Here are a few tips on how to draw out your suspension in CAD and use copies of it to see how the wheel behaves as the suspension is moved. To get started, you need the locations of the components shown in “Figure 1 Suspension Layout”. All measurements are taken in two dimensions projected against a vertical plain running through the steering knuckles on the steering arms. The circles U0, U1, L0, L1, S0, and S1 represent the center of the location of the ball joints and pivots. The wheel should be centered ½ the distance of the vehicle track from the chassis centerline.

Figure 1 Suspension Layout

Moving the Wheel

Many of the operations used when studying the dynamics of a suspension involve moving the wheel a particular amount. Since the wheel is fixed to the chassis at two locations, moving it is a complicated process. There are two ways to move the wheel. One is to find the virtual center of the suspension elements holding the wheel, moving it a small amount each time, finding the new center after each movement. The second, illustrated here, involves moving the wheel relative to one suspension arm, then correcting to match the second.

Figure 2 Preparing the wheel drawing

Referring to “Figure 2 Preparing the wheel drawing”, construction circles show the length of the control arms and the height of the upright. “arc1” is drawn centered on L0 to the radius defined by L0L1. “arc2” is centered on U0 of radius U0U1. “arc3” is centered on L1 with a radius of L1U1.

Rotate around L0

The wheel W, L1, U1, S1 (if used) and arc3 are rotated as a group on a circle centered on L0 in order to bring the middle of the wheel bottom as close to the target line “Tgt” as possible.

When the wheel group is moved only relative to L0, it is moved out of alignment with the upper control arm, U0U1 as can be seen in “Figure 3 Restoring U0U1”.

Figure 3 Restoring U0U1

Rotate around L1

To restore U0U1, the wheel group is moved by rotating it around the circle centered on L1. The center of U1 is moved to the intersection of arc3 and arc2, which represents the correct length of U0U1 and the upright.

Figure 4 Next Iteration

Moving the upright to restore U0U1 causes the wheel to once again move off of the target, as can be seen in “Figure 4 Next Iteration”. Repeating the procedure “Rotate around L0” will move the wheel closer to the target each time. The steps can be repeated until the wheel is as close to the target as is necessary for accuracy. See “Figure 5 Completed movement”.

Figure 5 Completed movement

Measuring Bump Steer

Bump steer is caused by the change in effective tie rod length as the wheel moves relative to the chassis and the steering rack. To measure bump steer, the wheel or wheels are moved using the technique as described in “Moving the Wheel”. Once located, the amount of bump steer is determined by measuring the difference between the tie rod length at rest and the distance between the steering rack pivot and the new location of the tie rod end.

Figure 6 A theoretical “ideal” steering rack location

Figure 7 Bump Steer

The wheel itself will pivot on the king pin axis, which is the line U1L1. To find the distance, measure the distance along the perpendicular of U1L1 through the center of S1 to where the perpendicular intersects the arc showing the length of the tie rod. See “Figure 8 Close up of tie rod end”. If the intersection is further from the chassis than S1 on the wheel, toe-out has occurred. If closer, toe-in has occurred. The toe angle is found by adding the amount of toe-in from both wheels (d):

Toe-in = 2 * invsin( d / 2 / steeringArmLength )

Figure 8 Close up of tie rod end

Checking Wheel Camber

Once the wheel has been moved, one further operation is to check the change in camber of the wheel. The camber is checked relative to the road surface, as that is the relationship that is important (rather than the relationship of the wheel to the chassis). The wheel camber is checked by measuring the angle between the wheel vertical centerline and absolute vertical from the midpoint of the bottom of the wheel. See “Figure 9 Checking Wheel Camber”.

The wheel camber changes in bump and droop can be found by moving the wheel the appropriate amount up or down relative to the chassis using the technique described in “Moving the Wheel”.

Figure 9 Checking Wheel Camber

Finding the Roll Center

The roll center of an a-arm independent suspension can be found using construction lines through the suspension links. The intersection of the construction lines is the instantaneous center of the wheel movement. The intersection between the lines from the instantaneous centers to the mid point of the corresponding wheel is the vehicle roll center.

“Figure 10 Instantaneous Center” shows how to find the instantaneous center of a wheel. The intersection of line 1 drawn through U0 and U1 and line 2 drawn through L0 and L1 is the instantaneous center. Line 3 through that intersection and the wheel center is used to find the roll center.

Figure 10 Instantaneous Center

Referring to “Figure 11 Finding the Roll Center” the roll center is found using both wheels. The height of the roll center is measured from the ground. It is possible for the roll center to be below the ground.

Figure 11 Finding the Roll Center

Checking Vehicle Roll

When side forces act on a vehicle, those forces will induce movement of the chassis. These movements will always be around the roll center of the vehicle. The amount of movement depends on the resistance of the springs and of the roll couple – the distance between the roll center and the center of gravity of the chassis.

As the vehicle moves on its suspension, the roll center changes. To properly determine the behavior of the wheels as the chassis rolls, the chassis must be moved incrementally. The new roll center is determined after each movement, and the next iteration rolls the vehicle around the new center.

First the roll center is found as in “Figure 11 Finding the Roll Center”. The entire vehicle is then rolled about the roll center the desired incremental amount, as in “Figure 12 Roll the vehicle about the roll center”.

Figure 12 Roll the vehicle about the roll center

Next both wheels are returned to the ground reference (see “Moving the Wheel”). Once the wheels are in the correct location, the new roll center can be computed (see “Figure 13 Roll operation complete”) and the vehicle rolled once again until the desired roll angle is achieved.

Figure 13 Roll operation complete

Finding the Wheel Rate

Wheel rate is found by determining the ratio of the wheel movement to the spring movement. This can be found geometrically by measuring the leverage of the spring along the lower control arm and the angle of the spring, but this might change as the wheel moves up and down. In “Figure 14 Spring mounts” the spring mounts can be seen marked as SL0 and SL1. The spring length is easily seen as 12” in this case. This is the ideal location for a shock that extends to 13” and compresses to 10”. Anticipate 2/3 motion bump to 1/3 droop, to avoid hitting the bump stops while a wheel is weighted.

Figure 14 Spring mounts

To achieve “Figure 15 Shock/spring compressed by 1"” a construction circle (arc4) is marked 11” from the center of SL0. The lower control arm pivots around the lower inner mount L0, so arc5 centered on L0 with a radius to SL1 is required. Select L1, SL1, U1 and the wheel, and move these elements as a group around the center L0 until SL1 is centered on the intersection of arc4 and arc 5. Next select a group comprised of U1 and the wheel. This group is translated around L1 until U1 is centered on the intersection of arc2 (centered on U0 radius U0U1) and arc3 (centered on L1 radius L1U1).

Figure 15 Shock/spring compressed by 1"

The wheel rate can now be determined. The shock and spring have compressed 1” while the wheel has moved 1.666”, a ratio of 1:1.666. The wheel rate is the spring rate times this ratio. For example the wheel rate for a 250lb spring rate is 250lb / 1.666 = 150lbs.

The spring can be compressed further as can be see in “Figure 16 Full compression”. Using incremental measurements, it can be determined whether the suspension locations are providing a rising, falling or neutral spring rate. For this increment, the effective rate is now 1:(3.317-1.666) = 1:1.651, and the wheel rate is now 250lb / 1.651 = 151lbs. The wheel rate rises slightly as the spring is compressed.

Figure 16 Full compression

Using the drawing “Figure 16 Full compression” also allows the full bump wheel travel to be determined directly. A second drawing at full extension of the shock can be used to determine the wheel distance to full droop.

Finding the Scrub Radius

Refer to Figure 17 "Scrub Radius”, the scrub radius (positive in this figure) is the distance between the king pin inclination and the wheel center, when both are brought to the ground reference.

Figure 17 Scrub Radius

Ackerman Steering

When a vehicle with wheels that are longitudinally spread in a line along the vehicle axis goes around a corner, the outer wheels track in a larger arc than the inner wheels.

What Causes Ackerman?

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How to Draw Ackerman Effects

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Accounting For Slip Angles

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