Interaction between parametric and forced oscillations in multidimensional systems

Journal of Technical Physics. 16, 2, 213— 225, 1975. Polish Academy o f Sciences. Institute o f Fundamental Technological Research, warszawa INTERACTION BETWEEN PARAMETRIC AND FORCED OSCILLATIONS IN MULTIDIMENSIONAL SYSTEMS NGUYEN VAN DAO (HANOI) This paper is devoted to the investigation of the interaction between parametric and forced non-linear oscillations in multidimensional system described by two non-linear differential equations of the second order. The two modes (.X, y) of the system considered are excited by sinusoidal forces. The two modes are coupled non-linearly by means of the product of their coordinates. Under certain conditions the oscillation of the first mode (x) excites parametrically the oscillation of the second one and so the two oscillations of the second mode (parametric and forced oscillations) may coexist and there exists some kind of interaction between them. We shall now consider the stationary oscillations of the modes and their stability. 1. Equations of Motion. Stationary Oscillations Let us consider the oscillations of the system with two degrees of freedom described by a set of two differentia] equations of the type x+ / 2 ^ + £/2(/i0ir + ccc3 + cv2 x) = Osinyf, (1 1) ' ỷ + (ư2y + eco2(hỷ + fỉy3 + bx2y) = eơ)2p cos(ví-f <5), where h0 > 0, h > 0, a, Cy b, Q,p > 0, /?, Ổ, are constants and £ is a small parameter. We assume the following relations between the frequencies cư,r, and y: (1.2) cư2 = Ơ2V2 + eơ)2A , y = ev, nX # m y , where Ơ, e are rational numbers, A is detuming of frequencies and m, n are integers. First, we transform the Eqs. (1.1) by means of the formulae X = <7siny/-f ax COSỠ!, X = yqcosyt— Aai sinfli, 0 -3 ) y — ỠCOSỚ, ỳ = -ơrasine, q = J ĩ ị p , where al , ỚJ, Ơ, 6 are the new variables which will be determined later. 214 Nguyen Van Dao Substituting (1.3) into (1.1) and transforming it, we obtain the following equations in the standard form: da ị dĩ dxp l = EẰ(h0x + ccx3 + cy2x)sinOi , (1.4) ai —J— = eXỌìqX + QLX3 -+■ cy^x )CO SỚỊ, at = eơv<Ị>(x,y,ỷ, 0 sinớ + 0 0 2), at ady ~dT = eơv&(x, y, ỳ, t)cosd + 0(e2), where where (1.5) &(x9 y, ỳ, 0 = Ay + hỳ + bx2y + Py3 — pcos(vt+ Ỗ), y)ị = ỚJ — At, rp = d — ơvt Averaging the right-hand parts of (1.4) over the time, we receive the equations of the first approximation for the unknowns a, al9y>: 1 ) for ơ # e, ơ # 1 . £//i0 - «1. (1.6) n 2 eơ~vz a = -y-ha+ b , 3 ỡ . 2~ + 4 ^ 8 a1 = — — h^ax (1.7) £V à = — ~Y [vha +/?sin(y>— ($)]+ axp = 3) for e = Ơ # 1 £T zla + ợ2ứ + — /fa3 -p cos(y — Ô) ỏi = ” 2 ^°ứl’ (1.8) à = — eơr ơvha + - 7- a2a sin 2 w 4 £ơ = -y r + ; Interaction between parametric and forced oscillations 215 4) for e = Ơ = 1 ■ EẢ u a\ — 2~ 0ứl» (1.9) a — — vha+ -^-q2asin2y)Jrpsin(y)—ỗ)\+ = b 3 b Aa+ ~Y q2a+ ậa3— -Ỵ q2acos2y)—pcos(y — Ô) + where the nonwritten terms disappear when flj = 0. By analogy with N e s s [1], the first case is called the non-resonant case, the second case — the harmonic resonant case, the third (1.8) — parametric resonant case and the last (1.9) — harmonic and parametric resonant case. Obviously, the most interesting is the last resonant case. We shall inves tigate it in more detail. The stationary solution of the system (1.9) is the one which is determined from equa tions = ả = ỹ) = 0 or flj = 0, (1.10) r / j f l - f <72 f l s i n 2 y j + / ? s i n ( Y > — Ô) = 0 , Aa-\- — q2a+ pa3— Ỵ q2acos2ĩp—pcos(y-ô) = 0. Eliminating \p in the two last equations of (1.10), we obtain the following relation for the amplitude a = a0 = const of the stationary oscillation of the coordinate y: (1.11) A/ = 0, where M = (w2 + u2 — V 2) 2 — /? 2 [ (w — acos2 <5)2 + (w*f Z>sin2<5)2], (1.12) u = vha0, v = ~ q 2a0, IV = - a0 ỊA + q2 + ậaị The relation (1.11) is expressed in the parameters of the initial system as al (1.13) -F I - * 2 b 2 3 l\ 2,2 b1 e + 2 q + 4 H + ^ 2 - T 6 ^ 1 —k2 b 3 o 2 b \ b 2 . . - —— h q2 + + -j-ợ2cos2<5 + I co/2 H ~ ọ sin2 Ỏ e I 4 4 / \ 4 = 0 , * = CO Following this formula the resonant curves are presented in Figs. 1-6. 2. Stability of Stationary Oscillations The equations in variations for the system (1.9) are dồQỵ Ằ [216] Interaction between parametric and forced oscillations 217 F i g . 3. dò (2.1) = —(u' +v' sin2y)0)ôa—[2vcos2y)0+pcos(y0-ỏ)]ỏy, it a0 j — = —(w' + vfcos2y)0)ôa+[2vsin2rp0+psin(yỈQ — ỏ)]ỏy>, where T = 2t/ev and primes denote the derivatives with respect to ŨQ. The characteristic equation of the system (2.1) takes the form: e+ -ị-h0)(a0Q2 + 2vha0ọ + R) = 0, [218] interaction between parametric and forced oscillations 219 where (2 .2 ) R = — (u +v' sin2 ^o)[2^sin2 ^o+/7sin(^o~ Ổ)] - ( w ' + V' COS 2\po) [ 2 v COS2 ^ 0 + p CO S(yj0 - Ô) ] . By using the relations (1.10), (1.12), the expression R may be rewritten in the form (2.3) R = 2 ~— 2 - -JT— • 2(w2 + u2 — V2) oa0 As h0 > 0, h > 0, a0 > 0, the stability condition of the stationary solution is (2.4) £ Ì ^ > 0 , ƠŨQ where E = w2 + u2— V2. The resonant curves (M = 0) divide the plane (a,ky into the regions, in each of which the expression M has a definite sign ( + or —). If moving up along the straight line parallel to the axis a0 we pass from the region M < 0 to the region M > 0 then at the point of intersection between the straight line and the resonant curve the deriv ative õMỊỗa0 is positive. So, this point corresponds to the stable state of oscillation if £ = vv2*f u2—v2 > 0 and to the unstable one if £ < 0. On the contrary if we pass from the region M > 0 to the region M < 0, then the point of intersection corresponds to the stable state of oscillation if E < 0 and to the unstable one if E > 0. Id the limit case when h = Ổ = 0 the equation for the stationary amplitude a0 (1.13) may be written in the form: / I - * 2 3bq2 3 . 2\ 2 l ] - k 2 bq2 3 . , \ 2 p2 (2.5) + ^ r + T fit) [ ( V 1 + 4~ + 4 ft*) - j r From here we obtain a double root (2.6) /:2 = 1 + A sbq2 + eậaổ, and two other roots (2.7) = 1 + 4 - w + ị epai + , 4 4 fl0 (2.8) p = l + 4 ^ + i - £^ _ J£ 4 4 ứ 0 Following these formulae the resonant curves are presented in Figs. 1-4, where the branch expressed by (2 .6) is shown by number 1 and the branches expressed by (2 .7) and (2.8) are shown by numbers 3 and 4, respectively. The parameters are chosen as (e/4)bq2 = 0.13, (3/4)e/3 = 0.1, ep = 0.1. In the shaded region (the region of parametric resonance) the expression E = w2 + u2— V1 is negative. On the heavy lines, EdM/da0 is positive, so that they correspond to the stable state of oscillations. The dotted lines corres pond to the unstable state of oscillations. The signs + and — in the figures are those of the expression M. On the branch 1 oT the resonant curves when h — 0, the stability of stationary oscillation is doubtful because on this branch E = 0. = 0. 220 Nguyen Van Dao If we take into account a small value of friction h (h > 0) the resonant curves have the form shown in Figs. 5, 6, where v2h2 = 0.005. These curves are obtained by solving the Eq. (1.13) on a digital computer. The branch 1 in the case of h = 0 (Figs. 1-4) changes into two either stable (Fig. 6) or unstable branches (Fig. 5). With larger values of friction (h) the resonant curves take the forms presented in Fig. 7 (v2h2 = 0.01) and Fig. 8 (v2h2 = 0.02). From the results obtained the following conclusions may be drawn: 1) Inside the region of the parametric resonance, the parametric excitation caused by the first mode (x) strongly influences the stability of the stationary forced oscillations of the second mode (j>). Some branches of the resonant curves of the second mode oo which are unstable for X = 0 now become stable and * vice versa. Outside the region of the parametric resonance, the mechanism of parametric excitation does not influence the stability of the forced oscillations. 2) The jump phenomenon of the amplitudes — in the case of a hardening character istic (Fie. 1, 5), when the frequency k decreases, and in the case of a softening character istic (Fies. 4,6), when the frequency increases — is observed quite clearly. The change of the stationary amplitudes follows the M-form. We assume now that the amplitude of the external force acting on the second mode (v) is not small, so that the equations of motion take the form where V ^ cư. On the assumption that there are resonant relations (1.2), we first transform the Eqs. (3.1) by means of the formulae: 3. System with a Large Amplitude of the External Force (3.1) x + ?.2X+ e?.2(h0x+ ccx3 + cy2x) = Qsinyt, ỷ+ơ)2y+eoj2(hỳ+fìy2 + bx2y) = Z)cos(r/+ (5), (3.2) X = qsiuyt + a1 COSỠj , X = yq cos y t — Áũ 1 sin 6!, y = jcos0 + c/cos(vr + <5), (3.3) (X)2—V2 r2 (ơ2 — 1) The equations for new variables a1,61, a,6 are (3 .4 ) aixpl = £Ẳ(/20i:+ ttx*+ cy2x)cosO I, à =* CƠT0! (*,}>, ỷ) sin 0 + ơ(£2), axp = £ ơ r0 i( x , j,ỷ )c o s ớ -f ỡ (£ 2), (x, y, ỳ) = Ay + hỷ+Py3+ bx1y ì y>ị = ỚỊ — Ằí, y = 6 — ơvt. [2211 222 Nguyen Van Dao By averaging the right-hand sides of (3.4) over the time, we obtain, 1) for Ơ = e = 3 (superharmonic and parametric resonance) SẰ' A0fli, <3.5) à = 3r»' 3 8 1 — I- r/ĩứ + i/ 3 sin (y) — 3 Ổ) — bq2a sin 2^ 2 o o ay = 3ev { — fl-f p 3 3 <f3 J3 + 0— cos (y* — 3 Ổ) o 4 o * 2 * 2 4- — q a— — qzaoosl\p 2) for Ơ = 3, e = 2 (superharmonic and combination resonance) • _ EĨ'2 u ai — 2— °Ơ1 ’ — \ - v h a + 4 -/ W 3sin(Y> —3 Ổ )- <7 2*/sin(y>+<5) 1 o o aỳ) = 3er +/5 3 3 d3 q Q3jt -r-d2a+ ~~ cos(yj-3<5) o 4 o -Ỵ ^ 2a - - Ỵ ^ dcos^ + *)Ị + ỡ(«i); 3) for Ơ = 3, e = 1 (superharmonic and combination resonance) e?.2 a, = - h0Qi * <3.7) à — 3er 3 8 h — vha + — d3 sin (v> - 3 Ổ) — q2d sin (V - Ổ) X o o dip = 3ei> |- y - — -3- q2dcos{xp-<5)> + 0{a^)\ 4) for ơ = -ị-, e = -i- (subharmonic-parametric and combination resonance) a, = - h0ait E V ị V 3 z? I <3.8)  = -y-j — -^-/ỉj+-ị-^íÌ72sÌD(3v>—ổ)“ -~ ợ 2[ữsin2^ + í/sin(^— + . _ ev ị Aa 3 2 +i> 3 3 3 -5- Ữ3 + -7- í/2ữ -f -5- dtf 2cos(3y> — (5) 8 4 8 + -7-qa q2[acos2y) + dcos(y— (5)]? -f 0{al)\ [...]... analysis of the systems (3.8) and (3.9) is rather complicated The system (3.5) has been investigated exhaustively in the previous paragraph As the study of the remain­ ing systems is not difficult we shall not consider them here References 1 D J N e s s , Resonance classification in a cubic system , A SM E A pplied M echanics C onference 1971 A sym ptotic methods in theory o f non-linear oscillations. . .Interaction between parametric and fo rced oscillations 1 223 2 5) for Ơ = - y , e = — (subharmonic and combination resonance) • — a\ _ _ £*2 hoai> 2 u - -y- [ —-^r ha + ^-fida2 sin(3y>—-ổ) + - -^-q2d cos(y>+ < )1 + 5 6) 0 0 !);... ro w n an drugiego rzẹdụ (1.1) O bie Dormalne w sp ỏ h zẹd n e (x , y ) sạ w zbudzane przez sity harm oniczne D w ie w spom niane w spóỉrzẹdne sạ zwiạzane z sobạ przez ich iloczyn Interaction between parametric and fo rced oscillations 225 Przy okreslonych w arunkach drgania picrwszej wspóỉrzẹdnẹj (*) param etrycznie w zbudzaja drgania drugiej wspofrz^dnej (>>), a wiẹc oba d rgania: param etrvczne i... ha + — pda2 sin(3v> —Ô ) 6 o £ ' I Aa , o T p r + i ị8 a 3+ -4ị for ơ = e # ^ > 3 • = ài _ (3.12) 1 2 d3 ơ 3 + - J í / 2ứ + ~ - c o s ( y > —3Ổ) o 7) for Ơ = -J, e # (3.11) 3 4 3 = 3ev I + 8 cos(3w —Ỏ + ) ^ a j + O O J; (parametric resonance) e/'2 J u h0a , à = eơr = £Ơ T — /lứ— g"#2^ sin2y^ + 0(ứi)> /3 3 3 4-P Ị — a3+ — d2a \I+ * 2 — è q2a cos2yj —2 9) for 2e —ơ—1 = 0, e # 2, -J- (combination resonance)... Streszczenie ODDZIALYWAN1E MIẸDZY DRGANIAMI PARAMETRYCZNYMI I WYMUSZONYMI w WIELOWYMIAROWYCH ƯKLADACH Niniejszy arty k u ỉ zostal poswiẹcony b ad an iu oddziatyw an m iẹdzy drganiam i param etrycznyrai i wym uszonym i w w ielow ym iarow ych ukJadach, k tó re opisyw ane sạ przez ukỉad dw och n ie lin io w y c h rốzniczkowych ro w n an drugiego rzẹdụ (1.1) O bie Dormalne w sp ỏ h zẹd n e (x , y ) sạ... 4-P Ị — a3+ — d2a \I+ * 2 — è q2a cos2yj —2 9) for 2e —ơ—1 = 0, e # 2, -J- (combination resonance) SẢ2 h o^i» 224 N guyen Van Dao (3.13) à = eơv - ~ h a - — q2d sin(v> + ô) Aa axp = eơv 10) d2a ữ3+ -Ỵ, 1 + -T- q2a - qzd C S(yi + O ô) (combination resonance) £^2 ^ M l ,i, ứ = £Ơ - ^ h a - — q2d $ m ( y - ồ ) + 0 ( a l) 9 T Aa />/ 3 2 +M 8 = eơv 11 Ị -jp for ơ —1 = ± 2 e , e • aì _ (3.14) + /ỉ - 3 ,... MOKiỊV HHMH HMeeTCH HeKOTOpbm BHU B33HM0HeỉÌCTBHH BblJIH HCCJIC^OBaHbl CTailHOHapHbie KOJieOaHHH KOOpjIHHaT H HX VCTOifMHBOCTb d e p a r t m e n t o f m a t h e m a t ic s a n d p h y s ic s POLY TECHN]cAL INSTITUTE, HANOI Received October 14, 1974 . (HANOI) This paper is devoted to the investigation of the interaction between parametric and forced non-linear oscillations in multidimensional system described by two non-linear differential equations. 1975. Polish Academy o f Sciences. Institute o f Fundamental Technological Research, warszawa INTERACTION BETWEEN PARAMETRIC AND FORCED OSCILLATIONS IN MULTIDIMENSIONAL SYSTEMS NGUYEN VAN DAO (HANOI) This. oscillation of the second one and so the two oscillations of the second mode (parametric and forced oscillations) may coexist and there exists some kind of interaction between them. We shall now