Vibrations in an elastic beam with nonlinear supports at both ends

Vibrations in an elastic beam supported by nonlinear supports at both ends under the influence of harmonic forces are analyzed in this study. It is hypothesized that the elastic Bernoulli-Euler beam is supported by cubic springs to simulate nonlinear boundary conditions. The dynamic behavior of the beam is described by using the Fourier expansion and the Bessel functions. The Hankel transform is then applied to obtain particular (nonhomogeneous) solutions. This study succeeds in describing the "jump" phenomenon (instantaneous transition of the system from one position to another) of the vibrating system at certain frequencies. Models based on linear boundary conditions are unable to capture this phenomenon. A larger modulus of elasticity in nonlinear supports increases the frequency of unstable vibrations in the first mode and also widens the frequency region of system instability. This influence is less prominent in the second mode, in which the largest amplitude is smaller than those observed in the first mode.