Analytical Vibration of Spinning, Elastic Disk-Spindle Systems

This work develops the dynamic equations of motion for a spinning disk-spindle system and casts them in a structured formulation that reveals the classical gyroscopic nature of the system. The disk and spindle are modeled as elastic continua coupled by a rigid, three-dimensional clamp. The inherent structure of the system is clarified with the definition of extended operators that collect the component disk, spindle, and clamp equations of motion into a compact analytical form. The extended operators are easily identified as the inertia, elastic bending stiffness, gyroscopic, and rotational stiffness operators, and they possess the symmetry and definiteness characteristics that define gyroscopic continua. Consequently, well-known analytical methods for gyroscopic systems are readily applied to disk-spindle systems. Qualitative eigensolution properties, an exact, closed-form response analysis, and the Galerkin discretization that follow naturally from the structured formulation are discussed. A free and forced vibration example is presented.