Nonlinear Oscillation of Electron on Axis of Charged Ring

The nonlinear oscillation of electron on the central axis of charged ring is studied.After setting up the motion equation,the exact solution and the period of free oscillation are achieved with elliptic integrals.The nonlinearity of oscillation are discussed and illustrated.Truncating expanded series leads to Duffing equation.The approximate expressions of solution and period are gained by Duffing oscillation and harmonic oscillation,which are compared with the exact ones.The appling significance of nonlinear oscillation solution in engineering is indicated.