On the non-linear vibrations of an inextensible rotating arm with setting angle and flexible hub

This work is concerned with the development of a dynamic model for a slender flexible arm undergoing relatively large planar flexural deformations, and clamped with a setting angle to a compliant rotating hub. The model is derived assuming an inextensible Euler–Bernoulli beam, but takes into account the axial inertia and nonlinear curvature. Two coordinate transformations were used to obtain the inertial position vector of a typical material point along the beam span. The inextensibility constraint is used to relate the axial and transverse displacements as well as velocities of the material point in the beam body coordinate system. The displacement and velocity vectors obtained are used in the system kinetic energy and exact curvature to eliminate dependence of the system Lagrangian on the beam axial deflection and velocity. The assumed mode method is used to discretize the system Lagrangian and to derive, directly, the nonlinear equivalent temporal solutions to the problem. The resulting dynamic model is a set of four strongly coupled nonlinear ordinary differential equations, which, in the present work, is solved only numerically. Examples of the results of the numerical solutions for the effects of setting angle and defined physical parameter ratios on the system dynamic response characteristics for a sinusoidal hub torque profile are presented and discussed.