Second-order approximation of angle-ply composite laminated thin plate under combined excitations

The influence of the quadratic and cubic terms on non-linear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations is investigated. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted and investigated at this approximation order. Two cases of the sub-harmonic resonances cases (Ω2 ≅ 2ω1 and Ω2 ≅ 2ω2) in the presence of 1:2 internal resonance ω2 ≅ 2ω1 are considered. The stability of the system is investigated using both frequency response equations and phase-plane method. It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.