## 4 Results
### Weave  
The weave mode was already captured in its essence by the very first motorcycle model by [PERSON] [13] (p. 195). Moreover, there is no evidence of parameters which produce a behaviour inversion of the damping curve, as can be observed for the wobble mode. This already suggests that the weave mode reproduces in some way the \"basic\" motorcycle behaviour. This idea was also proposed by [PERSON] [45] (p. 28), who describes the weave mode as a degenerated dynamic stabilisation process involving steering, roll and yaw oscillations. This can be shown with a simple time simulation where the upright riding motorcycle is excited with a lateral force impulse applied to the frame. Assuming that the wobble is well damped, the motorcycle reacts to this excitation with a weave oscillation, which may diverge or stabilise depending on the motorcycle's speed and parameters. Figure 9 shows the frequency and damping of the weave mode as a function of the speed. The effect of the rider is analysed in Section 4.3, so this figure refers to the case with a rigid rider; the other DoF are all present. In order to represent the saturation shown by the damping curve at high speed, the speed range differs from the one in Figure 4. The behaviour of the weave damping shows some interesting characteristics: below \(70\,\mathrm{km}\,\mathrm{h}^{-1}\) weave is stabilised with increasing speed; above it is destabilised. However, above \(220\,\mathrm{km}\,\mathrm{h}^{-1}\) the weave damping shows a plateau. This particular behaviour suggests that the weave mode changes with speed. Moreover, the speed dependency could be explained by a certain influence of the gyroscopic effects, which are also speed dependent. The gyroscopic effects are obtained with the whole left-hand side of the Euler equations, which is shown in Equation (10) for the front wheel. \[\mathbf{I}_{\mathrm{fw}}\dot{\mathbf{\omega}}_{\mathrm{fw}}\,+\,\mathbf{\omega}_{\mathrm{ fw}}\times(\mathbf{I}_{\mathrm{fw}}\mathbf{\omega}_{\mathrm{fw}})\,=\,\mathbf{M}_{ \mathrm{ext}}, \tag{10}\]  
where \(\mathbf{\omega}_{\mathrm{fw}}\) is the rotational velocity vector of the front wheel and \(\mathbf{I}_{\mathrm{fw}}\) is the front wheel inertia tensor. Figure 8: (a) Wobble damping for the 2 and 3-DoF models with variation of the tensor \(\mathbf{I}_{\mathrm{fw}}\) and compensation of \(\Delta v_{\mathrm{gc}}\). The curve with \(\mathbf{I}_{\mathrm{fw}}=1\,\%\) for the 2-DoF model is identical to that with \(\mathbf{I}_{\mathrm{fw}}=100\,\%\) so it was not plotted. (b) Figure 4b copied to facilitate the comparison. Figure 10 further illustrates this concept. In this figure, the weave eigenvector is shown with the so-called compass plot. The arrows are named phasors and represent how the motorcycle's DoF are involved in the weave oscillation. An arrow's length is the magnitude of the motion of the related DoF. In order to maintain the information about the absolute amplitude of the phasors, no normalisation is present. The phasor's angle shows the phase. In the following paragraphs the relative angle between the phasors is called relative phase. At low speed, (Figure 10a) the roll (\(\rho\)) and lateral displacement (\(y\)) show big amplitudes. The latter is due to the small gyroscopic effects, so that if weave is excited (for example, with a lateral impulse), the motorcycle describes a \"slalom\" at low frequency (see Figure 9) involving a significant lateral displacement. This can be described as a low-frequency \"self-stabilising\" motion, which the literature already considers as weave. As the speed increases (Figure 10b), the relative phase between the motions significantly changes, whereby the lateral displacement is no longer important. The other DoF show a similar amplitude. Finally, when the speed increases further (Figure 10c), the relative phase does not change remarkably. The most important observation, however, is the reduction of the roll (\(\rho\)) amplitude with increasing speed, which can be seen when comparing the related phasor in Figure 10b,\(c\).