Vehicle Dynamics - Basic Handling parameters
AIM: To solve the given conceptual numerical question based on 'Vehicle Handling'
Queston- A car has a weight of 1900 lb front axle and 1552 lb on the rear with a wheelbase of 100.6 inches. The tires have the following cornering stiffness values
OBJECTIVES: Determine the following-
1. Ackermann steer angles for 500, 200 and 50 ft radius
2. Understeer gradient
3. Characteristic speed
4. Lateral acceleration gain at 60 mph
5. Yaw velocity gain at 60 mph
6. Slideslip angle at the CG on an 800 ft radius turn at 60 mph.
7. Why is the calculated understeer gradient quite low (usually it is around 3-4 deg/g)
GIVEN:
Front axle weight, `W_f = 1900lb`
Rear axle weight, `W_r = 1552lb`
Wheelbase, `L = 100.6`in
Cornering stiffness values for different tire loads:
SOLUTION:
1. Ackermann steer angles for 500, 200 and 50 ft radius-
Ackerman steer angle, `delta = L/R*180/pi`
a) For R = 500ft
`delta = 100.6/(500*12)*180/pi = 0.961deg`
b) For R = 200ft
`delta = 100.6/(200*12)*180/pi = 2.4deg`
c) For R = 50ft
`delta = 100.6/(50*12)*180/pi = 9.61deg`
2. Understeer gradient-
The weight distribution on each wheel at front and rear will be 50/50 of the corresponding weights at front and rear respectively.
Therefore weight of each wheel:
`w_f = 950lb`
`w_r = 776lb`
Now to determine the understeer gradient we need cornering stiffness values at these loads. We use the interpolation formula to determine those values from the given table.
Therefore,
`C_(alpha_f) = 225 + (950-900)*((257-225)/(1125-900)) = 232.111(lb)/(deg)`
`C_(alpha_r) = 171 + (776-675)*((225-171)/(900-675)) = 195.24(lb)/(deg)`
Understeer gradient, `K = w_f/C_(alpha_f)-w_r/C_(alpha_r) = 950/232.111-776/195.24`
`K = 0.118(deg)/g`
3. Characteristic speed-
Characteristic speed is the speed at which the steering angle requied to negotiate the turn is twice the ackerman angle. It is given by:
`V_(char) = sqrt((57.3Lg)/K) = sqrt((57.3*100.6*32.2)/(12*0.118))`
`V_(char) = 362.05(ft)/(sec) = 246.85mph`
4. Lateral acceleration gain at 60mph-
`60mph = 88(ft)/(sec)`
lateral acceleration gain is given by:
`a_y/delta = (V^2/(57.3Lg))/(1 +(KV^2)/(57.3Lg)) = (88^2/(57.3*100.6/12*32.2))/(1+(0.118*88^2)/(57.3*100.6/12*32.2))`
`a_y/delta = 0.473g/(deg)`
lateral acceleration gain = `0.473g/(deg)`
5. Yaw velocity gain at 60mph-
`60mph = 88(ft)/(sec)`
yaw velocity (or yaw rate) gain is given by:
`r/delta = (V/L)/(1+(KV^2)/(57.3Lg)) = (88/(100.6/12))/(1+(0.118*88^2)/(57.3*100.6/12*32.2`
`r/delta = 9.911(deg//sec)/(deg)`
yaw velocity gain = `9.911(deg//sec)/(deg)`
6. Slideslip angle at the CG on an 800ft radius turn at 60mph-
The sideslip angle is given by:
`beta = 57.3c/R - (W_rV^2)/(C_(alpha_r)Rg)`
where, c is the distance of CG from the rear axle.
we know, `W_f = W_(Total)*c/L`
`=> c = (W_fL)/(W_(Total)) = (1900*100.6)/((1900+1552)*12) = 4.614ft`
`C_(alpha_r) = 257 + (1552-1125)*(300-257)/(1350-1125) = 338.60(lb)/(deg)`
`therefore beta = 57.3*4.614/800 - (1552*88^2)/(338.60*800*32.2)`
`=> beta = -1.047deg`
7. Why is the calculated understeer gradient quite low? (usually it is around 3-4 deg/g)
Answer`=>` We determined the understeer gradient from the study of a 2DOF model and we only considered the cornering stiffness which is tire property. But in real scenario, other systems such as the suspension, steering etc. will also contribute to the understeer gradient.