On the differential equation of a rapidly rotating slender rod

Abstract

In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod’s fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be “complete” in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these expansions across Stokes lines in the complex plane. A consistent approximation to the characteristic equation is obtained, and the limiting behavior of the spectrum is derived analytically.