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Calculation of the MOI Perpendicular-Axis Theorem Parallel-Axis Theorem Physical Pendulum & Direct Measurement References and Related-Literature

Calculation of the MOI

In computation of the moment of inertia, one can replace the summation shown in [2] of Inertia Tensor by an integration over the body:

   [1]

where r = the perpendicular distance from the particle to the axis of rotation, and dm = the mass of the particle which is a function of the density.

Let's now apply [1] to a thin uniform rod shown in Figure 1. The MOI of the rod about the Y axis is

    Figure 1

   [2]

since

   [3]

where r = the density of the rod, m = the mass of the rod, and L = the length of the rod.

 

The moment of inertia of the uniform circular ring shown in Figure 2 about the Z axis (the symmetry axis) is

   Figure 2

   [4]

since

   [5]

where dl = the length of the arc formed by dq. [5] is also applicable to a circular cylinder.

 

A uniform circular disc of radius R can be considered as a cascade of uniform circular rings as shown in Figure 3. Thus, from [5], the moment of inertia about the Z axis (the symmetry axis) becomes

    Figure 3

   [6]

since

   [7]

[6] can be used in computing the MOI of a circular bar about its longitudinal axis as well.

A uniform sphere can be considered as a cascade of uniform circular discs as shown in Figure 4. From [6], the moment of inertia about the vertical axis (Z axis) is

    Figure 4

   [8]

since

   [9]

See BSP Equations for the moment-of-inertia equations for the geometric shapes commonly used in the human body modeling.

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Perpendicular-Axis Theorem

Now, let's go back to the uniform circular disc (Figure 3). The moment of inertia of a uniform circular disc about its perpendicular axis (Z axis) can be expressed as

   [10]

since

   [11]

In other words, the MOI about the Z axis is equal to sum of those about the X and Y axes. [10] is true for any rigid lamina: the MOI of any rigid lamina about an axis normal to the lamina plane is equal to the sum of the MOIs about any two perpendicular axes lying on the plane and passing through the normal axis. This is the so-called perpendicular-axis theorem.

Since the circular disc has symmetric shape,

   [12]

and, from [6]:

   [13]

One can directly obtain [13] from [11] or

   [14]

[2] can be used in further developing [14] to obtain [13]. See BSP Equations for this approach.

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Parallel-Axis Theorem

Now, let's compute the MOI of a uniform circular column. Circular bar can be regarded as a cascade of circular discs as shown in Figure 5. From [13]:

    Figure 5

   [15]

since

   [16]

where m_disc = mass of the circular disc, and I_y'y'(disc) = the MOI of the disc about the Y' axis. [16] basically says that the MOI of a circular plate about the Y axis is equal to the sum of the MOI about the parallel axis on the disc (Y') and the mass of the disc times square of the distance between the two axes. This is the so-called parallel-axis theorem.

The MOI of the circular column is therefore

   [17]

since

   [18]

Comparing [18] with [2], one can clearly see the difference in the MOI between a thin rod and a thick rod (circular column). Similarly, [13] and [17] shows the difference in the MOI between a circular disc and a thick circular plate (circular column).

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Physical Pendulum & Direct Measurement

Unfortunately, the integration approach is possible only when the body has a known geometric shape. In the mathematical human body models such as Hanavan (1964) and Yeadon (1990), it is assumed that the body segments show a group of geometric shapes such as ellipsoid of revolution, elliptical solids, and stadium solids. See BSP Equations for the details.

If the body has a irregular shape, the integration approach has not much use and a direct measurement must be attempted. Figure 6 shows a body with irregular shape which is rotating freely about an axis passing through its one end. The X axis is the axis of rotation, thus, the center of mass (CM) of the body moves within the YZ plane.

    Figure 6

The torque produced by the weight of the body about the X axis is then

   [19]

where T_x = the torque about the X axis, I_xx = MOI of the body about the X axis, a = angular acceleration, m = the mass of the body, g = the gravitational acceleration (9.81 m/s²), and L = the distance between the axis of rotation to the body's CM. For a small q,

   [20]

and, from [19],

   [21]

Solving [21] for q, one obtains

   [22]

where q_o = the amplitude, f = the frequency of the pendulum, e = the phase angle, T = the period of the pendulum. As shown in [22], the MOI of the body about the X axis, after all, can be computed from the period of a small pendulum motion of the body. The MOI about the parallel axis, which passes through the CM of the body, can be also computed based on the parallel-axis theorem:

   [23]

See Chandler et al. (1975) for an example of this approach.

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References and Related Literature

Chandler, R. F., Clauser, C. E., McConville, J. T., Reynolds, H. M. and Young, J. W. (1975). Investigation of inertial properties of the human body. AMRL-TR-74-137, AD-A016-485. DOT-HS-801-430. Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio.

Hanavan, E. P. (1964). A mathematical model of the human body. AMRL-TR-64-102, AD-608-463. Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio.

Yeadon, M. R. (1990). The simulation of aerial movement-II. A mathematical inertia model of the human body. J. Biomechanics 23, 67-74.

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© Young-Hoo Kwon, 1998-