He's multiple scales method for nonlinear vibrations

He's multiple scales method is a couple of the homotopy perturbation method and the multiple scales technology in the classic perturbation method. This method has been proved to be a powerful mathematical tool to various nonlinear equations, and it is extremely effective for forced nonlinear oscillators. This paper shows that the method can be further improved by incorporating some known technologies, e.g., the parameter-expanding technology, the enhanced perturbation method and the homotopy perturbation method with an auxiliary term. Due to the wide application of the homotopy perturbation method, He's multiple scales method cleans solutions of nonlinear equations while the classic perturbation method fails.