## Calculating Moment of Inertia

#### CALCULATING MOMENT OF INERTIA

Moment of inertia ("MOI") is similar to inertia, except it applies to rotation rather than linear motion. Inertia is the tendency of an object to remain at rest or to continue moving in a straight line at the same velocity. Inertia can be thought of as another word for mass. Moment of inertia is, therefore, rotational mass. Unlike inertia, MOI also depends on the distribution of mass in an object. The greater the distance the mass is from the center of rotation, the greater the moment of inertia.

A formula analogous to Newton's second law of motion can be written for rotation:
F = Ma (F = force; M = mass; a = linear acceleration)
T = IA (T = torque; I = moment of inertia; A = rotational acceleration)

Choosing the Reference Axis Location

Three reference axes are necessary for calculating center of gravity, but only one axis is necessary to define moment of inertia. Although any axis can be chosen as a reference, it is generally desirable to choose the axis of rotation of the object. If the object is mounted on bearings, then this axis is defined by the centerline of the bearings. If the object flies in space, then this axis is a "principal axis" (axis passing through the center of gravity and oriented such that the product of inertia about this axis is zero (see discussion of product of inertia). If the reference axis will be used to calculate moment of inertia of a complex shape, choose an axis of symmetry to simplify the calculation. This axis can later on be translated to another axis if desired, using the rules outlined in the section entitled "Parallel Axis Theorem".

Polarity of Moment of Inertia

Values for center of gravity can be either positive or negative, and in fact their polarity depends on the choice of reference axis location. Values for moment of inertia can only be positive, just as mass can only be positive.

Units of Moment of Inertia

In the United States, the word "pound" is often misused to describe both mass and weight. If the unit of weight is the pound, then the unit of mass cannot also be a pound, since this would violate Newton's second law. However, for reasons which have been lost in antiquity, in the USA an object weighing 1 pound is often referred to as having a mass of 1 pound. This leads to units of moment of inertia such as lb-in2, where the "lb" refers to the weight of the object rather than its mass. Correct units of moment of inertia (or product of inertia) are: MASS x DISTANCE2 When lb-in2 or lb-ft2 are used to define MOI or POI, the quantity MUST be divided by the appropriate value of "g" to be dimensionally correct in engineering calculations. Again, dimensional analysis will confirm if correct units are being used. The following table shows some of the units in use today for moment of inertia and product of inertia:

lb-in2 lb = weight; must be divided by g = 386.088 in/sec2
lb-in-sec2 lb-in-sec2 = distance2 x weight/g; weight/g = mass; dimensionally correct
slug-ft2 slug = mass; dimensionally correct
kg-m2 Kg = mass; dimensionally correct
The most common units used in the U.S. are lb-in2 , even though this is dimensionally incorrect.

RULE 1. If moment of inertia or product of inertia are expressed in the following units, then their values can be used in engineering calculations as they are:
Slug-ft2, lb-in-sec2, kg-m2, lb-ft-sec2, oz-in-sec2

RULE 2. If moment of inertia or product of inertia are expressed in the following units, then their values must be divided by the appropriate value of "g" to make them dimensionally correct.
lb-ft2, lb-in2, oz-in2
Value of g : 32.17405 ft/sec2 or 386.088 in/sec2
Do not use local value of g to convert to mass!

Calculating the Moment of Inertia

MOI, sometimes called the second moment, for a point mass around any axis is: I = Mr2
where I = MOI (slug-ft2 or other mass-length2 units)
M = mass of element (Slugs or other mass unit)
r = distance from the point mass to the reference axis

The moment of inertia of any object about an axis through its CG can be expressed by the formula: I = Mk2
where I = moment of inertia
M = mass (slug) or other correct unit of mass
k = length (radius of gyration) (ft) or any other unit of length

The distance (k) is called the Radius of Gyration. The method of calculating radius of gyration is outlined in the following sections. Consider first the body consisting of two point masses each with a mass of M/2 separated by a distance of 2r. The reference axis is through a point equidistant from the two masses. The masses each have a MOI of Mr2/2. Their combined MOI is therefore Mr2. The second example shows a thin walled tube of radius r. By symmetry, the CG lies on the centerline of the tube. Again, all the mass is located at a distance r from the reference axis so its MOI = Mr2. In these examples, the radius of gyration is k = r. This leads to the definition: "The radius of gyration of an object, with respect to an axis through the CG, is the distance from the axis at which all of the mass of an object could be concentrated without changing its moment of inertia. Radius of gyration is always measured from CG."

Parallel Axis Theorem for Calculating Moment of Inertia

If in the example above we wanted to determine the MOI of the object about the axis Xa rather than the axis X, through the CG, then the value can be determined using the parallel axis theorem: Ia = I + d2 M, Since I = k2 M, then Ia = M (d2 + k2) where k is the radius of gyration. This parallel axis theorem is used very frequently when calculating the MOI of a rocket or other aerospace item. The MOI of each component in the rocket is first measured or calculated around an axis through its CG, and the parallel axis theorem is then used to determine the MOI of the total vehicle with these components mounted in their proper location. The offset "d" is the distance from the CG of the component to the centerline of the rocket.

Useful Approximations

Since the moment of inertia of an object displaced from its reference axis is proportional to (d2 + k2), we can make two observations that will simplify the job of calculating MOI: RULE 1. If the radius of gyration of an object is less than 1% of its offset distance "d", then the MOI of the object around its CG can be ignored when calculating total MOI, and the value becomes d2M. For example if a gyro with a mass of 0.1 slug is located near the outer surface of a rocket and the offset to the CG of the gyro is 3 feet while the radius of gyration of the gyro is only 0.02 ft., then the MOI about the center line of the rocket due to the gyro is d2M = 0.9 slug-ft2. The error using this approximation is less than 0.01%. RULE 2. If the radius of gyration of an object is more than 100 times its offset distance "d", then the offset of the object can be ignored when calculating total MOI, and the value becomes k2 M. For example if a rocket motor with a mass of 100 pounds is located near the center line of the rocket and the offset to the CG of the rocket motor is 0.100 inches, while the radius of gyration of the rocket motor is 12 inches, then the MOI about the center line of the rocket due to the rocket motor is k2 M = 14400 lb-in2 (or more properly 37.3 lb-in-sec2). Again the error of approximation is less than 0.01%. Rule 2 can also be applied to alignment errors when calculating or measuring MOI. If the offset or misalignment is less than 1% of the radius of gyration, then the alignment error is insignificant.