Moment of Inertia and area about Inclined Axis | Additional Study Material for Mechanical Engineering PDF Download

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 , I_x, I_y, I_xy 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:

Moments of Inertia about inclined axis,, continue

Given

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

• The change of axes yields

• 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;

• These equations show that I_x’ , I_y’ and I_x’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 I_x’ , I_y’ 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 θ = θp ;

By substituting for θ in I_x’ , I_y’ and I_x’y’ equations and simplifying, we obtain;

Summary