Instability of asymmetric shaft system

• The parametric equations of asymmetric shafts in inertial frame are formulated.
• Analytical expression of stability boundary is derived by a perturbation method.
• Stability boundary is presented in terms of non-dimensional parameters.
• Combined effect of bearing stiffness and mass on stability is presented.
• Material, geometry and bearing selection guidelines for stability are inferred.

In the present work, parametric instability of asymmetric shaft mounted on bearings is studied. Towards this end, four different models of increasing complexity are studied. The equations corresponding to these models are formulated in the inertial reference frame. These equations involve a periodically varying coefficient. This is similar to classical Mathieu equation but in a multi-degree of freedom context. As such, under suitable parameter combination these systems result in growing oscillation amplitudes or instability. For wider generalization, the equations and results are presented in a non-dimensional form. The unstable parameter regimes are found using the Floquet theory and perturbation methods. These results are also corroborated with existing results in the literature. The nature of the stability boundary and its dependence on various system parameters is discussed in elaborate detail. The stability boundary can be used to determine unstable operating speed ranges for different asymmetric shaft cross-sections. Further, material, geometry and bearing selection guidelines for ensuring stable operations can be inferred from these results.