Lyapunov Method For Motion Stability Of Single-Track Vehicles

Single-track vehicles,including bicycles and motorcycles,are widely used in transportation,competition and military,because of their flexibility and environmental protection.Although single-track vehicles carry the most dangerous road users,their handling stability has not been well studied compared with passenger cars and commercial vehicles.In this paper,the dynamic model of the single-track vehicle is established.Based on this model,the vehicle stability in straight running and equilibrium cornering condition are analyzed.Specifically,the following aspects of research are carried out.Firstly,based on the homogeneous coordinate matrix,an 11-DOF non-linear single-track vehicle dynamics model with a large roll angle "magic formula" tire is established.In order to solve the steady-state motion of the vehicle,a driver model based on preview following theory is established.The augmented method combined with Baumgarte stabilization method is used to transform the vehicle-driver model into state space form and solve it.The simulation comparison with commercial software Bikesim verifies the validity of the DOF11-Bike.Secondly,the problem of motion stability in straight running of the single-track vehicle is transformed into the problem of equilibrium stability through the Bopoheu equation,and the linearization equation of the system is obtained.By plotting the root locus,some important modes such as wobble,weave,capsize are obtained.According to Lyapunov stability theory,under the input vehicle parameters,the vehicle can maintain the stability of straight running at the speed of 9m/s-67m/s;if the speed is too low or too high,the disturbed vehicles will diverge in different forms.This conclusion is validated by the impulse response of the system,and the feasibility of transforming the motion stability problem of a single-track vehicle into the equilibrium stability problem is also proved.According to the obtained modes,a vehicle parameter optimization method based on eigenvalue is proposed.Furthermore,aiming at the problem that there is no operating point in the global coordinates when a single-track vehicle in equilibrium cornering condition,according to the principle of moving the equilibrium state of the free system to the spatial origin in Lyapunov stability theory,the operating point of the vehicle in equilibrium cornering condition is solved by coordinate transformation.Through root locus analysis,it is found that the front hop mode can easily lead to wheel jump and divergence of steering motion at high speed,which will cause vehicle instability.Then the influence of suspension damping on system stability is analyzed.The step response analysis of the system verifies the calculation of the root locus and the feasibility of the coordinate transformation method.The vehicle-driver dynamic model of the single-track vehicle and the stability analysis based on the inherent characteristics in this paper have certain theoretical significance for the study of single-track vehicles.