Summary
Quasi-periodic solutions exist widely in nonlinear dynamical systems. Multi-harmonic balance method (MHBM) and variable-coefficient harmonic balance method (VCHBM) are two popular methods for determining the quasi-periodic solutions semi-analytically, where MHBM uses a multi-dimensional Fourier series and VCHBM a variable-coefficient Fourier series. In this work, the intrinsic relationship between the two methods is unveiled in the case for the quasi-periodic solutions consisting of two frequency components. The transform matrices between the algebraic equations of the Fourier coefficients of the two methods are explicitly derived. Particularly, the transform matrices are also applied to the transform of the derivative of Fourier coefficients for nonlinear terms between them. An example is given to demonstrate validity of the transformation between the two methods.
