Free vibrations of elastic laminated beams, plates and cylindrical shells

In this chapter, based on the equivalent single layer model for thin laminated members, natural modes and corresponding eigenfrequencies for laminated elastic beams plates and cylindrical shells are studied taking into account shears. At first, elastic vibrations of laminated beams are analyzed in Sect. 4.1, the emphasis being made on non-uniformly stressed beams contacting with an elastic inhomogeneous medium. Then, in Sect. 4.2, the eigenmodes and frequencies of elastic rectangular plates are analyzed for two variants of boundary conditions: if all edges are simply supported and have diaphragms preventing shears, the boundary-value problem is solved in the explicit form; and if one of edges is free of a diaphragm, the solution of a corresponding boundary-value problem is constructed in the form of the superposition of the main stress-strain state and the edge effect integrals accounting for the edge shears. Section 4.3 is devoted to vibrations of a circular cylindrical shell of an arbitrary length with constant geometrical and physical parameters. In Sect. 4.4, the localized natural modes for a medium-length laminated cylinder is investigated. And finally, Sect. 4.5 contains the problem on free localized vibrations of a laminated cylindrical shell under axial forces no-uniformly distributed in the circumferential direction. In the last two sections, natural modes are constructed by using the asymptotic method. In all problems, the effect of shears on the natural frequencies is analyzed. Examples on free vibrations of laminated cylinders and panels assembled from different materials are considered.