Product of Inertia vs. Moment of Inertia

According to Newton’s Laws, objects will stay in their current state of motion unless something acts upon them. While Newton’s Laws are usually applied to linear motion, they also apply to rotation. A rotating object will continue to rotate unless a force acts on it. How quickly an object rotates depends on its mass properties and the amount of force applied to it.

Moment of inertia (MOI) describes the amount of force, or torque, required to change the rate of rotation of an object. Product of inertia (POI) reveals the ways that an object might be imbalanced. The two are similar but have different overall applications. Here’s what you need to know about MOI and POI.

What Is Product of Inertia?

Product of inertia (POI) is a feature of an object that describes an imbalance relative to a defined set of coordinate axes. If the mass of the object is distributed symmetrically with respect to the XY, YZ, and ZX planes, then the POI in all three planes is zero. However, if the mass is not distributed symmetrically, the result is a non-zero POI causing an imbalance when the object is rotated. One way to better understand POI is to think of a tire on a vehicle. Tires need to be balanced on both the inside of the tire as well as the outside — otherwise, the vehicle can be unsafe to drive. When a tire is perfectly balanced, the weight of the wheel is distributed evenly around it and between the outside of the rim and the inside.

A variety of factors can cause a tire to become imbalanced, such as uneven wear on the treads or damage to the rim of the wheel. In those instances, the tire is no longer perfectly round. When a person drives a vehicle with imbalanced tires, they are likely to notice vibrations and a bumpy ride when they drive.

POI is used to determine the principal axes of an object in flight. When the principal axes differ from the geometric axes, the flight pattern tends to be “wobbly”.

How to Calculate POI

When mass is distributed in a symmetrical way, the POI should be zero. It can be calculated using the formula  IXY = ∫ xy dA. In this formula, “A” is the area of the object and “d” is the distance between the x and y axes. Although the target POI is zero, it’s possible for the value to be positive or negative. Whether POI is negative or positive depends in large part on the reference axis used.

Often, it’s more accurate to measure the POI of an object rather than calculate it. There are several ways to measure POI and to make adjustments as needed. You can use a vertical two-plane spin balance machine to measure POI directly. With a spin balance machine, the payload spins around an axis, potentially producing imbalance forces.

In some cases, it may not be possible to spin an object, or it may not be possible to spin an object around all of the axes. Two examples of objects that might not be spun around all axes are missiles and satellites. When it’s not possible to measure POI using a spin balance machine, another option is to determine POI using the MOI method.

Using MOI to calculate POI isn’t as accurate as using a spin balance machine. The accuracy of the result depends on factors such as the characteristics of the payload, the quality of the MOI instruments used and the angles between the measurements that are used.

What Is Moment of Inertia?

Moment of inertia (MOI) is also known as rotational inertia. MOI or rotational inertia tells you how difficult it is to change the rotational velocity of an object on its axis. A baseball player swinging a bat is an example of rotational inertia, as is a ball swinging around a pole while attached by a tether.

The greater the mass of a particular point on the object and the greater the distance from the axis, the greater the MOI.

How to Calculate MOI

The formula for calculating MOI is I=mr², where “I” is inertia, “m” is mass and “r” is the radius or distance from the axis to the mass. Once you’ve calculated MOI, you can calculate the angular momentum of an object as well as its rotational kinetic energy.

Rotational kinetic energy uses the formula K = Iω², where “I” is the MOI and “ω” is the angular velocity of the object. Angular momentum uses the formula L = Iω. Another way to write the formula is T = IA, where “T” is torque,  “I” is inertia and “A” is rotational acceleration.

Using a formula to calculate MOI is often sufficient when the object is simple, such as a wheel or a single sphere. But when you are trying to calculate MOI for a complex object, such as an aircraft engine with multiple moving parts, a simple formula no longer suffices. You’d need to repeat the formula multiple times for each mass, then add the MOIs together to get an aggregate.

In those instances, it is often more efficient to use instruments to measure MOI.  Raptor Scientific manufactures more than 50 instruments designed to measure MOI on objects ranging in size from less than one gram to more than 10,000 kilograms. Our instruments measure MOI using the principle of the inverted torsion pendulum. The object rests on a table and is attached to low-friction bearings. The bearings restrict the motion of the object, allowing only for pure rotation. A digital counter connects to a sensing device to determine the period of oscillation.

What Is the Difference Between Product of Inertia and Moment of Inertia?

There are several notable differences between POI and MOI, the first of which is what they measure. POI refers to the symmetry or imbalance of an object on an axis. MOI is the figure that reflects how difficult it is to change the rotational speed of an object.

Another notable difference between POI and MOI is the value of each. POI can be zero, or it can be negative or positive. In contrast, MOI is always positive. One way to remember that MOI is always positive is to remember that the mass of an object is always positive.

Finally, MOI is referenced to an axis while POI is referenced to a plane, such as Xy, YZ, or ZX.

Applications of POI vs. MOI

One of the uses of MOI is to determine how a mass will behave in response to a known torque. Torque is the measurement of the force needed to make an object rotate on an axis. It’s a vector quantity and is only used to measure rotation. Torque is calculated by multiplying force times distance.

When you know the MOI of an object and the torque, you can divide torque into the MOI to find the angular acceleration.

Understanding POI allows you to correct asymmetry or imbalance in an object. To improve symmetry and ensure a smoother flight or ride, your goal is typically to get POI to zero.

Instruments Used to Measure MOI

A very crude method of measuring MOI is to hang the object from a wire, then oscillate it. While the object oscillates, an engineer can time how long it takes for one oscillation. While hanging and oscillating an object does let you measure the rate of acceleration, there are also many variables involved that affect the results. The object is likely to swing back and forth or bounce up and down, affecting the accuracy of the timing. Furthermore, large or unusually shaped objects might be difficult to suspend.

Fortunately, multiple instruments are available to measure MOI accurately and efficiently. An inverted torsion pendulum allows you to get an exact measurement of the oscillation period of the object. When using the instrument, you rest the object on a rotary table, which is at the top of the device. The table and object are supported by low-friction air bearings. These instruments allow for far more accurate and reliable measurements of MOI, especially for professional applications.

Among the benefits of using an inverted torsion pendulum to measure MOI are minimal fixturing, a well-defined axis and minimal computational techniques.

Using an inverted torsion pendulum is usually a multi-step process:

• The object is attached to the table and oscillated. A digital timer will count down the amount of time it takes for the object to oscillate. The total moment of inertia is calculated by multiplying the oscillation time by the machine’s calibration constant.
• The object is taken off of the table. The table is then oscillated on its own to determine the tare moment of inertia of the table itself.
• The tare moment of inertia is subtracted from the total moment of inertia with the object attached. The difference is the MOI of the object alone.

Raptor Scientific produces several MOI instruments that use an inverted torsion pendulum:

• XKR Series: Measure the MOI of objects ranging from 0.1to 2.3 kilograms (kg). They are extremely accurate, with an accuracy of 0.1%.
• XR Series: Measure the MOI of objects up to 115 kg. They have an accuracy of 0.25% and are designed for general use.
• GB Series: Measure the MOI of heavier objects between 68 kg and 6,000 kg. They are ideal for use in critical military and space applications. They have an accuracy of 0.1%.
• MP Series: Measure the MOI and center of gravity, as well as the weight, of objects up to 4,500 kg. They have an accuracy of 0.25%.
• KSR Series: Measure the MOI and center of gravity of objects up to 9,700 kg, with an accuracy of 0.1%.
• POI Series: Measure all types of mass properties, including MOI, of objects up to 10,500 kg. They have an accuracy of 0.1%.

Benefits of Measuring MOI

Using an instrument to measure MOI is often much faster than trying to calculate MOI. There are other reasons to measure rather than calculate MOI, such as :

• Lowered costs: Using an instrument to measure MOI takes less time than calculating, which means your team of engineers can spend less time on rote calculations and more time on the things that matter most to your company.
• Better accuracy: Instruments are often much more accurate than calculations, meaning you’ll enjoy fewer errors and instances of needing to return to the design process to fix issues.
• Improved quality control: Flight vehicles need to have a certain MOI to ensure high performance. The greater accuracy provided by measuring MOI is likely to mean fewer quality control issues.

Instruments Used to Measure POI

A spin balance machine can measure POI on certain types of objects and is often the most commonly used instrument for measuring POI. Spin balance machines rotate the object at a set speed and then measure the reaction forces on the upper and lower bearings. The machines have a computer that calculates POI automatically, using the height of the center of gravity of the object and the spacing between the two bearings.

There are several benefits and drawbacks of using a spin balance machine for POI. The machines can be very sensitive, minimizing air turbulence and improving results. A drawback of this type of machine is that it can’t measure the POI of objects such as satellites with large solar panels or control fins. In those cases, it might be more effective to use the MOI method.

Can You Use MOI to Measure POI?

In some cases, it’s not possible to use a spin balance machine to measure POI. These cases are usually due to the shape or size of the object. Some objects can’t be spun at all or can’t be spun on every axis. An alternative is to use MOI measurements to calculate POI. Using MOI for POI isn’t as accurate as measuring POI directly, but is often the best alternative solution.

To use MOI to measure POI, you need to measure the object in six positions on a torsion pendulum. Once you have all six of the measurements, you can calculate the POI using rotational angles.

While using this method lets you calculate POI on objects where you otherwise wouldn’t be able to, it does have some drawbacks. The process is long and arduous. It can take many hours to perform, which can add to its overall cost. On the other hand, it often costs less than a spin balancing machine and puts less stress on the object you’re measuring.

Measure POI and MOI With Raptor Scientific

The product of inertia and the moment of inertia are both critical calculations made by engineers in many industries. Raptor Scientific has more than five decades of experience with mass properties measurement. We manufacture POI and MOI instruments and offer mass properties measurement services in our state-of-the-art laboratory. To learn more, contact us today for a quote.