Vibration and stability of high-speed spinning cantilevered beams with shear deformation effects and an attached rigid body at its tip

This work investigates the vibration, critical speeds, and high-speed instabilities in shear-deformable beams, spinning about their longitudinal axis, that have a rigid body attached to their tip. A model is derived using Hamilton’s principle. The equations are cast into extended operator form, which exemplifies the system’s gyroscopic structure and facilitates Galerkin discretization for numerical solution. Numerical results are calculated for systems with identical inertia and stiffness properties in the two bending directions (called a symmetric system) and non-identical inertia and stiffness properties in the two bending directions (called an asymmetric system). Both systems have forward and backward orbit vibrations in single-mode, free response. Symmetric systems have material points along the span that move in circular orbits. The orbits become elliptical for asymmetric systems. Symmetric systems have degenerate stationary-system natural frequencies that split for non-zero speeds. All eigenvalues cross, without interaction, as the rotation speed varies. The symmetric system eigenvalues are purely imaginary except at critical speeds. Asymmetric systems, due to their differing inertia and stiffness properties in the two bending directions, have distinct stationary-system eigenvalues, distinct critical speeds, and regions of divergence instability. Because of shear deformation effects, eigenvalue veering occurs when any decreasing forward orbit eigenvalue comes into close proximity with an increasing backward orbit eigenvalue. Within veering regions the modes couple, creating forward orbits in some segments of the beam span and backward orbits in others. Shear deformation effects also lead to flutter instability at high speeds. Atypical instability behavior occurs at high speeds, including immediate transitions between divergence and flutter instability. The results from this work could improve the high-speed performance of resonator devices like MEMS gyroscopes.