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# Moment of Inertia and area about Inclined Axis Mechanical Engineering Notes | EduRev

## Mechanical Engineering : Moment of Inertia and area about Inclined Axis Mechanical Engineering Notes | EduRev

The document Moment of Inertia and area about Inclined Axis Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.
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Moments of Inertia about inclined axis

In structural and mechanical design, it is sometimes necessary to calculate the moment of inertia with respect to a set of inclined u, v, axes when the values of q , Ix, Iy, Ixy are known.

To do this we will use transformation equations which relates the x, y, and xâ€™, yâ€™ coordinates.

From Figure, these equations are:

Note:

Moments of Inertia about inclined axis,, continue

Given

we wish to determine moments and product of inertia with respect to new axes xâ€™ and yâ€™.

• The change of axes yields

• By adding the equations for Ixâ€™ and Ixâ€™ we can show that the polar moment of inertia about z axis passing through point O is independent of the orientation of xâ€™ and y;

 Jo = Ixâ€™ + Iyâ€™ = Ix + Iy

• These equations show that Ixâ€™ , Iyâ€™ and Ixâ€™yâ€™ depend on the angle of the inclination, Î¸, of the xâ€™, yâ€™ axes.

Principal Axes and Principal Moments of Inertia

We will now determine the orientation of these axes about which Ixâ€™ , Iyâ€™ are maximum and minimum. This particular axes are called principal axes By differentiating the first of Eqs. 10-9 with respect to q and setting the result to zero. Thus;

Therefore, at Î¸ = Î¸p ;

By substituting for Î¸ in Ixâ€™ , Iyâ€™ and Ixâ€™yâ€™ equations and simplifying, we obtain;

Summary

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