Semi-Analytical Model to Study Vibrations of High-Speed, Rotating Axisymmetric Bodies Coupled to Other Rotating/ Stationary Structures
The vibration of complex mechanical systems that include coupled rotating and stationary bodies motivates this work. A semi-analytical model is developed for high-speed, compliant, rotating bodies. Exploiting the axisymmetry of the rotating body, the developed semi-analytical model only discretizes the two-dimensional radial cross-section; Fourier series are used in the circumferential direction. The corresponding formulation for thin-walled, axisymmetric shells is given. Even though the body is axisymmetric, its deflection as well as external forces, constraints, and supports acting on the body are allowed to be asymmetric. These asymmetric elements can be stationary or rotating. The model includes Coriolis and centripetal effects. The prestress (or geometric) stiffness matrix that arises from external forces and constant centripetal acceleration has additional terms compared to the literature, and these terms can significantly change the natural frequencies. Discrete stiffness-damper elements, elastic foundations, and constraint equations are used to couple the rotating body to other rotating and stationary bodies. The model is developed in a stationary reference frame to avoid time-dependent coefficients in the equations of motion when coupled to stationary components. Surface constraints are developed using equivalent force relations between multiple points on the surface and a reference node. Discrete stiffness-dampers, asymmetric elastic foundation, and asymmetric constraints introduce non-axisymmetry in the system. The speed-dependent natural frequencies and complex-valued vibration modes, presence of multiple Fourier harmonics in each mode, changes to critical speeds, divergence and flutter instability phenomena, and eigenvalue veering are investigated for spinning systems with asymmetric features. The developed semi-analytical model is used for rotationally periodic systems, for example, planetary gears. Rotationally periodic systems consist of multiple vibrating, rotating central components and substructures. The model is developed in a reference frame rotating with the central component that supports the substructures. Structured modal properties of the cyclically symmetric systems and diametrically opposed systems are investigated. The modes of the spinning system are categorized into translational-tilting, rotational-axial, and substructure modes. Time-varying coupling elements act as parametric excitation in the system. Large strain energy in the coupling elements lead to large parametric instability regions. The analytical closed-form expression of the parametric instability bandwidth obtained using a perturbation method compares well with numerical results from Floquet theory.