Steering Geometry 1

Given,

Hardpoint	x	y	z
LCA Subframe Pivot Front	-200	-400	180
LCA Knuckle Ball Joint	0	-750	100
LCA Subframe Pivot Rear	200	-400	180
UCA Subframe Pivot Front	100	-490	525
UCA Knuckle Ball Joint	40	-675	525
UCA Subframe Pivot Rear	250	-490	500
Wheel Center	0	-800	300

1. i) Caster is calculated as the angle made by the kingpin/steering axis with the verticle as viewed from the side.

let θ be the angle made by the kingpin axis with the horizontal in the side view

We've slope given by: tan θ = (z2-z1)/(x2-x1)

Using coordinates for upper and lower ball joint,

tan θ = (525-100)/(40-0)

θ = 84.62 deg (angle made by the kingpin axis with the horizontal)

Thus, Caster = 90 - θ = 90 - 84.62 = 5.38 deg (rearwards inclination of kingpin axis i.e. positive caster)

ii)Equation of kingpin axis as viewed from the side is: z - z1 = [(z2-z1)/(x2-x1)](x-x1)

Using the coordinates of upper and lower ball joints,

z - 100 = [(525-100)/(40-0)](x-0)

Simplifying,

z = 10.625x + 100 ...... (I)

Now,

For Mechanical trail we use the z-coordinate at the ground and find the corresponding x-coordinate in the kingpin axis.

So, putting z = 0 in Equation (I),

x = -9.41

Now Kingpin lateral offset is calculated as the distance between the x-cordinates of the tire print/wheel center and the kingpin axis evaluated at the ground level.

Thus, Mechanical trail = |-9.41-0| = 9.41 mm (towards the front of the vehicle)

iii) Kingpin inclination is calculated in the front view.

let θ be the angle made by the kingpin axis with the horizontal in the front view

Slope is given by: tan θ = (z2-z1)/(y2-y1)

Using coordinates for upper and lower ball joint,

tan θ = (525-100)/(-675+750)

θ = 80 deg (angle made by the kingpin axis with the horizontal)

Kingpin inclination is the angle made by the kingpin axis with the vertical.

Thus, Kingpin inclination = 90 - θ = 90 - 80 = 10 deg (towards the inboard)

iv) Equation of kingpin axis as viewed from the front is: z - z1 = [(z2-z1)/(y2-y1)](y-y1)

Using the coordinates of upper and lower ball joints,

z - 100 = [(525-100)/(-675+750)](y+750)

Simplifying,

z = (17/3)y+4350 ....... (II)

Now,

For Scrub radius, we use z-coordinate at the ground and find the corresponding y-coordinate in the kingpin axis

So putting z = 0 in Equation (II),

y = -767.65

Now Scrub radius is calculated as the distance between the y-cordinates of the wheel center and the kingpin axis evaluated at the ground level.

Thus, Scrub radius = |-767.65-(-800)| = 32.35 mm (inboard)

v)For Kingpin lateral offset, we use the z-coordinate of the wheel center and find the corresponding y-coordinate in the kingpin axis.

So, putting z = 300 in Equation (II),

y = -714.7

Now Kingpin lateral offset is calculated as the distance between the y-cordinates of the wheel center and the kingpin axis evaluated at the wheel center height.

Thus, Kingpin lateral offset = |-714.7-(-800)| = 85.3 mm (inboard)

2. Pneumatic trail = 25 mm and Lateral force = 2000 N

Now,

Restoring moment acting at the point where the kingpin intersects the ground in side view = Lateral force x ( Mechanical trail + Pneumatic trail)

Thus, Restoring moment = 2000 x (9.41+25) = 68820 Nmm = 68.820 Nm

3. Scrub radius would be zero when the steering axis meets the ground at the tire contact as seen from the front view in the yz-plane.

In yz-plane, coordinate of tire contact = (-800,0)

We have, coordinate of upper ball joint = (-675,525)

Equation of steering axis if it were to pass through the above two points is:

z - z1 = [(z2-z1)/(y2-y1)](y-y1)

z - 0 = [(525-0)/(-675+800)](y+800)

Simplifying, z = 4.2y + 3360 ....... (III)

For current z-coordinate of the lower ball joint i.e. z = 100, from equation (III), we get y = -776.19

Thus, changing the y-coordinate of the given lower ball joint to -776.19mm would give us a ZERO scrub radius.

Further decrease in the y-coordinate (i.e. more negative value) makes the scrub radius negative (outboard).

So, Suggested LCA Knuckle Ball Joint coordinate = (0, -780,100) to change the sign of the scrub radius.