Final Project
Aim: To select roll bar and prepare an understeer budget sheet for a light weight sporty vehicle.
Inputs:
Target And Constraints:
Formula Used:
1. Roll Bar:
a. The CG distance from front and rear is calculated using the weight distribution.
Front axle to CG distance(a) = (Msr*WB)/m
Rear axle to CG distance(b) = b=(Msf*WB)/m
Where,
Msr =Rear sprung mass (lbs)
Msf = Front sprung mass (lbs)
WB = wheelbase (in)
m = Total sprung mass (lbs)
b. Sprung mass roll moment arm length.
`h(arm) = h-([Zrcf*b+Zrcr*a]/(WB))`
Where,
h = CG height (in)
Zrcf = Front Roll Centre height (in)
Zrcr = Rear Roll Centre height (in)
c. The lateral load transfer is calculated using the following procedure-
Total load transfer(LT) = `W*h/T`
Where,
W = Total weight (lbs)
T = Trackwidth (in)
Load transfer at front is needed to be 5% more for achieving understeer.
LTf = LT * (`(%Wf+5)/100`)
Front load transfer due to unsprung mass
LTu = `(Muf*rt)/T`
Where,
Muf = Front unsprung mass (lbs)
rt = tire radius (in)
Front load transfer due to forces on sprung mass,
LTs = Msf*(Zrcf/T)
Where,
Msf = Front sprung mass
Front load transfer due to roll,
LT(phi) = LTf - LTs - LTu
d. The roll moment has two components, lateral component and gravity component.
Lateral component(Ml) = Ms*h(arm)
where,
Ms = Total Sprung Mass
Gravity component(Mg) = Ms * h(arm)*`(phi/(Ay))*(pi/180)`
where,
`phi/(Ay)` = Roll Gradient
Total roll moment(M) = Ml+Mg
e. Roll stiffness
Total roll stiffness due to roll gradient `k = (M/phi)`
Front roll stiffness at tire kf = `LT*T/phi`
Rear roll stiffness at tire kr = k - kf
Roll stiffness due to wheel rate contribution `kphiw = kw*((TW^2)/(2))`
Roll stiffness due to tire rate contribution `kphit = kt*(T*W^2)/2`
Roll stiffness at wheel centre (removing tire effect) `kphiwc = (kphit*k)/(kphit-k)`
Auxiliary roll stiffness `karb = kphiwc - kphiw`
% auxiliary roll stiffness `%karb = karb*100/(kphi)`
f. Anti-roll stiffness calculations
Anti-roll stiffness = `(500000*Do^4)/(0.4422A^2B +0.2264C^3)`
Anti-roll stiffness for one wheel jounce = `(500000*(Do^4-Di^4))/(0.4422A^2B +0.2264C^3)*MR^2*[(TW^2)/2]`
Where,
Do and Di are outer and inner dia of roll bar
| INPUT DATA | ||||||
| Vehicle Weights and Dimensions | Spring | |||||
| Parameter | Units | Value | Parameter - Front | Units | Value | |
| Total weight | lb | 2750 | Motion ratio | in/in | 0.95 | |
| Weight distribution | % front | 60 | Spring rate | lb/ | 250 | |
| Front axle unsprung mass | lb | 187 | ||||
| Rear axle unsprung mass | lb | 121 | Parameter - Rear | |||
| Motion ratio | in/in | 0.8 | ||||
| Wheelbase | in | 98.42 | Spring rate | lb/in | 300 | |
| CG height | in | 21.62 | ||||
| Trackwidth | in | 58 | Tires | |||
| Parameter | Units | Value | ||||
| Front roll center height | in | 0.497 | Loaded radius | in | 12.27 | |
| Rear roll centre height | in | 0.896 | Vertical stiffness | lb/in | 1313 | |
| Anti-Roll Bar | Normal Load | Cornering Stiffness | Camber stiffness | |||
| Parameter - Front | Units | Value | lb | lb/deg | lb/deg | |
| Motion ratio | in/in | 0.64 | 500 | 167.23 | 8.37 | |
| Virtual torque arm (A) | in | 8.9 | 1000 | 262.17 | 17.76 | |
| Centre length (B) | in | 32 | 1500 | 301.75 | 25.3 | |
| Physical torque arm (C) | in | 10.7 | 2000 | 269.86 | 29.9 | |
| Parameter - Rear | Tire response | Units | Front | |||
| Motion ratio | in/in | 0.75 | Slip angle | deg/g | 3.88 | |
| Virtual torque arm (A) | in | 8.51 | Camber | deg/g | 4.55 | |
| Centre length (B) | in | 32.2 | Lateral force | N/g | 3753 | |
| Physical torque arm (C) | in | 8.59 | Aligning torque | Nm/g | 139.8 | |
Roll Bar Calculation:
| ROLL BAR & STIFFNESS | ||||||
| Weight Distribution | Lateral load transfer | |||||
| Parameter | Units | Value | Parameter | Units | Value | |
| Total sprung mass | lbs | 2442 | Total load transfer | lbs | 1025.086207 | |
| Front sprung mass | lbs | 1465.2 | Load transfer at front | lbs | 615.0517241 | |
| Rear sprung mass | lbs | 976.8 | Desired load transfer at front (+5%) | lbs | 666.3060345 | |
| Load transfer at rear | lbs | 358.7801724 | ||||
| Front corner sprung mass | lbs | 732.6 | ||||
| Rear corner sprung mass | lbs | 488.4 | Front load transfer due to unsprung mass | lbs | 39.56017241 | |
| Front load transfer due to forces on sprung mass | lbs | 12.55524828 | ||||
| Front corner unsprung mass | lbs | 93.5 | Front load transfer due to roll | lbs | 614.1906138 | |
| Rear corner unsprung mass | lbs | 60.5 | ||||
| Roll stiffness | ||||||
| Sprung mass calculations | Total roll stiffness at tire due to roll gradient | ft-lb/deg | 1197.10141 | |||
| Parameter | Units | Value | ||||
| Front axle to CG distance (a) | in | 39.368 | Front roll stiffness at tire | ft-lb/deg | 781.2073596 | |
| Rear axle to CG distance (b) | in | 59.052 | Rear roll stiffness at tire | ft-lb/deg | 415.8940501 | |
| Front roll centre heigth | in | 0.497 | Front roll stiffness (wheel rate contribution) | ft-lb/deg | 551.962194 | |
| Rear roll centre heigth | in | 0.896 | Rear roll stiffness (wheel rate contribution) | ft-lb/deg | 469.7030083 | |
| Sprung mass roll moment arm (h_arm) | in | 20.9634 | Front roll stiffness (tire rate contribution) | ft-lb/deg | 3212.083593 | |
| Rear roll stiffness (tire rate contribution) | ft-lb/deg | 3212.083593 | ||||
| Roll gradient | deg/g | 3.8 | ||||
| Front roll stiffness at wheel centre (removing tire effect) | ft-lb/deg | 1032.262897 | ||||
| Rates | Rear roll stiffness at wheel centre (removing tire effect) | ft-lb/deg | 477.7524678 | |||
| Parameter | Units | Value | ||||
| Front spring rate | lb/in | 250 | Front auxiliary roll stiffness at wheel centre | ft-lb/deg | 480.3007029 | |
| Rear spring rate | lb/in | 300 | Rear auxiliary roll stiffness at wheel centre | ft-lb/deg | 8.049459527 | |
| Front tire rate | lb/in | 1313 | Front auxiliary roll stiffness at wheel centre | % | 61.48184563 | |
| Rear tire rate | lb/in | 1313 | Rear auxiliary roll stiffness at wheel centre | % | 1.935459169 | |
| Front wheel rate | lb/in | 225.625 | Anti-roll stiffness | |||
| Rear wheel rate | lb/in | 192 | Parameter - Front | Units | Value | |
| Anti-roll stiffness | lb/in | 33.00479777 | ||||
| Front ride rate | lb/in | 192.5392 | Anti-roll spring rate | lb/in | 66.00959554 | |
| Rear ride rate | lb/in | 167.5056 | Anti-roll stiffness for one wheel jounce | lb/in | 22738.56 | |
| Roll moment | Parameter - Rear | Units | Value | |||
| Lateral component | ft-lb/g | 4266.05 | Anti-roll stiffness | lb/in | 21.20511913 | |
| Gravity component | ft-lb/g | 282.9352 | Anti-roll spring rate | lb/in | 42.41023825 | |
| Total roll moment | ft-lb/g | 4548.985 | Anti-roll stiffness for one wheel jounce | lb/in | 20062.69333 | |
| ARB Outer diameter | Front | Rear | Pneumatic Trail | in | ||
| mm | 14 | 12 | Front | 1.466544093 | ||
| in | 0.551181 | 0.472441 | Rear | 0.842105718 | ||
TIRE STIFFNESS PLOTS:
| UNDERSTEER BUDGET | |||||
| Tire Stiffnesses at given load (by interpolation method) | |||||
| Parameter | Weight (lbs) | Cornering Stiffness (lb/deg) | Camber stiffness (lb/deg) | Aligning Stiffness (lb/deg) | |
| Front | 1650 | 292.183 | 26.68 | 62.69 | |
| Rear | 1100 | 270.086 | 19.268 | 39.548 | |
| Compliance Coefficients | |||||
| Parameter | Unit | Front | Rear | ||
| Roll steer | % | 6.4 | -5 | ||
| Roll camber | % | 85 | 55 | ||
| Lateral force compliance steer | deg/kN | 0.025 | 0.03 | ||
| Lateral force compliance camber | deg/kN | 0.2 | 0.2 | ||
| Aligning torque compliance steer | deg/100Nm | 0.3 | 0.08 | ||
| Understeer Budget Sheet | |||||
| Understeer Source | Front Cornering Compliance | Rear Cornering Compliance | Units | Understeer gradient | % Contribution |
| Weight Distribution | 5.647145796 | 4.072776819 | deg/g | 1.574368976 | 58.5 |
| Aligning Torque | 0.040265307 | -0.075859922 | deg/g | 0.116125228 | 4.32 |
| Roll Steer | 0.2432 | -0.19 | deg/g | 0.4332 | 16.1 |
| Roll Camber | 0.294939815 | 0.149101101 | deg/g | 0.145838714 | 5.42 |
| Lateral Force Compliance Steer | 0.04125 | 0.033 | deg/g | 0.00825 | 0.31 |
| Lateral Force Compliance Camber | 0.068539265 | 0.033886614 | deg/g | 0.034652651 | 1.29 |
| Aligning Torque Compliance Steer | 0.4194 | 0.04064 | deg/g | 0.37876 | 14.07 |
| TOTAL | 2.691195569 | 100 | |||
Results:
| FINAL DELIVERABLES: | ||
| Roll Bar Diameter (mm) | ||
| Front | 14 | |
| Rear | 12 | |
| Roll Stiffness at wheel centre (ft-lb/deg) | Primary | Secondary |
| Front | 551.96 | 1032.26 |
| Rear | 469.7 | 477.75 |
| Auxiliary Roll Stiffness (%) | ||
| Front | 61.48 | |
| Rear | 1.94 | |
| Understeeer Budget | ||
| Understeer Source | % Contribution | |
| Weight Distribution | 58.5 | |
| Aligning Torque | 4.32 | |
| Roll Steer | 16.1 | |
| Camber Thrust | 5.42 | |
| Lateral Force Compliance Steer | 0.31 | |
| Lateral Force Compliance Camber | 1.29 | |
| Aligning Torque Compliance Steer | 14.07 | |