An equivalent linearization method for conservative nonlinear oscillators

Approximate analytical solutions for the oscillation of conservative nonlinear systems with odd nonlinearities are presented. The approximation is based on combining linearization of the governing equation with the analysis of the asymptotic behaviour of the exact period for small amplitudes. The approximate period for the nonlinear differential equation is the exact period for the equivalent linear differential equation and depends on one parameter obtained by comparing term-by-term the power-series expansions of the approximate and exact expressions for the period. We make use of the fact that the period of a conservative oscillator can be easily obtained and expressed as an improper integral. Four examples are presented to illustrate the usefulness and effectiveness of the proposed technique.