Parametric excitation of a thin ring under time-varying initial stress; theoretical and numerical analysis

Initial stress in rings is one of the destructive effects which is almost inevitable due to various reasons such as being subsystems of a shrink-fitted joint, imperfections in the manufacturing, assembly or misalignment of the supporting mounts, and unbalancing in rotating condition. So, in this paper we focus on the effect of the initial stress and its variation with time on the dynamics of the pre-stressed ring. For this purpose, the equation of motion for in-plane bending vibration of a thin ring is derived using Hamilton's principle. It is assumed that the initial stress is due to the distributed radially time-varying pressure. By representing the dynamic initial stress in the coefficients of the equation of motion; the equation is converted to Mathieu's equation. The strained parameters method has been used to obtain the stability regions of motion and transition curves. Furthermore, to validate the obtained stability regions, numerical solutions of the equation of motion and Floquet theorem are used in some selected values of the parameters of the initial stress (magnitude of static and dynamic components of the initial stress). The fourth-order Runge-Kutta algorithm is used for numerical analysis of the equation of motion. The results show that the parameters of initial stress have direct impact on the stability of dynamic response. The obtained results from theoretical and numerical methods which are notably consistent with each other demonstrate that the initial stress, which has been almost always neglected in dynamic models of the systems, has a significant effect on the dynamics of the system, and it may even lead to an unstable dynamic response, while the excitation frequency is far enough from resonance region. So this paper can present the other application of modal analysis to non-destructive measure of initial stress.