# A Screw-Axis Approach to the Stability of Two-Wheeled Vehicles
## 1 Introduction  
The lateral stability of two-wheeled vehicles has been investigated extensively in the last 50 years, both numerically and experimentally. It is well known that there are two main vibration modes, namely weave and wobble [1, 2, 3, 4, 5, 6]. Weave is characterized by steering oscillations combined with yaw, roll and lateral motion of the vehicle, with frequencies usually below 4-5 Hz, while wobble is dominated by steering oscillation, with frequency usually in the range 6-10 Hz. Weave is usually lightly damped at high speed, say above 100 km/h, while wobble tends to be lightly damped at low to medium speed, although such range is affected by the structural stiffness of the vehicle [7, 8]. The effect of a rider on the vehicle stability has also been investigated. Historically, the rider has been assumed to be rigidly attached to the chassis, with the inertial properties estimated using prediction-regression formulas from biomechanical databases. A number of different sources are available in the literature, with a review of the most used given in [9]. Starting from the mid 1980s, the rider's passive motion on the saddle has been modelled using spring-damper elements tuned to give the typical modal properties obtained from experiments [10, 11, 12, 13]. More recently, the effect related to 'hands-on' and 'hands-off' the handlebar has been investigated. Experiments aimed at the identification of the rider-steering interaction has been carried out [14, 15]. The numerical simulations and the experiments suggest that with hands on the handlebar wobble becomes more stable, while weave destabilizes [16, 17, 18]. Most of the analyses mentioned above focus on the effect that the main vehicle, tyre and rider parameters have on the frequency and damping of the vibration modes. Such modes are usually presented in terms of the real and imaginary part of the eigenvalues--which are related to the damping and frequency of the vibration modes--as a function of the vehicle speed. In contrast, the focus of this work is on the shape of the vibration modes. The underlying idea is that, given the same combination of frequency and damping ratio, the vehicle may be perceived to be more or less stable by the rider depending on the shape of the vibration modes. In order to characterize the mode shape, the screw-axes associated to the weave and wobble are computed, in addition to the compass diagrams, which are the standard to describe the mode shape. It is recalled that each motion in the space, e.g., that of the chassis of a two-wheeled vehicle, can be represented (instantaneously) as a rotation around and a translation along a screw-axis, also called Mozzi axis [19; 20]. The direction of the screw axis coincides with the (instantaneous) direction of the angular velocity vector. The intersection between the screw axis and a reference plane gives the velocity centres of the motion projected on that reference plane. Two planes are considered in this work: transverse plane (motion observed from the top of the vehicle) and frontal plane (motion observed from the front of the vehicle). The side view is not interesting in the case of straight-motion analysis. Indeed, it is well know that in such a condition weave and wobble modes do not involve either suspension oscillations or tyre radial oscillations, as long as the vehicle is symmetric (which is the case on most models, including the one considered in this work). The screw axis has been previously employed to analyse the motion of motorcycles during slalom [21], lane-change and turning [22], and lap simulation [23]. More recently, velocity centres have been used to investigate the mode shape of the weave mode [24]. In this work, such analysis is generalized through the screw-axis and extended to include also the front view and wobble. The work is organized as follows. Section 2 describes the vehicle model employed and discusses its vibration modes, in order to show its consistency with other results reported in the literature.