Dynamics of an Unbalanced Shaft Interacting with a Limited Power Supply

Abstract

Rotation of an elastically mounted unbalanced shaft may be in general affected by its lateral vibration, due to a “vibrational torque”. This interaction is nonlinear, and can be neglected only in case of an unlimited power supply. Whenever the available power of the drive is comparable with power consumption due to vibration, various nonlinear phenomena may be observed, the most well-known of these being the so called Sommerfeld effect – slowing down or complete capture of the shaft at resonance. The corresponding steady-state motions and their stability can be studied by asymptotic methods, as applied to the governing nonlinear set of two second-order equations. However, study of transient motions in general requires numerical solution. This numerical solution is obtained here, and extensive parametric studies are performed of the Sommerfeld effect. In particular, the influence is evaluated of damping ratio and slope of the torque-speed curve of the drive on a passage/capture threshold. The results of numerical simulation, as well as experiments with a physical model, also demonstrate the effect of smooth passage through resonance with a limited power supply, based on using a “switch” of suspension stiffness from a certain artificially increased value to the design one. A brief description is presented also of time-variant components of the resonant amplitude and rotational frequency responses in the case of capture, as observed both in numerical simulation studies and in experiments. Whilst these components are small compared with the corresponding constant ones, i.e. steady-state vibration amplitude and rotational frequency, the above observations indicate the possibility for periodic or chaotic nonstationarity in the system's response.