Nonlinear vibrations and buckling of a flexible rotating beam: A prescribed torque approach

This work is concerned with theoretical and numerical analyses of the nonlinear transverse free vibrations of a flexible rotating arm attached with a setting angle to a rigid hub. The system is assumed to be subjected to an external torque and a rotation dissipation load which collectively referred to as perturbations. A set of two strongly coupled second-order ordinary nonlinear differential equations, with an ignorable coordinate, is used to model the dynamics of the individual mode in plane flexural deflection of the arm and its overall rotational motion. A special coordinate transformation is used to demonstrate that for small perturbations the system model is essentially a slowly varying nonlinear oscillator where the arm single mode flexural vibration amplitude is described by a single degree of freedom nonlinear oscillator with an angular momentum parameter of the system that varies slowly in time according to a first-order differential equation. For the unperturbed system, the system angular momentum parameter, not the hub rotational speed, becomes a specified constant of motion. This use of the realistic torque-controlled approach allows for considering the beam deflections effect on the overall motion in addition to the overall motion effect on the beam deflections. A straightforward first mode stability analysis shows that at some critical value of the system angular momentum parameter, the straight beam configuration becomes unstable and undergoes a classic pitchfork bifurcation in which two new stable, simple buckling, symmetric dynamic equilibrium states appear. The structure of the corresponding one parameter Hamiltonian system phase space shows the existence of a family of homoclinic motions for the system angular momentum parameter above the critical value. Also, presented is an approximate expression for the fundamental natural frequency of the system obtained using a two-term harmonic balance (2THB) approach which shows a good agreement, after assuming a constant angular hub speed, with those available in other investigations. Examples of the results of numerical simulation over a selected range of system parameters, which show the effect of hub radius to arm length ratio, setting angle ψ and system angular momentum at the end of the starting torque period on the beam buckling behavior, are presented to verify the obtained theoretical ones.