Stability and primary simultaneous resonance of harmonically excited non-linear spring pendulum system

摘要

The considered system is a harmonically excited, non-linear spring-pendulum, which simulates the ship roll motion. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the fourth order approximation. All possible resonance cases were extracted at this approximation order. This work is limited only to the primary resonance cases. The stability of the system is investigated using both frequency response equations and phase-plane method. The effects of the different parameters on system behavior are studied numerically. Variation of some equations parameters leads to the bending of the frequency response curves and hence to the jump phenomenon. It is quite clear that some of the simultaneous primary resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported.