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As shown in [6] in Inertia Tensor, the angular momentum of a rigid body with respect to the origin of the local reference frame is expressed as
If, by chance, all the off-diagonal terms of the inertia tensor shown in [1] become zero, [1] can be further simplified to
This can happen when one aligns the axes of the local reference frame in such a way that the mass of the body evenly distribute around the axes, thus, the product-of-inertia terms all vanish. The non-zero diagonal terms of the inertia tensor shown in [2] are called the principal moments of inertia of the object.
As shown in [1], there is no guarantee that the angular momentum vector has the same direction to that of the angular velocity vector. This causes a problem: if the direction of the angular momentum keeps changing, it develops a torque which eventually forces the axis of rotation to move. This is the main reason that causes wearing and vibration in machinery with rotating parts. But in some special cases, the following condition may hold so that the angular momentum and velocity vectors show the same direction:
where I = the equivalent scalar moment of inertia of the body about the axis of rotation. Any axis of rotation of the body which suffices [3] is called a principal axis. There are a group of principal axes (theoretically 3) in a three-dimensional body. For example, there are three perpendicular principal axes for the system shown in Figure 1.
[3] basically says that the inertia tensor can be replaced with a single scalar moment of inertia when the axis of rotation is a principal axis. Diagonalization of the Inertia Tensor From [3]:
Or [4] can be simplified to
where 1 = the identity matrix. I shown in [5] is called an eigenvalue while w is the eigenvector. [5] is the eigenvalue equation. In order for [5] to have a nontrivial solution the determinant of the coefficients should vanish:
[6] leads to the secular equation which is basically cubic, thus provides three roots (eigenvalues): I1, I2 & I3. Each root corresponds to a moment of inertia about a principal axis. In fact the three roots are the principal moments of inertia of the rigid body introduced in [2]:
Once the eigenvalues are known, the principal axes can be computed. Let
where n = the unit vector of the principal axis, thus,
From [4] & [8]:
For each eigenvalue, one can compute the corresponding nx, ny & nz from [9] & [10]. One must pay attention to the direction of the eigenvector in this process. In motion analysis, the principal moments of inertia can be obtained from the inertial properties of the body segments. I1, I2 & I3 of each segment are generally known. The data are available in the form of the radius-of-gyration ratios (ratio of the radius of gyration to the segment length), regression equations, and scaling coefficients. One can also compute the principal moments of inertia of the body segments through modeling using some geometric shapes. See Individualized BSP Estimation for the details.
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© Young-Hoo Kwon, 1998- |
[1]
[2]
[3]
Figure 1
[4]
[5]
[6]
[7]
[8]
[9]
[10]