Analytical Bifurcation Tree of Period-1 to Period-4 Motions in a 3-D Brushless DC Motor with Voltage Disturbance

In this paper, the bifurcation trees of analytical solutions of a 3-D brushless DC motor with the voltage disturbance are obtained through the generalized harmonic balance method. The electrical and mechanical model of the 3-D brushless motor is transformed to the dynamic system of coefficients of finite Fourier series. Stable and unstable analytical solutions of the 3-D brushless motor are solved based on such a Fourier series coefficient system. Bifurcation trees of analytical solutions of period-1 to period-2 and period-1 to period-4 motions are achieved. Stability and bifurcations of the analytical solutions of the 3-D brushless motor are determined by the eigenvalues of Jacobian matrix of the coefficient dynamic system. Frequency-amplitude characteristics of periodic motions are presented for a better understanding of the motion complexity in frequency domain. Numerical illustrations are completed for comparison of the analytical solutions with numerical results. The complex dynamics of the 3-D brushless motor are exhibited through the bifurcation trees of analytical solutions.