How to Find MOI Through CG Without Knowing the CG Location

Why measure Moment of Inertia (MOI) through the Center of Gravity (CG)?

When an object is free to rotate, it will rotate around an axis passing through its center of gravity. Therefore it is essential to know moment of inertia through center of gravity to assess the flight characteristics of a payload.

The MOI about an axis A passing through CG is the smallest MOI around any axis parallel to A. Once you know MOI through CG you can extrapolate to MOI through any axis parallel to it providing you know the distance between CG and your axis. The formula is:

MOI through axis A = MOI through CG + Md²

Where:
M is the mass of the object
d is the distance between the CG and the axis A

How to measure MOI through CG without knowing the CG location?

Of course the easiest way to measure moment of inertia through center of gravity is to use an instrument that measures both CG and MOI. Our KSR series are high accuracy instruments that measure CG and MOI with 0.1% accuracy. On this instrument, one payload setup allows to measure two coordinates of center of gravity location and one moment of inertia. The instrument gives moment of inertia results directly through the center of gravity.

Our spin balance machines (POI series) also measure CG and MOI, and therefore give moment of inertia through center of gravity results. Our MP series are instruments that also measure both CG and MOI, although with less accuracy.

But it is not necessary to know the center of gravity position in order to measure moment of inertia through center of gravity. By measuring MOI about multiple parallel axes, one can calculate MOI through CG.

The optimum number of moment of inertia measurements (best compromise between accuracy and time) is 6. More measurements will not provide much more accuracy. Fewer measurements will reduce accuracy significantly.

Moment of inertia accuracy depends on several factors, including:

• The positions used for each measurements
• The accuracy of the instrument
• The accuracy of the fixture

Depending on your payload, you will get an accuracy in the order of 1.5 to 3 times worse than your instrument accuracy. In other words, if your moment of inertia measurement instrument has 0.1% accuracy, you will obtain MOI through CG with 0.15% to 0.3% accuracy.

From one measurement to the next, the payload must be translated in a horizontal plane, without changing its orientation. Moment of inertia measurements give best results when the center of gravity of the payload is located close to the machine centerline. Therefore, measurement positions must be chosen carefully to stay within the tolerance of the instrument for overturning moment, but to provide as great a variation as possible from one position to the next.

Fixturing is crucial for these measurements. The use of a two axis translation table allows measurements without refixturing the payload.

Can I find center of gravity location with this method? What accuracy can I expect?

This method will give you a rough estimate of center of gravity location, but the uncertainty is very large. Do not rely on this method to give you a center of gravity location that you can use in calculation or for balancing purposes.

Accuracy depends on your payload. On a golf club head for example, you can get a center of gravity location to +/- 0.25 inches (+/- 6 mm). On other payloads, the uncertainty can be +/- 1 inch (+/- 2.54 cm) or more.

How does Raptor Scientific implement this method?

Raptor Scientific’s moment of inertia instruments support the multiple MOI measurements method of finding MOI through CG. We design custom software and fixtures that automate the measurement and calculation process.

Our moment of inertia measurement instruments also allow manual center of gravity inputs and translation of MOI results to any axis.