## 2 Related Work  
Vehicle dynamics is a very popular research topic which features in many books and publications. Most of the literature is focused on four-wheelers; nevertheless, there is also a huge collection of research on two-wheeled vehicles. The different relevant authors are mentioned below. Back in 1971, [PERSON] [3] published a paper about the driving stability of two-wheeled vehicles. This research is considered the starting point of the modern motorcycle dynamics. In fact, thanks to a (linearised) tyre model it was possible to detect the wobble mode, which was not present in the even earlier and less complex motorcycle models by [PERSON] [13] (p. 195). [PERSON] [2] (p. 278) [5] and [PERSON] [6] presented a solution to overcome the previously mentioned incoherence between reality and rigid model simulations when analysing wobble. A fork bending compliance with a lumped stiffness was added: a revolute joint in the front fork, with its rotating axis (named twist axis [14]) perpendicular to the steering head axis. A certain stiffness and a damping are assigned to this joint. Lumped parameter models are described in [15]. This kind of modelling is also found in other sources, especially when a more complex non-rigid model is developed [16; 17; 18; 19; 20; 21]. This additional fork bending results in a more realistic wobble-damping characteristic: wobble-instability at low speed and a clear stabilisation when increasing the speed. [PERSON] [2] (p. 280) and [PERSON] [6] explain this change of characteristic with the combination of two opposed effects caused by the fork bending compliance:  
1. The bending compliance itself reduces the stability. 2. The combination of wheel-spin and the rotation around the twist axis produces a gyroscopic moment about the steering axis, leading to stabilisation. Since the gyroscopic moment is proportional to the speed, the first effect dominates at low speed, causing lower stability in this speed range. The second effect becomes dominant with increasing speed, which explains the restored stability at high speeds. Section 4.1 investigates the influence of the fork bending compliance considering another phenomenon caused by the bending compliance itself. There are several parameters affecting a motorcycle's wobble stability; a complete list can be found in [2; 3; 17; 21]. For this work, besides the fork bending compliance, two tyre parameters are of particular importance: the relaxation length and the cornering stiffness. Some research [3; 16; 17; 20] has underlined that a tyre model with cornering stiffness is necessary in order to simulate the wobble mode. This also suggests an important consideration: the principal cause of the wobble is given by the tyre's response. When the wheel is subject to an outer disturbance steering the wheel itself, the tyre reacts with a side force which, thanks to the mechanical trail, produces an aligning moment around the steering axis. When returning to the equilibrium position, the wheel starts oscillating about the steering axis, thereby triggering the wobble mode [22]. Depending on the speed of the motorcycle, this oscillation will diverge (unstable behaviour) or converge to the equilibrium (stable behaviour). The other important tyre parameter influencing wobble is the relaxation length [3; 13; 23]. Several works [3; 16; 17; 20; 24] investigated its effect on wobble damping, underlying how the relaxation length destabilises the wobble mode, since it generates a delay between the wheel steering and the tyre-lateral force generation. As a consequence, [PERSON] [3] (p. 323) clarifies that both tyre sideslip and tyre relaxation properties are fundamental for a proper representation of a motorcycle's dynamic properties. Like most of the tyre parameters, the relaxation length also depends on several factors. [PERSON] et al. [25] and [PERSON] [1] report that the relaxation length increases with increasing speed. Moreover, a dependence on the vertical load is present.