AN INTRINSIC MULTIPLE-SCALE HARMONIC-BALANCE METHOD FOR NONLINEAR VIBRATION AND BIFURCATION PROBLEMS

An intrinsic multiple-time-scale harmonic balance method for the analyses of non-linear periodic oscillations and bifurcation problems is developed. The new approach combines selected features of the intrinsic harmonic balancing (introduced earlier) and multiple-time-scaling in an effort to produce a more efficient technique for non-linear analysis. This is achieved by retaining advantageous aspects of both methods and eliminating some of the observed disadvantages. Thus, the new approach has the following features: (1) it produces ordered forms of consistent approximations for the solutions of a non-linear problem such that the number of perturbations required for a particular approximation is less than that required by the intrinsic harmonic balancing; (2) it leads to simplified differential equations governing the local dynamics which readily yield the stability properties of the solutions; (3) it does not require the solution of differential equations at each perturbation step, as in the conventional multiple-time-scaling method, and the need to eliminate secular terms does not arise. The application of the new procedure is illustrated on several examples.