Asymptotic integration of nonlinear equations of cylindrical panel vibrations (Q757147)

Complete asymptotic expansions of the solution of the two-dimensional problem of the nonlinear vibrations of a cylindrical panel with free curvilinear boundaries are constructed using the boundary layer method [e.g.: \textit{A. L. Gol'denvejzer}, Theory of thin shells (1976; Zbl 0461.73052)] in the case when the parameter \(\delta\), equal to the ratio between the lengths of the clamped and free sides, is fairly small. The principal term of the expansion for the deflection function is determined from the known nonlinear integrodifferential equation of arch vibrations. The discrepancies in satisfying the boundary conditions on the clamped boundaries turn out to be higher-order infinitesimals in \(\delta\) and are compensated by boundary layer functions that are determined from linear boundary value problems for a biharmonic operator in a half-strip. Calculations are performed by using finite differences for elastic, elasto-plastic cylindrical panels, arches, and rectangular plates subjected to an instantaneously applied transverse step load, and the limits of applicability are established for a monomial expansion.