Improved Forced Response Analysis via Frequency Detuning in the Parameter-Splitting Multiple-Scales Method

The parameter–splitting multiple–scales (PSMS) method is widely used to analyze strongly nonlinear mechanical systems and offers high accuracy. However, it often demands considerable computational resources, particularly as the number of splitting parameters increases. This paper presents a detuned parameter-splitting multiple-scales (d-PSMS) method, which incorporates frequency detuning from the direct normal form (DNF) approach into the PSMS framework. The d-PSMS method significantly improves computational efficiency compared with the classical PSMS method while retaining comparable accuracy. It also provides notably higher accuracy than the conventional detuned multiple-scales (d-MS) method. Comparative studies including two single-degree-of-freedom oscillators and a two-degree-of-freedom system with one-to-three internal resonance based on frequency-amplitude responses obtained using PSMS, d-MS, numerical continuation, and the d-PSMS method show that the proposed approach achieves a favorable balance between efficiency and accuracy. The d-PSMS method thus serves as a practical tool for analytically approximating of forced responses in nonlinear mechanical systems.