## 2 Related Work
A more recent paper by [PERSON] [12] proposes a tyre model based on the Pacejka magic formula [26] where dependencies on both speed and vertical load are reproduced. This model and the related parameters are also used in the present work. In summary, some minimal requisites for a correct analysis of the wobble mode can be defined. [PERSON] [26] and [PERSON] [16] suggest that the tyre properties have the greatest influence on motorcycle stability. The Pacejka magic formula [26] coupled with a first order dynamic response, as described in the previous paragraph, is necessary in order to produce reliable wobble stability analyses [13] (p. 324). This tyre model allows an accurate description of the tyre forces and moments if the frequency of the external forces is lower than 15 Hz [13]; these conditions are generally satisfied for common stability analysis, therefore the magic formula is used in almost all the latest researches based on multibody analysis [2; 7; 12; 27; 28; 29]. In addition to the tyre modelling, the lateral compliance at the front of the machine is necessary when simulating wobble [13; 30]. This can be obtained with a bending compliance of the front fork [5; 6; 7; 8; 9], or by adding a lumped torsional stiffness of the main frame at the steering head joint with its rotation axis perpendicular to the steering axis [12; 26; 27; 28]; a combination of both stiffnesses can also be implemented, as in the present work. Besides the wobble mode, many of the mentioned sources also analyse vehicle \"weave\" [2; 3; 5; 7; 17; 19; 21]. The weave mode can also be qualitatively represented with a motorcycle model having no flexibilities [13] (p. 195). However, the literature underlines that some additional degrees of freedom (DoF) notably influence the weave mode. In particular, [PERSON] [17] (p. 17) and [PERSON] [6] (p. 28) underline that the fork bending compliance destabilises the weave mode. [PERSON] [23] and [PERSON] [7] analyse the effect of the swingarm torsional and bending compliance. Reference [23] simulated two motorcycle types with significantly different stiffness values: a super-sport motorcycle (higher stiffness), where the two swingarm compliances had almost no influence on weave, and an enduro-motorcycle, where the swingarm bending compliance slightly destabilised the weave mode, which was at the same time slightly stabilised by the torsional compliance of the swingarm. The results of [7], on the contrary, show the destabilising effect of both swingarm bending and torsional compliance; however, the influence was also small in this case. Furthermore, the rider has proven to greatly affect the motorcycle's stability behaviour. In fact, [PERSON] [31] points out that the stability analysis of a riderless motorcycle can be very misleading. Significant changes in the weave stability due to the rider have also been reported by [PERSON] [26] (p. 535). The modelling of the rider's passive response and its influence on stability is a broad topic which will be summarised in Sections 3.3 and 4.3. An interesting argument strictly connected to weave is the so called \"cornering weave\". During cornering, the in-plane eigenmodes (bounce, pitch) become increasingly coupled with the out-of-plane eigenmodes (weave, wobble), whereby the former shows degrees of freedom typical of the latter and vice versa [9; 13; 32]. Concerning cornering weave, the eigenvector also shows front and rear suspension travels which are not present in the straight-running weave [32]. This mode-coupling is critical because it provides a signal transmission path from road undulations to lateral motions [33]. Moreover, under cornering conditions, bounce and weave have similar eigenfrequencies in the middle to high speed range (70-120 km h\({}^{-1}\)) [9; 24; 32; 34], thereby leading to mutual triggering [13; 33; 34]. From these considerations it should be clear that the suspension system must also be modelled when carrying out stability analysis of motorcycles under cornering conditions [13; 34].