## 5 Numerical Results and Comparison  
To compare the modal discretization and spatial discretization techniques, a simply supported beam with ten uniformly distributed gyroscopes was studied as a demonstrating example. The length, density, cross section radius, Young's modulus, and shear modulus were 10 m, 1200 kg/m\({}^{3}\), 0.1 m, \(7.84\times 10^{6}\) Pa, \(2.667\times 10^{6}\) Pa, respectively. The length, density, inner and outer radius for the each gyroscope were 0.082 m, 8000 kg/m\({}^{3}\), 0.1 m, 0.2 m, respectively. In Figure 4, the first four pairs of natural frequencies computed by 121 order modal discretization and 121-element spatial discretization are presented with varying angular momentum of the uniformly distributed gyroscopes. With the supplement of the gyroscopes, any one of the natural frequencies, denoting the planar modes, bifurcates into two, denoting the lower backward whirling (BW) and the higher forward whirling (FW) of three dimensional complex modes. The first four orders of the complex modes of both backward whirling and forward whirling are demonstrated in Figure 5. Similar phenomena on the frequency and complex mode appeared in [11], but the angular momentum was assumed to be continuously distributed. Figure 4: The varying natural frequencies with increasing angular momentum. **(a)** The results of the modal discretization. **(b)** The results of the spatial discretization. Figure 5: \(Cont\). The varying frequencies with zig-zag configurations are related to the veering phenomenon, which has been discussed in gyroscopic structures such as rotors, blades, and gears [17; 32; 33; 34]. In the current study, we did not consider the veering phenomenon, but focused on the numerical methods that have the power to show the gyroscopic dynamics. To show the convergence of the two methods, the results from the different discretization orders are listed in Tables 1-17. The frequency unit in all tables is expressed as 'rad/s'. In Tables 1-9, the natural frequencies for the different momentum of gyroscopes are presented to show the accuracy with the increasing modal discretization order \(k\). It can be found that the results are satisfactory when the discretization order \(k\) is two times higher than the maximum mode being studied. If only lower vibration modes are used, the lower discretization order can be adopted to save computation time consumption. The modal discretization method has been shown to be efficient and powerful when dealing with a regular structure whose modal functions without attachments are explicit. \begin{table}
\begin{tabular}{c c c c c c} \hline  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Order**} & \multicolumn{1}{c}{**1 BW**} & \multicolumn{1}{c}{**1 FW**} & \multicolumn{1}{c}{**2 BW**} & \multicolumn{1}{c}{**2 FW**} & \multicolumn{1}{c}{**3 BW**} & \multicolumn{1}{c}{**3 FW**} & \multicolumn{1}{c}{**4 BW**} & \multicolumn{1}{c}{**4 FW**} \\ \hline
11 & 0.0138 & 0.0545 & 0.126 & 0.230 & 0.401 & 0.898 & 2.749 & 5.880 \\
33 & 0. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Order**} & \multicolumn{1}{c}{**1 BW**} & \multicolumn{1}{c}{**1 FW**} & \multicolumn{1}{c}{**2 BW**} & \multicolumn{1}{c}{**2 FW**} & \multicolumn{1}{c}{**3 BW**} & \multicolumn{1}{c}{**3 FW**} & \multicolumn{1}{c}{**4 BW**} & \multicolumn{1}{c}{**4 FW**} \\ \hline
11 & 0.00832 & 0.0330 & 0.0763 & 0.140 & 0.249 & 1.404 & 3.144 & 8.741 \\
33 &