Random vibrations of laminated planar frames

Abstract

In this study, the normal mode method is used to determine the response of a generalized laminated planar frame structure subjected to random loading conditions. The model, which is based on the Timoshenko–Ehrenfest beam theory, accounts for the effects of shear deformation, rotary inertia and elastic coupling due to structural anisotropy. The free vibration analysis of the frame is formulated and solved, and the orthogonality condition derived. Following this, the normal mode method is applied to determine expressions for the mean square longitudinal, transverse and rotational displacements in terms of the excitation spectral density. Although the proposed model is capable of analyzing a wide range of problems, the numerical analysis conducted in this paper is centered on investigating the response of inclined two-member and three-member frames excited by a point force, whose spectrum is band-limited white noise. It is shown that in cases where the modal cross-correlation is negligible and the boundary conditions are symmetric, the spatial response of the mean square displacements are nearly symmetric about the center point of the frame. Contrarily, as the influence of the modal cross-correlation on the overall response increases, the featured spatial symmetry vanishes. An interesting characteristic of frame structures, is that the spacing between the natural frequencies is dependent on the frame angle. Consequently, modal cross-correlations can have a much greater effect on certain frame angles compared to others, even when the damping is fairly low. The study also investigates the effects of bending-extension material coupling on the random response of asymmetrically laminated cross-ply frames. The numerical results presented in this work are verified with independent finite element simulations conducted in the software package, Ansys® Parametric Design Language (APDL).