# A Screw-Axis Approach to the Stability of Two-Wheeled Vehicles
## 2 Multibody Model
It is noted that the controller is disengaged before linearising the system, in order to obtain the open-loop dynamics, i.e., the dynamics of the bike and not those of the bike-controller combination (in addition, one would have more states in (3) in the case the PID is still engaged at the time of the linearization). The procedure is automated using an Adams _macro_, which is then called from a Matlab script. The vibration modes consist of the (complex) eigenvalues of the state matrix \(A\), with the mode shape being the related (complex) eigenvectors. Figure 1 shows the root-locus (imaginary vs. real part of the eigenvalues) of the vehicle under investigation, for speeds between 5 (circles) and 50 m/s (squares). It goes without saying that there are as many eigenvalues as the number of states in (3)--as usual only the upper half of the root-locus diagram is shown because it is symmetric with respect to the \(x\)-axis. Two 'high' frequency modes can be seen above 13.5 Hz (85 rad/s), corresponding to the front and rear wheel hop, respectively--the former mainly involves the bouncing of the front wheel on the road, while the latter mainly involves the bouncing of the rear wheel on the road. They are marginally affected by the speed variations, the rear hop mode being mainly affected in terms of frequency (which slightly decreases at high speed), the front hop mode being mainly affected in terms of damping (which slightly decreases with speed). The rear wobble mode is visible below the hop modes, with a damping increasing with the speeds, while its frequencies decrease with speed from 9.5 Hz (60 rad/s) to 0 Hz (non-vibrating). This mode is usually not very interesting because it is always stable. Wobble is a mode mainly involving steering oscillations, with frequencies decreasing from 8.5 Hz (53 rad/s) to 5.5 Hz (35 rad/s) in this vehicle. Weave is a mode involving the lateral degrees-of-freedom of the vehicle, and has frequencies rising with speeds from 0.5 Hz (3 rad/s) up to 5 Hz (31 rad/s). On the horizontal axis there is the capsize mode: It does not vibrate, is lightly unstable at low speeds (in this vehicle), and mainly involves the roll and lateral degrees of freedom. Finally, the modes associated with the suspension dynamics are visible at lower frequencies: Bounce (or heavy) has frequencies around 2.4 Hz (15 rad/s), while pitch has frequencies around 3.2 Hz (20 rad/s). As a recap, the modes of the vehicle in straight running can be divided into two categories: In-plane modes (namely bounce, pitch and hops) and out-of-plane modes (namely capsize, rear wobble, wobble and weave). The latter modes are the most interesting for the current analysis, because they have implications on the running stability of two-wheeled vehicles. Weave is also one  
of the most investigated modes, because it is lightly damped in many vehicles when the speed is high enough. Figure 2 shows the compass plots of the weave mode shape: The length of each arrow represents the magnitude of the associated degree of freedom, with the related phase being the angle with respect to the horizontal axis. It is recalled that, as is common in vibration analysis, only the ratio between the magnitudes and the relative angles matter. The shape of the weave mode at high speed is similar to other results reported in the literature, with yaw almost in phase opposition with respect to the steering, roll in between the steer and yaw, while the steer lags with respect to the lateral motion [8; 12; 24]--an SAE conventions is employed, i.e., the \(z\) axis (both yaw and steer) points downward and the \(x\) axis points forward, while the \(y\) (lateral) axis completes the orthogonal triad. At low speeds all the four main degrees of freedom involved have similar magnitudes, with the roll (\(\phi\)) prevailing on the others, while at high speeds the steer angle (\(\delta\)) becomes dominant (however, there are still visible components of the other degrees-of-freedom).