1. Geometry and Linkage Lecture 1 Day 1-Class 1

2. References  Gillespie, T., The Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA, 1992.  Milliken, W.F. and Milliken, D.L., Chassis Design Principles and Analysis, Society of Automotive Engineers, Warrendale, PA, 2002.  Hunt, D., Farm Power and Machinery Management, Iowa State University Press, Ames, IA, 2001.

3. Ackerman Geometry  Basic layout for passenger cars, trucks, and ag tractors  δ o = outer steering angle and δ i = inner steering angle  R= turn radius  L= wheelbase and t=distance between tires δoδo δiδi L t R Figure 1.1. Pivoting Spindle Turn Center Center of Gravity δoδo δiδi (Gillespie, 1992)

4. Cornering Stiffness and Lateral Force of a Single Tire  Lateral force (F y ) is the force produced by the tire due to the slip angle.  The cornering stiffness (C α ) is the rate of change of the lateral force with the slip angle. t α V FyFy Figure 1.2. F y acts at a distance (t) from the wheel center known as the pneumatic trail (Milliken, et. al., 2002) (1)

5. Slip Angles  The slip angle (α) is the angle at which a tire rolls and is determined by the following equations: W = weight on tires C α = Cornering Stiffness g = acceleration of gravity V = vehicle velocity (2) (3) (Gillespie, 1992) t α V FyFy Figure 1.2. Repeated

6. Steering angle  The steering angle (δ) is also known as the Ackerman angle and is the average of the front wheel angles  For low speeds it is:  For high speeds it is: (4) (5) α f =front slip angle α r =rear slip angle (Gillespie, 1992) t δiδi L R Center of Gravity δoδo δiδi δoδo Figure 1.1. Repeated

7. Three Wheel  Easier to determine steer angle  Turn center is the intersection of just two lines δ R Figure 1.3. Three wheel vehicle with turn radius and steering angle shown

8. Pivoting Single Axle  Entire axle steers  Simple to determine steering angle δ R Figure 1.4. Pivoting single axle with turn radius and steering angle shown

9. Both axles pivot  Only two lines determine steering angle and turning radius  Can have a shorter turning radius δ R Figure 1.5. Both axles pivot with turn radius and steering angle shown

10. Articulated  Can have shorter turning radius  Allows front and back axle to be solid Figure 1.6. Articulated vehicle with turn radius and steering angle shown

11. Aligning Torque of a Single Tire  Aligning Torque (M z ) is the resultant moment about the center of the wheel do to the lateral force. (6) t α V FyFy MzMz Figure 1.7. Top view of a tire showing the aligning torque. (Milliken, et. al., 2002)

12. Camber Angle  Camber angle (Φ) is the angle between the wheel center and the vertical.  It can also be referred to as inclination angle (γ). Φ (Milliken, et. al., 2002) Figure 1.8. Camber angle

13. Camber Thrust  Camber thrust (F Yc ) is due to the wheel rolling at the camber angle  The thrust occurs at small distance (t c ) from the wheel center  A camber torque is then produced (M Zc ) F yc tctc M zc (Milliken, et. al., 2002) Figure 1.9. Camber thrust and torque

14. Camber on Ag Tractor Pivot Axis Φ Figure 1.10. Camber angle on an actual tractor

15. Wheel Caster  The axle is placed some distance behind the pivot axis  Promotes stability  Steering becomes more difficult (Milliken, et. al., 2002) Pivot Axis Figure 1.11. Wheel caster creating stability

16. Neutral Steer  No change in the steer angle is necessary as speed changes  The steer angle will then be equal to the Ackerman angle.  Front and rear slip angles are equal (Gillespie, 1992)

17. Understeer  The steered wheels must be steered to a greater angle than the rear wheels  The steer angle on a constant radius turn is increased by the understeer gradient (K) times the lateral acceleration. (7) (Gillespie, 1992) t α V ayay Figure 1.2. Repeated

18. Understeer Gradient  If we set equation 6 equal to equation 2 we can see that K*a y is equal to the difference in front and rear slip angles.  Substituting equations 3 and 4 in for the slip angles yields: (8) Since (9) (Gillespie, 1992)

19. Characteristic Speed  The characteristic speed is a way to quantify understeer.  Speed at which the steer angle is twice the Ackerman angle. (10) (Gillespie, 1992)

20. Oversteer  The vehicle is such that the steering wheel must be turned so that the steering angle decreases as speed is increased  The steering angle is decreased by the understeer gradient times the lateral acceleration, meaning the understeer gradient is negative  Front steer angle is less than rear steer angle (Gillespie, 1992)

21. Critical Speed  The critical speed is the speed where an oversteer vehicle is no longer directionally stable. (11) Note: K is negative in oversteer case (Gillespie, 1992)

22. Lateral Acceleration Gain  Lateral acceleration gain is the ratio of lateral acceleration to the steering angle.  Helps to quantify the performance of the system by telling us how much lateral acceleration is achieved per degree of steer angle (12) (Gillespie, 1992)

23. Example Problem  A car has a weight of 1850 lb front axle and 1550 lb on the rear with a wheelbase of 105 inches. The tires have the cornering stiffness values given below: Load lb/tire Cornering Stiffness lbs/deg Cornering Coefficient lb/lb/deg 225740.284 4251150.272 6251560.260 9252180.242 11252600.230

24. Determine the steer angle if the minimum turn radius is 75 ft  We just use equation 1. Or 6.68 deg

25. Find the Understeer gradient  The load on each front tire is 925 lbs and the load on each rear tire is 775 lbs  The front cornering stiffness is 218 lb/deg and the rear cornering stiffness 187 lb/deg (by interpolation)  Using equation 7:

26. Find the characteristic speed  Use equation 8 plugging in the given wheelbase and the understeer gradient

27. Determine the lateral acceleration gain if velocity is 55 mph  Use equation 10