## 4 Results
### Wobble
Even if the curve with compensated \(\Delta v_{\text{y}_{\text{JC}}}\) in Figure 6 assumes the same qualitative behaviour as the 9-DoF model, the damping values are still significantly different. In order to get a curve similar to that of the 9-DoF model, the gyroscopic moment introduced by fork bending must be compensated. The moment acts about the steering axis and can be calculated with Equation (9) (this formula differs from the complete Euler equations (compare to Equation (10)), whose left-hand side defines the whole gyroscopic effects. Equation (9) is used here because its simple structure allows one to compensate the gyroscopic effects about the steering axis as external moments in the used multibody simulation software), as shown by [PERSON] [5]. This formula is also well-known in the literature, as it is present in the book on vehicle dynamics by [PERSON] [44]. The symbol \(\Delta\) in Equation (9) indicates that the formula only refers to the gyroscopic moment due to the fork lateral flexibility. \[\Delta M_{z}\,=\,\dot{\alpha}_{\text{ff}}\,\omega_{y_{\text{fw}}}\,I_{yy_{ \text{fw}}}, \tag{9}\]  
where \(\omega_{y_{\text{fw}}}\) is the front wheel rotation speed and \(I_{yy_{\text{fw}}}\) is its polar moment of inertia. The dash-dotted curve in Figure 6 shows the effect of compensating \(\Delta M_{z}\). As expected, the damping curve gets very  
Figure 6: Effect of compensating \(\Delta v_{y_{\text{JC}}}\) on wobble damping. In the dotted curve only \(\Delta v_{y_{\text{JC}}}\) is compensated. In the dash-dotted curve both \(\Delta v_{y_{\text{JC}}}\) and \(\Delta M_{z}\) (with Equation (9)) are compensated. Figure 7: Variation of the lateral velocity of the contact point \(v_{y_{\text{JC}}}\) due to front fork flexibility. Time simulation at \(50\,\text{km}\,\text{h}^{-1}\) and with a steer torque impulse as excitation. close to the 9-DoF model. The remaining offset can be attributed to other secondary phenomena, which have not yet been completely identified. In summary, the influence of the fork bending compliance on wobble can be explained by considering two main phenomena:  
1. Gyroscopic moment;
2. Variation of the lateral speed of the wheel contact point. Both of them originate through the rotation \(\alpha_{\mathrm{ff}}\) and its time derivatives. When observing the curve of the 10-DoF case in Figure 6, it is evident that it is less stable than the 9-DoF case only up to about 80 km h\({}^{-1}\). Above this speed, the higher stability of the 10-DoF model compared to the 9-DoF model can be justified with the same consideration made by [PERSON] [5]: the gyroscopic moment \(\Delta M_{z}\) about the steering axis stabilises the wobble with increasing speed. At this point another question may be raised: Which phenomenon is responsible for the decreasing wobble damping shown by the 9-DoF model and the model with compensated \(\Delta v_{y_{\mathrm{C}}}\) compared to the 10-DoF one? To answer this question, a model containing only the front fork and front wheel has been used. With this model two cases are studied: two DoF characterised by the rigid fork and three DoF with a flexible fork; the first model correlates with the 9-DoF model, and the second with the 10-DoF model. Figure 8a shows the damping curves for these two models. Different variations were made: the wheel inertia tensor \(\mathbf{I}_{\mathrm{fw}}\) was set to 1% of the original value (to eliminate the gyroscopic moments) and \(\Delta v_{y_{\mathrm{C}}}\) has been compensated as in Figure 6. Some important conclusions can be drawn:  
* The 2-DoF model did not show a decrease in damping with increasing speed, as did the 9-DoF model. * The damping of the 3-DoF model grew more rapidly than the one of the 10-DoF model. * The case \(\mathbf{I}_{\mathrm{fw}}=1\,\%,\Delta v_{y_{\mathrm{C}}}=0\) is very close to the curve of the 2-DoF model, as was expected by to theoretical observations. In fact, the 2-DoF model had no gyroscopic moments about the steering axis, as no rotation about the fork x axis was present.