In computation of the moment of inertia, one can replace the summation shown in  of Inertia Tensor by an integration over the body:
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.Thin Rod
See BSP Equations for the moment-of-inertia equations for the geometric shapes commonly used in the human body modeling.
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
In other words, the MOI about the Z axis is equal to sum of those about the X and Y axes.  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,
and, from :
One can directly obtain  from  or
 can be used in further developing  to obtain . See BSP Equations for this approach.
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 :
where mdisc = mass of the circular disc, and Iy'y'(disc) = the MOI of the disc about the Y' axis.  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
Comparing  with , one can clearly see the difference in the MOI between a thin rod and a thick rod (circular column). Similarly,  and  shows the difference in the MOI between a circular disc and a thick circular plate (circular column).
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.
The torque produced by the weight of the body about the X axis is then
where Tx = the torque about the X axis, Ixx = 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,
and, from ,
Solving  for q, one obtains
where qo = the amplitude, f = the frequency of the pendulum, e = the phase angle, T = the period of the pendulum. As shown in , 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:
See Chandler et al. (1975) for an example of this approach.
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.
© Young-Hoo Kwon, 1998-