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Weak self sustained system under the actions of less weak excitations

It has been known that, in several cases, to study quasi~linear oscillating system, the degrees of smallness of various factors must be distinguished in detail [2-7]. To affirm again this interesting remark, we shall examine a weak (of order e2 ) self-sustained system subjected to less weak (of order e:) excitations in resonance cases. It will be seen that the system considered is enhanced. | Journal of Mechanics, NCNST of Vietnam T. XVI, 1994, No 3 (5 - 10) WEAK SELF-SUSTAINED SYSTEM UNDER THE ACTIONS OF LESS WEAK EXCITATIONS NGUYEN VAN DINH Institute of Mechanics, NCNST of v~·etnam SUMMARY. It has been known that, in several cases, to study quasi~linear oscillating system, the degrees of smallness of various factors must be distinguished in detail [2-7]. To affirm again this interesting remark, we shall examine a weak (of order e 2 ) self-sustained system subjected to less weak (of order e:) excitations in resonance cases. It will be seen that the system considered is enhanced. §1. SYSTEM UNDER CONSIDERATION AND ITS APPROXIMATE SOLUTION Let us consider a quasi-linear oscillating system described by the following differential equation: (1.1) f(x,wt) = { + 3p cos 2wt j, = (3x 2 fz = 2pxcoswt (1.2) where: x - an oscillatory variabb; e > 0 - small parameter; overdots denote differentiation with respect to timet; {3, 1- coefficients of the quadratic and cubic non-linearities, respectively; ho > 0 - damping viscous coefficient; h > 0, k > 0, p 2 0- constants; 1 - the natural frequency, e: 2 D. - the de tuning parameter assumed to be of order e 2 • If p = 0, we have a "pure~ self-sustained system with the positive friction force (hX- kX 3 ). If p > 0, the mentioned system is subjected to the external excitation 3p cos 2wt in subharmonic resonance of order one-half or to the parametric one 2px cos wt in principal resonance (by fundamental we mean the cases where the natural frequency is near that of the external excitation or one-half of that of the parametric one). . The damping (negative frictiori) force is introduced to facilite the analyses andj as it will be shown below, the quadratic non_-)inearity {Jx 2 is necessary in the case of external excitation. Using the asymptotic method fl], the solution of the differential equation (1.1) will be found in the form: x= acos,P+w,(a,O,¢) +s2u2(a,O,,P), a= eA1(a,O) + 0 Bao aBo B6o .