The effect of quintic nonlinearity on the investigation of transversely vibrating buckled Euler-Bernoulli beams

A new formulation of vibrations of the axially loaded Euler-Bernoulli beam with quintic nonlinearity is investigated in the present study. The beam nonlinear natural frequency as a function of the initial amplitude is obtained. In this direction, modern powerful analytical methods namely He's Max-Min Approach (MMA) and Amplitude-Frequency Formulation (AFF) are employed to approximate the frequency-amplitude relationship of the beam vibrations. Afterwards, it is clearly shown that the first term in the series expansions is sufficient to produce a highly accurate approximation of the nonlinear system. Finally, preciseness of the present analytical procedures is evaluated in contrast with numerical calculation methods.