LONGITUDINAL VEHICLE MODEL
In practical terms, a vehicle not only travels on a level road but also up and down the slope of a roadway as well as around corners. In order to model this motion, the description of the roadway can be simplified by considering a straight roadway with two-dimensional movement. This two dimensional model will focus on vehicle performance, including acceleration, speed, and grade ability, as well as braking performance.
Figure 2.1 shows the forces acting on a vehicle as it travels at a given speed along a roadway with a specific grade. Fundamental principles of mechanical systems can be used to express the relationship between the vehicle acceleration and the forces acting on the vehicle body as:
ma = Ft − Fw − Fg – Fr ……………. (2.1)
where m is the vehicle mass, a is the acceleration of the vehicle. Ft is the total tractive force acting upon the vehicle body, Fw is the aerodynamic drag force, Fg is the grading resistance force, and Fr is the rolling resistance force.
As air travels over the body of the vehicle, it generates normal pressure and shear stress on the vehicle’s body. The external aerodynamic resistance is comprised of two components, shape drag and skin friction. The shape drag arises from high-pressure areas in front of the vehicle and low-pressure areas behind the vehicle that are created as the vehicle propels itself through the air. These high-and low-pressure zones act against the motion of the vehicle, while the skin friction is due to the shear stress in the boundary layer on the surface of the body of the vehicle. In comparison, shape drag is much larger in magnitude than skin friction and constitutes more than 90% of the total external aerodynamic drag of a vehicle. Aerodynamic drag is a function of effective vehicle frontal area, A, and the aerodynamic drag coefficient, Cd, which are highly dependent on the design of the vehicle body:
where ρ is the air density, V is the vehicle longitudinal speed, and Vw is the wind speed.
As a vehicle travels up or down an incline, gravity acting on the vehicle produces a force which is always directed downward, as shown in Figure 2.1. This force opposes the forward motion during grade climbing and aids in the forward motion during grade descending LONGITUDINAL VEHICLE MODEL. In typical vehicle performance models, only uphill operation is considered as it resists the total tractive force. The equation for this force is a function of the road angle θ, vehicle mass m, and the gravitational acceleration g:
For a relatively small angle of θ, tan θ = sin θ. Using this approximation, the grade resistance can be
approximated by mg tan θ, or mgG, where G is the slope of the grade.
Rolling resistance force is a result of the hysteresis of the tire at the contact patch as it rolls along the roadway. In a stationary tire, the normal force due to the road balances the force due to the weight of the vehicle through the contact patch which is in line with the center of the tire. When the tire rolls, as a result of tire distortion or hysteresis, the normal pressure in the leading half of the contact patch is higher than that in the trailing half. The normal force due to the road is shifted from the center of the tire in the direction of motion. This shift produces a moment that exerts a retarding torque on the wheel. The rolling resistance force is the force due to the moment, which opposes the motion of the wheel, and always assists in braking or retarding the motion of the vehicle LONGITUDINAL VEHICLE MODEL. The equation for this force is a function of the normal load Fz and the rolling resistance coefficient fr, which is derived by dividing the distance the normal force due to the road is shifted by the effective radius of the tire rd.
TOTAL TRACTIVE FORCE
Equation 2.1 shows the factors affecting vehicle performance with a particular interest in the overall tractive force of the vehicle.
By rearranging Equation 2.1 we arrive at an equation that expresses longitudinal vehicle motion as a combination of total tractive effort minus the resistance. In order to determine the total tractive effort, the normal forces, Fzf and Fzr, need to be determined. The front and rear tire contact points should satisfy the equilibrium equations for moments:
Where Fzf and Fzr are the normal forces on the front and rear tires, lf and lr are the distances between the front and rear axles and vehicle center of gravity, respectively. hw is the height for effective aerodynamic drag force and h is the height of vehicle center of gravity. For simplicity, usually hw is assumed to be equal to h. Equations 2.7 and 2.8 can be rearranged to solve for the normal forces on the front and rear tires:
The total tractive force can be expressed as the tractive forces acting on each tire:
where Fxf and Fxr are the longitudinal forces on the front and rear tires, respectively. The friction generated between the tire–road contact patch creates the longitudinal force. Therefore, the longitudinal force generated on each tire can be represented as a function of the tire friction coefficient and the normal force:
Fxf = μf Fzf , Fxr = μrFzr………(2.12)
where Fxf and Fzr are the normal forces on the front and rear tires given by Equations 2.9 and 2.10 and μf and μr are the friction coefficients on the front and rear tires, respectively.