Mode localization by structural imperfections in one-dimensional continuous systems

Longitudinal free oscillations of a beam with small axial rigidity, continuously restrained by imperfect elastic springs, are analyzed. It is shown that small imperfections can localize the first modes of vibration in a restricted region of the beam. Systems with single localized defects are first considered; these provide physical insights and make clear that the phenomenon is governed by a turning point mathematical problem. Analogies with similar problems in quantum mechanics are emphasized. Periodic or nearly periodic imperfections are then analyzed by applying the Floquet theory, and finally non-periodic imperfections are numerically treated. It is shown that when the exponential decay of the solution is strong, the modes localize in the neighbourhood of each defect, so it is possible to limit oneself to a local analysis. The problem of a beam of infinite length with a single imperfection is then solved by applying the asymptotic WKB method. An application relative to a parabolic imperfection is analytically developed.

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