25 Vehicle Mechanics y F y β 100 m x F FIGURE 2.2 Grade of the roadway. The average roadway percent grade is the vertical rise per 100 horizontal distance of roadway with both distances expressed in the same unit. The angle β of the roadway associated with the slope or grade is the angle between the tangent vector and the horizontal axis x F as shown in Figure 2.2. If ∆ y is the vertical rise in meters, then ∆ % grade = y 100% = ∆ y %. 100 m ∆ y The tangent of the slope angle is, tan β = . 100m The percent grade or β is greater than zero when the vehicle is on an upward slope and is less than zero when the vehicle is going downhill.
Exercise 2.1 A straight roadway has a profile in the x F y F plane given by f ( ) x x F F = ≤ 3.9 for 0 x F ≤ 2 miles . x F and y F are given in feet. (a) Plot the roadway, (b) find β ( ) x F , (c) find the percent grade at x F = 1 mile and (d) find the tangential road length between 0 and 2 miles. 1.95 Ans. (b) tan − 1 ; (c) 2.68% and (d) 10,580 ft. x F 2.2 LAWS OF MOTION Newton’s second law of motion can be expressed in equation form as ∑ F m i = a i where ∑ F i is the net force, m is the mass and a is the acceleration.
The law is applied to the vehicle by i considering a number of objects located at the several points of contact of the vehicle with the outside world on which the individual forces act. Examples of such points of contact are the front and rear wheels touching the roadway surface, the frontal area that meets the force from the air resistance, etc. We shall simplify the problem by merging all these points of contact into one location at the center of gravity (cg) of the vehicle, which is justified since the extent of the object is immaterial. For all the force calculations to follow, we shall consider the vehicle to be a particle mass located at the cg of the vehicle.
The cg can be considered to be within the vehicle as shown in Figure 2.3. The particle motion is described by the particle velocity and acceleration characteristics. For the position vector r for the particle mass on which several forces are working as shown in Figure 2.4, the velocity v and acceleration a are