Analysis of collision vibration in two-degree-of-freedom system without damping subjected to periodic excitation (Derivation of analytical solutions for 2nd order super harmonic resonance)

Collision vibration systems are usually modeled as a nonlinear spring whose characteristics are described by the broken line model. These systems are called piecewise-linear systems. A piecewise-linear system is highly nonlinear, and it is usually difficult to predict the system response using any general analytical solution. If the effects of design parameters such as clearance size and dynamic nonlinearity of the systems are known, the structures can be designed to be safer and more comfortable. This paper deals with forced collision vibration in a mass-spring system for two-degree-of-freedom. The analytical model is mass-spring system having two masses in which one mass is subjected to an exciting vibration with arbitrary functions. Then the restoring force, which has characteristics of an asymmetric piecewise-linear system, collides elastically to another mass when amplitude of the mass increases farther than clearance. In order to analyze resulting vibration for the super harmonic resonance, the Fourier series method is applied and analytical solutions for this system are derived. Next, following the analytical solutions, numerical calculations are performed, and the resonance curves are constructed by using resulting vibration. Effects of amplitude ratio of excitation, nonlinearity of the system and mass ratio on the resonance curves are shown numerically. For verification of the analytical solutions, numerical simulations are performed by the Runge-Kutta method, and numerical results based on analytical solutions are compared with numerical simulation results on the resonance curves. The analytical results are in a fairy good agreement with the numerical simulation results.