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Stability of the equilibrium regime of a system of two degrees of freedom in amplitude phase variables

In this dephase has been explained as that of such motion called characteristic and this explanation permits us to propose a lightly modification in Hans Kauderer's method. The same problem will be examined below for the equilibrium regime of an oscillating system of two degrees of freedom. It will be shown that the results obtained in can be applied without difficulty. | T~p chi Ccr h9c Journal of Mechanics, NCNST of Vietnam T. XVII, 1995, No 1 (28- 34) STABILITY OF THE EQUILIBRIUM REGIME OF A SYSTEM OF TWO DEGREES OF FREEDOM IN AMPLITUDE-PHASE VARIABLES NGUYEN VAN DINH Institute of Mechanics, NCNST of Vietnam In [1] Hans Kauderer has used the amplitude-phase variables to study the stability of the equilibrium regime which is considered as a special oscillation of amplitude r = 0 and of constant dephase e·. In [2] this dephase has been explained as that of such motion called characteristic and this explanation permits us to propose a lightly modification in Hans Kauderer's method. The same problem will be examined below for the equilibrium regime of an oscillating system of two degrees of freedom. It will be shown that the results obtained in [2] can be applied without difficulty. §1. SYSTEM UNDER CONSIDERATION AND ITS AVERAGED EQUATIONS Let us consider a quasi-linear oscillating system of two degrees of freedom described by the following differential equations: (p = (Ll) I, 2) where x 11 x 2 - oscillatory variables; e > 0- small parameter; overdot denotes time derivative; w 11 exciting frequencies near the natural ones 1 respectively; Jill, j( 2 l- functions of the form: w2 - N 2 fit•) = 2 2.= {xv[A~'") + 2.= 2.=(C~';"'Icosmw,t+S~';"'Isinmw,t)] +xv[:4~'" 1 + v=l N + nL=l r=l 2 " ' "'(-(~m) ~ ~ cvr cos rnwrt + -(wn) s vr sin mw t l]} + (. } (!" = 1,2) (1.2) rn=l r=l . . mteger; . AI~O) sit•"') t coe ffi Cients, · ( . . . ) represents t h e with: N - a pos1t1ve v , clt•m) vr vr • • • - constan 1 1 terms of powers equal or greater than 2 relative to x 1 , :i: 1 , x 2 , :i: 2 . It is assumed that w 1 , w2 don't satisfy the relations of type: (L3) In other words, the system considered is riot in internal resonant situation. Introducing slowly varying variables either of type (a, b) or ( r, 8) we put, respectively: (L4) 28 'I or ( I ~,I The correspon.ding avtTag:t>d systems are ' .