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The poincare method for a strongly nonlinear duffing oscillator

In the present article, the mentioned Duffing oscillator is examined by another modified Poincare method. Practically acceptable results are obtained. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 25, 2003,' No 1 (19 - 25) THE POINCARE METHOD FOR A STRONGLY NONLINEAR DUFFING OSCILLATOR NGUYEN VAN DINH Institute ofMechanics It is well-known that the classical Poincare method is limited to weakly nonlinear systems and for extending the range of validity of this method to strongly non - linear systems, various modifications have been developed. Some authors have replaced the original parameter c by a new a; the later is chosen such that the values a are always kept small regardless of the magnitude of E. High degree of accuracy have been obtained. For instance in [2], for an undamped Duffing oscillator having large cubic non-linearity the free oscillation period evaluated by the "a method" in the fourth approximation (solution to O(a 5 )) is identical with that given by the exact solution, even for large c:a 2 (= 1000) . In the present article, the mentioned Duffing oscillator is examined by another modified Poincare method. Practically acceptable results are obtained. §1. System under consideration. Period from the exact solution Consider an oscillator governed by the differential equation X+ X + EX 3 = (1.1) 0, where x is an oscillatory variable; overdots denote differentiation with respect to time t; E is the cubic non-linearity coefficient which is assumed to be positive and arbitrary (c: needs not to be small) The attention is focused on the period of free oscillation satisfying initial conditions x(O) x(O) =a, = (1.2) 0. The exact period is given by the formulae J rr/ 2 ~x= 4 \.h + Ea 2 0 dB J1 - msin 2 Ea e' 2 m = - - - 2- 2(1 + c:a (1.3) ) and the value Tex as function of c:a2 was given in [2] (see the second column of the table) 19 §2. Period from the standard Poincre method For the sake of comparison, the standard Poincre method [1 J is used first for the case of small c. Let w be the frequency to be evaluated. Introducing the new time r = wt, the equation (1.1) can be .