## 4 Results
### Weave
This can also be observed in the time simulation. If the motorcycle is excited by a lateral impulse at high speed, the resulting weave oscillation does not show a significant roll motion, while the front wheel and the frame rotate  
Figure 10: Compass plot of the weave eigenvector at different speeds. \(\delta\) is the steering angle, \(\psi\) is the yaw angle, \(\rho\) is the roll angle and \(y\) is the lateral displacement. Figure 9: Weave eigenfrequency and damping as a speed function. around the steering and vertical axis almost in opposition of phase, as shown by the vectors \(\delta,\psi\) in Figure 9(c). The reduction in roll motion with increasing speed is also shown by a free rolling wheel. This phenomenon can be explained considering the gyroscopic effects which increase with speed, thereby preventing the wheel (or the motorcycle) capsizing. Which are the possible causes of the peculiar damping behaviour shown in Figure 8(b)? Some useful knowledge can be derived by an eigenvalue analysis with reduced wheel inertia. The correlation to Figure 9 is shown in Figure 10, for which the total wheel inertia tensors \(\mathbf{I}_{\mathrm{fw}},\mathbf{I}_{\mathrm{rw}}\) was set to \(10\,\%\) of the original tensors. The first consideration resulting from the comparison is that in Figure 11 the relative phase between the phasors no longer changes with speed, as it happens in Figure 10. The plot at \(30\,\mathrm{km}\,\mathrm{h}^{-1}\) is not present, as the weave mode is no longer available at this speed. It is important to point out that, because of this massive inertia reduction, the gyroscopic effects almost disappear. Two main observations can be made:  
1. The missing weave mode at low speed can be attributed to the (almost) missing gyroscopic effects. The time simulation at \(30\,\mathrm{km}\,\mathrm{h}^{-1}\) with a lateral force impulse shows that the motorcycle does not react to the excitation with a weave oscillation, but it capsizes. With the original inertia values the gyroscopic effects are present, despite the low speed. They prevent the motorcycle from capsizing immediately after the excitation and producing the low-frequency \"self-stabilising\" motion. 2. The gyroscopic effects are also the main cause for the change in the weave eigenvector shown in Figure 10. In fact, when they are very small, as in Figure 11, the relative phase between the eigenvector components is no longer speed dependent. The correlation with Figure 9 is shown in Figure 12. The speed range starts now from \(80\,\mathrm{km}\,\mathrm{h}^{-1}\), as for the following reasoning only this range is necessary. Moreover, the weave with reduced wheel inertia is not present up to about \(50\,\mathrm{km}\,\mathrm{h}^{-1}\) and in the range \(50\!-\!80\,\mathrm{km}\,\mathrm{h}^{-1}\) it is extremely well damped (real part \(>25\,\mathrm{rad}\,\mathrm{s}^{-1}\)). Similarly to the full inertia case, the damping continues to decrease with increasing speed in the small inertia case shown in Figure 11(b). As the gyroscopic effects are almost missing, the reason for this behaviour can be determined by investigating in the tyre response. The tyre-lateral force is proportional to the slip angle which is in turn proportional to the ratio between \(v_{y_{C}}\) and \(v_{x_{C}}\) (Equation (2)). The authors verified that \(v_{y_{C}}\) mainly depends on the kinematic steering angle (see [2] for its definition), which is proportional to the steering angle. Figure 11 shows that above a certain speed (\(100\,\mathrm{km}\,\mathrm{h}^{-1}\)) the amplitude of steering motion in the weave eigenvector remains fairly constant. The numerator of \(\alpha_{\mathrm{ss}}\) in Equation (2) is therefore almost constant with speed. The denominator of \(\alpha_{\mathrm{ss}}\) is given by \(v_{x_{C}}\), which clearly increases with speed. As a result, \(\alpha_{\mathrm{ss}}\) decreases with increasing speed and so does the tyre-lateral force. The tyre-lateral forces tend to bring the motorcycle back to the equilibrium position, thereby stabilising the weave.