## 6 Numerical Simulations
### Dynamic Meshing Forces  
Figure 8 illustrates the time-domain responses of the dynamic meshing force on the right-side planet-sun gear pair, which is represented as \(F_{sp1}^{R}(t)\) (\(F_{sp1}^{R}(t)=\Theta_{sp1}^{R}\cdot k_{sp1}^{R}\), where \(\Theta_{sp1}^{R}\) and \(k_{sp1}^{R}\) are the same as those in Equations (5) and (18)), for two cases of with and without planet-gear eccentric error. For the sake of clarity, just one side of the dynamic meshing forces of a part of the time interval for the HPGT is demonstrated in Figure 8. It can be observed from Figure 8 that because the input torque of the system \(\mathrm{T_{in}}\) = 100 KN\(\cdot\)m is constant, the dynamic meshing forces of the right-side meshing pair possess no low-frequency components for both models with and without the planet-gear eccentric error excitations. The average values of the dynamic meshing forces for the models with and without the planet-gear eccentric error excitations are approximately 57 KN as shown by the red dot-dash lines in Figure 8a,b. A comparison of Figure 8a,b indicates that the meshing force fluctuation amplitude from the model with the planet-gear eccentric error excitation \(\mathrm{E_{P1}}\) = 100 \(\mathrm{\SIUnitSymbolMicro m}\) is obviously larger than that without error excitations. That is to say, the planet-gear eccentric error excitation increases the meshing force amplitude significantly in contrast to the model without manufacturing error excitations. The similar tendency is also able to be discovered for the meshing forces on one side of internal (planet-ring) meshing pair. In short, the planet-gear eccentric error excitation prominently increases the dynamic meshing force fluctuation amplitude for herringbone planetary gears. Figure 7: Definition of the planet–gear eccentricity. Figure 9 presents the dynamic meshing force \(F_{sp1}^{R}(t)\) frequency spectra on the right-side external (planet-sun) meshing pair, where (a) and (b) show the results for the model with and without the planet-gear eccentric error excitations, respectively, which are acquired using the signal processing method of fast Fourier transform (FFT) to transform the time domain response as shown in Figure 8 into the frequency domain. For the model with the planet eccentricity, it is found from Figure 9a that some main frequencies of 27.3 Hz, 133 Hz, 162 Hz, 213 Hz, 232 Hz, 507 Hz, 613 Hz, 1187 Hz, and 1523 Hz appear in the right-side dynamic meshing force \(F_{sp1}^{R}(t)\) frequency spectra. The peak value of \(F_{sp1}^{R}(t)\) frequency spectra at the frequency of 0 Hz is maximum, and it is the direct current component of \(F_{sp1}^{R}(t)\). As is shown in Figure 9a, the dynamic meshing force \(F_{sp1}^{R}(t)\) amplitude at the frequency of 0 Hz is 57 KN, which is exactly the same as the average value of the dynamic meshing force \(F_{sp1}^{R}(t)\) in the time domain shown in Figure 8. In addition, at 162 Hz is associated with the 7 th order natural frequency in HPGT, and the peak value of \(F_{sp1}^{R}(t)\) frequency spectra is the second highest. The 133 Hz, 213 Hz, 232 Hz, 507 Hz, 613 Hz, 1187 Hz, 1523 Hz, and 27.3 Hz are associated with the 5-, 8-, 9-, 14-, 16-, 22-, 26-th natural frequencies and the mesh frequency of the HPGT system, respectively. Therefore, it can be concluded that in the model with the planet eccentricity, as shown in Figure 9a, the dynamic meshing force \(F_{sp1}^{R}(t)\) frequency spectrum demonstrates the appearance of larger amplitudes at the meshing frequency and some natural frequencies of HPGT. The frequency spectrum characteristics of dynamic meshing force are in fairly good agreement with the results of modal analysis. Figure 9b displays the \(F_{sp1}^{R}(t)\) frequency spectrum from the model without error excitations. By comparing Figure 9a,b, it is noted that the variation amplitudes of dynamic mesh forces with the planet eccentricity excitation are remarkably greater than those without error excitations.