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### The Timescale Function Method for Solving Free Vibration of Nonlinear Oscillator

#### Abstract

new method for obtaining an approximate solution of free vibration with high accuracy for nonlinear oscillators with strong nonlinearity is introduced. By using the first integral of motion the exact free vibration frequency is obtained. The nonlinear dynamical equation can be converted into first order differential equation, named timescale differential equation, which is derived from equation of phase trajectory by expressing the free vibration solution in a harmonic function of timescale function, viz. x = Acos τ (t) . It is found that timescale function should contain linear and periodical terms of time. A fit expression of timescale function is selected consequently, in which undetermined coefficients are fixed by solving conditions. Then a new type of free vibration solution was gained, which is error-free at the equilibrium and limit displacement points. Since no approximate hypothesis relating to the weak nonlinearity is adopted in this method, the solution has high accuracy, for quintic oscillator, the maximum relative error of free vibration displacement is less than 0.25%.