Harmonic Balance Based Averaging: Approximate Realizations of an Asymptotic Technique

Averaging is a classical asymptotic technique commonly used to studyweakly nonlinear oscillations via small perturbations of the harmonicoscillator. If the unperturbed oscillator is autonomous and stronglynonlinear, but with a two-parameter family of periodic solutions, thenaveraging is allowed in principle but typically not considered feasibleunless (a) the required family of unperturbed periodic solutions can befound in closed form, and (b) the averaging integrals can be found inclosed form. Often, the foregoing requirements cannot be met. Here, itis shown how both these difficulties can be bypassed using the classicalbut heuristic approximation method of harmonic balance, to obtain approximate realizations of the asymptotic analytical technique. Theadvantages of the present approach are that (a) closed form solutions tothe unperturbed problem are not needed, and (b) the heuristic andasymptotic parts of the calculation are kept conceptually distinct, withscope for refining the former, while preserving the asymptotic nature ofthe latter. Several examples are provided, including oscillators with astrong cubic nonlinearity, velocity dependent nonlinear terms (includinga strongly nonconservative system), a nondifferentiable characteristic,and a strongly nonlinear but homogeneous function of order 1; dynamicphenomena investigated include damped oscillations, limit cycles, forcedoscillations near resonance, and subharmonic entrainment. Goodapproximations are obtained in each case.