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

Given Data: weight on front axle $w_{f}$ $w_f$ = 1900 lb weight on rear axle $w_{r}$ $w_r$ = 1552 lb wheelbase $l$ $l$ = 100.6 in The tires have the following stiffness values: Deliverables: 1) Ackermann steer angles for 500, 200 and 50 Ackerman Steering Angle $δ = (\frac{180}{π}) \cdot \frac{L}{R}$ $delta = (180/pi)*L/R$ For the given values of radius: Radius (ft) Radius…

Umang Tyagi
updated on 14 Jun 2021

Given Data:

weight on front axle $w_{f}$ $w_f$ = 1900 lb

weight on rear axle $w_{r}$ $w_r$ = 1552 lb

wheelbase $l$ $l$ = 100.6 in

The tires have the following stiffness values:

Deliverables:

1) Ackermann steer angles for 500, 200 and 50

Ackerman Steering Angle

δ = (\frac{180}{π}) \cdot \frac{L}{R}

$delta = (180/pi)*L/R$

For the given values of radius:

Radius (ft)	Radius (in)	Ackerman Angle $δ$ $delta$
500	6000	0.9607
200	2400	2.4016
50	600	9.6066

2) Understeer Gradient

Understeer gradient

k = \frac{W_{f}}{C_{α f}} - \frac{W_{r}}{C_{α r}}

$k = W_f/C_(alphaf)-W_r/C_(alphar$

$W_{f}$ $W_f$ is the weight on the front axle (one tire)

$W_{r}$ $W_r$ is the weight on the rear axle (one tire)

$C_{α f}$ $C_(alphaf$ is the front cornering stiffness

$C_{α r}$ $C_(alphar$ is the rea cornering stiffness

$W_{f} = \frac{1900}{2} = 950$ $W_f = 1900/2 = 950$ lb

$W_{r} = \frac{1552}{2} = 776$ $W_r = 1552/2 = 776$ lb

Calculating $C_{α f}$ $C_(alphaf$ using interpolation:

$\frac{1125 - 950}{1125 - 900} = \frac{257 - C_{α f}}{257 - 225}$ $(1125-950)/(1125-900) = (257-C_(alphaf))/(257-225)$

$\Rightarrow$ $=>$ $C_{α f}$ $C_(alphaf$ = 232.11 lb/deg

Calculating $C_{α r}$ $C_(alphar$ using interpolation:

$\frac{900 - 776}{900 - 675} = \frac{225 - C_{α r}}{225 - 171}$ $(900-776)/(900-675)=(225-C_(alphar))/(225-171)$

$\Rightarrow$ $=>$ $C_{α r}$ $C_(alphar$ = 195.24 lb/in

Implementing the formula for understeer budget:

$k = \frac{950}{232.11} - \frac{776}{195.24}$ $k = 950/232.11-776/195.24$

$k = 0.1183$ $k = 0.1183$ deg/g

3) Characteristic speed

Characteristic Speed

u_{c h a r} = \sqrt{57.3 \cdot L \cdot \frac{g}{k}}

$u_(char) = sqrt(57.3*L*g/k)$

$g = 386.08858267562$ $g = 386.08858267562$ in/sec^2

$u_{c h a r} = \sqrt{57.3 \cdot 100.6 \cdot \frac{386.08858267562}{0.1183}}$ $u_(char) = sqrt(57.3*100.6*386.08858267562/0.1183)$

$u_{c h a r}$ $u_(char$ = 4337.4 in/sec = 110.17 m/s

4) Lateral acceleration gain at 60 mph

v = 60 mph = 88 ft/sec

g = 32.2 ft/sec^2

Lateral Acceleration Gain

\frac{a_{y}}{δ} = \frac{\frac{v^{2}}{57.3 L g}}{1 + k \frac{v^{2}}{57.3 L g}}

$a_y/delta = ((v^2)/(57.3Lg))/(1+kv^2/(57.3Lg)$

$\frac{a_{y}}{δ} = \frac{\frac{88}{57.3 \cdot \frac{100.6}{12} \cdot 32.2}}{1 + (0.118 \cdot \frac{88^{2}}{57.3 \cdot \frac{100.6}{12} \cdot 32.2})}$ $a_y/delta = (88/(57.3*100.6/12*32.2))/(1+(0.118*88^2/(57.3*100.6/12*32.2)))$

$\frac{a_{y}}{δ} = 0.473$ $a_y/delta = 0.473$ g/deg

5) Yaw velocity gain at 60 mph

Yaw Velocity Gain

\frac{r}{δ} = \frac{\frac{y}{L}}{1 + k \frac{v^{2}}{57.3 L g}}

$r/delta = (y/L)/(1+kv^2/(57.3Lg))$

$\frac{r}{δ} = \frac{\frac{88}{\frac{100.6}{12}}}{1 + 0.118 \cdot \frac{88^{2}}{57.3 \cdot \frac{100.6}{12} \cdot 32.2}}$ $r/delta =(88/(100.6/12))/(1+0.118*88^2/(57.3*100.6/12*32.2))$

$\frac{r}{δ} = 9.911$ $r/delta = 9.911$ deg/sec/deg

6) Slideslip angle at the CG on an 800 ft radius turn at 60 mph.

Body Side Slip

β = 57.3 \frac{c}{R} - \frac{w_{r} v^{2}}{C_{α r} g R}

$beta= 57.3c/R-(w_rv^2)/(C_(alphar)gR)$

R = 800 tf

$w_{f} = (w_{f} + w_{r}) . \frac{c}{L}$ $w_f = (w_f+w_r).c/L$

$\Rightarrow c = \frac{1900.100 .6}{(1900 + 1552) \cdot 12} = 4.614$ $=> c = (1900.100.6)/((1900+1552)*12) = 4.614$ ft

Calculating $C_{α r}$ $C_(alphar$ using interpolation:

$\frac{1552 - 1350}{1350 - 1125} = \frac{C_{α r} - 300}{300 - 257}$ $(1552-1350)/(1350-1125)=(C_(alphar)-300)/(300-257)$

$\Rightarrow$ $=>$ $C_{α r} = 338.60$ $C_(alphar)= 338.60$ lb/deg

$β = 57.3 \cdot \frac{4.614}{800} - \frac{1552 \cdot 88^{2}}{338.60 \cdot 800 \cdot 32.2}$ $beta = 57.3*4.614/800-(1552*88^2)/(338.60*800*32.2)$

$β = - 1.047$ $beta = -1.047$ deg

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

There are various factors that affect the understeer gradient including the steering and suspension system but here the understeer gradient is being calculated using only a 2 degree of freedom model and cornering stiffness.