This study analyzed vibrations in an elastic beam supported by nonlinear supports at both ends under the influence of harmonic forces. We hypothesized that the elastic Bernoullis-Euler beam was supported by cubic springs to simulate nonlinear boundary conditions. We described the dynamic behavior of the beam using Fourier expansion and simulated mode shapes using the Bessel function. We then applied the Hankel transform to obtain particular (non-homogeneous) solutions. This study succeeded in describing the jump phenomenon of nonlinear frequency response, from which we derived the resonance frequency and the frequency region(s) of system instability. 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 vibration in the 1st mode and also widens the frequency region of system instability. This influence is less prominent in the 2nd mode, in which the largest amplitude was smaller than those observed in the 1st mode. The model and the analytical approach presented in this study are widely applicable to construction problems, such as shock absorber systems in motorcycles, suspension bridges, high speed railways, and a variety of mechanical devices. According to our results, system stability is influenced by the frequency regions of system instability as well as the nonlinear modulus of elastic supports.
Journal of Applied Mechanics and Technical Physics 56(2), pp.337-346