TAILIEUCHUNG - Interaction between parametric and forced oscillations in fundamental resonance
The present paper is devoted to examine the interaction between parametric and forced oscillations in fundamental resonances. Some remark about the resolution of the equations determining the stationary oscillations will begiven, some particularities of the resonance curve will be described. | T\'P chi CO' h9c Journal of Mechanics, NCNST of Vietnam T. XVII, 1995, No 3 (12- 19) INTERACTION BETWEEN PARAMETRIC AND FORCED OSCILLATIONS IN FUNDAMENTAL RESONANCE NGUYEN VAN DINH Institute of Mechanics, NCNST In nonlinear systems, the interaction between different oscillations is complicated and has attracted the attention of a lot of researches [1, 2}. Interesting results have been obtained, some· aspects of this phenomenon can be found in a recent work [3]. The present paper is devoted to examine the interaction between parametric and forced oscillations in fundamental resonances. Some remark about the resolution of the equations determining the stationary oscillations will be .given, some particularities of the resonance curve will be described. §1. SYSTEM UNDER CONSIDERATION AND THE AVERAGING METHOD "' Let us consider a quasi-linear oscillating system governed by t}le differential equation x+ w 2 z = •{- h:i:- 7:>:3 + + 2pzcos2wt + qcos(wt + u)} () where x- an oscillatory Variable, e: > 0- a small parameter, h ;::: 0- the damping viscous coefficient, 7 - the cubic nonlinearity coefficient, 2p > 0, q > 0 and 2w, w - inten~ities and frequencies of the parametric and external excitations, respectively, (w 2 - 1) - the detuning parameter (1 - the natural frequency), cr(O $ cr 0. L If D f. 0, from (), we deduce: sinO=- ab{hwcosu- W:a 2 -a) -p] sino-} cosO= ab{ hwsinu + [C41 a 2 - a)+ p] () COSO"} and the amplitude - frequency relationship is of the form: W1 = a%~ 2 { (hw coso-- [ or c: a 2 2 -a) -p] sino-) +(hwsinu+ [ w, = a2q~2 { [ c41a 2 - 2 a) + pcos 2u] + c: a 2 a) +p) coso-) 2 } -1 = 0 () - [hw + psin 20"] 2 } - 1 = 0 () n> = 0 () As it has been in §2, under condition D = 0, () can be replaced by w= 2 q {[( 3 2 7 a - a)+ pcos2ur + [hw + psin2u] 2 }- a2 2. If () the two linear algebraic equations () is in "critical" situation respectively by denote the coefficient matrix and the
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