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An initial-value transformation approach is presented for strongly nonlinear vibratory systems.The periodic solutions of a nonlinear vibrator can be expressed in the form of basic harmonics and bifurcate harmonics.Thus, A nonlinear oscillation system which is described as a second order ordinary differential equation, can be expressed as a set of non-linear algebraic equations with an angular frequency, amplitudes as the independent variables using Ritz-Galerkins method.