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

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

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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 x’, y’ coordinates.

From Figure, these equations are:

Note:

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

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

Moments of Inertia about inclined axis,, continue

Given

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

we wish to determine moments and product of inertia with respect to new axes x’ and y’.

  • The change of axes yields

 

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

 

  • By adding the equations for Ix’ and Ix’ we can show that the polar moment of inertia about z axis passing through point O is independent of the orientation of x’ and y;

 

Jo = Ix’ + Iy’ = Ix + Iy

 

  • These equations show that Ix’ , Iy’ and Ix’y’ depend on the angle of the inclination, θ, of the x’, y’ axes.

Principal Axes and Principal Moments of Inertia 

We will now determine the orientation of these axes about which Ix’ , Iy’ 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;

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

Therefore, at θ = θp ;

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

By substituting for θ in Ix’ , Iy’ and Ix’y’ equations and simplifying, we obtain;

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

Summary

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

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

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

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

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