Handling

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

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

4. Under Steer Path R > R0 Original Path/ Neutral Steer Path Over Steer Path R < R0 R R0 R R R R V High Speed Turning

5. Tire Slip Angle

6. Tire Cornering Stiffness

7. Factors affecting cornering stiffness

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

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

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 Vchar = √57.3Lg/K Critical speed = speed at which steer angle required to negotiate a turn is 0 Vcrit = √-57.3LgK Lateral acceleration gain ay/δ = V2/57.3Lg(1+ KV2/57.3Lg) Yaw velocity gain r/δ = V/L(1+ KV2/57.3Lg)

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

13. Effect on Yaw velocity gain

14. Slip Angle Calculation (primary tire effect) 1. Calculate front and rear vertical wheel loads Wf and Wr 2. Assume lateral acceleration ay/gas % (g). 3. Lateral tire force (front & rear) Fyf = Wf*ay and Fyr = Wr*ay 4. From tire data find slip angles for all 4 tires, use extrapolation 5. Find average slip angle for front and rear αf and αr 6. Calculate under steer αf – αr 7. Do calculations for ay/g from 0.1 to 1.0

15. Effect of Body Roll W Fz0 > Fzi

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 Mφ = Wh1 [ V2/(Rg) + φ] Mφ = Mφf + Mφr = (Kφf+Kφr) φ Hence, φ = Wh1V2/[Rg(Kφf+Kφr-Wh1)] Roll rate Rφ = dφ/day = Wh1/[Kφf+Kφr-Wh1] Where φ = roll angle, Kφ = roll stiffness, h1 = distance between C.G. & roll ctr. Vertical load difference between outside and inside wheel (Fzof –Fzif)tf = Kφf*φ + WfhfV2/Rg and (Fzof +Fzif) = Wf (Fzor –Fzir)tr = Kφr*φ + WrhrV2/Rg and (Fzor +Fzir) = Wr Where hf and hr = roll center height front and rear

18. Slip Angle Calculation (roll effect) 1. Calculate front and rear vertical wheel loads Wf and Wr 2. Assume lateral acceleration ay/gas % (g). 3. Lateral tire force (front & rear) Fyf = Wf*ay and Fyr = Wr*ay 4. Calculate roll rate and find roll angle φ 5. Calculate Fzi and Fzo 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 αf and αr 8. Calculate under steer αf – αr 9. Do calculations for ay/g from 0.1 to 1.0

19. F = 1000 lb z 200 Zero Slip Angle g 150 Lateral Force (lb) 100 50 0 0 1 2 3 4 5 6 7 8 9 Camber Angle (deg) C g Camber Coefficient, Cg/Fz (lb/lb/deg) Camber Thrust • Tires produce a lateral force (camber thrust) when inclined • Characterized by camber stiffness, Cg • Camber coefficient • Radials are lower • Bias-ply are higher

20. Camber Thrust Lateral Tire load due to camber Fyc = Cγ*γ = Cγ*(dγ/dφ)*(dφ/day)*ay = Cγ*(dγ/dφ)*roll rate*ay γg = γb + φ Where γg = camber w.r.t. ground γb = camber w.r.t. body φ = roll angle γ-φ relationship Lateral tire force causing tire slip = W*ay - Fyc

21. Slip Angle Calculation (roll/camber effect) 1. Calculate front and rear vertical wheel loads Wf and Wr 2. Assume lateral acceleration ay/gas % (g). 3. Calculate roll rate and find roll angle φ 4. Calculate Fzi and Fzo 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) Fyf = Wf*ay-Fycf and Fyr = Wr*ay-Fycr 8. From tire data find slip angles for all 4 tires, use extrapolation 9. Find average slip angle for front and rear αf and αr 10. Calculate under steer αf – αr 11. Do calculations for ay/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 Deflection Understeer Deflection Oversteer Turn Turn Yaw center Cornering Force Cornering Force 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 Mass, C.G. Roll Inertia Tread Vehicle Lateral Acceleration Under-steer Over-Steer Stability • Roll Stiffness • Roll Stiffness Distribution • Roll Center Height • Tire Capacity • Steering Geometry • Camber

28. Vehicle Roll-over Safety

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

30. Roll-over Threshold t/2h

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

32. Roll-over Threshold

33. Roll-over Forces on a Suspended Vehicle M0=0= Msayh-Msg[t/2 - φ(h-hr)] φ = Rφ*ay Hence, max acceleration ay/g = t/{2h[1+Rφ(1-hr/h)]}

34. Roll-over Threshold for Suspended Vehicle

35. Transient Roll-over in Step Steer Iφφ”+ Cφφ’ + [Kφ-Mg(h-hr)] φ=W ay(h-hr)/g Where Iφ = Roll moment of inertia Cφ= Roll damping Kφ= Roll stiffness h = C.G. height hr = roll center height W = vehicle weight ay = lateral acceleration Roll-over condition ay/g = t/{2h[1+Rφ(1-hr/h)]} where Rφ = φmax/(ay/g)

36. Step Steer V2/R Lateral Acceleration R L / V time V L

37. Roll Response to Step Steer

38. Effect of Damping

39. Transient Roll-over in Sinusoidal Steer Iφφ”+Cφφ’+[Kφ-Mg(h-hr)]φ=Way(h-hr)sinωt/g Where Iφ = Roll moment of inertia Cφ= Roll damping Kφ= Roll stiffness h = C.G. height hr = roll center height W = vehicle weight ay = lateral acceleration Roll-over condition ay/g = t/{2h[1+Rφ(1-hr/h)]} where Rφ = φmax/(ay/g)

40. Sinusoidal Steer Y = Y0 sin (π*V*t/L) and lateral accn Y” = (π*V/L)2Y0 sin (π*V*t/L) V 2L Y0

41. Sinusoidal Steer

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