27 Vehicle Mechanics 2.3 VEHICLE KINETICS The tangential direction of forward motion of a vehicle changes with the slope of the roadway. In order to simplify the equations, a tangential co-ordinate system is defined below so that the forces acting on the vehicle can be defined through a one-dimensional equation. Let u x T F ( ) be the unit tangent vector in the fixed co-ordinate system pointing in the direction of increasing x F . Therefore, df i + j T x ( ) F dx F u x T F ( ) = = F F F . T x F F ( ) df 2 1 + dx F The tangential co-ordinate system shown in Figure 2.6 has the same origin as the fixed co-ordinate system.
The z -direction unit vector is the same as in the fixed co-ordinate system, but the x - and y -direction vectors are constantly changing with the slope of the roadway. Newton’s second law of motion can now be applied to the cg of EV in the tangential co-ordinate system as d v ∑ F m T T = = a m T dt where m is the total vehicle mass. In terms of the components of the co-ordinate system, d v ∑ F m xT = xT (compoment tangent to the road), dt ∑ d v F m = yT yT (component normal to the road), dt ∑ d v F m zT = = zT 0 (since motion is assumed confined to the – x y plane). dt v xT is the vehicle tangential velocity.
The gravitational force in the normal direction is balanced by the road reaction force, and hence, there will be no motion in the y T normal direction. In other words, the tire always remains in contact with the road. Therefore, the normal velocity v yT is zero. The vehicle motion has been assumed confined to the x F y F or x T y T plane, and hence, there is neither force nor velocity acting in the z -direction. These justified simplifications allow us to use a one- directional analysis for vehicle propulsion in the x T - direction.
It is shown in the following that the ) ( F T x u cg Roadway T j F j T i F i T F k k = FIGURE 2.6 Tangential co-ordinate system and the unit tangent vector on a roadway. yT j T xT i zT T k T yT j T xT xT zT