Viscoelastic modelling of rotor—shaft systems using an operator-based approach

Abstract

Damping exists in every material in varying degrees, so materials in general are viscoelastic in nature. Energy storage, as well as dissipation in varying degrees, accompanies every time-varying deformation, with the effect that stress and strain in a material get out of phase. This work presents the development of equations of motion of a rotor—shaft system with a viscoelastic rotor after discretizing the system into finite elements. Subsequently, these equations are used to study the dynamics of the rotor—shaft system in terms of stability limit of spin speed and time response of a disc as a result of unbalance. The primary inspiration for a viscoelastic model arises from the need to capture the influence of broad band spectral behaviour of rotor—shaft materials, primarily polymers and polymer composites, which are principally the materials of light rotors, on the dynamics of rotor—shaft system. For this, the material constitutive relationship has been represented by a differential time operator. Use of operators enables one to consider general linear viscoelastic behaviours, represented in the time domain by multi-element (three, four, or higher elements) spring—dashpot models or internal variable models, for which, in general, instantaneous stress and its derivatives are proportional to instantaneous strain and its derivatives. Again such representation is fairly generic, in a sense that the operator may be suitably chosen according to the material model to obtain the equations of motion of a rotor—shaft system. The equations so developed may be easily used to find the stability limit speed of a rotor—shaft system as well as the time response when the rotor—shaft system is subjected to any dynamic forcing function.