Abstract: The dynamics model and the motion of string-bound two-body particles are studied.By using the Taylor series expansion method and the Runge- Kutta numerical method，the approximate and numerical solution of differential equations are obtained，respectively.Calculations of the dynamic equation give the evolution of position vector as a function of time，motion trajectory and phase diagrams. The motion of particle depends on the initial position and transverse velocity.For a larger or smaller initial transverse velocity，the motion is non-linear oscillation. There is a critical initial velocity at which the two-body system is in mechanical equilibrium，and will perform harmonic vibration under small perturbation.The results are valuable for teaching and research on the mechanics 　　problems of micro-vibrations.

Key words: string - bound, non - linear oscillations, numerical solution, analytical solution, central force