# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 4 Numerical Illustrations
The harmonic amplitudes expressed in the frequency spectrum are \(a_{01}=\), \(A_{1}\approx 2.35206\), \(A_{3}\approx 0.35233\), \(A_{5}\approx 0.04062\), \(A_{7}\approx 4.88\)\(\times\)\(10^{-3}\), \(A_{9}\approx 5.89\)e \(\times\)\(10^{-4}\), \(A_{11}\approx 7.1\)\(\times\)\(10^{-5}\), \(A_{13}\approx 8.6\)\(\times\)\(10^{-6}\), and \(A_{15}\approx 1.03\times 10^{-6}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Figure No.** & \(\Omega\) & **Initial Condition** & **Types** \\ \hline Figure 6a,b & 6.5 & (5.3227832, 65.525293, 19.302422) & stable P-1 \\ Figure 6c,d & 6.5 & (9.7357535, 55.891501, 13.299756) & stable P-2 \\ Figure 6e,f & 6.5 & (7.5583327, 63.739675, 18.706304) & unstable P-1 \\ Figure 7a,b & 6.46 & (10.279434, 58.625938, 15.979369) & stable P-4 \\ Figure 7c,d & 3.16 & (0.8259358, 53.755447, 0.2334629) & stable P-1 \\ Figure 7e,f & 3.14 & (1.2720598, 52.714859, 1.8667075) & stable P-2 \\ \hline \end{tabular}
\end{table} 
Table 2: Input data for numerical illustrations. Parameters: (\(v_{q}=0.168\), \(\rho=60\), \(Q_{0}=10\), \(\delta=0.875\), \(v_{d}=20.66\), \(\sigma=4.15\), \(\eta=0.26\), \(\overline{T}_{L}=0.53\)). Figure 6: Dynamic characteristics of periodic motion when \(\Omega=6.5\): **(a,b)** stable P-1 motion with (\(x_{1}=5.3227832\), \(x_{2}=65.525293\), \(x_{3}=19.302422\), \(m=1\)); **(c,d)** stable P-2 motion with (\(x_{1}=9.7357535\), \(x_{2}=55.891501\), \(x_{3}=13.299756\), \(m=1\)); **(e,f)** unstable P-1 motion with (\(x_{1}=7.5583327\), \(x_{2}=63.739675\), \(x_{3}=18.706304\), \(m=1\)). Parameters: (\(v_{q}=0.168\), \(\rho=60\), \(Q_{0}=10\), \(\delta=0.875\), \(v_{d}=20.66\), \(\sigma=4.15\), \(\eta=0.26\), \(\overline{T}_{L}=0.53\)). Figure 7a,b represents the dynamic characteristics of stable period-4 motion when \(\Omega=6.45\). The initial conditions for numerical integration are \(x_{1}=10.279434\), \(x_{2}=58.625938\), and \(x_{3}=15.979369\). The analytical solution for this period-4 motion is based on 80 harmonic terms. In 3D displacement, the analytical and numerical solutions match very well. The motion repeats every four excitation periods, which indicates that the motion of the system  
is period-4 motion. The harmonic amplitudes are \(a_{01}\approx 0.1808\), \(A_{1/4}\approx 0.0306\), \(A_{1/2}\approx 0.3646\), \(A_{3/4}\approx 0.0738\), \(A_{1}\approx 6.3280\), \(A_{5/4}\approx 0.0758\), \(A_{3/2}\approx 0.9903\), \(A_{7/4}\approx 0.2108\), \(A_{2}\approx 3.8709\), \(A_{9/4}\approx 0.1220\), \(A_{5/2}\approx 1.3008\), \(A_{11/4}\approx 0.2486\), \(A_{3}\approx 1.6423\), \(A_{13/4}\approx 0.0653\), \(A_{7/2}\approx 0.9600\), \(A_{15/4}\approx 0.1633\), \(A_{4}\approx 1.6915\), \(A_{17/4}\approx 0.0312\), \(A_{9/2}\approx 0.7286\), \(A_{19/4}\approx 0.1521\), \(A_{5}\approx 0.3028\), and \(A_{k/m}<1\times 10^{-1}\) (\(k=21\)-\(80\); \(m=4\)). The dynamic characteristics of stable period-1 motion are represented in Figure 7c,d when the disturbance frequency is 3.16. The initial conditions for a numerical solution are \(x_{1}=0.8259358\), \(x_{2}=53.755447\), and \(x_{3}=2.0334629\). The number of harmonic terms increases with a lower disturbance frequency. Therefore, the analytical solution for this period-1 motion is based on 40 harmonic terms. It can be inferred from the 3D displacement that the numerical solution is almost the same as the analytical solution. There is one clear loop in 3D displacement, which indicates that the motion is stable period-1 motion. The harmonic amplitudes are \(a_{01}\approx 0.0557\), \(A_{1}\approx 3.4044\), \(A_{2}\approx 0.4410\), \(A_{3}\approx 6.0229\), \(A_{5}\approx 0.9966\), \(A_{6}\approx 0.8270\), \(A_{7}\approx 2.3835\), \(A_{8}\approx 0.5349\), \(A_{9}\approx 0.6293\), \(A_{10}\approx 0.4421\), \(A_{11}\approx 0.4750\), \(A_{12}\approx 0.2319\), \(A_{13}\approx 0.2430\), \(A_{14}\approx 1570\), and \(A_{k/m}<1\times 10^{-1}\) (\(k=15\)-\(40\); \(m=1\)). Figure 7e,f describes the dynamic characteristics of stable period-2 motion when \(\Omega=3.14\).