Moment of Inertia
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Moment of Inertia of a System of Particles
Moment of Inertia of a Rigid Body
Ellipsoid of Inertia

Moment of Inertia of a Systems of Particles

Newton's first law of motion says "A body maintains the current state of motion unless acted upon by an external force." The measure of the inertia in the linear motion is the mass of the system and its angular counterpart is the so-called moment of inertia. The moment of inertia of a body is not only related to its mass but also the distribution of the mass throughout the body. So two bodies of the same mass may possess different moments of inertia.

A rigid body can be considered as a system of particles in which the relative positions of the particles do not change. The moment of inertia of a single particle (I) can be expressed as

   [1]

where m = the mass of the particle, and r = the shortest distance from the axis of rotation to the particle (Figure 1).

wpe16.jpg (2715 bytes)     Figure 1

As shown in [1], moment of inertia is equal to mass times square of the distance and it is also referred to as the second mass moment. Mass times distance, mr, is called as the first mass moment. This concept of first mass moment is normally used in deriving the center of mass of a system of particles or a rigid body. See Center of Mass-System of Particles for the details.

Expanding [1] for a system of particles:

   [2]

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Moment of Inertia of a Rigid Body

Based on [2], one can obtain the moment of inertia of a rigid by shown in Figure 2:

wpe17.jpg (6477 bytes)    Figure 2

   [3]

where ri = the position of particle i, and n = the unit vector of the axis of rotation. Note here that the axis of rotation passes through the local reference frame, the OXYZ system. Let

   [4]

and

   [5]

where cosa, cosb & cosg = the three direction cosines of vector n to the XYZ system. Substituting [4] & [5] into [3] leads to

    [6]

where

   [7]

Ixx, Iyy & Izz are called the moments of inertia while Ixy, Iyx, Iyz, Izy, Izx, & Ixz are the products of inertia. For a rigid body, the relative position of the particles do not change and one can write [7] as:

   [8]

When the shape and the density distribution of the rigid body is precisely known, one can use [8] to compute the moments and products of inertia. (See BSP Equations for the MOI equations of the typical geometric shapes commonly used in human body modeling.) Otherwise, it is difficult to compute them through integration. Rather, the moment of inertia must be measured directly from the object. See Measuring MOI for the details.

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Ellipsoid of Inertia

The moments and products of inertia shown in [7] and [8] are basically specific to the local reference frame defined and reflect the mass distribution within the body in relation to the local reference frame. As shown in [6], the actual moment of inertia of a rigid body about an axis of rotation is a function of not only the moments and products of inertia for a given reference frame but also the orientation of the axis of rotation, a, b & g. Thus, it would be more accurate to say that the moment of inertia of a rigid body reflects the mass distribution within the body with respect to the axis of rotation.

As the axis of rotation changes, so does the moment of inertia. To show this point clearly, let

   [9]

Substituting [9] into [6] yields

   [10]

Interestingly, [10] suffices the general form of the ellipsoid with its center at the origin of the reference frame. When Ixy = Iyz = Izx = 0, the ellipsoid defined by [10] definitely becomes symmetric about the three axes.

Since

   [11]

the distance from the center of the ellipsoid to the surface is 1 divided by the square root of the moment of inertia of the rigid body for a given orientation, a, b & g. The ellipsoid defined by [10] is called the ellipsoid of inertia since it describes the moment of inertia of an object as a function of the orientation of the axis of rotation.

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Young-Hoo Kwon, 1998-