Dynamics of a Euler–Bernoulli beam on nonlinear viscoelastic foundations: a parameter space analysis

Abstract

The dynamics and spectrum of vibrations of an uniform Euler–Bernoulli beam on nonlinear viscoelastic foundations are investigated in parameter space. An alternative method of approach based on mean-field approximation is proposed. Spatial and temporal solution are obtained in an interactive way. The spectrum is reached directly from spatial solutions. It was shown that the system exhibits sudden birth and death of chaotic attractors (crises) and that there are periodic windows immersed in chaotic regions. It was determined the regions in the parameter space where chaotic and regular attractors occur. Moreover, the model presents the coexistence of many attractors. The numerical simulations found geometrical structures known as shrimps, and these periodic structures are involved by chaotic regions. In this work it was investigated the dynamics of a beam of Euler–Bernoulli with viscous and dynamic support. It is presented, then, a model to treat this type of problem, which is nonlinear and non-autonomous. The dynamics of the model presents a complex behavior, being in some cases sensitive to the initial conditions. Changing properly the parameters the dynamics goes from chaotic behavior to periodic regime. As the system is forced and dissipative, complex structures in the parameter space, known as shrimps, arise. Also, it was found that the spatial solution is not sensitive to the dynamics, although they are coupled. However, the dynamics is very sensitive to the mean quadratic deformation of the beam and, therefore, it is related to the vibration modes obtained in the spatial solution.