The vehicle is symmetrical about the x-z axis; 2. The vehicle’s total mass is lumped; 3. The vehicle’s roll axis is fixed and ; 4. The road’s surface conditions are consistent throughout the modelling; 5. Small angle approximations apply to the vehicle’s motion. The dynamics of the 4WS vehicle system can be divided broadly into three categories which are: 1. Tyre side forces; 2. Yaw moments; 3. Roll moments. These aspects of the steering system will be investigated separately based on three kinds of steering systems which are the 2WS (two wheel steering) with front wheel steering, 4WS under 40 km/h where the wheels are steered in opposite phases and 4WS over 40 km/h where the wheels are steered in the same phase. The three modes of steering and the relevant dynamics and motion investigation are discussed below. 2. Vehicle Dynamics Where: The variables , and all represent various kinds of disturbances that may affect the lateral, yaw and roll directions such as drag effects, side wind gusts, braking on ice, modelling uncertainties, loads, a flat tyre, an uneven road etc. These external disturbances may exert a sizeable influence in certain circumstances but for the sake of this investigation these disturbances will be neglected. The longitudinal forces are related to the wheels’ rotational model and these forces can be described by denoting them as: Where is the effective rotational inertia that includes all related drive train effects. The lateral forces are non-linear functions that can be described using the magic formula as below: Where are six different coefficients that depend on the vehicle load and the camber angles and . In order to study the dynamic behaviour of the vehicle model being investigated under the assumptions that the side slip angle is small under regular driving conditions, the equation listed above can be linearized and written as: The tyre slip angles presented in the notation above can be written as: If the roll angle is assumed to be small then and . In this case the equations listed above can be linearized as below: Similarly: And: If the vehicle is considered to be travelling under constant velocity conditions in a steady state fashion then = 0 and the longitudinal force displayed by the wheels can be approximated as: The dynamics of the involved actuator can also be represented as a linear first order lag system that can be described as: Where: And: Using the equations listed above a descriptor system can be obtained that possesses the form listed below: Where: The matrix inverse operation can then easily be used in tandem with the matrices presented above to produce a linear time invariant system that is described as below: The system above can now be easily modelled as a state space system in MATLAB in order to see how the system behaves when subjected to different steering angles and speeds. The parameters of interest are the lateral velocity, the yaw rate and the roll angle. 3. MATLAB Simulation The space state system was simulated as such in MATLAB for a host of combinations. The input steering angles were investigated for two wheel steering (2WS) and four wheel steering (4WS) systems for the yaw rate, the lateral velocity and the roll angle for limits of vehicle speed above and below 40 kilometres per hour. The 2WS system was investigated as such both above and below 4
1. Introduction Typical vehicle models for 4WS (four wheel steering) are composed of steering as well as roll motion, drive train and tyre dynamics. Some models for vehicles are modelled along the 2 DOF (degrees of freedom) bicycle model but such models fail to compensate for roll angle modelling and only relate lateral acceleration and yaw rates…
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Wheel Nut with Mechanical Locking/Unlocking Device.
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