It can be hanged to rotate about a vertical axis. Three arbitrarily chosen different rotating axes would give you a way to get an answer.

Not exactly. To get the PRINCIPAL axes you must know the inertia moments against 3 axes. Till now what was written is correct but "rotating" does NOT give the inertai value. The principle is to hang the part with a torsion spring (a wire) and measure its own frequency. Knowing the wire stiffness it is easy to compute the inertia. You also MUST know precisely the angles and relative positions (in space) between the 3 axes. the rest is computing.

Thanks for the suggestions. But how do frequency helps in defining the principal axes? And there are many modes involved. How are we sure of capturing most of the participating modes?

Frequency does NOT help to determine the axes BUT ONLY to determine the VALUES of the inertia around the different axes. If you have a look in a book about mechanics (classical you not need to go too far) you find the definitions of the inertia and how it does relate to a center and rotational axis. You need for your solution only to know the lowest frequencies.

I think and please do not misunderstand me that you need to review your knowledge about inertia definitions and basic mechanical equations. According to your questions I think it is the very first step you should do.

The steps should be:

what is an inertia ?

what are the principal inertia axes ?

When you obtained the answers you will know how to proceed.

Thanks for the response. Appreciate it.

For any irregular shapes, i can measure CG location, Ixx, Iyy & Izz about the CG. But where are the principal axes and how to locate them experimentally/physically. Thanks again.

Hi Yee,

Here is a common way to do this:

Step 0:

Set a random body coordinate frame (xyz) for your part.

Step 1:

Use a bifilar pendulum, which basically hangs your part with two parallel strings.The moment of inertial along the vertical axis is

I = mg*T^2*W^2/(16*pi^2*L).

where T is the period of motion, W is the distance between the strings and L is the length of each string. you need to repeat this experiment 3 times to get Ix, Iy, and Iz. These are not the principal moment of inertia. They are only diagonal elements of the inertial tensor.

Step 3:

The next step is to measure the off-diagonal elements i.e. Ixy, Iyz, and Ixz. The off-diagonal elements couple the dynamic of the object about one axis to its dynamic about other axes. You can take advantage of this coupling and compute the off-diagonal elements. For example spinning the part around x-axis with omegaX generates two reaction torques about y and z axes proportional to Ixy and Ixz respectively. Thus you can simply compute Ixy = TourqeY / omegaX. And similar formulas for Ixz and Iyz.

Step 4:

Having all 6 values for the inertia tensor described in xyz frame, one can find eigenvectors of such a tensor. The eigenvectors are the principal axes.

Depending on the complexity of your part, you may want to design a mounting frame for your part and at the end subtract the moment of inertial of the frame from the combined moment of inertia.

Hope that was helpful.

Yazdian, 

This procedure looks fine.  Do you know some book where we can find details? 

Thank you

The 3 x 3 inertia tensor is symmetrical, so only 6 elements need to be determined after which the tensor can be diagonalized, i.e rotated so that the off diagonal elements are zero, thus defining the orientation of the principal axes. In principle you therefore have to measure the moments of inertia about 6 non parallel axes (unless there are planes of symmetry of the object). This can be done by a compound pendulum method, see Dezso Szoke and Norbert Horvath Measurement of Tensor of Inertia with tetrahedon method, by a bifilar or trifilar suspension see M W Green Measurement of the Moments of Inertia of full scale Aircraft, Hinrichsen Bifilar Suspension Measurement of Boat Inertia Parameters, Huw Williams Measuring the Inertia Tensor or by FRF modal analysis. These references can all be found on the web A good book is Classical Mechanics by R Douglas Gregory, Cambridge University press.