DIRECT TREATMENT AND DISCRETIZATIONS OF NON-LINEAR SPATIALLY CONTINUOUS SYSTEMS

✍ Scribed by W. Lacarbonara

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 192 KB
• Volume: 221
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1998.2049

✦ Synopsis

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-dierential equations and boundary conditions produces the same approximate dynamics as those obtained by applying the reduction method to the full-basis Galerkindiscretized system (using the complete set of eigenfunctions of the associated linear system) or to convenient low-order recti®ed Galerkin models. As a corollary, it is shown that, due to the eects 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 recti®ed 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.