Reducing 3x3 Inertia Tensor to Scalar Value

[frasool]

How to reduce a 3 by 3 inertia tensor to a scalor value?
Hi
I have the following tensor and i need to reduce it to a scalor quantity:
3 by 3 matrix
4150470.48 , 317.64, -353.42
317.64, 2047101.07,-1407556.61
-353.42, -1407566.61, 2284136.55

Please its urgent and any help would be greatly appreciated!
Regards
Faizan

 

[Cyrus]

You cannot reduce a matrix into a scalar value, this question doesn't even make any sense. What, exactly, are you trying to do here?

 

[Brian_C]

My guess is that he's looking for some sort of tensor invariant.

 

[frasool]

Hey.
Im looking to moment of inertia calculations for a caster wheel. I have defined paramtetric analysis for the wheel. I desgined the wheel on solid works and it gives the moment of inertia in a 3 by 3 matrix. however I want to use a value in my parametric analysis which requires to input a value of moment of inertia. So i don't know how to use this 3 by 3 matrix in my formulae.
so its basically being used in a formulae! but i can't use a 3 by 3 matrix with a bunch of scalor values can I. I hope that explains my problem !. Any help would be greatly appreciated.

Vc, Desired shimmy speed
T, Current caster offset
Iw, Mass Moment of Inertia
Ks, Tire Slip Coefficient
B, tread separation
Cd, Torsional Damping

Cd= (Vc/ T)*Iw

 

[Cyrus]

The inertia value you want to 'puck' out of your Solid Works tensor is the one that has an axis along the axis of rotation (Most likely the case here, based on your equation).

More importantly, do you understand why?

 

[frasool]

hmmm
Well solidworks gives me three different types of values:
1. Principal Axes of inertia and principal moments of inertia
2. Moments of inertia taken at Center of mass and aligned with the output coordinate system
3. Moments of inertia taken at output co-ordinate system

Could you explain to me why please?. And my axis of rotation should be along the y-axis i think!:S. Its not a symmetric object so you can't take one axis as your axis of rotation, that's what i think

 

[Cyrus]

You should consult the Solid Works help documentation for that, but I think they are pretty self evident. As for knowing what axis of rotation - that's your job to figure out. You should know from how you derived your equations of motion. Really, sit down and spend some time thinking about your problem.

 

[frasool]

my friend if i knew wht the solution to my problem is i would not be asking you. I have spent days on this problem and have to get it done real soon that's why i asked. I can't figure out what the significance of these values is. So I am going to do my own calculations now. One question for applying the parallel axis law, wht would you do if your body is a rigid shaped body. Like what would be your Icm, which is about the centroidal axis?
I= Icm + md^2. D being the perpednicular distance to the axis of rotation.

 

[Cyrus]

The significance of the values are explained in the output screen:

1. Principal Axes of inertia and principal moments of inertia
2. Moments of inertia taken at Center of mass and aligned with the output coordinate system
3. Moments of inertia taken at output co-ordinate system

Really, these are descriptions. For example, No. 2 is the inertia values taken at an origin at the center of mass, aligned with your output coordinate system: as described in the sentence. All I can say is go reread them carefully a second time.

I don't mean to be rude here, but I can't understand half of what you write. Please take some time in your posts. I get the impression that you just want to plug and chug into formulas and use software without having any fundamental appreciation for what they describe. In terms of the parallel axis theorem, your textbook should describe that in detail. Have you looked there yet? Or, do you know why we even use this theorem?

 

[frasool]

im in my second yr of mechanical engineering. i took physics course a year ago so don't remember that much but i do know what it signifies. Its similar to Steiners Theorem. I know its sued for rigid bodies and its application
Point is I don't know how to use the 3 by 3 matrix in my calculations that's all.
ok tell me if I am following the right approach.
1.Im going to make a reference axis.
2. Apply parallel axis law for each part of the wheel with reference to that axis
3. Add the individual inertias up.
4. Get moment of inertia

Problem still is how to use this in 3d!

 

[Cyrus]

The equation you provided in post number 4 is a scalar equation, so you are not using an inertia matrix. As per my earlier reply, you need to find out which inertia value to pick out from this matrix that corresponds to the inertia value as described in your equation.

The point of solid works is that you can specify an origin and have it calculate the inertia values so that you do not have to use the parallel axis theorem (Thats the third item in the list.).

 

[frasool]

hmm. i require it about its vertical axis.
So i just consider the z values then i suppose.

 

[Cyrus]

Look at what axis the inertia value is calculated based on the derivation of your equation. You want to have Solid Works calculate the inertia matrix about a similar reference frame. Then you can simply pick out the value that corresponds to the axis you are interested in. Without a derivation of your equation, or a picture of your CAD file, I have no idea what the z values correspond to, or what that means.

Side: Please capitalize the word "I" in sentences.