Free vibrations of axisymmetric shells: Parabolic and elliptic cases

摘要

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lame system) are determined by an asymptotic analysis as the thickness (2 epsilon) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar model that depends on two parameters: the angular frequency k and the half-thickness epsilon. Optimizing k for each chosen epsilon, we find power laws for k in function of e that provide the smallest eigenvalues of the scalar reductions. Corresponding eigenpairs generate quasimodes for the 3D Lame system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lame system.