Problem Set-1b Week 11Challenge : Basic Handling parameters

Determine the following: (2 points each except for the first question)

1. Ackermann steer angles for 500, 200 and 50 ft radius (3 points)
2. Understeer gradient
3. Characteristic speed
4. Lateral acceleration gain at 60 mph
5. Yaw velocity gain at 60 mph
6. Slide slip angle at the CG on an 800ft radius turn at 60mph
7. Why is the calculated understeer gradient quite low (usually it is around 3-4 deg/g)

SOLUTION

1. Ackermann steer angles for 500, 200 and 50 ft radius (3 points)

Steering angle = l / r => l=100.6 inches = 2.54 m

Radius= r (ft) Radius =r (m) Ackermann steer angles

500 152.4 = 2.54 / 152.4 = 0.01667 rad = 0.01667*180/3.142 = 0.954 degrees

200 60.96 = 2.54 / 60.96 = 0.0416 rad = 0.0416*180/3.142 = 2.383 degrees

50 15.24 = 2.54 / 15.24 = 0.1667 rad = 0.1667 * 180 / 3.142 = 9.54 degrees

2) FA Mass at each tire at front = 1900 lb = 1900 / 2 = 950 lb

RA Mass at each tire at front == 1552 lb = 1552 / 2 = 776 lb

By interpolation stiffness for load 950 lb =

1125-900=225 ;; 257-225 = 32

32/225 = 0.142 * 50 +225 = 232.12 lb/ deg for 950 lb cornering stiffness = 232 lb/deg = Cf

llly for 776 lb

900-675 = 225 ;; 225-171 = 54

54/225 = 0.24 * 101 + 171 = 195.24 lb/ deg for 776 lb cornering stiffness = 195.24 lb/deg =Cr

Under steer gradient : k = Wf / Cf – Wr / Cr

=950/232-776/195.24= 0.12 deg /deg

Under steer gradient = 0.12 deg / g

3) Characteristic speed

U² = 57.3 * (100.6 *0.0254/) * 9.81 / 0.12 => U = 109.3 m/s

4) Lateral accln gain at 60mph

U=60mph = 26.8m/s

a_y / δ =U²/ (57.3 * l *g ) / ( 1+ ( k*U² / 57.3* l * g))

= 26.8² / 57.3 * 2.554 *9.81 ) / ( 1+ ( 0.12* 27.8²/57.3*2.554*9.81))

= 0.5 /1.06

a_y / δ = 0.47 g/deg

5) Yaw velocity gain at 60 mph

r / δ = ( V / L ) / (1 + ( k*V² / 57.3* l * g))

= (26.8 / 2.554) / (1+(0.12*26.8² /57.3*2.554*9.81))

= 10.49 / 1.06

r / δ =9.899 deg/s / deg

6) Side slip angle

Mf = 1900 lbs Mr = 1552 lbs M = Mf +Mr = 3452 lbs

% mass on front = 1900/3452 = 0.55 lf = 0.45 * 100.6 = 45.27 in

% mass on rear = 1552/3452 = 0.45 lr = 0.55 * 100.6 = 55.37 in = 1406.398mm

At low speed,

Body side slip angle = (lr/l)*ackermann steer angle

For R= 500ft, slip angle = (55.37 / 100.6)* 0.954 = 0.525deg

For R=200ft, slip angle = (55.37 / 100.6)* 2.383 =1.311 deg

For R=50ft, slip angle = (55.37 / 100.6)* 9.54 = 5.25 deg

At high speeds,

Side slip angle,(lr/R)-Br, Br=ar=Wr/Car*ay

For R=800ft and U=60mph

55.37 / 9600 – (1552 /195.24)*0.47 = -3.73 deg

For R=500ft and U=60mph

55.37 / 6000 – (1552 /195.24)*0.47 = -3.72 deg

-sign indicates direction of slip of the rear wheel

R=200ft,

55.37 / 2400 – (1552 /195.24)*0.47 = -3.71 deg

R=50ft,

55.37 / 600 – (1552 /195.24)*0.47 = -3.64 deg

7. Why is the calculated understeer gradient quite low (usually it is around 3-4 deg/g)

Increased under steer gradient results in more steering, Under steer is also dependent on cornering stiffness.