Moment of Inertia - Moment of Inertia Tensor

Moment of Inertia Tensor

The moment of inertia for a rigid body moving in space is a tensor formed from the scalars obtained from the moments of inertia and products of inertia about the three coordinate axes. The moment of inertia tensor is constructed from the nine component tensors,

where ei, i=1,2,3 are the three orthogonal unit vectors defining the inertial frame in which the body moves. Using this basis the inertia tensor is given by

This tensor is of degree two because the component tensors are each constructed from two basis vectors. In this form the inertia tensor is also called the inertia binor.

For a rigid system of particles Pk, k=1,...,N each of mass mk with position coordinates rk=(xk, yk, zk), the inertia tensor is given by

where E is the identity tensor

The moment of inertia tensor for a continuous body is given by

where r defines the coordinates of a point in the body and ρ(r) is the mass density at that point. The integral is taken over the volume V of the body. The moment of inertia tensor is symmetric because Iij= Iji.

The inertia tensor defines the moment of inertia about an arbitrary axis defined by the unit vector n as the product,

where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as I12 is obtained by the computation

and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis.

The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,

 = begin{bmatrix}
I_{11} & I_{12} & I_{13} \
I_{21} & I_{22} & I_{23} \
I_{31} & I_{32} & I_{33}
end{bmatrix}=begin{bmatrix}
I_{xx} & I_{xy} & I_{xz} \
I_{xy} & I_{yy} & I_{yz} \
I_{xz} & I_{yz} & I_{zz}
end{bmatrix}.

It is common in rigid body mechanics to use notation that explicitly identifies the x, y, and z axes, such as Ixx and Ixy, for the components of the inertia tensor.

Read more about this topic:  Moment Of Inertia

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