Abstract
Forced vibrations of duffing equation with damping is considered. Recently developed Multiple Scales Lindstedt-Poincare (MSLP) technique for free vibrations is applied for the first time to the forced vibration problem in search of approximate solutions. For the case of weak and strong nonlinearities, approximate solutions of the new method are contrasted with the usual Multiple Scales (MS) method and numerical simulations. For weakly nonlinear systems, frequency response curves of both perturbation methods and numerical solutions are in good agreement. For strongly nonlinear systems however, results of MS deviate much from the MSLP method and numerical simulations, the latter two being in good agreement.
Highlights
While a complete review of the attempts to validate perturbation solutions for strongly nonlinear oscillators is beyond the scope of this work, a partial list will be given
The justification for combining both methods is that Multiple Scales is better in determining transient solutions while Lindstedt Poincare method may be better under some circumstances in determining steady state solutions [16]
For undamped and damped duffing oscillators, results of the new method are in good agreement with the numerical simulations for strong nonlinearities
Summary
Perturbation methods are well established and used for over a century to determine approximate analytical solutions for mathematical models. For undamped and damped duffing oscillators, results of the new method are in good agreement with the numerical simulations for strong nonlinearities This recently developed method is applied to an equation with quadratic and cubic nonlinearities [17]. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results whereas amplitude and frequency estimations of classical Multiple Scales yield high errors. The expansions of natural and external frequencies to obtain valid solutions are nontrivial and the outline of the method is given for the forced vibrations of a duffing equation with damping. Developed Multiple Scales Lindstedt Poincare method [15] will be applied to the forced vibrations for the first time to obtain approximate expansions.
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