FREE NON-LINEAR VIBRATION OF A ROTATING THIN RING WITH THE IN-PLANE AND OUT-OF-PLANE MOTIONS

Free non-linear vibration of a rotating thin ring with a constant speed is analyzed when the ring has both the in-plane and out-of-plane motions. The geometric non-linearity of displacements is considered by adopting the Lagrange strain theory for the circumferential strain instead of the infinitesimal strain theory. By using Hamiltonʹs principle, the coupled non-linear partial differential equations are derived, which describe the out-of-plane bending and torsional motions as well as the in-plane bending and extensional motions. During deriving the equations of motion, we discuss how to model the circumferential stress and strain in order to consider the geometric non-linearity. Four models are established: three non-linear models and one linear model. For the four models, the linearized equations of motion are obtained in the neighbourhood of the steady state equilibrium position. Based on the linearized equations of the four cases, the natural frequencies are computed at various rotational speeds and then they are compared. Through the comparison, this study recommends which model is appropriate to describe the non-linear behaviour more precisely.