## 1 Introduction  
Motorcycle dynamics have always been an extensive research topic due to the complexity and the influence on the riders' safety. The knowledge used to develop a stable motorcycle chassis comes, to a certain extent, from experience or sensitivity analysis which sometimes lack physical and theoretical understanding of the hidden phenomena. This paper wants to clarify these phenomena, in particular with regard the understanding of the effect of the front fork bending compliance and of the weave damping behaviour with changing speed.  
Two of the principal eigenmodes of a motorcycle are known as wobble and weave. The first one includes almost solely the rotation of the front assembly about the steering axis whose frequency is in the range between 7 and 10 Hz depending on the motorcycle's parameters [1]; the second one is more complex, and when affected by this mode, the motorcycle shows roll, yaw, steering-head rotation and lateral displacement [2].  
The starting point for this work is an observation resulting from the comparison of experience in reality and simulation results. Using the eigenvalue analysis, the earlier motorcycle models, such as the model of [PERSON] [3] which only includes a type model but no compliances, show a stable wobble mode in the lower speed range (up to 80 km h\({}^{-1}\)), and this mode becomes unstable in the higher speed range. However, the experience with a real motorcycle shows an almost opposite behaviour: lower stability (or even instability) at low speed and increasing stability at higher speed. This was demonstratedthrough measurements on a real motorcycle by [PERSON] [4]. Previous research addressed this apparent incompatibility between simulation and reality, concluding that the front fork bending compliance is a key parameter when studying wobble [5; 6; 7; 8; 9], as discussed further in Section 2. [PERSON] [10] conducted an analysis where the motorcycle stiffnesses were progressively eliminated (i.e., set to infinity); in this case also, a remarkable variation in the motorcycle response was shown.  
For weave mode, the situation is different: the well-known correlation between eigenmode-damping and speed is also reproduced by models without chassis compliances [2; 5; 6]. In this case, up to a certain speed, the weave mode becomes more stable with increasing speed. After a tipping point, the stability degenerates again when the speed increases. This behaviour is unexpected, as the increasing gyroscopic moments of the wheel should stabilise the motorcycle, as is the case, for example, with a gyroscope. This is a complex system to analyse because there are many possible gyroscopic moments and their combined influence is not trivial. This peculiar behaviour is further analysed in Section 4.2.  
The different models that will be presented within this paper were produced using open source multibody simulation software named \"MBSim\" (www.mbsim-env.de) which was developed at the Institute of Applied Mechanics of the Technical University of Munich. An introduction [11] on the functionalities of this software is provided. Multibody simulation is common for vehicle dynamics, and was used, for example, by [PERSON] in multiple publications [2; 5; 7; 12]. It allows the building of the desired model through a CAD-like interface, by connecting the bodies with joints and assigning them stiffness and damping when needed. The present motorcycle model is described in Section 3.  
To summarise, this work deeply investigates the influence of the front fork bending compliance on the wobble mode. Starting from the point reached by previous authors [1; 5; 6], Section 4.1 provides a different justification for the well-known effect of the fork bending compliance on wobble damping. Moreover, in Section 4.2 the weave mode is further analysed and a possible justification for its damping behaviour is provided. Finally Section 4.3 investigates the influence of the rider on wobbling and weaving.