Handling Lowspeed turning Highspeed turning Understeer Lowspeed Turning

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

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

Low-speed Turning do L = ----L d i = tan-1 ----- R-t/2 L =

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

High Speed Turning Under Steer Path R > R 0 Original Path/ Neutral Steer

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

Tire Slip Angle

Tire Slip Angle

Tire Cornering Stiffness

Tire Cornering Stiffness

Factors affecting cornering stiffness

Factors affecting cornering stiffness

High-speed Turning • NSL force and moment analysis • Geometry for steer angle vs.

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

Understeer Gradient, K • Positive – understeer • Zero – neutral steer • Negative

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

Steer Angle vs. Speed

Steer Angle vs. Speed

Speeds & Gains Characteristic speed = speed at which steer angle required to negotiate

Speeds & Gains Characteristic speed = speed at which steer angle required to negotiate a turn is 2 times Ackerman angle Vchar = √ 57. 3 Lg/K Critical speed = speed at which steer angle required to negotiate a turn is 0 Vcrit = √-57. 3 Lg. K Lateral acceleration gain ay/δ = V 2/57. 3 Lg(1+ KV 2/57. 3 Lg) Yaw velocity gain r/δ = V/L(1+ KV 2/57. 3 Lg)

Effect on Lateral Acceleration Gain • Understeer – Very controlled gain with speed •

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

Effect on Yaw velocity gain

Effect on Yaw velocity gain

Slip Angle Calculation (primary tire effect) 1. Calculate front and rear vertical wheel loads

Slip Angle Calculation (primary tire effect) 1. Calculate front and rear vertical wheel loads Wf and Wr 2. Assume lateral acceleration ay/g as % (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

Effect of Body Roll W Fz 0 > Fzi

Effect of Body Roll W Fz 0 > Fzi

Effect of Body Roll No roll: For 800 lb load on each wheel 760

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

Effect of Body Roll Overturning moment Mφ = Wh 1 [ V 2/(Rg) +

Effect of Body Roll Overturning moment Mφ = Wh 1 [ V 2/(Rg) + φ] Mφ = Mφf + Mφr = (Kφf+Kφr) φ Hence, φ = Wh 1 V 2/[Rg(Kφf+Kφr-Wh 1)] Roll rate Rφ = dφ/day = Wh 1/[Kφf+Kφr-Wh 1] Where φ = roll angle, Kφ = roll stiffness, h 1 = distance between C. G. & roll ctr. Vertical load difference between outside and inside wheel (Fzof –Fzif)tf = Kφf*φ + Wfhf. V 2/Rg and (Fzof +Fzif) = Wf (Fzor –Fzir)tr = Kφr*φ + Wrhr. V 2/Rg and (Fzor +Fzir) = Wr Where hf and hr = roll center height front and rear

Slip Angle Calculation (roll effect) 1. Calculate front and rear vertical wheel loads Wf

Slip Angle Calculation (roll effect) 1. Calculate front and rear vertical wheel loads Wf and Wr 2. Assume lateral acceleration ay/g as % (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

Camber Thrust • Tires produce a lateral force (camber thrust) when inclined • Characterized

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

Camber Thrust Lateral Tire load due to camber Fyc = Cγ*γ = Cγ*(dγ/dφ)*(dφ/day)*ay γg

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

Slip Angle Calculation (roll/camber effect) 1. Calculate front and rear vertical wheel loads Wf

Slip Angle Calculation (roll/camber effect) 1. Calculate front and rear vertical wheel loads Wf and Wr 2. Assume lateral acceleration ay/g as % (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. Calculateral 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

Roll Steer • All suspensions steer with roll • Steer to the outside is:

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

Lateral Force Compliance Steer • All suspensions steer due to a lateral force •

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

Constant Radius Understeer Test

Constant Radius Understeer Test

Constant Speed Understeer Test

Constant Speed Understeer Test

Process for Calculating Cornering Response • • • Decide on the lateral acceleration requirement

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

Suspension Design for Handling Mass, C. G. Roll Inertia Tread Lateral Acceleration Vehicle •

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

Vehicle Roll-over Safety

Vehicle Roll-over Safety

Roll-over Forces M*ay*h - M*g*θ*h + Fzi*t – M*g*t/2 = 0 ay/g = (t/2

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 Mgθ Roll-over condition ay/g = t/2 h + θ Where θ is the cross-slope Road super-elevation angle θ

Roll-over Threshold t/2 h

Roll-over Threshold t/2 h

Roll-over Forces M*ay*h + M*g*φ*h + Fzi*t – M*g*t/2 = 0 ay/g = (t/2

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/2 h - φ Mgφ Vehicle roll angle φ Where φ is the vehicle roll angle

Roll-over Threshold

Roll-over Threshold

Roll-over Forces on a Suspended Vehicle M 0=0= Msayh-Msg[t/2 - φ(h-hr)] φ = Rφ*ay

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

Roll-over Threshold for Suspended Vehicle

Roll-over Threshold for Suspended Vehicle

Transient Roll-over in Step Steer Iφφ”+ Cφφ’ + [Kφ-Mg(h-hr)] φ=W ay(h-hr)/g Where Iφ =

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/{2 h[1+Rφ(1 -hr/h)]} where Rφ = φmax/(ay/g)

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

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

Roll Response to Step Steer

Roll Response to Step Steer

Effect of Damping

Effect of Damping

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=

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/{2 h[1+Rφ(1 -hr/h)]} where Rφ = φmax/(ay/g)

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

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

Sinusoidal Steer

Sinusoidal Steer

Suspension Design to Prevent Roll-over Mass, C. G. Roll Inertia Tread Step & Sinusoidal

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