TAILIEUCHUNG - Interactionbetween the forced and parametric excitations with different degrees of smallness
The nonlinear system under consideration in this paper has a specification which can be stated as an interaction between the first order of smallness nonresonance parametric excitation and the second order of smallness resonance forced excitation. In the first approximation these excitations have no effect. However, they do interact one with another in the second approximation. | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 2 (11- 17) INTERACTIONBETWEEN THE FORCED AND PARAMETRIC EXCITATIONS WITH DIFFERENT DEGREES OF SMALLNESS NGUYEN VAN DAO Vietnam National University, Hanoi ABSTRACT. The nonlinear system under consideration in this paper has a specification which can be stated as an interaction between the first order of smallness nonresonance parametric excitation and the second order of resonance forced excitation. In the first approximation these excitations have no effect. However, they do interact one with another in the second approximation. The equations for the amplitude and phase of oscillation are found by means of the asymptotic method. The stationary oscillations and their stability are of special interest. 1. The equation of motion and asymptotic solutions Let us consider : nonlinear system governed by the differential equation x + w2x = e-pxcoswt + e- 2 [~x- 2hi:- (3x 3 + rcos(wt -I))], () where () e is a small dimensionless parameter, 1 is natural frequency, ~ is detuning parameter, p, h, (3 , r, 1), w are constants and overdots denote differentiation with respect to time t. We look for the solution of the equation () in the form: x = acos6 + e-u1(a,,P,6) + e- 2u2(a,,P,6) + . , () where 6 = wt + 1/1, u;(a, '1/1, 6) are periodic functions with period 211" with respect to both angular variables 1/1 and 8, and a and 1/1 are functions of time which will be determined from the equations: da ( 2 dt =e-A1 a,'I/J)+e- A2(a,,P)+ . , d,P = e-B1 (a,,P) dt () 2 + e B 2(a,,P) + 11 these equations A; (a, .p), B; (a, .P) are periodic functions of the angular variable with period 2rr. Substituting the expressions () and () into the equation () and comring the coefficient of c 1 we obtain -2wAtsin0-2waBlcos0+w 2 (a;;l +u1) =apcos()cosO. () >mparing the harmonics in () gives: At= B1 = 0, () u 1 = -pa2 [ cos .P- -1 cos(20- .p) ] . 2w 3 Comparing the .
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