Non-linear forced periodic oscillations of laminates with curved fibres by the shooting method

• We model variable stiffness composite laminates by a p-version finite element.
• We find a reduced order model for the equations of motion using modal coordinates.
• Periodic solutions of the reduced order model are obtained using shooting method.
• Frequency-response and bifurcations of laminates before damage onset are studied.
• Steady-state stresses are studied using phase plane and Fourier spectra.

Large-amplitude forced vibration, before damage onset, of variable stiffness composite laminated plates with curvilinear fibres are studied. The fibre paths considered change linearly in relation to one Cartesian coordinate. The plates are rectangular and with clamped edges. The displacement field is modelled by a third order shear deformation theory and the equations of motion, in the time domain, are obtained using a p-version finite element method. The in-plane inertia is neglected, still taking into consideration the in-plane displacements, and the model is statically condensed. The condensed model is transformed to modal coordinates in order to have a reduced model with a smaller number of degrees-of-freedom. A shooting method using fifth-order Runge–Kutta method, as well as adaptive stepsize control, is used to find periodic solutions of the equations of motion. Frequency-response curves of composite laminates with different curvilinear fibre angles and various thicknesses are plotted and compared. Tsai–Wu criterion is employed in order to predict the damage onset. When it is detected that damaged started, the continuation method is interrupted and no further points of the response curve are computed. The reason behind this interruption is that the model does not include the effects of damage. Examples of bifurcations are presented and studied in detail, using projections of trajectories in a phase plane and Fourier spectra. The time histories and frequency spectra of steady-state stresses are plotted for VSCL plates with different fibre angles. The steady-state stresses are also displayed for bifurcated branches of the solutions.