Abstract

The transverse dynamics of a high speed spinning disc which is clamped at the inner radius and rotating with a time-varying spin rate is examined in a space fixed frame of reference. The general nonlinear equations of motion governing the disc dynamics are systematically derived using displacement as well as stress function formulations. These equations of motion include the effects due to inherent bending rigidity, membrane stresses arising from centrifugal forces, non-axisymmetry of the in-plane and transverse displacements, geometric nonlinearities, aerodynamic damping arising from air stationary and moving with respect to the disc, parametric excitation due to time varying spin rate, etc. For the constant rotation case, the linearized equations of motion are solved by taking both membrane as well as flexural stiffness effects into account. This leads to a power series solution for the radial shape functions and harmonic solutions for the circumferential shape functions. The two-dimensional eigen-functions thus obtained can describe a disc mode with any number of nodal diameters and nodal circles, and the resulting eigen-frequencies match very well with the numerical results. The nonlinear and non-axisymmetric in-plane response is also determined, and a 2-DOF system of ODEs is obtained which governs the dynamic variation of the amplitudes of traveling waves associated with the dominant mode of the transverse motion.

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