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Mixed regime in a quasi-linear system

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In the system under consideration there appears mixed oscillation due to the interaction between derived excitations. Various forms of the resonance curve were identified. The stability study was based on an abbreviated form of the second stability condition [2]. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 22, 2000, No 4 (205 - 211) MIXED REGIME IN A QUASI-LINEAR SYSTEM NGUYEN VAN DINH - TRAN KIM CHI Institute of Mechani'cs, NCST of Vietnam ABSTRACT. A quasi-linear system with cubic nonlinearity under two external excitations in subharmonic responces of order 1/2 and 1/3 was examined. In the system under consideration there appears mixed oscillation due to the interaction between derived excitations. Various forms of the resonance curve were identified. The stability study was based on an abbreviated form of the second stability condition [2]. 1. Introduction The present article deals with the effect of two external excitations in subharmonic resonances on an oscillator which has weak (order c-) cubic non-linearity. The attention is focused on mixed regimes due to supplementary excitations of c--order, which are introduced by the "original" excitations through the cubic nonlinearity. The asymptotic method [1] and some remarks given in [2] are used . The so-called associated equations are established; the analytical identification of the resonance curves is done; the classification of the resonance curves is based on the location of the critical part; stable branches of the resonance curve are determined by an abbreviate form of the second stability condition. 2. System under consideration - Original and Associated equations Consider a quasi~linear ·system governed by the differential equation: x+x = c-(-hx -1x 3 ) - 3bw 2 cos 2wt - 8cw 2 cos(3wt +a), (2.1) where x is an oscillatory variable, over dots denote derivatives relative to time t, c- > 0 is a positive small parameter, h ~ 0, b > 0, c > 0, w ~ 1, 0 ~ a 0), the critical part C" (if it exists) is reduced to a single point. Let us examine in detail the case u = 0. The resonance curves shown in Figures 5, 6, 7, 8, 9 correspond to h = 0.001 , 0.002753 , 0.0033, 0.00357 , 0.00362 . In the case h = 0.001 the resonance curve Co only consists of the .