Suppression of Sommerfeld effect in a non-ideal discrete rotor system with fractional order external damping

Abstract

Sommerfeld effect often causes dynamic instability in high speed rotors driven through energy source with limited power. As a result, the system dynamics changes in a very unusual manner while exceeding a critical value of input power with sudden nonlinear jump of rotor speed proximity to the system resonance. Such instability phenomena arising out of strong nonlinear rotor-drive interactions need to be attenuated to ensure smooth operation and industrial safety. A novel approach is presented here to study the attenuation of this kind of instability in an internally damped DC motor driven discrete rotor system with the adjustment of fractional order parameter in its external damping term. Using Lagrangian formulation, the equations of motion of the discrete rotor system are derived. The well-known Caputo model is adopted for the fractional order external damping modelling. Following, a characteristic equation in the form of a fifth order polynomial is obtained through steady-state energy balance approach which leads to obtain various amplitude frequency responses with a few values of fractional order of the external damping. As the fractional order increases gradually, the Sommerfeld effect is found to be attenuated accordingly. A few steady-state results are found to be in good agreement with some results reported earlier. Additionally, root-loci technique is employed to obtain a specific value of fractional order for which the complete disappearance of Sommerfeld effect is achieved through the merging of the break points. Finally, a transient analysis also confirms the steady-state results numerically.