TAILIEUCHUNG - Associated equations and their corresponding resonance curve
In the theory of nonlinear oscillations, in order to identify the resonance curve we usually try to eliminate the dephase in the equations of stationary oscillations. We obtain thus a certain frequency-amplitude relationship. | Vietnam Journal of Mechanic•, NCST of Vietnam Vol. 21, 1999, N~ S (147 - 155) ASSOCIATED EQUATIONS. A N D THEIR CORR ESPONDING RESONANCE CURVE NGUYEN VAN DINH Institute of Mechanics In the theory of nonlinear oscillations, in order to identify curve we usually try to eliminate the dephase fJ in the equations of stationary oscillations.' We obt ain thus a certain frequency-amplitude relationship. In simple cases when the oned equat ions contain only and linearly the first h armonics (sin 8, cos 8) the elimination of 8 is elementary, by using the trigonometrical identity sin 2 8 + cos 2 8 = 1. In general, high harmonics (sin 28, cos 28, etc.) are present. Consequently the expressions of sin 8, cos 8 are cumbersome or do not exist and the analytical elimination of 8 is quite inconvenient or impossible. For this reason, to identify' the resonance curve of complicated systems, we use the numerical method. Below, intending to develop the analytical method, we shall propose a procedure enabling us to transform the "original" complicated equations of stationary oscillations into the so-called associated ones, only and linearly containing sin 81 cos 8. The equivalence of the original and associated equations will be treat ed an 0 is a small formal parameter; Po, Qo, So, Cot, Rob Kot are polynomials in w, a. Constant amplitude and dephase of stationary oscillations satisfy the equations: fo =Po+ Sot sinO+ Col cos 0 + M sin28 = 0, () Uo = Qo + Qol sin fJ + Ko1 cos fJ + M cos 20 = 0. The equations () will be called "original" ones. They determine the "true" "original" resonance curve-denoted by C 0 • We use the following two step procedure to eliminate (sin2fJ,cos2fJ): First, we form the equations, equivalent to () and of the same structure as {} It= focosfJ- uosinfJ = ··P 1 + Su sinO+ Cu cos 0 + S 12 sin20 + C 12 cos 20 91 = 0, = /osinO + go cosO= = Q1 +Ru sinO + Ku ccsO + R12sin20 + K12 cos20 = 0, where: P1 1 = 2(Co1 ~ .
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