Can You Simplify Inertia Tensor Calculations for 3D Objects?

[oasamostexianu]

I'm trying to write a quick and dirty computer simulation for 3D rotational dynamics. This is the first time I've ever seen the sort of physics needed for doing this, so please bear with me.

I've decided that I don't need a huge amount of accuracy in the final simulation, so I was hoping to represent any rotating object as an aggregation of several smaller, simpler objects. For example, a rotating ice cream cone I could fudge as a cone and a sphere. With this method, I was hoping to get around a lot of slow and complex mathematics.

Here was my idea: If I know the inertia tensors for simple objects like spheres and rods around their primary axes, I can just shift my moments and sum them up into a final tensor, eliminating the need to transform coordinates and find eigenvalues.

For example: if I had an object that was composed of a solid sphere and a solid circular cylinder, with the center of mass of the sphere 5 units to the left of the object, and the CoM of the cylinder 2 units down from the CoM of the object, I could say that

(sphere)
Ixx = 2/5mr^2 + 25m
All other moments stay as they were; the products of inertia still remain zero.

(cylinder)
Izz = 1/2mr^2 + 4m
All other moments stay as they were; the products of inertia still remain zero.

Then I could take these two inertia tensors and add them together to get the final tensor of my aggregate object.

Will this work? Can I do the parallel axis shifts like that in 3D?

Thanks in advance!

 

[jamesrc]

Originally posted by oasamostexianu
For example, a rotating ice cream cone I could fudge as a cone and a sphere.

Mmmm... fudge swirl ice cream...

Sorry about that: what you're saying sounds good. If you've got parallel axes, you can use the parallel axis theorem to recalculate inertia.

 

[M98Ranger]

Your method of aggregating smaller, simpler objects to represent a rotating object is a valid approach and can help simplify the calculations involved in rotational dynamics. However, there are a few things to consider when calculating the inertia tensor using this method.

Firstly, it is important to note that the inertia tensor is not a simple sum of individual objects' inertia tensors. The inertia tensor takes into account the distribution of mass within an object, and simply adding the inertia tensors of individual objects may not accurately represent the true inertia of the entire object. It is recommended to use the parallel axis theorem to calculate the inertia tensor of an aggregate object, which takes into account the distance of each individual object from the overall center of mass.

Secondly, it is also important to ensure that the individual objects used in your aggregation method are rigid bodies. This means that the objects must have a fixed shape and size, and their mass must be distributed evenly. If the objects are not rigid bodies, their inertia tensors may not accurately represent their rotational behavior.

In addition, it is also important to consider the orientation of the individual objects in your aggregation. The inertia tensor is dependent on the orientation of an object in space, so if the individual objects are not aligned in a specific way, their inertia tensors may not accurately represent their rotational behavior.

Overall, your approach can work in certain cases, but it is important to keep in mind the limitations and considerations mentioned above. It is always recommended to use the parallel axis theorem to calculate the inertia tensor of an aggregate object, as it takes into account the distribution of mass and the orientation of the individual objects. I hope this helps and good luck with your simulation!