Finding the rotational moment of inertia of a multihinged arm

[Dudd]

Homework Statement

I'm trying to calculate the moment of inertia of a multihinged arm, with one joint at the shoulder and a second joint at the elbow. Each joint has a set of local axis, with y being a vector pointing down the shaft of the cylinder, z being a vector pointing up, and x being a vector pointing towards the right. The forearm is free to rotate about the local z axis of the elbow, and the entire arm is free to rotate in all three directions at the shoulder joint. The two arm segments are being modeled as solid cylinders with masses m1 and m2, lengths l1 and l2, and radii r1 and r2. Given a rotation theta around the elbow z axis, I need to find the overall moment of inertia around the shoulder joint. A crudely paint drawn diagram of the system is shown below:

http://img5.imageshack.us/img5/1122/momentofinertia.jpg

Homework Equations

Moments of inertia around the center mass:

I_y = mr² / 2
I_x = I_y = 1 / 12 * m * ( 3r² + L²

Parallel axis theorem to get I_y and I_z around the joint:

I_end = I_cm + MD², where D is equal to L/2.

The Attempt at a Solution

When theta is a multiple of 90, I believe I can simply again use the parallel axis theorem to get the moment of inertia of the forearm around the shoulder joint, keeping in mind that I will not always be adding Iy and Ix together depending on whether theta is a multiple of 90 or 180. However, when the rotation is any value in between and the axis are no longer parallel, I cannot use the parallel axis theorem to find the moment of the second segment. So, this is the point where I'm stuck, and I'm hoping there is some equation for a cylinder rotated an arbitrary angle that I have been unable to find. Thanks for your help.

 

[anathema]

Thank you for your question. Calculating the moment of inertia for a multihinged arm can be a bit tricky, but there are a few equations and principles that can help guide you in your calculation. First, I would recommend breaking down the arm into separate segments and calculating the moment of inertia for each segment individually. This will make the overall calculation easier and more accurate.

For each segment, you can use the equation I = mr^2 to calculate the moment of inertia around its center of mass. In this case, the center of mass will be at the midpoint of each segment. You can then use the parallel axis theorem to calculate the moment of inertia around the joint, using the distance from the center of mass to the joint as the distance in the equation.

When it comes to calculating the moment of inertia for a cylinder rotated at an arbitrary angle, you can use the equation I = 1/12 * m * (3r^2 + h^2), where h is the height of the cylinder. This will give you the moment of inertia around the axis of rotation.

In order to calculate the overall moment of inertia for the multihinged arm, you will need to add together the moment of inertia for each segment, taking into account the rotation of the arm and how it affects the moment of inertia. This may involve using different equations depending on the angle of rotation, as you mentioned in your post.

I hope this helps guide you in your calculation. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your research!

 

[nlightner]

I would suggest using the general formula for the moment of inertia of a solid cylinder, which is I = (1/2)mr^2 + (1/12)ml^2. In this case, you have two cylinders, one for the upper arm and one for the forearm, so you would need to calculate the moment of inertia for each one separately and then add them together to get the total moment of inertia for the multihinged arm.

To do this, you will first need to find the center of mass for each cylinder. This can be done by using the formula for the center of mass of a uniform solid, which is given by x = (1/2)l and y = (1/2)l. Once you have the center of mass for each cylinder, you can use the parallel axis theorem to find the moment of inertia around the shoulder joint. This would involve adding the moment of inertia of each cylinder at its center of mass to the product of its mass and the distance between its center of mass and the shoulder joint (which would be half the length of the cylinder).

If the rotation is not a multiple of 90 degrees, you can still use the parallel axis theorem, but you would need to take into account the rotation of the forearm around the elbow joint. This would involve adding the moment of inertia of the forearm at its center of mass to the product of its mass and the distance between its center of mass and the elbow joint (which would be half the length of the forearm). This would give you the total moment of inertia for the multihinged arm.

I would also suggest considering the direction of rotation and the orientation of the axes when performing these calculations to ensure that the correct moments of inertia are being added. Additionally, you may want to consider using a 3D coordinate system to better visualize and calculate the moments of inertia in all three dimensions.

I hope this helps and good luck with your calculations!