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On the dephase angle in a variational system of the equilibrium regime

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In the quasi-linear theoxy of vibrations, the initial variable X is transformed either to the variables (a, b) or to the amplitude-phase ones ( r, 8) respectively by the formulae. | T~p Journa.l of Mechanics, NCNST of Vietnam T. XVI, 1994, No 2 (29- 35) chi Ccr h9c ON THE DEPHASE ANGLE IN A VARIATIONAL SYSTEM OF THE EQUILIBRIUM REGIME NGUYEN VAN DINH Institute of Mechanics NCNST of Vietnam SUMMARY. In the quasi-linear theoxy of vibrations, the initial variable X is transformed either to the variables (a, b) or to the amplitude-phase ones ( r, 8) respectively by the formulae x=acoswt+bsinwt, X= rcos(wt- e), (0.1) (0.2) :i:=-wasinwt+wbcoswt X= -wrsin(wt- e) where W is near the natural frequency of the system under consideration. These two types of v~riables are equivalent if the oscillatory regime is concerned. However, for the equilibrium regime, the situation is more complicated. In {lj (pp. 211-213), to study the stability of the equilibrium regime, the variables transformed into the ones (a, values ao = 0, bo = b). (r,O} are The equilibrium ~egime corresponds to the couple of determinate 0 so that the signification of the perturbations Sa= a- ao, Ob = b- bo is evident and the variational system can be easily established~ Other author [2] {pp. 617-624) has used the amplitude-phase variables (r, 0) for seeking the stability condition of the equilibrium regime which coresponda to the "zero" amplitude To dephase angle f) remains indeterminate. A certain constant (unlmown) value 8o = 0. The is assigned to the equilibrium regime. By this manner, the "classical" process of studying the stability by introducing the perturbations Or= r - To, SO = (}- Oo and the variational equations can be applied. As it will be seen below, the variational system obtained has an "anormal" form. In the present paper, the significatiOn of the mentioned deph 0 and 2w are the intensity and the frequency of the parametric excitation, respectively, w ~ w 0 , h > 0 is the damping linear coefficient; '1 is ,the coefficient of .the cubic non-linearity, e > 0 is a small parameter. .In the phase plane Oxi:, the solution x = 0 (equilibrium .