Free vibration of multi-layered spherically isotropic hollow spheres

A three-dimensional analysis is carried out for the free vibrations of a multi-layered spherically isotropic hollow sphere using a state-space method. By the introduction of three displacement functions and two stress functions, two independent state equations with varying coefficients are derived. Taylorʹs expansion theorem is then employed to obtain solutions to the two state equations and relationships between the state variables at the upper and lower surfaces of each lamina are established. A variable substitution technique is particularly used to make the derivation more natural and simpler. By virtue of the continuity conditions between two adjacent layers, two sets of linear algebraic equations about the boundary variables at the inner and outer surfaces of a multi-layered hollow sphere are obtained. Frequency equations for the free vibrations are then presented.