Problem Set-1b Week 11Challenge : Basic Handling parameters
Inputs
Front Axle Weight = 1900 lb
Rear Axle Weight = 1552 lb
Wheelbase = 100.6 in =
|
Load (lb) |
Cornering Stiffness (lb/deg) |
|
225 |
67 |
|
450 |
121 |
|
675 |
171 |
|
900 |
225 |
|
1125 |
227 |
|
1350 |
300 |
Solutions:
- Ackermann steer angles:
Delta = L/R*180/pi
i. R = 500 ft = 6000 in
Delta = 100.6/6000*180/pi
= 0.96 deg
ii. R = 200 ft = 2400 in
Delta = 100.6/2400*180/pi
= 2.40 deg
iii. R = 50 ft = 600 in
Delta = 100.6/600*180/pi
= 9.61 deg
- Understeer Gradient:
Weight on one wheel at front, Wf = Front Axle Weight / 2
= 950 lb
Weight on one wheel at rear, Wr = Rear Axle weight / 2
= 776 lb
Using interpolation method to find the cornering stiffness values from the table of values given.
Interpolation formula => y = y1+(x-x1)*((y2-y1)/(x2-x1))
At Front,
x = 950, x1 = 900, x2 = 1125, y1 = 225, y2 = 227
Cornering Stiffness, Cf = 225+(950-1125)*((227-225)/(1125-900)
= 225.44 lb/deg
At Rear,
x = 776, x1 = 675, x2 = 900, y1 = 171, y2 = 225
Cornering Stiffness, Cr = 171+(776-900)*((225-171)/(900-675)
= 195.24 lb/deg
Understeer gradient, K = Wf/Cf)-Wr/Cr
K = 0.24 deg/g
- Characteristic Speed:
V_char = sqrt((57.3*L*g)/K)
= sqrt((57.3*100.6*32.2)/(12*0.24))
= 254.24 mph
- Lateral acceleration gain:
V= 60 mph = 88 ft/s
lateral acceleration gain, Ay/ delta = (V^2/(57.3*L*g))/(1 +(K*V^2)/(57.3*L*g))
= (88^2/(57.3*(100.6/12)*32.2))/(1+(254.24*88^2)/(57.3*(100.6/12)*32.2))
= 0.48
- Yaw velocity gain:
V= 60 mph = 88 ft/s
Yaw velocity gain, r/delta = (V/L)/(1+(K*V^2)/(57.3*L*g))
= (88/100.6)/(1+(254.24*88^2)/(57.3*100.6*32.2))
= 9.37 (deg/s)/deg
- Slideslip angle:
V= 60 mph = 88 ft/s
R = 800 ft
Slideslip angle , beta = 57.3*c/R - (Wr*V^2)/(Cr*R*g)
Where, c = (Wf*L)/(W)
= (1900*100.6) / ((1900+1552)*12)
= 4.61 ft
Interpolation : x = 1552, x1 = 1125, x2 = 1350, y1 = 257, y2 = 300
Cornering Stiffness, C = 257 + (1552-1125)*(300-257)/(1350-1125)
= 338.6 lb/deg
beta = 57.3*c/R - (Wr*V^2)/(Cr*R*g)
= 57.3*4.61/800-(1552*88^2)/(338.6*800*32.2)
= -1.04 deg
- The understeer gradient was calculated from the study of a model with 2 DOF and we consider only cornering Stiffness, which is a tire level property. Whereas in the real scenario, other systems like suspension, steering, etc will contribute to the Understeer Gradient.