2. Handling Low-speed turning High-speed turning Understeer

3. Low-speed Turning   o i R L t R+t/2 Turn Center R+t/2  tan -1  L o L R-t/2 L i L For large radii, R >> t/2  Ack L R

4. High Speed Turning V R R R R R R0R0 Original Path/ Neutral Steer Path Under Steer Path R > R 0 Over Steer Path R < R 0

5. Tire Slip Angle

6. Tire Cornering Stiffness

7. Factors affecting cornering stiffness

8. NSL for force and moment analysis Geometry for steer angle vs. radius From Newton’s Second Law From tire properties From the geometry: Understeer Gradient High-speed Turning ryf f rfz f r r r f αf αr αf αr

9. Positive – understeer Zero – neutral steer Negative – oversteer –Has a critical speed –Vehicle is unstable Oscillatory Divergent Understeer Gradient, K Understeer Gradient

10. Steer Angle vs. Speed

11. Speeds & Gains Characteristic speed = speed at which steer angle required to negotiate a turn is 2 times Ackerman angle V char = √57.3Lg/K Critical speed = speed at which steer angle required to negotiate a turn is 0 V crit = √-57.3LgK Lateral acceleration gain a y /δ = V 2 /57.3Lg(1+ KV 2 /57.3Lg) Yaw velocity gain r/δ = V/L(1+ KV 2 /57.3Lg)

12. Understeer – Very controlled gain with speed Neutral steer – Increasing gain with speed Oversteer – Increases dramatically with speed 108 in wheelbase Stability limit 88 mph SW Angle/g 5 deg 6 deg 10 deg 20 deg 40 deg Effect on Lateral Acceleration Gain

13. Effect on Yaw velocity gain

14. Slip Angle Calculation (primary tire effect) 1. Calculate front and rear vertical wheel loads W f and W r 2. Assume lateral acceleration a y /g as % (g). 3. Lateral tire force (front & rear) F yf = W f *a y and F yr = W r *a y 4. From tire data find slip angles for all 4 tires, use extrapolation 5. Find average slip angle for front and rear α f and α r 6. Calculate under steer α f – α r 7. Do calculations for a y /g from 0.1 to 1.0

15. Effect of Body Roll W F z0 > F zi

16. Effect of Body Roll No roll: For 800 lb load on each wheel 760 lb of lateral force at 5 deg slip angle Body Roll: In hard cornering inside & outside wheel loads can be 400 & 1200 lb with average lateral force of 680 lb, requiring more slip angle to maintain the turn

17. Effect of Body Roll Overturning moment M φ = Wh 1 [ V 2 /(Rg) + φ] M φ = M φf + M φr = (K φf +K φr ) φ Hence, φ = Wh 1 V 2 /[Rg(K φf +K φr -Wh 1 ) ] Roll rate R φ = dφ/da y = Wh 1 /[K φf +K φr -Wh 1 ] Where φ = roll angle, K φ = roll stiffness, h 1 = distance between C.G. & roll ctr. Vertical load difference between outside and inside wheel (F zof –F zif )t f = K φf *φ + W f h f V 2 /Rg and (F zof +F zif ) = W f (F zor –F zir )t r = K φr *φ + W r h r V 2 /Rg and (F zor +F zir ) = W r Where h f and h r = roll center height front and rear

18. Slip Angle Calculation (roll effect) 1. Calculate front and rear vertical wheel loads W f and W r 2. Assume lateral acceleration a y /g as % (g). 3. Lateral tire force (front & rear) F yf = W f *a y and F yr = W r *a y 4. Calculate roll rate and find roll angle φ 5. Calculate F zi and F zo for front and rear 6. From tire data find slip angles for all 4 tires, use extrapolation 7. Find average slip angle for front and rear α f and α r 8. Calculate under steer α f – α r 9. Do calculations for a y /g from 0.1 to 1.0

19. 0123456789 Camber Angle (deg) 0 50 100 150 200 Lateral Force (lb) F = 1000 lb Zero Slip Angle z C   Tires produce a lateral force (camber thrust) when inclined Characterized by camber stiffness, C  Camber Thrust Camber coefficient –Radials are lower –Bias-ply are higher Camber Coefficient, C  /F z (lb/lb/deg)

20. Camber Thrust γ g = γ b + φ Where γ g = camber w.r.t. ground γ b = camber w.r.t. body φ = roll angle Lateral Tire load due to camber F yc = C γ *γ = C γ *(dγ/dφ)*(dφ/da y )*a y = C γ *(dγ/dφ)*roll rate*a y γ-φ relationship Lateral tire force causing tire slip = W*a y - F yc

21. Slip Angle Calculation (roll/camber effect) 1. Calculate front and rear vertical wheel loads W f and W r 2. Assume lateral acceleration a y /g as % (g). 3. Calculate roll rate and find roll angle φ 4. Calculate F zi and F zo for front and rear 5. Calculate γ-φ relationship from suspension data 6. Calculate lateral tire force due to camber for each tire 7. Lateral tire force for slip (front & rear) F yf = W f *a y -F ycf and F yr = W r *a y -F ycr 8. From tire data find slip angles for all 4 tires, use extrapolation 9. Find average slip angle for front and rear α f and α r 10. Calculate under steer α f – α r 11. Do calculations for a y /g from 0.1 to 1.0

22. Roll Steer All suspensions steer with roll Steer to the outside is: –Understeer on front –Oversteer on rear Solid axle on a trailing arm: –Arm angle determines understeer –Angled down is oversteer –Angled upward is understeer

23. Lateral Force Compliance Steer All suspensions steer due to a lateral force Minimize compliance steer Yaw center Cornering Force Deflection Understeer Turn Cornering Force Deflection Oversteer Turn Yaw center

24. Constant Radius Understeer Test

25. Constant Speed Understeer Test

26. Process for Calculating Cornering Response Decide on the lateral acceleration requirement Calculate roll-stiffness based on the suspension properties Calculate roll rate Calculate left and right tire vertical loads for the max lateral acceleration Choose tire to minimize understeer or oversteer Determine camber vs roll angle relationship for your suspension Make adjustments to understeer/oversteer Calculate critical speed Calculate yaw velocity and lateral acceleration gains

27. Suspension Design for Handling Vehicle Roll Stiffness Roll Stiffness Distribution Roll Center Height Tire Capacity Steering Geometry Camber Mass, C.G. Roll Inertia Tread Under-steer Over-Steer Stability Lateral Acceleration

28. Vehicle Roll-over Safety

29. Roll-over Forces M*a y* h - M*g*θ*h + F zi* t – M*g*t/2 = 0 a y /g = (t/2 + θ*h – F zi t/Mg)/h When θ=0 and a y =0, F zi = M*g/2 When θ=a y /g, F zi = M*g/2 Roll-over condition a y /g = t/2h + θ Where θ is the cross-slope Road super-elevation angle θ Mgθ

30. Roll-over Threshold t/2h

31. Roll-over Forces M*a y* h + M*g*φ*h + F zi* t – M*g*t/2 = 0 a y /g = (t/2 - φ*h – F zi t/Mg)/h When φ=0 and a y =0, F zi = M*g/2 When φ=a y /g, F zi = M*g/2 Roll-over condition a y /g = t/2h - φ Where φ is the vehicle roll angle Vehicle roll angle φ Mg φ

32. Roll-over Threshold

33. Roll-over Forces on a Suspended Vehicle M 0 =0= M s a y h-M s g[t/2 - φ(h-h r )] φ = R φ *a y Hence, max acceleration a y /g = t/{2h[1+R φ (1-h r /h)]}

34. Roll-over Threshold for Suspended Vehicle

35. Transient Roll-over in Step Steer I φ φ”+ C φ φ’ + [K φ -Mg(h-h r )] φ=W a y (h-h r )/g Where I φ = Roll moment of inertia C φ = Roll damping K φ = Roll stiffness h = C.G. height h r = roll center height W = vehicle weight a y = lateral acceleration Roll-over condition a y /g = t/{2h[1+R φ (1-h r /h)]} where R φ = φ max /(a y /g)

36. Step Steer L R V time Lateral Acceleration L / V V 2 /R

37. Roll Response to Step Steer

38. Effect of Damping

39. Transient Roll-over in Sinusoidal Steer I φ φ”+C φ φ’+[K φ -Mg(h-h r )]φ=Wa y (h-h r )sinωt/g Where I φ = Roll moment of inertia C φ = Roll damping K φ = Roll stiffness h = C.G. height h r = roll center height W = vehicle weight a y = lateral acceleration Roll-over condition a y /g = t/{2h[1+R φ (1-h r /h)]} where R φ = φ max /(a y /g)

40. Sinusoidal Steer V Y0Y0 2L Y = Y 0 sin (π*V*t/L) and lateral accn Y” = (π*V/L) 2 Y 0 sin (π*V*t/L)

41. Sinusoidal Steer

42. Suspension Design to Prevent Roll-over Vehicle Roll Stiffness/stabilize bar Roll Stiffness Distribution Roll Center Height Tire Capacity Mass, C.G. Roll Inertia Tread Roll Angle Rollover Threshold Step & Sinusoidal Steer