Study On Coupled In-plane And Transverse Vibration And Rigid-elastic Analysis Of A Variable Accelerated Flexible Spinning Annular Disk

The modeling, theoretical formulation and dynamical response of flexible spinning disc are described in this paper. Based on the Kirchhoff plate theory and the von Karman strain theory, considering the angular acceleration of the disc and the geometric nonlinearity of the displacements, the nonlinear displacement-strain relations, strain-stress relations and internal force in xyz coordinate system are derived, then transform to rθz coordinate system. Dynamics stiffening effects are taken into account, from the expressions of potential and kinetic energies of the system, Hamiltion's principle is employed to obtain non-linear governing partial differential equations of motion, which are not only coupled equations between the large rigid-body motion and small elastic deformation encountered in constrained multibody systems, but also coupled equations between the radial, tangential and transverse displacements.The non-linear governing partial differential equations of motion are discretized by using assumed modes method. First, the full nonlinear equations are simplified to simpler static state in-plane and transverse vibration differential equations neglecting the rotatory inertia effect and the spinning velocity of the flexible disc. Then the simpler equations are solved, from the boundary conditions and the normalizing conditions, the first four natural frequency and assumed modes functions of radial, tangential and transverse vibration are obtained through modal truncation. Assumed modes functions of in-plane vibration are linearity superposition of the first order Bessel functions and the second order Bessel functions, assumed modes functions of transverse vibration are linearity superposition of the first order Bessel functions , the second order Bessel functions and distortional the first order Bessel functions, distortional the second order Bessel functions.From the discretized equations, the dynamics responses are computed by Wilson-θmethod.