Effects of weak nonlinearity on dispersion relations and frequency band-gaps of periodic structures

Abstract

The analysis of the behaviour of linear periodic structures can be traced back over 300 years, to Sir Isaac Newton, and still attracts much attention. An essential feature of periodic struc-tures is the presence of frequency band-gaps, i.e. frequency ranges in which waves cannot propagate. Determination of band-gaps and the corresponding attenuation levels is an im-portant practical problem. Most existing analytical methods in the field are based on Floquet theory; e.g. this holds for the classical Hill’s method of infinite determinants, and the method of space-harmonics. However, application of these for nonlinear problems is impossible or cumbersome, since Floquet theory is applicable for linear systems only. Thus the nonlinear effects for periodic structures are not yet fully uncovered, while at the same time applica-tions may demand effects of nonlinearity on structural response to be accounted for. The present work deals with analytically predicting dynamic responses for nonlinear continuous elastic periodic structures. Specifically, the effects of weak nonlinearity on the dispersion re-lation and frequency band-gaps of a periodic Bernoulli-Euler beam performing bending os-cillations are analyzed. Modulation of the beam structural properties is not required to be small or piecewise constant. Various sources of nonlinearity are analyzed, namely, nonlinear (true) curvature, nonlinear inertia due to longitudinal motions of the beam, nonlinear materi-al, and also nonlinearity associated with mid-plane stretching. A novel approach, the Method of Varying Amplitudes, is employed. This implies representing a solution in the form of a harmonic series with varying amplitudes; however, in contrast to averaging methods, the amplitudes are not required to vary slowly in space. As a result, a shift of band-gaps to a higher (or lower) frequency range is revealed, while the width of the band-gaps appears rela-tively insensitive to (weak) nonlinearity. The results are validated by numerical simulation, and explanations of the effects are suggested.