How To Calculate Centre of Gravity Position

Finding your centre of gravity position (referred to as COG as well in this post) is very important when you want to start understanding how your car handles and why it handles the way it does. Finding your COG position is the first step in determining why your car rolls as much as it does and helps to calculate your roll moment too. It is also one of the stages required if you wish to calculate the ideal spring rates for your race car. Below is the technique used to calculate the static centre of gravity position of your car.

Equipment Required

In order to accurately place your centre of gravity position you will need the following equipment:

• A set of corner weights
• A tape measure
• A hydraulic jack or lifting equipment

Before You Begin

In order to get an accurate result you must carry out the following steps before you begin:

• Prepare rigid metal lock out bars that will effectively lock out your front dampers. These will be required later on.
• Ensure all fluids are full
• Place mass in the driver’s seat equivalent to the mass of the driver
• Disconnect the anti-roll bar

Items To Be Measured

There are a few parameters that must be measured either with the tape measure or with the corner weights in order to calculate the COG position.

The table below displays all of the items that need measuring and can be used along the way to keep track of measurements that will be used in the equations later on below.

Before continuing you must make sure that all your ride heights are set to their desired heights and will not be altered after this calculation. Also, similar to when you are corner weighting your car, the anti-roll bars must be disconnected from side to side to remove any static torsion from the suspension system.

The Measurement Stage

The first measurements to take from the car are the track width and wheels base. The easiest way to measure this is to run a tape measure from the centre of a front wheel, down the side of the car to the centre of the rear wheel. This will give you the wheel base. Record this in the above table in meters. Next measure the track width. To do this run the tape measure from the centre of the tyre from a front view across the width of the car to the centre of the tyre on the other side of the car. If your front wheels are wider apart than your rear wheels or vice versa, record front and rear track width in the table above.

With those measurements complete and recorded in the table it’s time to measure some weights.

Place your car on the corner scales ensuring that all tyres pressures are set to what you will run on track. Also, ensure that all other set up checks are complete as mentioned in the “How to Corner Weight Your Car” article. This will ensure accurate measurement.

The corner weights will show 5 things for definite:

• Front Right Mass
• Front Left Mass
• Rear Right Mass
• Rear Left Mass
• Total Mass

Take these readings from the scales and put them into the above table making sure that the mass is being measured in KG. If not then convert your readings to KG before inputting the data in the table.

Some scales will also calculate the front to rear percentage mass split and the left to right percentage mass split. If that is the case then put the front percentage mass in the above table as a percentage and put the left percentage mass in the above table as a percentage.

If your scales do not have the capability to calculate percentage split and only tell you the mass on each corner then the following steps can be taken to get the required percentage figures.

To make this easier some example figures can be seen below taken for a single seater racing car:

• Front Right Mass = 92Kg
• Front Left Mass = 88Kg
• Rear Left Mass = 123Kg
• Rear Right Mass = 122Kg
• Total Vehicle Mass = 425Kg

Using these figures we can determine that:

• Total Front Mass = 180Kg
• Total Rear Mass = 245Kg
• Total Left Mass = 211Kg
• Total Right Mass = 214Kg

Calculating COG Position Front to Rear and Side to Side

We now need to calculate the front weight percent and the left weight percent. Therefore we need to select our total front mass and our total mass and put the numbers into the below equation:

Therefore:

Now for the left percentage split we will need the total left mass and the total vehicle mass to be used in a similar fashion in the below equation:

Therefore:

These figures can now be put into the above table.

The next stage to locating the COG position front to rear and left to right is to use the earlier recorded measurement of track width and wheelbase. It does not matter which track width figure is used for this so we will use the front track width measurement.

For the purposes of the future calculations the example figures taken from the single seater racing car will be used in the table below in order to do worked examples to show how to use the equations and sums.

In order to calculate the left to right position of the COG we can use the following equation using the track width measurement and the left mass percentage:

Therefore:

Now we need to locate the centre line of the axle by dividing the track width by 2:

We can now subtract our first result from the above result to give the distance of the COG from the centre line:

Therefore:

If the answer is positive, it means that the COG is off centre by that amount towards the right wheel. If the value is negative it tells you that the COG is off centre by that amount towards the left wheel. Therefore, in this example the COG is off centre by 5.2mm towards the right wheel.

Next we need to locate the COG position front to rear. This is done in a similar way to above but instead uses the wheel base and the front mass percentage.

Therefore:

Now we need to locate the centre line of the car by dividing the wheel base by 2.

Now we can subtract our first result from the above result to determine how far away from the centre line of the car the COG sits front to rear:

Therefore:

If the answer is positive it means that the COG is off centre by that amount towards the rear wheels. If the answer is negative it means that the COG position is off centre by that amount towards the front wheels. In this case the COG position is off centre by 203mm towards the rear wheels.

The diagram below from the top view shows the current location of the COG position of this car.

Calculating the Height of the COG Position

To calculate the height of the COG position is very important if you want to understand the roll characteristics of your car and if you want to calculate your ideal spring rates.

With the car still on the corner weight scales the rear of the car needs to be raised a certain amount so that the car is ideally at an angle of between 15 and 20 degrees from horizontal with the front wheels still on the scales.

To raise the rear of the car by the correct amount we need to use a mathematic principle called SOHCAHTOA. We will be using the SOH part which stands for Sin (angle), opposite length and Hypotenuse length. The hypotenuse length is the same as the wheel base measurement. For this example we will use an angle figure of 18 degrees as this sits nicely in the middle of our ideal range. The angle used for the Sin function is 18 degrees.

To calculate how high you need to lift the rear of the car the following equation can be used:

Therefore in our example:

This shows that we need to raise the rear of the car up by 81.9cm vertically to achieve the correct angle. This figure will change depending on the wheel base figure that you have measured for your particular car.

The easiest way to do this accurately is to insert lock out bars on the front dampers to prevent them from compressing when the rear is lifted. Also increase the front tyre pressure to remove sidewall flex from the measurements. Then place a jack between both rear wheels on the subframe or the differential in the centre of the car. Jack the hydraulic jack up until it makes contact with the subframe but before it begins to lift the car. Measure the height of the jack head from the floor. Now jack the car upwards until the jack head is 81.9cm higher up than when it was just in contact with the subframe. This requires a high lift jack to achieve such a high level of raise. If you have a regular trolley jack then you might have to raise the car up in stages to achieve this height by jacking it up to the maximum travel, placing the car on axle stands then putting wood between the jack and the car and going again until the height is achieved. In the image above a ramp has been used to achieve the lift required for accurate measurement. If you select a similar method then take care ensuring the rear axle is very securely fixed to the ramp. If you are going to struggle to reach the height then use the 15 degrees at the lower end of the range which will require less lift.

What If I Can’t Lift My Car That High

If you are unable to lift your car high enough to elevate it to an angle of 15 degrees then a lesser height and angle can be used but it will be less accurate than if you are inside the range of 15 to 20 degrees. However, if you are unable to lift that high then this will still give you a close approximation of your centre of gravity height.

The absolute minimum that you must raise your vehicle by is 26cm. Anything less than this will produce an inaccurate result. Therefore, it is best to raise the car to the highest that you possibly can and take note of the vertical height raised. Now the following equations can be used to produce a value for “Tan θ” that we need for our COG height calculation later.

First we need to calculate the adjacent length below the car for the height raised using Pythagoras theorem.

Where:

• WB = Wheel Base (m)
• Z = Vertical Height Raised at The Rear (m)

Now with this value we can use it to get a value for “Tan θ” for our main equation below. Using SOHCAHTOA we can determine that:

Where:

• Z = Vertical Height Raised at The Rear (m)
• Adj = Adjacent Length Calculated in The Above Equation (m)

The number for “Tan θ” will be a decimal number. This can be kept and inserted into the main equation below entirely in the place of “Tan θ”. With this section complete we can now re-join the original procedure.

The Equation

With the car in this position take a note of the mass that is now present at the front wheels on the corner weight scales. Add the two figures together to produce the total front mass on the scales. In this example the total front mass with the rear lifted was 188Kg.

Therefore our front raised mass is:

Now we just need to deduct this value from our earlier measured total front axle mass figure to produce the front axle mass change:

Therefore:

The front axle mass change can be a negative value as well as a positive value. It is irrelevant whether it is negative or positive and just needs recording as a value.

These values can all now be recorded in the previous table for the final stage of calculations. The updated example table can be seen below:

With the above table now complete we can insert our figures into one final equation which will tell us the vertical height above our axles of the COG position. The equation is:

Therefore:

This height figure is the height of the COG position above the axle height. In order to get the height from the ground one final measurement must be taken. With the car sat back on all four wheels measure the diameter of the front tyres. If you have different sized front wheels or tyres then take an average. However in most cases the front wheels will both be the same diameter. With the diameter reading divide it by 2 to create the radius of the front tyre. This radius value is your front axle height.  In the case of the single seater the front tyre radius is 0.24m. This value is simply added to the previously calculated height to give the height of the COG position above ground level for the car.

Therefore:

This height is the final figure required to accurately locate our centre of gravity position on your car. The diagrams below show the COG position of the example car from top view, side view and front view.

Front View:

Side View:

Top View:

What Next?

Now that you have accurately located your centre of gravity you will be able to begin making adjustments to your car to attempt to lower it to reduce the amount of roll. You can alter ride heights or begin removing mass from certain areas and adding it in others to shift the COG position to get it as central as possible. Also common is installing lighter carbon fibre parts at high points such as the roof, bonnet or doors. You can re-measure using this process after each alteration to see how effective your modifications have been.