Based on the Lagrange function, a new nonlinear relative rotational equation with Davidenkov hysteresis is established. Firstly, the bifurcation characteristics of the hysteretic relative rotational autonomous function and non-antonomous function are discussed. Secondly, the approximate solution of the nonlinear function under periodic force excitation is obtained by KBM method. At last, by numerical simulation, several bifurcation structures are obtained, and the comparisons result indicate the approximate solution of KBM method has higher accuracy and better reflects the dynamic characteristic of equation effectively.