A WAVELET-GALERKIN PROCEDURE TO INVESTIGATE TIME-PERIODIC SYSTEMS: TRANSIENT VIBRATION AND STABILITY ANALYSIS

A wavelet-Galerkin procedure is introduced in order to obtain transient and periodic solutions of multi-degree-of-freedom (d.o.f.s) dynamical systems with time-periodic coefficients. Numerical comparisons, achieved with a Runge–Kutta method, emphasize that the wavelet-based procedure is reliable even in the case of problems involving both smooth or non-smooth parametric excitations and a relatively large number of degrees of freedom. The procedure is then applied to study the vibrations of some theoretical parametrically excited systems. Since problems of stability analysis of non-linear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, the method is shown to be effective for the computation of the Floquet exponents that characterize stable/unstable parameters areas and consequently allows estimators for stability/instability levels to be provided. Stability diagrams of some theoretical examples including a single-degree-of-freedom Mathieu oscillator and a two-degree-of-freedom parametrically excited system, illustrate the relevance of the method. Finally, future studies are outlined for the extension of the wavelet method to the non-linear case.