On the extinguishing of parametric vibrations

The essence of this method consists in the following: maintaining the given working regime of external force, we change by introduction of the supplementary load, the equivalent natural

I n s titu te o f F u n d a m e n ta l T e c h n ic a l R e se a rch , P o lish A c a d e m y o f S c ie n c e s

ON T H E EX T IN G U ISH IN G O F PARAM ETRIC VIBRATIO NS

N G U Y E N VAN D A O (H AN O I)

In the present paper a method is proposed for extinguishing of parametric vibrations The essence of this method consists in the following: maintaining the given working regime

of external force, we change by introduction of the supplementary load, the equivalent natural frequency of the vibrating system and therefore we lead it from the resonant state

1 The Case of Simple Parametric Vibration

Let us consider the vibrating system described by the equation of form:

(1.1) x+fxbx+(co2+fic cos yt)x+fj,qx3 = 0, q > 0.

Suppose that we have the resonant relation

*

where // is a small parameter.

ĩn the first approximation, the solution of Eq (1.1) may be represented in the form

(1.3)

1

6 = — yt+y).

The unknown quantities r and xpsatisfy the averaging equations

yr = by+c sin 2y))r,

(1.4)

yrxp = f i \ — Ơ+ qr2+ - ~ c cos 2y) \ r.

The stationary vibrations are determined by the correlations

r = ỳ = 0.

Hence we receive the following equation for the stationary amplitudes r0:

It can easily be shown that from two forms (1.5) only the form

is stable.

Now, we raise the following question: In what manner is the amplitude of stationary

v ib r a tio n s r 0 c h a n g e d i f in Eq ( 1 1 ) X is r e p la c e d b y * 4 P s i n vt , w h e re V a n d (X) a re lin e a r

independent—i.e., between them there is no relation of form— nv + mo) = 0; Tt, m are the integers.

W ith th a t s u b s t it u t io n o f th e v a r ia b le , th e e q u a tio n fo r X r e d u ce s to

(1.7) jc +- pbx f (co2 + /accos y t)x + p q (x + P s in v t)*

4 [ — Pv2 sinv/ + ịibvP COS vt + P(cu2 + fxc cosy/) sin vt] = 0.

As in the past, we shall seek solution of this equation in the form (1.3) It is easily verifiable that the variables r and 6 satisfy the equations

— f = / i I — ơ r c o s ỡ — s i n ỡ - f r c COS y t COS 9

+ q(r cos 0 f Psinw)31 sin d f [—Pv 2 sin vt + ỊibvP cos vt f P(<o 2 -ị-/ẮCCOS y t ) sin vt ] sin 0,

(1 8 ) y ró = r [ oj 2 c o s 2 d r sin 2 0 Ị + / 4 Ị — sin 9 + r c COS y t COS 9

-\ -q(r COS Q 4 - P s i n w ) 3 j COS 0 f [— P v 2 s in v t + p b v P C O S v t

+ P(ọ )2 + / X C COS y t) sin v t] COSỚ.

Averaging in time the right-hand parts of this system, we obtain in the first approxi­ mation the following equations

yr = y / i r ( —iy + c sin 2y>), (1.9)

— Ơ+ y q P 2+ 4 2 cco s2 y )*

Now, the stationary amplitudes are:

(1.10) r * = -2f>2+ - i - ( 2 < r ± | / 7 ^ * v )

F r o m t h e f o r m u l a e (1 5 ) a n d (1 1 0 ), w e m a y c o n c lu d e t h a t w h e n X is r e p l a c e d b y X f

+ />sinw the amplitude of stationary vibrations decreases If the constant p 2 strives for

- j - ( 2 < 7 + j /c 2 —b 2y 2) t then the amplitude /ọ o f the stable vibration vanishes.

On the extinguishing o f param etric vibrations 225

2 The Extinguishing of Connected Vibrations

Now consider a nonlinear connected system o f the form [2]:

X VX\x = t* [-h x f (a y + b y2)x + c x J]t

y-\ \ \ y =•- q sin aj/f/i/O r, y).

The external force qsinrưí directly excites the vibration of coordinate y With the well-

k n o w n c o n d itio n s [2 ] in th e sy stem u n d er c o n s id e r a tio n in te n s e v ib r a tio n 01 the c o o r d in a te

X m a y o cc u r In m a n y c a s e s , th is a s s o c ia te d v ib r a t io n is u n d e sir a b le a n d c a lls fo r liq u i­

d a tio n

We assume that there exists a resonant relation

Then the solution of Eqs (2.1) in the first approximation we find in the form:

Ị 2" * + > X = — ỵOLCO sill I y / 4 ,

y = q* sin (X)t T-/4C0S ớ, ỳ = q * 0 ) COS ojt— X2 A sin ớ , q* = ợ/( ^2 ~ 0j2)-

The unknown quantities a, (f, A , Ớ satisfy the following averaging equations:

X — Of cos

(2.3)

COOL — — Y Iiz(hcư + aq* cos2ọ?),

Hence we obtain the stationary values of the amplitude o f vibration of coordinate x:

It can easily be shown that only the form

2

a2 = - y - (2ơ—ồ<7 * 2 + sign c y ' ^ q * 2 — h2to2),

is stable In order to extinguish the corresponding stable vibration, wc can affect in addition

coordinate y by a force y 0co$vt.

In fact, now the system (2.1) takes form:

X + Ằ ị x = I i [ - h x + ( a y + b y 2) x + c x * h

= q sin U) t+y 0 COS v t + f i / l x , y).

We assume that CO and V are incommensurable Then, with the resonant condition (2.2), the solution of the system (2.6) can be written in the form:

o t c o s Ị y / I -93Ị , i = — ỳacosinỊ-^-z + yj,

y = q*sin 0)t r yi COS vt \ A COS 6, (2.7)

ỳ = g*c/j cos co/— sin v t — X2 A sinớ, q* = ?/(*2- w 2), yS = y0K*i-V2).

Substituting (2.7) into (2.6), we receive the following equations for the new variables

a , Ay d:

0)

(2.8)

2 á = ỊẰ{ a ơ c o s 0 - [— (ứ>> + + CJC3!} sin

Averaging the right-Jiand parts in time, we obtain:

(2.9)

LLOL

co<x= Y (hcủ+aq*cos2<p)>

I b y ỉ 2 b q *2 aq* 3ca2\

F « A* (<x- — 2 - 22 + ' 22 “‘*“ r sin 2<p~ 44 /• )

The stationary values of amplitude a and phase 9? are determined from the equations

(X = ip = 0.

Hence we find that

Comparing (2.5) and (2.10), we see that in the case be > 0, by the introduction o f the

supplementary force y 0 c o s v t the amplitude o f vibration of coordinate X decreases.

3 The Extinguishing by a Constant Force

We note that sometimes, for extinguishing associated vibrations, we can in addition

affect coordinate y by a constant force.

We consider again the system (2.1) in the principal resonant case

The solution o f system (2.1) we shall find in the form:

y =* 0*sina>f+/í COS0, ỷ = g*cocoseoí— Ẩ2^ sin 0

On the extinguishing o f parametric vibrations 227

It can easily be verified that in the first approximation, p and y satisfy the following averaging equations:

(3.3)

ơ)ậ = — ^ - ị h c o - ị - — - sin 2y>|,

= y ^ - ^ -h —4 — cos 2 ^ — y32 J

The stationary values of /9 are given by the formula

(3.4)

ues of /9 are given by the formula:

If a constant force y 0 affects coordinate y, then we have the equations of motion:

ý + ^ \ y = qs i n c o / f >-0-I- f ự ( x , y )

We obtain the solution o f t h i s system in t h e form:

x = P c o s V, x = — (ìcưúnV, v = 0)t-ị-y)t •

y = q* sin c o t-f.yg-Mcos0, ỳ = CO coscot — Ằ2 Asindi y% =

J'oMI-Substituting these values X, x , y f y into (3.5), we receive the following equations for

the u n k n o w n functions /?,* y>, Ay 6 :

cưặ — ụ {fiecos V— [—hx + ( ay + b y 2)x + cx3]}sin Vy (ỉcúỳ = fx{(iecosV— [—h x + ( a y + b y 2) x i - c x * ] } c o s V 9

(3.8)

Averaging the right-hand parts in time, we have:

— — - 2~ (Aco+ ~ 4 sin 2^|»

coý) = J i ị e - y Ị ( a + b y S ) ~ - ^ Y - + b q * C0S

Hence we find the stationary values of amplitude /3 o f vibration o f the coordinate X

Comparing (3.4) and (3.9) we see that if the value y 0 is so chosen that cy*(a + b y i ) > 0

then the amplitude o f vibration of coordinate Xdecreases.

References

1 N N B o o o u u b o v a n d Y u A M it r p o l s k ii , Asymptotic methods in the theory o f nonlinear vibrations,

M oscow 1963.

2 N g u y e n v a n D a o , Nonlinear connected oscillations o f rigid bodies, Proc V ibr P ro b l., 1 0 , 3, 1969.