Nonlinear vibrations and buckling of a flexible rotating beam: a prescribed torque approach (Q886360)

The authors investigate the dynamic behavior of a flexible beam attached with a setting angle to a rotating rigid hub. The rotating beam is described by a strongly coupled two-degree-of-freedom nonlinear model with an ignorable coordinate, derived in [\textit{M. N. Hamban} and \textit{A. H. El-Sinawi}, J. Sound Vib. 281, 375--398 (2005)]. The degrees of freedom in this model are the arm first mode in-plane flexural vibration amplitude and overall angle velocity. A special coordinate transformation is used to demonstrate that for small perturbations (e.g. small net applied torque) the system model is essentially a single-degree-of-freedom (SDOF) nonlinear oscillator with an angular momentum parameter that varies slowly in time according to a first-order differential equation. On the other hand, for the unperturbed (e.g. the net applied torque is zero) system the model becomes a SDOF nonlinear oscillator where the angular momentum parameter, not the hub rotational speed, is a specified constant of motion. Phase plane portraits and a straightforward stability analysis of equilibrium solutions of the unperturbed system are carried out to determine the critical value of angular momentum parameter for the existence of homoclinic motions. An expression for the first natural frequency of the 2DOF unperturbed system is obtained using the approximate analytic two-term harmonic balance method, and is compared with results available in the literature. The authors present also some results obtained by numerically integrating the autonomous SDOF model for selected values of system parameters, e.g. hub radius to arm length ratio, setting angle and system angular momentum at the end of the starting torque period. Analytical and numerical investigations of dynamic behaviour and chaotic motions in the nonautonomous case are left for a future undertaking.