Elliptic harmonic balance method for two degree-of-freedom self-excited oscillators

Elliptic harmonic balance (EHB) method as an analytical method is widely used for strongly non-linear oscillators with a single degree-of-freedom (DOF). The oscillation equations are expressed by elliptic functions, and then the expressions are expanded as harmonics of elliptic type while only the first harmonic is retained. To the best of our knowledge, however, it seems that the EHB method has not been found applications in two or multiple DOFs systems. One possible reason is that the EHB method may cause a difficult problem that the number of equations obtained by harmonic balancing is not equal to that of unknowns. To this end, in the present paper, the EHB method is therefore extended to study a class of strongly self-excited oscillators with two DOFs. Prior to harmonic balancing, an additional equation is established to tackle the aforementioned problem. Illustrative examples show that solutions of limit cycles obtained by the proposed method are in good agreement with the numerical solutions.