Non-linear dynamic model of an inextensible rotating flexible arm supported on a flexible base

A mathematical model for a flexible arm undergoing large planar flexural deformations, continuously rotating under the effect of a hub torque and supported by a flexible base is developed. The position of a typical material point along the span of the arm is described using the inertial reference frame via a transformation matrix from the body co-ordinate system, which is attached to the flexible root of the rotating arm. The condition of inextensibility is employed to relate the axial and transverse deflections of the material point, within the beam body co-ordinate system. The position and velocity vectors obtained, after imposing the inextensibility conditions, are used in the kinetic energy expression while the exact curvature is used in the potential energy. Lagrangian dynamics in conjunction with the assumed modes method is utilized to derive, directly, the non-linear equivalent temporal equations of motion. The resulting non-linear model, which is composed of four coupled non-linear ordinary differential equations, is discussed, simulated and the results of this simulation are presented. The effects of the base flexibility are explored by comparing the resulting simulation results, for various flexibility coefficients, with previously published works of the authors. Moreover, the numerical results show that the base flexibility has a very important effect on the stability of rotating flexible arms that should be accounted for when simulating such systems.