The effect of rotatory inertia on the frequency and normal mode equations of non-uniform rotating beams

This paper presents a dynamic model for the vibration of rotating tapered beams including rotary inertia. The governing differential equations of motion of the beam in free vibration are derived using Lagrange’s equations and include the effect of an arbitrary hub radius. Three linear partial differential equations are derived. Two of the linear differential equations are coupled through the stretch and chordwise deformation, the other equation is an uncoupled one for the flapwise deformation. An approximate method based on the Rayleigh-Ritz solution is proposed to solve the natural frequency of very slender rotating beam at high angular velocity. The parameters for the hub radius, rotational speed, tapered ratio, rotary inertia and slenderness ratio are incorporated into the equation of motion. The theory is valid for a wide range of applications in rotating machinery design