Natural Frequency and Mode Shape of a Shaft With Elastically Mounted Rotary Inertia From Governing Equations With Dirac Delta Function

Abstract

This paper addresses the exact solution of a shaft’s free vibrations with a concentrated rotary load within its intervals. A new approach in deriving the governing equation of motion of such systems is demonstrated with the fundamental difference from the classic approach being that the dynamics of the concentrated rotary load is taken into account in the partial differential equation rather than the boundary conditions. The properties of Dirac delta functions are used to represent the concentrated loads. The Dirac delta function appears as a coefficient in the governing differential equations. The specific technique to solve such differential equations is presented. The solution derived using this technique is fundamentally identical to the solution of the classic method; however, the proposed approach offers a simplified and more straight-forward route to the derivation of the characteristic equation. As an example of the application of the proposed method, the characteristic equation, natural frequencies, and mode shapes of a shaft with an elastically attached flywheel are derived.