Complex Moment of Inertia Calculation

[k2kyo]

Homework Statement

I'm just looking for the proper equation to use to find the moment of inertia of a complex assembly of bodies. I'm not a student and this isn't homework, but it seemed the most relevant place to pose the question.

One of my hobbies is making toys and I'm trying to calculate the moment of inertia for a given section of an assembly as it relates to the whole system. All my parts are round, revolved around the same X-axis. The assembly is mirrored on either side of the Y-axis. Therefor, I know all the centers of mass are y = 0, and I can calculate all the x distances (dx in my drawing).. what I'm unsure of is how to relate that x distance into the moment of inertia formula so that I can know what each body is contributing to the whole.

Taking a section of the drawing.

Body 3)
Area = 146.27 mm^2
Mass = 20.12 grams
Volume = 15545.56 mm^3
Moment of Inertia of the single body (Ix) = 4416.93
d(x) = 12.53

Now using solidworks, I know the result..

Moment of Inertia for body 3 in relation to the whole system (Ixx) = 7576.69

But I don't know how it got there.. am I correct in assuming the variable is the distance x to the center of its mass? If so, how do I integrate that into the equation?

Drawing -
http://www.nothingtoseehere.info/Example_Problem1s.jpg

Homework Equations

This is the part I'm looking for :) I don't need any help working the equation, just a point in the direction of what I should use.

The Attempt at a Solution

I can calculate the individual moment of inertia for a given part. What I'm trying to work out is a system where I could swap out body 3 or 4 for any other similar design, and recalculate the moment of inertia for the whole assembly without having to build a new assembly in solidworks. So if I can figure which formula to use, I can use the variables to calculate the Ixx for each section 3 or 4 I want to use, and swap those out.

I hope that makes sense, this problem has been driving me nuts. I use to do this type of stuff in college but I fear it's been too many years.

Any help would be greatly appreciated

Thanks,
Kyle

 

[Orodruin]

Have you looked up the definition for moment of inertia? In particular I would point you to http://en.wikipedia.org/wiki/Moment_of_inertia#Calculating_moment_of_inertia_about_an_axis
You might also want to look up the parallel axis theorem.

 

[CWatters]

I've deleted my reply as I was confused by which axis the object rotates around.

 

[k2kyo]

Have you looked up the definition for moment of inertia? In particular I would point you to http://en.wikipedia.org/wiki/Moment_of_inertia#Calculating_moment_of_inertia_about_an_axis
You might also want to look up the parallel axis theorem.

I had, but had not examined parallel axis theorem for mass moment I had only seen area.

That part if parallel is exactly what I needed, thanks.

Problem solved.

Kyle

 

[HalfLostAndSearching]

As a fellow scientist, I can understand your frustration with this complex moment of inertia calculation. This is a common challenge in engineering and physics, and there are many different equations that can be used depending on the specific system and geometry.

One approach that may be helpful in your situation is to use the parallel axis theorem, which allows you to calculate the moment of inertia of a body about a different axis by adding the moment of inertia about the center of mass and the product of the mass and the square of the distance between the two axes. In your case, you could use this theorem to calculate the moment of inertia of each individual body about the x-axis, and then add them together to get the total moment of inertia of the entire assembly.

Another useful equation for calculating moment of inertia is the perpendicular axis theorem, which allows you to calculate the moment of inertia of a 2D object rotating about an axis perpendicular to the plane of the object by using the moment of inertia about two other perpendicular axes. This may be helpful for calculating the moment of inertia of your mirrored assembly.

Ultimately, the best equation to use will depend on the specific geometry and configuration of your system, so it may be helpful to consult with a physics or engineering textbook or seek assistance from a colleague or professional in the field. Good luck with your project!