Nonlinear free vibration analysis of imperfect cylindrical panels with discontinuous unilateral elastic base

Abstract

In this work, the nonlinear free vibration analysis of the imperfect cylindrical panels in contact with discontinuous unilateral elastic base are conducted, evaluating the influence of the type of contact, unilateral or bilateral, and the contact area of the elastic base on the natural frequencies and nonlinear frequency-amplitude relations. For that, the structural cylindrical panel model considers the Donnell's nonlinear shallow shell theory to obtain the partial differential equilibrium equation and the compatibility and continuity equation. To represent the unilateral capability of the elastic base, a signum function is inserted in the reaction force of the elastic base to describe its dependence of the transversal displacement field of the cylindrical panel. To apply the elastic base in certain subdomain of the cylindrical panel, a Heaviside-type function is also inserted into elastic base's reaction forces. The nonlinear equilibrium equation is discretized by the Galerkin method, using a consistent transversal displacement field that it was obtained from a perturbation technique. The discretized set of equilibrium equations is employed to obtain the natural frequencies and the nonlinear frequency-amplitude relations, evaluating the influence of the unilateral contact and the contact region of the elastic base on these results. Depending on the signal of the imperfection's magnitude and the type of contact of the elastic base, the imperfect cylindrical panel can display an initial gap between the cylindrical panel and the elastic base, and this possibility is also investigated in this work. From the numerical results, it is observed that the unilateral elastic base applies less structural stiffness than the bilateral elastic base, decreasing the natural frequencies for the same imperfect cylindrical panel. The localization of the elastic base in the cylindrical panel also affects the natural frequencies, increasing or decreasing them depending on the modifications on the structural stiffness. The frequency-amplitude relations are strongly influenced by both unilateral contact of elastic base and the type of the contact consideration between the cylindrical panel and elastic base, exhibiting an intricated nonlinear behavior.