How to Calculate Product of Inertia From Moment of Inertia Measurements

Product of inertia (POI) is the characteristic of an object that causes dynamic unbalance. Knowing the POI of an object is essential to determine some of its flight characteristics, such as its tendency to wobble. Knowing POI in all planes allows determining the principal axes of a payload.

How to measure product of inertia?

Product of inertia can be measured directly by using a vertical two plane spin balance machine (POI series). Spin balance machines spin the payload about a desired axis and detect any unbalances generated. This is by far the most accurate method of determining product of inertia.

But some payloads cannot be spun (some satellites for example), or cannot be spun about all axes (missiles for example). For these payloads, product of inertia can be calculated from moment of inertia measurements (which do not require spinning). Three moment of inertia measurements per plane are required, at widely spaced angles (ideally 0, 45, and 90 degrees). Some moment of inertia measurements can be used in several orthogonal planes, so a total of only 6 measurements is required to get product of inertia in 3 planes. Product of inertia is directly related to these moment of inertia measurements.

How accurate is this method?

Calculating product of inertia based on moment of inertia measurements is not as accurate a method as directly measuring product of inertia by spinning the payload.

Accuracy of this method depends on several factors, including:

• Accuracy of the moment of inertia measurement instrument used
• Angles used between measurements
• Payload characteristics (payloads with large products of inertia will be easier to measure)

How can Raptor Scientific help me?

Raptor Scientific developed a piece of software that is available with its combined mass properties instruments (KSR Series, MP Series, and POI series). It allows selecting several moment of inertia measurements made in the same plane and angles at which these measurements were made. The control system outputs product of inertia in that plane.

Where do I find more technical information?

For more technical information on this method, we recommend our technical paper Using the “Moment of Inertia Method” to Determine Product of Inertia, which was presented at the 51st annual conference of the Society of Allied Weight Engineers (SAWE).