Residual calculation in He’s frequency–amplitude formulation

He’s frequency–amplitude formulation and its modifications mainly depend upon the residual calculation. A suitable choice of a residual leads to an ideal result. This paper discusses some effective methods for the residual calculation, and a modification with a free parameter is suggested to effectively estimate the frequency of a nonlinear oscillator. Furthermore, an energy-based residual calculation is also suggested, which is deduced from a variational principle.