A State Space Viscoelastic Shaft Finite Element for Stability and Response Analysis of Rotors with Structural and Frequency Dependent Damping

Abstract

The present work proposes a new Euler–Bernoulli shaft element for structurally damped and viscoelastic rotors in a spinning frame. The Maxwell–Wiechert linear viscoelastic material model, having one elastic branch and several parallel Maxwell branches, is used. This model introduces additional internal displacement variables between elastic and viscous elements in the Maxwell branches. Here, the stress depends not only on the elastic strain and elastic strain rate but also on additional fictitious strains and their rates. In the present work, it is assumed that these additional strains can be derived from continuous fictitious displacement variables in the same way as the elastic strains are derived from the actual displacement variables. These continuous fictitious displacements in turn are interpolated from their nodal values using the conventional beam shape functions. Therefore, in addition to the standard degrees of freedom, extra degrees of freedom are defined at the nodes. The viscoelastic shaft element is then used in time-domain analysis of rotors with structural damping or frequency-dependent damping. Parameters of the Maxwell–Wiechert model are so selected that they appropriately represent structural damping of a mild steel rotor and frequency-dependent storage modulus and loss coefficient of a typical viscoelastic rotor. Both stability and time response analyses are performed. The results obtained through dynamic analysis of two different rotor models using both structurally damped mild steel and a typical viscoelastic material PPC as shaft material are similar to those available in the literature and justify the methods applied.