Why Is Moment of Inertia Important?
The moment of inertia (MOI) is one of the many mass properties that describe an object’s stability and the forces needed to alter its motion. For aerospace engineering, stability is a crucial element in designing and manufacturing air and spacecraft. Knowing the MOI about various axes is vital to determining how a device can hold up to external and internal forces.
While engineers can calculate the MOI for a given device, measuring with the right tools can offer greater accuracy for component design and provide the trusted results needed in the aerospace industry.
What Is Mass Moment of Inertia?
The mass moment of inertia reflects an object’s amount of resistance to any change in its rotational rate about an axis. Mass MOI derives from Newton’s first law of motion regarding inertia. This law says that objects at rest will resist being put into motion, and objects in motion will resist stopping without external forces acting upon them. For linear motion, the force needed to move something is the product of its mass and acceleration (F=ma). Higher masses require more force to change their motion from standing still or moving.
Applying this concept to rotational motion results in the mass MOI correlating to the linear mass. The torque required comes from multiplying the mass moment of inertia by the angular acceleration. Therefore, torque in rotational motion is similar to linear force. Angular acceleration is the rotational movement equivalent of linear acceleration. Similar to linear inertia, greater mass MOI requires more torque to reach a specific angular acceleration.
The mass MOI has different forms depending on the object’s shape and the distribution of mass around a given rotational axis. For example, considering a single-point mass, the MOI is the product of the mass times the distance from the axis squared. This type of MOI works with evenly weighted ring shapes with a rotational axis perpendicular to the ring. Point mass MOI can also help calculate the value for an object with distributed masses by using the point mass for each distribution.
Since most objects don’t have a single, evenly distributed mass around the edge, this measurement alone will not suffice. Calculating MOI in objects with distributed mass requires a different approach. For distributed masses, the MOI is the sum of each of the point masses.
The Importance of Moment of Inertia
The MOI of an object determines how much torque an object needs to reach a specific angular acceleration. When calculating torque, or rotational force, you need to know the mass MOI. Multiplying the MOI by the required angular acceleration will give you the required torque value necessary to achieve that angular acceleration. Larger MOI values correspond to greater torque required to achieve a given acceleration.
The ability to precisely determine these values allows the designer to properly size their components while achieving the high performance demands the aerospace and defense industries need. The designer can optimize the trade-off between size, weight and power to achieve the mission goals.
In other cases, the measurement of MOI can be used to verify that manufacturing and assembly tolerances and processes goals are nominal. In yet a third example, higher performance can be achieved by minimizing MOI while not sacrificing strength.
How to Calculate Mass Moment of Inertia
It is possible to calculate the mass moment of inertia, but the approach will depend on the shape, distribution of mass and the rotational axis. Calculations can be broken down into two categories — simple objects and complex objects
Calculating Mass MOI for Simple Objects
Simple objects are general shapes like spheres, cubes and rods. Calculating the mass MOI for simple objects requires the point mass or multiple point masses. To find the point mass, multiply the squared distance between the rotational axis and the mass by the mass at the given point. Some objects with a single mass evenly distributed around the axis will have the mass MOI equal to this point mass.
Shape also makes a difference when finding the MOI. The general formula for mass MOI is I=Mk^2, where M is the mass of the object and k is the distance between the rotational axis and the radius of gyration. For example, a narrow rod will have a much smaller radius of gyration than a large sphere or cylinder, impacting the results of the calculation.
Calculating Mass MOI for Complex Objects
Complex objects may be composed of multiple simple objects, like a series of cylinders and rods. However, these objects might lack clear radii of gyration, making it more challenging to achieve an accurate calculation.
For calculating the mass moment of inertia for more complex objects, the summation of all the point masses of the piece gives the mass MOI. However, instead of calculating several point masses, calculus could help by using the integral of the squared radius of the mass of an infinitesimally small area of the object (dm).
Calculating the mass MOI presents some issues, especially in highly complex designs. However, considering this value’s importance, engineers and manufacturers of any aerospace craft should know it. Another way of determining the mass moment of inertia is through measurement.
How to Measure Moment of Inertia
Measuring the mass moment of inertia can take much less time than calculation, making it valuable to engineers working with strict timelines. Measurements can also offer more accuracy, particularly for complex shapes that lack clear dimensions for the point mass formula.
At Raptor Scientific, we carry a broad selection of instruments for measuring MOI. When selecting the right instrument among our products, consider the required accuracy for a project, the object’s mass and any additional measurements required. While some of our MOI instruments provide only MOI measurements, others include the center of gravity and other mass properties.
All Raptor Scientific systems use the inverted torsion pendulum method to measure mass MOI around a rotational axis. As a leader in MOI measurement, our products are the originators of the torsion flexure measurement used for our instruments.
With the inverted torsion principle, the object sits on a table attached to precision, low-friction bearings that constrain the object’s motion to pure rotation. A connected sensing device produces time pulses that trigger a digital period counter to determine the oscillation period. This measurement contributes to the calculation of MOI.
We offer the following mass MOI measuring instruments.
For small objects weighing from 0.1 kg to 2.3 kg (0.2 lbs. to 5 lbs.), the XKR series uses air bearings to provide extreme MOI measurement accuracy. Measurement accuracy of better than 0.1% is an option for instruments in this series.
Designed for general use, the XR series has a load capacity of up to 115 kg (250 lbs.). Accuracies for measurements of MOI with this series go up to 0.25%.
The GB series tackles the heaviest of objects. It can handle pieces weighing between 68 kg to 6,000 kg (150 lbs. and 13,000 lbs.). Due to its exceptional ability to measure MOI on large objects, the military and aerospace industries frequently use instruments in the GB series. It can produce measurements with an accuracy of up to 0.1%.
The KSR series combines the center of gravity and MOI measurement capability. Instruments in this series can hold objects up to 9,070 kg (20,000 lbs.). Additionally, accuracy is extremely high, with options of up to 0.1% measurement accuracy for both the center of gravity and moment of inertia.
The MP series is a set of general-use instruments that can accommodate heavier loads than the XR series. Instruments in this series can measure payloads up to 4,500 kg (10,000 lbs.). This series goes beyond the XR series by also measuring the center of gravity and weight with the moment of inertia. Accuracy for this series is 0.25%.
The POI instrument series combines center of gravity, MOI and product of inertia (POI) measurements with dynamic balancing capability. POI instruments offer the highest measurement accuracy available. This series holds payloads measuring up to 10,500 kg (23,000 lbs.) Products in this series take all measurements of mass, including the moment of inertia, the center of gravity, dynamic imbalance and the product of inertia. The instruments in the POI series are highly accurate, up to 0.1%.
Calculating vs. Measuring the Moment of Inertia
Finding the mass MOI through calculations alone could cause problems with the final results. While some objects may lend themselves to obvious measurements for the general formula or point mass calculation, many will not in real-world contexts. Calculating the MOI may not be the best option when accuracy is essential, as in the air and space industry.
Calculations, especially those based on sums of point masses, are only as reliable as the components used to find the result. Therefore, a single miscalculation in finding a point mass could cause a significant error in finding an object’s total mass moment of inertia, magnifying any mistakes in initial measurements.
Using calculations could cause errors of 30% compared to the measured MOI value. Such a drastic difference in the calculated versus measured values could create problems during manufacturing or practical use of the object.
Another significant problem with calculating values comes from the cost wasted. Calculating the MOI, finding out the value is incorrect and returning to engineers for another calculation squanders time. Doing the math takes far more time than MOI instruments take to measure. With highly accurate results of up to 0.1%, devices save time and increase precision over calculating.
Lastly, determining accurate MOI values is essential for many aerospace or military projects. Correct measurements of the MOI and other mass values ensure the project’s quality. Knowing the mass moment of inertia from measurements provides predictability of flight characteristics for air or spacecraft or performance metrics for other devices.
Accurately Measure the Mass Moment of Inertia With Instruments From Raptor Scientific
Accuracy of mass measurements, including the moment of inertia, is crucial for industries that need precision components for precision systems. Calculating the MOI is always an option, but the likelihood of inaccuracies is high due to human error and low-precision measuring tools.
At Raptor Scientific, we have a range of moment of inertia measurement instruments for measuring devices between 1 gram and 10,000 kg. These devices provide precision for mass MOI up to 0.1% to satisfy military and aerospace applications. Explore our range of measurement instruments online or contact us at Raptor Scientific for a free quote on any of our instruments.