Dynamic absorbers for systems with distributed parameters

From these it follows:great... 20 NGUYEN VAN DAO and NGUYEN VAN DINHd yn am ic absorber w ith m ean values of À... This is the case, for exam ple, when the friction force A is sufficient

th e effect of q u en ch in g will be achieved only w ith Borne m ean values of th e friction force.

Let US c o n s id e r t h e v ib r a t io n of th e string w it h two fixed en d s at 2 = 0 an d X = t ( Fig.

1) It is a s s u m e d t h a t th e strin g is acted on by the e x t e r n a l force alon g its w h o l e le n g t h w i t h the

w h i c h e x c i t e s t h e v i b r a t i o n o f the strin g in the y -d ir ec tio n To q u e n c h th is v ib r a t io n one c a n U9e

4 NGUYEN van Dao and NGUYEN Van DINH

here ỈVn (t) are u n k n o w n f u n c t io n s of tim e w hich are to be d e t e r m in e d

B y m u lt ip ly in g tw o s i d e s ' o f (1.2) w ith s i n — x d z and in te g r a t in g on X from G t ) £ the

fo llo w in g e q u a t io n s for Wn and u w ill be o b ta in ed

6 NGUYEN VAN DAO and NGUYEN Va n DINH

T h u s , in th e first a p p r o x im a t io n the strin g v ib r a t e s CIS

We c o n s id e r now th e weak d y n a m i c absorber It is a s s u m e d t h a t t h e stiffn ess c and the m ass

m of t h e a b s o r b e r are sm a ll T h e n , instead of (1.2 ) w e have t h e follow in g e q u a t i o n s o f m o t io n

- > r dy(b, t )'

dt

u — d y j b t dt

mu 4- c | u — y ( 6 , i ) j = — A u

-T h e e q u a t i o n s for Wn and u are

w n + f i w n = <1p , + c„u - cnYn(b)Wn], mil + ÀÚ + cu = Yn(b)(cWn + AWn ),

From the e x p r e s s io n s ( 1 2 2 ) , (1.2 3) it follow? that

1 T h e a m p li tu d e of v ib r a t io n of the strin g d e p e n d s e s s e n t i a lly on the coefficient n f friction A of

th e absorber If X = 0 or A is very hifch the absorber d o c s not w ork effectively T h e o p tim u m

3 If c 5= mpn t h e a m p l i t u d e curve takes the form s h o w n by “0 * in Fig 3 and o b v iou sly, the

d y n a m i c a b s o r b e r in t h is case b e c o m e s more effectiv e.

8 N GU YEN VAN DAO and NGUYEN VAN DINH

u(t) = Y ^ U n Wn(t),

n = 1

here W n [ t ) are u n k n o w n f u n c t i o n s of t , y n ( x ) are u n k n o w n fu n c t io n s of X a n d Un are c o n s t a n t s

s a t i s f y i n g th e b o u n d a r y c o n d i t i o n s (1 3 ) and a lso th e o r t h o g o n a li t y r e la tio n s

To find th e e q u a t i o n s for VVn ( t) , Yn(x)i Un we s u b s t i t u t e th e e x p r e s s io n s (2 2 ) i n t o ( 2 1 ) and

E q u a t i o n for Y„(x) is of form:

T0Y::(z) + nu* Yn{I ) + c\un - Y„(b)\6(x - 6 ) = 0 and its s o l u t i o n w it h b o u n d a r y c o n d i t i o n s (1.3 ) if found u n d er the ex p a n sio n

N o w , t h e s o l u t i o n o f t h e in it ia l e q u a tio n s (1.2) w ill be f o u n d in t h e series:

1 0 NGU YEN VAN DAO and NGUYEN VAN DINH

oo

y ( x , t ) = ^ 2 Y n(x)Wn(t,e)

n= 1oo

u ( t ) = Y ^ u nw n(t,e),

n= 1

( 2 1 3 )

where ^ ( x ) , Un are of f o r m s ( 2 7 ) , ( 2 1 1 ) and Wn(t}e) are u n k n o w n f u n c t io n s o f t and € S u b s t i ­

tu tin g ( 2 1 3 ) i n t o (1 2 ) t h e n m u lt ip ly in g th e terms w i t h Yj (x)dx an d i n t e g r a t i n g t h e m from 0 t o £

w e ob tain t h e f o llo w in g e q u a t i o n s for W n ftjc ):

E q u a t i o n s ( 2 1 4 ) can be tr a n sfo r m e d in to the s t a n d a r d form by m e a n s o f f o rm u la e

Wj = a ; costfy, Wj = —ayu>y sin 6 j t ( 2 1 6 )

where n e w v a r i a b le s ay, 9j s a t is f y the eq u a tio n s

U) 1 t h e n in t h e first a p p r o x i m a t i o n the follow in g averaged e q u a t i o n s w ill be o b t a in e d :

T h e t r iv ia l so lu t io n CLi = 0 of ( 2.18) is s t a b l e if h\ < 0 or if t h e fric tion A is suffic ie ntly b ig W h e n

is s t a b l e ii

(2 ) „, n A2 > -^1-^3

2 hl

T h e s o lu t io n a 2 — 0 a ^ 0 d e t e r m in e d by

^ — - I , ' 1), ,2 „2 •J A 3 a-'jaj — rlj 4

is s t a b le if

^2‘^3 2/iị 1)

12 N G U Y E N VAN DAO and NGUYEN VAN DINH

w e h a v e for t h e w e a k a b so rb er (c,Ao are small):

I dy ( d y \ 3 f d 2z 2^< z \

1ã í _ 3 V y ~ * \ d t * + l d ĩ * )

(3.3)

W h e n € = 0 t h e s y s t e m of e q u a t io n s (3.3 ) has the solu tion:

y = sin —nx(an coscưnt + òn s i n cưn í ) , n

2 = ^ s i n ^ n x ( a n coseJn t -f p n sin a;n i),

S u b s t i t u t i n g e x p r e s s i o n s (3.4 ) into (3.3 ) and c o m p a r in g the coefficie nts o f h a r m o n ic s s in -~nx w e

o b t a i n e q u a t i o n s for ân , 6n T h e n c o m b in i n g t h e m w ith (3.6) w e get t h e e q u a t io n s fo r ả n , 6fl in

t h e s t a n d a r d fo rm s For ã ị and bị w e have

u>iài = t 1 /?■! a i - — h ^ a ỵ ( á ị + b \ ) -f ei ( ^ ) ” ^ sin2 -h

— € I 1/11^1 — — OJ'I h^biịâị + èỊ) — Z\ ị co s 2 +

A v e r a g i n g t h e rig h t h a n d sides of (3.7 ) over th e t im e w e get

“ lài = ị I uiỵhi - $d 2 ịe] [j'j - U>ỶJ

14 NGUYEN VAN DAO and NGUYEN VAN DINH

th e vib ration o f th e b e a m If the vib ration s w ith high frequencies &re neglected , the eq u a tio n s

and th e so lu tio n (3 1 1 ) is alw ays stable w ith p o sitiv e values o f A.

We have m et w ith the form ula of type (3.11) and we have seeji w h a t the m in im u m o f A achieves w ith an averaged value o f A (see Fig 3) It is worth m entioning th a t w hen tw o low est frequencies of tw o b ea m s are equal:

then

Wi = V u < A = e 2 ( j ) ( Ị ) + 1

(e? - e 2) ( 7 ) + 7 = 0

absorber at its m ost effective G enerally, to quench th e vib ration of the b eam at a frequency u>y it

For a strong absorber (c is n ot sm all) the m o tio n equations are of form:

+ E I j £ + c(y - z ) = e f u

P1S1^ 5 - + E i h Ỵ - Ị + c(z - y) = e/a,

(3.13)

w ith th e b o u n d a r y c o n d itio n s (3.2), here

N on trivial so lu tio n of (3 2 1 ) corresponds to the values u>„ w hich satisfy the frequency equ ation s:

1 6 NGUYEN VAN DAO and NGUYEN VAN DINH

= 0 ( 3 2 2 )

j = 1, 2 , , oo T h u s w e have a set of values of frequencies u ị, ,cô ị > d ep en d in g on j

For qu asilm ear eq u a tio n s (3.13) we put

oo

5 3 P i 5 i Z n ( z ) [ f „ + w ^ r n Ị = e / 2

( 3 2 4 )

n — 1

In the first ap p ro x im a tio n we can replace the right hand sides of (3.28) by th e ừ averaged

From these it follows:

great In th is case, follow ing the formulae (3.23), (3.27) the vibration of the beam does n ot occur.

= hm L > 0

if

o L * L ( 1 )

• ấ l ), 2 A 2 ^ 2 3 - J UJI A1 ^

4) T h e v ib ra tio n a l regim e w ith tw o frequencies U>1 and u>2 is alw ays un stab le.

It is w orth m en tion in g th a t the effect of absorber w ill disappear if the coefficien t o f A in the

4 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 O F T H E P L A T E

1 8 NGUYEN VAN DAO and NGUYEN VAN DINH

T he problem presented in the previous paragraph m ay be considered for a more com p licated

sy stem , nam ely, for th e p late Self-excited vibration of a p late (D , /Ắ, u) under the action o f the

sm a ll force

edges we have the follow ing boundary conditions:

-We sh all find t h e s o lu t io n of (4 4) w ith b' nndary c o n d iti o n s (4 2 ) in th e form:

u = sin - n x sin j m y ( a nm COS wnmt + i „ m 8Ín w nm t),

&nm C08 UJnrnt 4" bnrn sin UJfimt = 0,

z = y ^ s i n - n x sin 7 m y ( a nfn COSUnmt + Pnm sinu;n m t),

w 2 nm =

I/ 2 =

^ n m

/7T n \ 2 / ? r m \ ( a ) + ( 6 )

the v ib r a t in g c o m p o n e n t s w i t h h igher frequencie s and s o lv i n g th e e q u a t i o n s (4.5 ) a n d (4 7 ) w i t h

r esp ect to a i l and fcu w e have:

20 NGUYEN VAN DAO and NGUYEN VAN DINH

d yn am ic absorber w ith m ean values of À In general, the effectiveness of th e absorber for q u en ch in g

o f th e p l a t e s c o i n c id e

case the m o tio n o f th e plates is governed by the equations:

w ith th e boundary con d ition s (4.2).

00

x i y) = y * ! Unm(Xị y ) T nm (t),

n,m = 1 oo

ind the o r th ogon ality relations

+ e - u?m - cZ”m = 0 ,

(4 2 2 ) + c - z?n - cUZl = 0.

T h e n o n t r iv i a l s o lu t io n of th e se alg ebraic eq u a tio n s corresp ond s to th e v a lu es o f U)nm) for w h i c h

S u b s t i t u t i n g these expressions into (4.14) we obtain

FVom the first equation of (4.31) it follows that the equilibrium A ll = 0 w ill be sta b le if

H i < 0 This is the case, for exam ple, when the friction force (A) is sufficiently great.

S A T o n d l, Q u e n c h in g of •elf-excited v ib ra tio n s, J Sound and V ibrations: E q u ilib riu m A sp ects, Vol 42, N o 2,

252, 1975, O n e an d tw o freq u en cy v ib ra tio n , Vol 42, No s, 261, 1975

4 A T o n d l, Q u en ch in g of te lf-e x c ite d v ib ratio n s: Effect of Dry Friction, J Sound v ib ra tio n s, V ol.45, 285, 1976

5 A T o n d l, A p p lic a tio n of tu n e d ab so rb ers to Belf-excited system s w ith several m asses, P ro ceed in g s o f th e

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IS N guyen V an D ao , D y n am ic a b so rb e r for drilling in stru m en t, Proceedings of IC N O -X I, B u d a p e s t 1987

14 N N B ogoliubov, Y u A M itro p o lsk i, A sy m p to tic m ethods in th e theory of n o n lin e a r v ib ra tio n s, M oscow 1963

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