A comprehensive study of the non-linear euler-bernoulli rotating beam

Abstract

The main goal of this work is to analyze the behavior of Euler-Bernoulli rotating beam with constant angular velocity around Z-axis. Our dynamic model is analyzed considering only the in-plane transverse deformation. Thus the model is defined by an ordinary differential equation with one degree of freedom (O.D.F) and the governing differential equation is Duffing's equation with a variable coefficient for the linear term. When the damping coefficient and the amplitude of the excitation force are zeros, the system is autonomous with an explicitly known homoclinic orbit. The homoclinic orbits is calculated. Melnikov function due to the homoclinic orbit is calculated to detect the transverse homoclinic orbit. Also a dynamic numerical simulation methods are used to obtain the time history, phase portrait, Laypunov exponent, power spectrum, Poincare’ maps and their fractal dimensions. A comparison has been done between this O.D.F. model and the 2nd D.F. model that we have analyzed in [23]. The numerical results showed the occurrence of regular motion and chaotic motion in both models. Thus, by reducing the order of the two-coupled Duffing's equations into an ordinary differential equation with O.D.F. a more simplified model for the flexible rotating beam is achieved.