Insightful and comprehensive formularization of frequency-amplitude formula for strong or singular nonlinear oscillators

The comprehensive formularization of the frequency-amplitude formula is the main interest in this work to cover higher powers of the restoring force which are not limited to cubic powers. Three equivalent styles of the generalized frequency amplitude have been performed. In addition, the restoring force is not restricted to an odd function in which the non-secular forces are included. Three forms of the non-secular forces have been formulated for the first time and treated as non-homogenous of the linearized Duffing oscillator. The modified style of He's formula has been applied to the singular oscillator using the enhanced potential function. The simplicity of the present approach provides extra advantages for high nonlinearity vibration. This method enriches the analysis with more details in new dimensions.