Proceed ings o f Vibration Problems, 12, 2, p p. 2 23 -2 2 8 , 1971 Ins titu te o f Fu nda m ental T ech n ical R esearch, P olish A c ad em y o f Sciences ON TH E EX T ING U ISH ING O F PARA M ETRIC VIBRATIONS N GU YE N VAN D A O (HAN OI) 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 * (1.2) -Z__co2 =/«r, where // is a small parameter. ĩn the first approximation, the solution of Eq. (1.1) may be represented in the form X = rcos0, x = — ~ r y sin 6 , (1.3) 1 6 = — yt+y). The unknown quantities r and xp satisfy the averaging equations yr = by+c sin 2y))r, (1.4) / 3 1 \ yrxp = fi\—Ơ+ qr2+-~ c cos 2y) \ r. 4 2 The stationary vibrations are determined by the correlations r = ỳ = 0 . 224 Nguyen Vũrt Dao Hence we receive the following equation for the stationary amplitudes r0: (1.5) rỉ = - ^ (2 ơ ± y /c2- b 2y2). It can easily be shown that from two forms (1.5) only the form (16) rẩ = ^ - ( 2<r + } / ^ V ) is stable. Now, we raise the following question: In what manner is the amplitude of stationary vibrations r0 ch an ge d i f in Eq. (1.1 ) X is replaced by * 4 Psin vt, w here V an d (X) are 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 that su b stitu tio n of th e v ariab le, th e eq u ation for X redu ces to (1.7) jc +- pbx f (co2 + /accos yt)x+pq(x+Psinvt)* 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 — ơrcosỡ— sinỡ-frc COS yt COS 9 + q(r cos 0 f Psinw)31 sin d f [—Pv2 sin vt + ỊibvP cos vt f P(<o2 -ị-/ẮC COS yt) sin vt] sin 0, (1.8) y ró = r [oj2 cos2 d r sin2 0Ị + /4 Ị — sin 9 + rc COS yt COS 9 -\-q(r COS Q 4 -P sin w )3j COS 0 f [—Pv2 sin vt+pbvPCOS vt + P(ọ)2+/XC COS yt) sin vt ] 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) ( 3 3 1 — Ơ+ y qP2+ 4 2 ccos2y)* Now, the stationary amplitudes are: (1.10) r* = -2f>2 + - i- ( 2 < r ± |/ 7 ^ * v ). F ro m th e fo rm u la e (1 .5) a n d (1.10), w e m a y co n c lu de th a t w hen X is re p la 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 —b2y2)t then the amplitude /ọ. o f the stable vibration vanishes. On the extinguishing o f parametric vibrations 225 2. The Extinguishing of Connected Vibrations Now consider a nonlinear connected system of the form [2]: X VX\x = t*[-hx f (ay+by2)x+cxJ]t y-\ \\y =•- q sin aj/f/i/Or, y). The external force qsinrưí directly excites the vibration of coordinate y. With the well- know n co n d ition s [2] in the system u nd er co n sid er a tio n inten se vib ration 01 the co ord in ate X m ay occur. In m any ca ses, this assoc iated v ib ra tio n is u ndesirable and calls for liq u i d atio n . We assume that there exists a resonant relation <2.2 ) = 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?), Í1 _ / bq*2 aq* . 3ca2\ (2.4) a)ọp= /zỊơ b — sin2<?-—^—1, ^ 0 = A ỉ+ M a* <p, A, t). Hence we obtain the stationary values of the amplitude of vibration of coordinate x: (2-5) a 2 — (2 ơ—bq*2± ]/a 2q*2—h2ơ)2). It can easily be shown that only the form 2 a2 = - y - (2 ơ—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 y0co$vt. In fact, now the system (2.1) takes form: X+Ằịx = Ii[-hx+(ay+by2)x+cx*h = q sin U)t+y0 COS vt+fi/lx, y). 226 Nguyen van Dao 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/— sinvt—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 cos 0 - [— (ứ>> + + CJC3!} sin a ỳ ~ ^{aơcos 0 —[—/Ú + (ay + fty2)*! cx3]}cos #>, <I> = r+ 9?. Averaging the right-Jiand parts in time, we obtain: (2.9) LLOL co<x= Y (hcủ+aq*cos2<p)> I byỉ2 bq*2 aq* . 3ca2\ F « A* (<x- - 2 + 2 sin 2<p~ 4 ) — 2 2 ' 2 “‘*“ r 4 /• The stationary values of amplitude a and phase 9? are determined from the equations (X = ip = 0. Hence we find that (2.10) a 2 = ~ ( —b y ĩ2 + 2ơ—bq*2± —A2co2) . Comparing (2.5) and (2.10), we see that in the case be > 0, by the introduction of the supplementary force y 0cosvt the amplitude of 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 (3.1) x\—a>2 = fxe. The solution of system (2.1) we shall find in the form: ^ X = /ÍC O S (cof+ y), X = — /fo>sin(aư+Y')» y =* 0 *sina>f+/í COS0, ỷ = g*cocoseoí— Ẩ2 ^sin0. 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 = — ^-ịhco-ị-— - sin 2 y>|, . __ I bq*2 bq*2 3c ớ2\ = y ^ ^ h —4— cos 2^— y3 2 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: ^ X + Ấ\x = n[-hx+(ay+by2)xJrcx3], ý + ^ \ y = q sinco/ f >-0 -I- f ự ( x , y ) . We obtain the solution of th is system in th e form: x = PcosV, x = — (ìcưúnV, v = 0)t-ị-y)t • y = q* sin cot-f.yg-Mcos0, ỳ = CO coscot — Ằ2 Asindi y% = J'oMI- Substituting these values X, x,yfy into (3.5), we receive the following equations for the u n k n ow n functions /?,* y>, Ay 6 : cưặ — ụ. {fiecos V— [—hx + (ay+ by2)x + cx3]}sin Vy (ỉcúỳ = fx{(iecosV— [—hx+(ay+by2)xi-cx*]}cosV9 (3.8) Averaging the right-hand parts in time, we have: — — - 2~ (Aco+ ~ 4 sin 2^|» coý) = J iịe - y Ị ( a + b y S ) ~ - ^ Y - + bq* C0S Hence we find the stationary values of amplitude /3 of vibration of the coordinate X Comparing (3.4) and (3.9) we see that if the value y0 is so chosen that cy*(a + byi) > 0 then the amplitude of vibration of coordinate X decreases. References 1. N. N. B o o o u u b o v and Yu. A. M it r p o ls k ii, Asymptotic methods in the theory of nonlinear vibrations, Moscow 1963. 2. N guyen v a n D ao, Nonlinear connected oscillations of rigid bodies, Proc. Vibr. Probl., 1 0, 3, 1969. . 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. />sinw the amplitude of stationary vibrations decreases. If the constant p 2 strives for -j-(2 < 7+ j/c 2 —b2y2)t then the amplitude /ọ. o f the stable vibration vanishes. On the extinguishing. vibration vanishes. On the extinguishing o f parametric vibrations 225 2. The Extinguishing of Connected Vibrations Now consider a nonlinear connected system of the form [2]: X VXx = t*[-hx f (ay+by2)x+cxJ]t