Direct treatment and discretizations of non-linear spatially continuous systems

Approximate analytical methods for the study of non-linear vibrations of spatially continuous systems with general quadratic and cubic non-linearities are discussed. The cases of an external primary resonance of a non-internally resonant mode and of a sub-harmonically excited two-to-one internal resonance are investigated. It is shown, in a general fashion, that application of the method of multiple scales to-the original partial-differential equations and boundary conditions produces the same approximate dynamics as those obtained by applying the reduction method to the full-basis Galerkin-discretized system (using the complete set of eigenfunctions of the associated linear system) or to convenient low-order rectified Galerkin models. As a corollary, it is shown that, due to the effects of the quadratic non-linearities, all of the modes from the relevant eigenspectrum, in principle, contribute to the non-linear motions. Hence, classical low-order Galerkin models may be inadequate to describe quantitatively and qualitatively the dynamics of the original continuous system. Although the direct asymptotic and rectified Galerkin procedures seem to be more "appealing" from a computational standpoint, the full-basis Galerkin discretization procedure furnishes a remarkably interesting spectral representation of the non-linear motions. (C) 1999 Academic Press.