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Inertia Tensor

The inertia tensor is a 3×3 matrix that governs the rotational behavior of a rigid body. This matrix is symmetric: elements with reciprocal indices have the same value. That is:

Ixy=Iyx, Iyz=Izy, Izx=Ixz

Because the inertia tensor is symmetric, it requires only six elements. Three are the moments of inertia and three are the products of inertia. The complete inertia tensor has the form:

Specifying Inertia Tensor

You can specify the inertia tensor manually, using one of two blocks: Solid and Inertia. To do this, in the block dialog box select Custom from the Inertia > Type drop-down menu. In the new set of parameters that appears, specify the inertia tensor in terms of the moments and products of inertia.

Moments of Inertia

The moments of inertia are the three diagonal terms of the inertia tensor:

In the Moments of Inertia dialog box parameter, enter the three diagonal elements as a row vector. Enter the elements in the order [Ixx, Iyy, Izz]. These are the moments of inertia of the solid with respect to a frame whose axes align with the block reference frame, and whose origin coincides with the solid center of mass.

Products of Inertia

The products of inertia are the three unique off-diagonal elements. Because the inertia tensor is symmetric, each off-diagonal element appears twice in the matrix.

In the Products of Inertia dialog box parameter, enter the three unique off-diagonal elements. Enter the elements in the order [Iyz, Izx, Ixy]. One easy way to remember the element order is to think of the missing subscript component: x, y, and z respectively. The elements are the products of inertia of the solid with respect to a frame whose axes align with the block reference frame, and whose origin coincides with the solid center of mass.

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