## 4 Results
### Wobble  
Section 2 describes how the front fork lateral flexibility (bending-compliance about the x axis) strongly influences the stability of wobble. This difference is shown in Figure 4. In order to isolate the influence of the fork compliance, the curve relative to the rigid fork was obtained with a 9-DoF model, wherein the DoF used to represent the motorcycle's stiffnesses, as described in Section 3.1, have been eliminated. Moreover, the rider's DoF are not present. In the curve relative to the flexible fork, only the DoF \(\alpha_{\text{ff}}\) has been added, so that the DoF are 10 in total. With the parameter set used in the present work and in the speed range considered, the wobble mode remains well damped. This is due to the relatively high value of the steering damper. In the following part of this section, the reason for such a drastic behaviour change between rigid and flexible fork is addressed. In particular, a possible cause is presented which partially differs from the explanation provided by [PERSON] [2] (p. 280) and [PERSON] [6]. A key concept described in Section 2 is the correlation between tyre properties and wobble. In particular, a tyre model with cornering stiffness is necessary to reproduce wobble [3; 16; 17; 20]. Therefore, the tyre and specifically the lateral force due to the slip angle is a fundamental factor when considering wobble stability. However, what is the reason for the behaviour inversion in Figure 4 and how can it be attributed to the tyre response? Equation (2) shows how the slip angle depends on the  
Figure 3: Model validation with eigenvalue analysis: comparison between the current model (parametrised with the same data of the [PERSON] model) and the model of [PERSON]. lateral velocity of the wheel contact point \(v_{y_{\rm{C}}}\). The fork lateral flexibility allows some lateral motion of the wheel because of the rotation about the revolute joint axis (fork x axis), thereby influencing \(v_{y_{\rm{C}}}\). This concept is shown schematically in Figure 5. The variation of \(v_{y_{\rm{C}}}\) due to the fork lateral flexibility can be expressed as:  
\[\Delta v_{y_{\rm{C}}}\;=\;h\,\dot{a}_{\rm{ff}}. \tag{8}\]  
To prove the effect of \(\Delta v_{y_{\rm{C}}}\), it is compensated by subtracting its value from \(v_{y_{\rm{C}}}\) in Equation (2). Figure 6 shows the corresponding eigenvalue analysis with the dots-marked curve. The compensation of \(\Delta v_{y_{\rm{C}}}\) remarkably stabilises wobble. Moreover the curve shows the same behaviour for the 9-DoF model. At this point a conclusion can be drawn: the fork lateral flexibility influences the wobble stability through its effect on the lateral velocity \(v_{y_{\rm{C}}}\) of the contact point. In a subsequent step, it is of particular interest to observe whether \(\Delta v_{y_{\rm{C}}}\) increases or reduces the value of \(v_{y_{\rm{C}}}\). Figure 7 shows a time simulation at \(50\,\mathrm{km}\,\mathrm{h}^{-1}\) with a steer torque impulse as excitation. The full line represents \(v_{y_{\rm{C}}}\) obtained from Equation (5); the dashed line was obtained with \(v_{y_{\rm{C}}}-\Delta v_{y_{\rm{C}}}\), whereby \(\Delta v_{y_{\rm{C}}}\) was derived from Equation (8). The subtraction of \(\Delta v_{y_{\rm{C}}}\) increases the value of \(v_{y_{\rm{C}}}\): \(\Delta v_{y_{\rm{C}}}\), which is normally included in \(v_{y_{\rm{C}}}\), thereby reduces the amplitude of \(v_{y_{\rm{C}}}\). The tyre-lateral force is proportional to the slip angle, which is proportional to \(v_{y_{\rm{C}}}\) (Equation (2)). This force tends to  
Figure 4: Influence of front fork flexibility on wobble. Figure 5: Variation of the lateral velocity at the contact point due to lateral fork flexibility. oppose the wheel motion due to the wobble oscillation. Therefore, the fork flexibility introduces a \(\Delta v_{\text{y}_{\text{JC}}}\), which reduces the tyre-lateral force. This explains why the curve with \(\Delta v_{\text{y}_{\text{JC}}}=0\) in Figure 6 is significantly more stable than the 10-DoF case with \(\Delta v_{\text{y}_{\text{JC}}}\
eq 0\).