Natural vibration of unsymmetric cross-ply laminates

This study considers the linear vibration characteristics of square [ 0 n / 90 n ] T laminates relative to their room-temperature static equilibrium configurations. A Rayleigh–Ritz approach combined with Hamiltonʹs principle is used to provide approximate solutions to this vibration problem. The vibration mode shapes are assumed to have the same spatial dependence as used in past investigations to study the room-temperature configurations of these laminates, and are thus assumed to be perturbations on the static equilibrium configurations. Hamiltonʹs principle then results in the so-called zero- and first-order equations. The zero-order equations lead to the classic static equilibrium results of past investigations, presented here in nondimensional form with analytical solutions at the bifurcation point. The first-order equations, combined with zero-order results, lead to the vibration characteristics for each zero-order static configuration. Interest centers on the lowest natural frequency and the associated mode shape for laminates clamped at their midpoints, with special attention as to how these vibration characteristics depend on the laminate side-length-to-thickness ratio. With an imaginary-valued frequency, the static saddle configuration for side-length-to-thickness ratios larger than the critical value is correctly assessed as unstable. A finite element model is also used to study the vibration characteristics and to compare with the findings for the developed analysis. The qualitative comparisons between the developed analysis and the finite element model are generally good, and the quantitative comparisons are also satisfactory.