excited vibration of system with tw o degrees of freedom [ 6 ] 19

2.2 Strong absorber for the system with two degrees of fr e e d o m 20

2.3 Weak absorber for the system with tw o degrees o f fr e e d o m 22

2.4 Self-cxcited vibration of system with n degrees of freedom [6, 8 ] 24

2.5 Weak absorber for system with n degrees of fr e e d o m 26

3 D ynam ic absorber for system with distributed p ara m eters 27

3.1 D ynam ic absorber for vibrating string [ 1 0 ] 28

3.2 D ynam ic absorber for beam [ 1 3 ] 33

3.3 Absorber for self-excited vibration o f the plate [ 1 3 ] 36

C o n c lu s io n 39

R e fe r e n c e s 39

S u m m a r y 40

Pc3fO M e 40

Introduction

T h e t h e o r y o f d a m p i n g f o r a l i n e a r s y s t e m u n d e r t h e a c t i o n o f e x t e r n a l h a r m o n i c

f o r c e is wel l 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 t h i s s y s t e m :

ị Nguyen Van Dao Nguyen Van Dinh

d e a l 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 i c t i o n s a n d w h e n t h e free f r e q u e n c y o f t h e

i b s o r b e r is e q u a l t o t h a t o f t h e e x t e r n a l force, t h e v i b r a t i o n o f t h 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 t o v i b r a t e a n d c r e a t e s t h e i n e r t f o r c e 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 t h e e x c i t a t i o n a l fo rce H e n c e t h e d y n a m i c a l b a l a n c e is t h e Drinciple o f 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

In t h e p r e s e n t w o r k , t o 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 s el f -e x ci te d s y s t e m w e l imit o u r s e l v e t o 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 c l a s s 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 t h e h e l p o f a well

Dynamic absorber effect for self-excited systems 5

Nguyen Van Dao Nguyen Van Dinh

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 h i l e 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 w e s h al l find t h e s o l u t i o n o f E q (1.9) in t h e f o r m :

Nguyen Van Dao Nguyen Van Dinh

Dynamic absorber effect for self-excited systems 9

3 V i b r a t i o n o f t h e s e c o n d n o r m a l m o d e £ 2 w i t h f r e q u e n c y a n d a m p l i t u d e

/7 d e t e r m i n e d by

a = 0, B = - h Q i b2 = h , — d ị / , /;, — </v > 0

4

10 Nguyen Van Dao Nguyen Van Dinh

F r o m t h e r e l a t i o n s (1.20) a n d (1.22) it f o l l o w s t h a t t h e a m p l i t u d e o f sel f-e xci te d

v i b r a t i o n d e c r e a s e s w h e n t h e coef f ic ien t Ả o f f r i c t i o n f o r c e i n c r e a s e s (Fig 4).

Dynamic absorber effect for self-excited systems 1 1

(1.24)

l ộ = L ( ộ) — H (Ộ) — r T ,

M X J 4" c jX J + c 2 (x J x 2) — T ,

m x 2 + c 2 ( x 2 — X j ) + Ả x 2 = 0.

I Nguven Van Dan Nguyen Van Dinh

Dynamic absorber effect for self-excited systems 13

Nguyen Van Dao, Nguyen Van Dinh

T h e d e p e n d e n c e o f A 0 ( t he a m p l i t u d e o f sel f-e xci te d v i b r a t i o n ) o n t h e c oef fic ient

Dynamic absorber effect for self-excited systems 15

6 Nquyen Van Dao, Nguvcn Van Dinh

Dynamic absorber cffcct for seif-excited systems

> Nguyen Van Dao, Nquycrt Van Dinh

Dynamic absorber effect for self-excited systems 1 9

Fig 10

>0 Nguyen Van Dao, Nquyen Van Dinh

Dynamic absorber effect for self-excited systems 21

22 Nguyen Van Dao, Nguyen Van Dinli

Dynamic absorber effect for self-excited systems 23

= 0 CO2 = 1 (Fig 12) a n d h i g h f r e q u e n c y o r = ú ị , A 2 — 0: to2 = 6 (Fig 13) F r o m

Fig 12

24 Nquven Van Dao Nguyen Van Dinh

Dynamic absorber effect for self-excited systems 25

26 Nguyen Van Dao, Nguyen Van Dinh

lynamic absorber is acted on effectively with a suitable choice of friction force (/.).

Dynamic absorber effect for self-excited systems 27

3 D y n a m ic absorber for system with distributed parameters

As w e see b e l o w , t h e d y n a m i c a b s o r b e r m a y b e u s e d t o d a m p t h e self -excited

28 Nguven Van Dao, Nguyen Van Dinh

Dynamic absorber effect for self-excited systems 29

30 N q uyen Van Dao N g u y e n Van Dinh

D ynam ic a b so rb e r effect for self-excited s yste m s 31

32 N g u y e n Van D a o , N q u v en Van Dinh

[ c o n t ] b ^n~dt = ~ l][ p 1 + c"u ~ c"Y"{b) COSọ ■

D y n a m ic a b so r b e r effect for se lf-e x c ite d s y s te m s 33

34 N g uyen Van D a o N g u ye n Van Dinh

D yna m ic ab so rb e r effect for self-excited s y s te m s 35

36 N g u ye n Van D a o N g u y e n Van Dinh

D y n am ic ab so rb e r effect for self-excited s y s te m s 37

38 N g u y e n Van D a o , N g u ven Van Dinh

D ynam ic a b so rb e r effect for self-excited s y s te m s 39

3 ] \ H a g e d o r n , U b er die Tilgung selbsterrergter Schwingunqen, Z A M P vol 29, 1978 815 821.

4 w M M a n s o u r , Quenching o f limit cycles o f a Van der Pol oscillator, J Sound Vibration, 25, 1972

395 405.

5 NtiUVTN Van Dinm, The tuned absorber in self-excited system , J Mechanics, Hanoi, 3 4, 1979.

6 N g u y e n Van D in h , The dynamic absorbers in quasi-Iinear systems, Dissertation, Hanoi, 1980.

7 N g u y e n V an D a o , a note on the dynamic absorber, J Mechanics, Hanoi, 2, 1982.

8 N g u y e n V an D a o , N g u y e n Van D in h, Dynamic absorber for self-excited systems Proceedings of

XIVth conference Dynam ics of machines, Prague, September 1983.

L) N g u y e n Van D a o , Quenching o f self-excited Oscillations o f mechanical system, IC N O — X Bulgarie,

40 N g u y e n Van D a o , N gu ven Van Dinh

12 N g u y e n V an D a g , Dynamic absorber for self-excited system Willi limit energy resource J Mechanics

H anoi, 3, 1989.

13 N g u y e n V a n D a o , N g u y e n V a n D i n ' h , Dynamic absorber f o r beam and plate under the action o f friction fo r c e s, J Mechanics, Hanoi 1, 1990.

14 A T o n d l , Quenching o f self-excited vibrations, J Sound and Vibrations 4 2 : Equilibrium Aspects

251 260, 1975; One and two frequency vibration, 261-271, 1975.

15 A T o n d l , Quenching o f self-excited vibrations: Effect o f dry friction J Sound and Vibrations 45

2 8 5 -2 9 4, 1976.

16 A T o n d l , Application o f tuned absorbers to self-excited systems with several masses, Proceedings of the

X Ith conference D ynam ics of Machines Prague 1977.

S U M M A R Y

D y n a m ic a b so r b e r e f f e c t fo r s e lf - e x c it e d s y s te m s

The dynam ics absorber effect has been investigated for the systems with one and several degrees of freedom and also for the systems with distributed parameters In all the cases under consideration the friction forces (A) play an important role in damping the self-excited vibrations Here, the role of the friction forces is neither in energy dissipation nor in dynamic balance as in the case of forced vibrations For strong absorbers, increasing the friction force leads to the decrease of the amplitude of self-excited vibrations while for weak absorbers their effectiveness arc found in the suitable choice of the friction force.

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