Nonlinear vibration analysis of functionally graded material beams with non-uniform cross section

Nonlinear vibration analysis of axially functionally graded material variable cross-section beams is performed. The beam is modeled according to Euler–Bernoulli beam theory. The beam material properties vary axially along the beam according to power law, exponential law, trigonometrically and polynomially. Three different cases are considered for the beam cross-section: linear variation of the width, linear variation of the height, and simultaneous variation of the width and height. The mathematical model of the beam is obtained using Generalized Hamilton's Principle. Linear vibration analysis is carried out via the Differential Quadrature Method; linear natural frequencies and critical loads of the beam are calculated, and mode shapes and frequency–critical load graphs are obtained. The nonlinear model of the beam is analyzed using Method of Multiple Scales (Perturbation Method) and the free vibration, primary resonance and parametric resonance analyses are carried out. The most significant novelty of the present study is that the perturbation method is applied to the beam vibration analysis integrated with the DQM. The nonlinear natural frequencies of the beam are obtained for free vibration where there is no loading. Primary resonance analysis of the beam subject to external excitation is done and frequency-phase modulation equations are obtained. Stability analysis is carried out by determining the steady-state solutions. Parametric resonance analysis of the beam under axially parametric excitation is performed. Stability of the steady-state solutions are investigated by deriving amplitude-phase modulation equations. The stable and unstable regions of the trivial solution are determined. Frequency-response graphs of the beam are obtained both for external excitation and for parametric excitation.