Nonlinear Dynamics of a Spinning Shaft with Non-Constant Rotating Speed

Abstract

Reported research on spinning shafts is mostly restricted to cases of constant rotational speed without examining the dynamics thatoccursduringtheir spin-up or spin-down operation. In suchcases, the motion is described by a nonlinear system of Partial Differential Equations (PDEs) coupled with an Integro-Differential Equation (IDE). The nonlinear system of PDEs with IDE, projected onto the infinite basis of the modes of the underlying linear system, results in a system of nonlinear Ordinary Differential Equations (ODEs). In this articleis appliedthe multiple scales perturbation method for dynamic analysis and the system in first order approximation takes the form of two coupled sets of pairedequations. The first pair describes torsional and rigid body rotation whilst the secondconsists of the equations describing the two lateral bending motions. Although in this system non-conservative forces are not considered in terms of damping or explicit externally applied load (torques/forces), the solution of the Is'order approximation of the first set of equations indicates that there are no periodic motions. The solution of the second set of equations of 1st order approximation coincides with the case of constant rotating speed. It isshown, that the Normal Modes in bending motions are the critical speeds of the shaft. It is shown that the frequencies in the Campbell diagram coincide with the frequencies associated with the 1st order solution of the nonlinear system. Moreover, the analytical solution of the first pair of equations is in good agreement with direct numerical simulations. This work paves the way for the development of the Nonlinear Campbell diagram that can be used to determine the dynamic behaviour of rotating structures during spin-up or spin-down operation.