A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches

• Modern asymptotic approaches are applied on the quintic nonlinear beam vibration.
• The frequency–amplitude relation of nonlinear vibration of quintic beam is achieved.
• The influences of system parameters on the quintic beam behavior are investigated.

This paper intends to promote the application of modern analytical approaches to the governing equation of transversely vibrating quintic nonlinear beams. Four new studied methods are Stiffness analytical approximation method, Homotopy Perturbation Method with an Auxiliary Term, Max–Min Approach (MMA) and Iteration Perturbation Method (IPM). The powerful analytical approaches are used to obtain the nonlinear frequency–amplitude relationship for dynamic behavior of vibrating beams with quintic nonlinearity. It is demonstrated that the first terms in series expansions of all methods are sufficient to obtain a highly accurate solution. Finally, a numerical example is conducted to verify the integrity of the asymptotic methods.