Towards the determination of a nonlinear Campbell diagram of a spinning shaft with non constant rotating speed

Overview & Synopsis

In the literature it is reported only the studies for the steady states of spinning shafts without any information of the shaft's dynamics during spin-up, spin-down operation. The derivation of the system's discrete equations of motion is required for the analysis of the behavior of spinning shafts during this type of operation. The equations describing the motion of a spinning shaft with nonconstant rotating speed forms a system of nonlinear Partial Differential Equations (PDEs) coupled with an IntegroDifferential (ID) equation which describes the rigid body motion of the shaft. Towards the determination of the Nonlinear Campbell diagram of the spinning shaft the equations has been discretized by projecting the dynamics to the infinite base of the linear mode shapes of the underlying linear PDEs and the resulted discrete equations forms a nonlinear system in a Non-Cauchy format. Further on, based on the derived formulation the Campbell diagram has been determined by considering constant rotating speed and the results have been compared with those obtained by Finite Element analysis and they are in very good agreement. This work paves the way for the determination of the Nonlinear Campbell diagram by considering the periodic motions of the derived nonlinear discrete system using either numerical (shooting method with continuation) or analytical techniques. It should be mentioned that the same approach can be used also for spin-up, spin-down dynamic analysis with the determination of Nonlinear Campbell diagram in case of rotating blades with nonconstant rotating speed with application in turbomachinery, wind turbines, pumps and many other mechanical applications with rotating components.