Second moment of area of Launch Vehicle adapter model

[R.enR]

I'm designing a model launch vehicle adapter (hollow truncated cone) with top diameter 60mm and lower diameter 80mm with a height of 1000mm. The skin thickness vary between options of 1.2mm 1.5mm and 2mm and I am trying to identify natural frequency. The issue is with second moment of inertia I have looked around and it mainly focuses on cones and hollow cylinders. I am unsure as how to identify values and any help would be much appreciated.

 

[tiny-tim]

Welcome to PF!

Hi R.enR! Welcome to PF!

I'm designing a model launch vehicle adapter (hollow truncated cone) … The issue is with second moment of inertia…

Every type of https://www.physicsforums.com/library.php?do=view_item&itemid=31" is additive (and subtractive) …

a hollow cone is a solid cone minus a shorter solid cone …

a truncated hollow cone is a hollow cone minus a shorter hollow cone …

find the moments of inertia, and do a bit of subtraction. ​

 

[R.enR]

Hi tiny-tim,

Thank you for the help I like the fact it was so simple when you think about it like that I must admit I was worried until now. I will attempt to work it out and then tell you if I am able to find the answer.

Thank you for your help :) wish me luck lol

 

[R.enR]

Hi again,

I am once again confronted by a stumbling block which I am hoping I have got the correct idea with. I have values for Ix, Iy and Iz however I am looking for I itself or the resultant I within the conical structure. I realize second moment of inertia is similar to Momentum so Imagine/hope that it is similar to how Mx, My and Mz can all be squared and added together and then square root the result to get the resultant momentum.

I respect the fact this is a physics forum and I may be embarrassing myself here but I would appreciate any feedback.

Apart from that once I can get this value I'm fairly confident I can identify natural frequency and other equations which would be great thanks to earlier comment thanks again tiny tim

 

[Unrest]

Can I ask how you intend to use these 2nd moments of inertia?

If you want to consider them as components of a vector, then sure it's fine to find a modulus. But I can't think how that would help in finding the vibration modes. The moment of inertia Ix tells you the stiffness to bending about the x axis.

I suppose you expect the 1st few modes to be be bending like a tapered beam? Are you using a tapered beam model or approximating it as a uniform cylinder with the same 2nd moment all along the length?

There will also be modes that aren't bending. I just did a rough and ready finite element model and it showed up bending as the lowest modes, closely followed by various twistings and squashings -
[PLAIN]http://dl.dropbox.com/u/21857463/taperedtubemodes.png

 

[tiny-tim]

Hi R.enR!

(just got up :zzz: …)

I have values for Ix, Iy and Iz however I am looking for I itself or the resultant I within the conical structure. I realize second moment of inertia is similar to Momentum so Imagine/hope that it is similar to how Mx, My and Mz can all be squared and added together and then square root the result to get the resultant momentum.

I don't know what you mean by "I itself".

There is no such thing as an "overall" moment of inertia (of any type) …

there is of course a moment of inertia matrix (tensor), but (in a symmetric case like this) that is just I_x I_y and I_z along the diagonal.

(technically, components of momentum make up a vector, but components of moment of inertia make up a tensor)

That's the maths … as to the physics, you'd better be guided by Unrest ​

 

[R.enR]

Hi

Yeah I was up late last night after a 9 hour shift at work so i was a bit tired. I think I realize this now that its a symmetric shape and that due to this I can use the Iz component for what I need, I been stressing over nothing probably. But to identify natural frequency there is an equation at first mode for sinusoidal vibration where I is required it's on a space design book with a design concept by fireSAT satellite. I am applying this to identify my models natural frequency to identify limits where the part must not be oscillating at. Ideally it will have a low natural frequency similar to most spacecraft . I should be good now I hope so anyway now that I see second moment of inertia as an adding subtracting basis it's easier to resolve.Thanks again for the help Tiny-tim and unrest =)

 

[spanky489]

after listening to tiny-tim's advice you can also model the object in a 3D CAD program and check out the moment of inertia it calculated for backup.

 

[Unrest]

Yea, you can just use one of the two identical I values for bending.

l have a low natural frequency similar to most spacecraft . I should be good now I hope so anyway now that I see second moment of inertia as an adding subtracting basis it's easier to resolve.

Hmm, I know you sound confident, but I'm still deeply suspicious! It sounds like you're using a beam model which would only find bending modes. Any structure will have many natural frequencies, getting higher and higher. This shape appears to have lots of them nearby each other. Most of those modes can't be found with a beam model using I. So it won't help much to just avoid bending resonances at, say 50Hz if there's a flappy-panel mode waiting at 60Hz, a twisting mode at 70Hz, etc.

Then again it might not matter anyway, even when you hit a resonance most systems can usually survive fine because of unavoidable damping.


Tiny Tim, so they're tensors! I guess that's where the double subscripts come from. Now I want to go find out what all the other components are and how this generalizes.

 

[R.enR]

I might have a look to reinforce my result that I have identified but I have input these values into several equations already so hopefully CAD will back my resolution up thank you for the help spanky489