Inertia Tensor
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Angular Momentum of a Rigid Body
Inertia Tensor

Angular Momentum of a Rigid Body

Angular momentum of a rigid body (Figure 1) can be obtained from the sum of the angular momentums of the particles forming the body:

   [1]

    Figure 1

 

where ri = the position vector of particle i, and w = the angular velocity vector of the rigid body. Now, let

   [2]

    [3]

Note here that a local reference frame, the XYZ system, is defined and fixed to the body at O and ri & w are described in this frame. Substituting [2] & [3] into [1]:

   [4]

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Inertia Tensor

Let

   [5]

Substituting [5] into [4]:

   [6]

   [7]

where I = the inertia tensor. The angular momentum of a rigid body rotating about an axis passing through the origin of the local reference frame is in fact the product of the inertia tensor of the object and the angular velocity. The diagonal elements in the inertia tensor shown in [7], Ixx, Iyy & Izz, are called the moments of inertia while the rest of the elements are called the products of inertia. Also see Moment of Inertia & Ellipsoid of Inertia for more details of the moments and products of inertia. As shown in [7], the inertia tensor is symmetric.

The 3 x 3 matrix in [7] suffices the requirements of a tensor of the 2nd rank:

   [8]

where i, j, k & l = 1 to 3, tij = an element of the orthogonal transformation matrix, and I'ij = an element of the transformed inertia tensor. This property is explained in detail in Transformation of the Inertia Tensor. In fact, a scalar is a tensor of zero rank:

   [9]

where S = a scalar, S' = the transformed scalar. In addition, a vector is a tensor of the first rank since its transformation follows

   [10]

[10] is identical to [2] shown in Transformation Matrix. A 2-dimensional symmetric matrix is not necessarily a tensor of the 2nd rank. It must suffice [8] to be a tensor.

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Young-Hoo Kwon, 1998-