# Instant Centre for any Suspension Linkage

## What is an Instant Centre

A suspension linkage’s Instant Centre is an “imaginary” pivot around which the hub is “rotating” at a given moment. This is because when the hub moves up and down relative to the body, there is normally some rotation around the longitudinal and lateral axis. This coupling of motion is related to the jacking force applied to the chassis, when horizontal force/torque applied at the contact patch. The term can either refer to a pivot in the pitching plane (X-Z) or roll plane (Y-Z).

Multibody suspension models automatically account for this in the force and torque summations; however, the linkages instant centre is a metric which is often used to help categorize a suspension’s behaviour.

In parametric vehicle simulations, the location of the instant centre is often used to estimate the magnitude of the jacking force. But methods for determining the location of the instant centre are generally limited to double wishbone or MacPherson strut type linkages. In both cases the location of the joints (e.g. ball joints and wishbone mounts) define 2 planes, the intersection of which provides the location of what we call the conventional geometric instant centre.

## A more general way of calculating the instant centre

But, if the location of the motions of the hub is directly related to the location of instant centre then the motions of the hub can be used to find that centre.

The instant centre is considered to be the centre of the circle that the hub is moving around. So, the translational motion and the rotation of a hub can be used to find the centre of rotation. If the hub is considered to be moving on an arc then the centre of that arc is perpendicular to the motion of the hub and the speed of movement and speed of rotation can be used to find the radius of the arc.

Therefore, a new sensor was created that uses the translational and angular velocity of the hub relative to the chassis to find the point at which it is rotating. This means that it can be attached to any quarter-car suspension system to find the instant centre as long as the hub is moving in bump.

While the sensor can be applied to a vehicle and used while the vehicle is in motion, it requires bump velocity to generate the position. Therefore it is most useful when used in an isolated bench test, providing the position relative to the chassis or world origin.

When trying to determine the IC for linkages such as multilink or trapezoidal one struggles to find a dependable canned geometrical formulation. With the hub dynamics approach, the instantaneous centre of any independent linkage is easily measured without needing the forces involved or a specific geometric calculation. Above is a hybrid multilink suspension with intentionally weird geometry to demonstrate the use on a difficult linkage.

## Comparing to conventional IC calculation methods

We then compared the hub dynamics IC calculation to the conventional geometric method on a double wishbone linkage.

To test the accuracy of each prediction a jacking experiment was created as shown above. This test involved the original linkage with the jacking force measured and a longitudinal force applied at the contact patch with the hub being held vertically. In this test, the multibody linkage will generate the correct amount of jacking force and is thus used as the datum.

The 2 instant centre positions (geometric and dynamic based) were measured and then a rotational joint placed at each of those positions was added. This creates a pivot around which the longitudinal force would be rotated into a jacking force.

The result is shown below with the axis of motion, perpendicular motion in blue; the force applied in green; the pivots at the 2 instant centres in red with the pivot arm between the pivot and contact point in grey.

When bump steer was minimal, the results were essentially identical. However, with a weird geometry set (abnormally large amount of bump steer) the departure of the conventional method from reality was clear. Note – this is generally not a concern as suspension designers only allow a limited amount of the bump steer in a suspension. It is merely being used as an example here to demonstrate the simplicity of the traditional approach relative to that of the hub dynamics approach.

The comparison for the weird geometry revealed a substantial difference in jacking force produced by the linkage and that of the geometric instant centre. The linkage (considered the benchmark) was creating -446.4N of jacking force. In one case the geometric instant centre prediction actually created +4N of jacking force, a 101% error! In comparison, the jacking force generated by the hub dynamics instant centre was -452.6N, an error of just 1%.

The marked difference between the geometric and hub dynamics based instant centre in this case can be put down to the geometric calculations’ lack of inclusion of the load carried by the steering or toe link. The calculations use an axis between the upper and lower ball joints in place of the upright. But if the linkage has some bump steer the wheel centre moves in an arc causing the instant centre to move differently than if there was no bump steer.

The hub dynamics based sensor is attached directly to the hub at the wheel centre so the bump steer motion is considered the same as motion introduced by any other method. Therefore, the instant centre generated by this method incorporates all the hub motions, regardless of how they are introduced or how the force is transferred to the chassis.

With this new sensor, previously unmeasurable linkages are easily characterized without needing to calculate forced based roll centres or other approximations. Instantaneous centres for linkages such as multilink or trapezoidal suspension can now be generated and used.

## How to use the new sensor

This new sensor will be included in the VeSyMA – Suspensions Library from the 2021.2 release.

It can be attached to any quarter car linkage at the wheel centre position and to the chassis frame to get the relative motion between them. That’s all that needs to be done.

## Closing

Conventional geometric instantaneous centre calculations provide a useful approximation for double wishbone and MacPherson strut type linkages, but fall down with other types of suspension linkages. The generic nature of the new hub dynamics based approach can come in quite handy for evaluating any type of independent suspension. Simply hook the sensor to the hub and you will be provided the instantaneous centre location with no need for additional geometric data inputs. Add to that the more accurate result achieved with this sensor and it can provide great benefits.

Written by: David Briant – Project Engineer

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