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

'roceedings of the NationỉU Centre for SdentiSc Research of Vietnam, Vol.. T he Lanchester damper effect for quenching both free and forced self-excited vib rations of the mechanical sy

'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 S T E M S

Ng u y e n Va n Da o

Institute of Mechanics NCSR of Vietnam

S u m m a r y T he Lanchester damper effect for quenching both free and forced self-excited vib rations of the mechanical sy stem s with one, two and m any degrees of freedom is in vesti­ gated by m eans of the a sy m pto tic method of averaging Many q uantitative estim ations for the station a r y amplitudes of vibrations and theừ stability are given.

In r e c e n t y e a r s a lo t o f p a p e r s c o n c e r n e d w i t h th e d y n a m i c a b s o r b e r effec t for q u e n c h ­ ing s e l f - e x c i i e d v i b r a t i o n s o f m e c h a n i c a l s y s t e m s w i t h fin ite d e g r e e s o f f r e e d o m a n d a ls o

4 NGUYEN VAN DAO

Usually, the “negative” friction force is of the form

h e re a n d s u b s e q u e n t ly e is a s m a ll p o s it iv e p a r a m e t e r s c h a r a c t e r iz in g t h e s m a lln e s s o f t h e

t e r m s T h e c o e f fic ie n t hi is a c o m b in a t io n o f lin e a r f r ic t io n a c t in g o n m a s s M a n d th e lin e a r p a r t o f th e e x c it in g “ n e g a t iv e ” f r ic t io n fo rc e

LAN CHESTER DAMPER E FFE C T FOR QUENCHING

Fig 2 The dcpcndcncc of amplitude of vibrating mass hi from the damping coefficient X in

the case of a strong Lanchcstcr damper

These equations belong to the standard form to which the averaging technique is applied

their averaged values over one cycle of vibration:

8 NGUYEN VAN DAO

£1 = a cos 0, (i = —aO 1 sin 0, Ớ = H it -t- t/>,

£0 = 6 cos ry, £0 = —6 02 sin »7, rj = fio t -r

M l

Mo Qnbặ) = - *77“ (^*1 •+■ f**) cos ĨỊ.

Ml = hỵ — (ơi — l )2/ i i2 < 0)

ỵ 2 = hi — (ơn — 1 )2 /ijl2 < 0

If t h e s e c o n d i t i o n s are n o t s a t i s f i e d s i m u l t a n e o u s l y t h e n t h e e q u i l i b r i u m o f t h e m a s s e s w ill

b e u n s t a b l e a n d v i b r a t i o n s m a y occur.

The vibration of the first mode with frequency ill and amplitude Oo is determined

LANCHESTER* DAMPER E FF E C T P O R QUENCHING 1 1

u = V,

m i) -f Xv = A Ì2,

w h e r e f u n c t i o n s Fi a n d F3 a r e o f f o r m ( 2 5 ) T h e d if fe r e n c e b e tw e e n th e e q u a t io n ( 2 1 ) a n d ( 4 2 ) is t h a t i n t h e la s t e q u a t io n o f th e s y s t e m ( 4 2 ) t h e r e is n o s m a ll p a r a m e t e r c

LANCHESTER DAMPER E FF E C T FOR QUENCHING 13

Ị - / i i + (ơi - l ) 7h í2 + - h s t í ị a * + - h & ị o ĩ + Ơ1 m2Q2 ^ ^2 I »

Ị —Al + (ơ2 — 1)2^12 + Ị^3^2a2 + “ /l3^ia? + ^ m2n2 + À* } *

16 NGUYEN VAN DAO

LANCHESTER DAMPER E F F E C T POE QUENCHING 1 7

18 NGUYEN v a n Da o

A FORCED SELF-EXCITED VIBRATING SYSTEM

L A N C H E S T E R D A M P E R E F F E C T FOR QUE NC HING

22 NGUYEN VAN DAO

c a se t h e f u n c t i o n u s a t i s f i e s t h e d iffe re n tia l e q u a t i o n

m ủ + Aủ = Ai = — Aa*) sin 'P,

a n d t h e r e f o r e ,

u = T 7 T 3 ? cos p A2 4- Tri~i \ 2 V'Z ? 2 sin A2 -f m -7 2 V9 3 )

E q u a t i o n s for a and \f> n o w are o f th e fo r m

t M