Free geometrically nonlinear oscillations of perfect and imperfect laminates with curved fibres by the shooting method

Large-amplitude free vibrations of composite laminated plates with curvilinear fibres are studied. The fibre angle in a ply changes linearly in relation to one Cartesian coordinate. The plates are rectangular with geometric imperfections (out-of-planarity). The edges of the laminates are clamped, except in comparison studies, where simply supported conditions are applied. The displacement field is modelled by a third-order shear deformation theory, and the equations of motion (full model), in the time-domain, are obtained using a p-version finite element method. When possible, the model is statically condensed neglecting inplane inertias but still taking into account the in-plane displacements. The condensed model is transformed to modal coordinates in order to have a model with fewer degrees of freedom (reduced model). Backbone curves are found by the shooting method, using Runge-Kutta-Fehlberg method modified with Cash-Karp method to control the error with adaptive stepsize. Backbone curves of composite laminates with different curvilinear fibre angles are plotted and compared. In addition to the fundamental backbone curve, the method is able to find bifurcations leading to other branches, as shown in some examples. Oscillations of some points are studied in detail using phase-plane plots and Fourier spectra of deflection. Finally, the effects of geometrical imperfection on the backbone curves are analysed.