Harmonic balance based averaging: Approximate realizations of an asymptotic technique

Averaging is a classical asymptotic technique commonly used to study weakly nonlinear oscillations via small perturbations of the harmonic oscillator. If the unperturbed oscillator is autonomous and strongly nonlinear, but with a two-parameter family of periodic solutions, then averaging is allowed in principle but typically not considered feasible unless ( a) the required family of unperturbed periodic solutions can be found in closed form, and (b) the averaging integrals can be found in closed form. Often, the foregoing requirements cannot be met. Here, it is shown how both these difficulties can be bypassed using the classical but heuristic approximation method of harmonic balance, to obtain approximate realizations of the asymptotic analytical technique. The advantages of the present approach are that ( a) closed form solutions to the unperturbed problem are not needed, and ( b) the heuristic and asymptotic parts of the calculation are kept conceptually distinct, with scope for refining the former, while preserving the asymptotic nature of the latter. Several examples are provided, including oscillators with a strong cubic nonlinearity, velocity dependent nonlinear terms ( including a strongly nonconservative system), a nondifferentiable characteristic, and a strongly nonlinear but homogeneous function of order 1; dynamic phenomena investigated include damped oscillations, limit cycles, forced oscillations near resonance, and subharmonic entrainment. Good approximations are obtained in each case.