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On the interaction between forced and parametric oscillations in a system with two degrees of freedom

In the present paper, the so called "harmonic and parametric case" [1, pp. 335-341] will be examined. Critical singnlar points [2] will be used to classify different forms of the resonance curve. The case in which the indirectly- excited parametric oscillation is not very intense will be analyzed in detail. As it will be shown, the resonance curve has either a loop or an oval. | Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 1 (12 - 21) Tx3 + cy2 x) ii + y + e(hy + /3y 3 + bx 2 y) For simplicity, h 0 , !3 = Qsin vt, = epcos(vt + 5). (1.1) and b are assumed to be positive. The oscillations are found in the forms: x = q sin vt +a, cos(At + ¢,); y=acos(vt+¢), :i; = vq cos vt- Aa 1 sin(At + ¢ 1 ), !i=~vasin(vt+¢), q q= AZ-vZ' (1.2) and the averaged differential equations in the first approximation are: (1.3) 12 Evidently, the intensity of the indirect parametric excitation is characterized by the coefficient b 2 4q. Stationary oscillations are determined from the equations: a1 = 0, fo = 0, (1.4) go = O, among them, the last two ones can be replaced by their equivalents: . f=fosm!/!-gocos,P=vhasin,Pg = f 0 cos,P [(3-f3a2 +-q b 2 +/),. ) --q b 2] acos,P+pcos6=0, 4 2 4 + g0 sin,P = [(~f3a 2 + ~q2 + /),.) + ~_q']asin,P + vhacos,P- psin6 = (1.5) 0. The resonance C (frequency-amplitude characteristic is defined as the ensemble of representing points ( v 2 , a) whose ordinate a is the amplitude of stationary oscillations corresponding to the frequency v. In general, C consists of two parts: the ordinary part G1 and the critical one G2 • 2. System without damping in the mode y We shall first examine the system without damping in the mode (y). In this case h = 0 and the system of equations (1.5) becomes sinlple: (2.1) In the ordinary region where: (2.2) from (2.1), sin ,P and cos ,P can be calculated without difficulty and the ordinary part G1 is easily obtained: W!(v2' a2) = 3 [ ( -f3a2 4 p2 cos 2 0 b 2 b + -q2 + t).) _ -q2] 2 + 2 4 a2 3 [( -f3a2 4 p2 sin 5 b b 2 4 + -q2 + /),.) + -q2] 2 a2 - 1 = 0. (2.3) The critical region is characterized by the equality: (2.4) It presents two curve in the plane (v 2 , a2 }: 3 2 2 ) b2 4f3a = (v - 1 - 2q b2 - 4q ' (2.5a) 3 2 = (2 b2 -f3a v - 1) - -q 4 2 b2 . + -q (2.5b) Oasin,P=psin6. (2.6) 4 Along (2.5a), the system (2.1) .