A Brief Study on Dynamics of Viscoelastic Rotors – An Operator Based Approach

Abstract

Viscoelasticity, as the name implies, is a property that combines elasticity and viscosity or in other words such materials store energy as well as dissipates it to the thermal domain when subjected to dynamic loading and most interesting the storage and loss of energy depends upon the frequency of excitation. Modeling of viscoelastic materials is always difficult whereas modeling the elastic behavior is easy, modelling the energy dissipation mechanism possess difficulty. This work attempts to study the dynamics of a viscoelastic rotor-shaft system considering the effect of internal material damping in the rotor. The rotation of rotors introduces a rotary damping force due to internal material damping, which is well known to cause instability in rotor-shaft systems. Therefore, a reliable model is necessary to represent the rotor internal damping for correct prediction of stability limit of spin speed and unbalance response amplitude of a rotor-shaft system. An efficient modelling technique for viscoelastic material, augmenting thermodynamic field (ATF) has been found in literature.
Here the material constitutive relationship has been represented by a differential time operator. Use of operators enables to consider general linear viscoelastic behaviours, represented in the time domain, for which, in general, instantaneous stress and its derivatives are proportional to instantaneous strain and also its derivatives. The operator may be suitably chosen according to the material model. The constitutive relationships for ATF approach is represented in differential time operator to obtain the equations of motion of a rotor-shaft system after discretizing the system using beam finite element method. The equations thus developed may easily be used to find the stability limit of spin speed of a rotor-shaft system as well as the time response as a result of unbalance when the rotor-shaft system is subjected to any kind of dynamic forcing function.
In this work dynamic behavior of an aluminium rotor is predicted through viscoelastic modelling of the continuum to take into account the effect of internal material damping. To study the dynamics of an aluminium rotor-shaft system stability limit of spin speed, unbalance response amplitude and time response are used as three indices. It is observed that, the operator based approach is more suitable for finding the equation of motion of a viscoelastic rotor which is used to predicts the dynamic behaviour of that continuum.