Modeling and Analysis of Spinning Disk-Spindle Vibration

Abstract

Abstract This work develops the equations of motion for a spinning disk-spindle system. Furthermore, a formulation of the governing equations in terms of extended operators reveals the classical gyroscopic nature of the system. The disk and spindle are modeled as elastic continua coupled by a rigid clamp. System kinematics and dynamics are derived for steady rotation about the spindle axis. The analytical model consists of three partial differential equations for the disk and spindle vibration coupled by four dynamic momentum balances for the clamp translation and rotation. The inherent structure of the system is clarified with the definition of extended operators that collect the component equations into a compact analytical form. The extended operators are easily identified as the classical inertia, elastic bending stiffness, gyroscopic, and rotational stiffness operators. These extended operators possess the symmetry and definiteness characteristics that define gyroscopic continua. Consequently, classical methods for gyroscopic systems are readily applied to disk-spindle systems. The disk-spindle eigensolution, closed-form modal analysis, and discretization procedures that follow naturally from the structured formulation are discussed.