Torsional stiffness of rotor with multiple radii 3

[ortabe]

Hello,
I'm writing some code to do rotor dynamic analysis(compressor trains). The rotor is read from a file and stiffness is calculated as follows:
k = G*I/(l/1000) # /1000 because input is in mm
But given the following rotor (all mm)
dia len
300 300
900 300
400 300
the calculated stiffness of the middle part will be much higher than it truly is. What would be a good rule of thumb to determine the 'equivalent' stiffness of this rotor?

 

[ishvaaag]

Ki=G·Ji/Li stiffness of the segment
Phi_i=M/Ki rotation of a segment subjedt to M
PHI=sum(Phi_i)=sum(M/Ki)=M/K

divide by M all numerators

1/K=sum(1/Ki)

 

[ortabe]

But how can you find an appropriate stiffness for the middle segment? I was thinking something along these lines:
r1 300 l1 300
r2 900 l2 300
r2_effective = r1 + l2/2 = 450 mm
and use r2_effective to calculate the stiffness of the middle segment. Would that do?
Thanks.

 

[TERIO]

Why do you think that calculating the stiffness of the middle segment as K_2=G*J_2/L_2 in inappropriate? Assuming it was inappropriate for some reason why would would your r2_effective term not depend at all on what r2 actually is?

 

[desertfox]

Hi ortabe

If your final goal is to calculate overall angular deflection,I think you can calculate the individual torsional stiffness of each section and consquently the angular deflection for each section, the total angular deflection will just be the sum of the individual ones.
I am making the assumption of course that the torque applied is a constant,

desertfox

 

[TERIO]

Also, your r2_effective term increases with increasing length (l2). This would predict that the stiffness would increase if you made the section longer, but increasing length should decrease stiffness.

 

[ortabe]

Thanks for your input.

> Why do you think that calculating the stiffness of the
> middle segment as K_2=G*J_2/L_2 in inappropriate?
While that formula works for a shaft with one radius I think I read somewhere that a shaft segment has less stiffness than its nominal value if neighboring segments have a much smaller radius.

> Also, your r2_effective term increases with increasing
> length (l2). This would predict that the stiffness would
> increase if you made the section longer, but increasing
> length should decrease stiffness.
The assumption is that the stiffness of the middle segment is proportional to the radii of the neighboring segments and its own length. If r2_effective turns out to be larger that 900 (in this example) I'll just use 900.

> If your final goal is to calculate overall angular
> deflection,I think you can calculate the individual
> torsional stiffness of each section and consquently
> the angular deflection for each section, the total
> angular deflection will just be the sum of the individual
> ones.
I'm calculating eigenfrequencies, mode shapes and later on I'll add forced response. So a good value for k is needed to get accurate results.

Perhaps I'm on the wrong track and there's no such thing as the lesser-that-nominal stiffness.

 

[desertfox]

Hi ortabe

Well this site might be of some help:-

http://www.freestudy.co.uk/d225/t13.pdf

desertfox

 

[desertfox]

Hi again

http://books.google.co.uk/books?id=LEA9AAAAIAAJ&pg=PA252&lpg=PA252&

dq=calculating+torsional+stiffness+of+stepped+shafts&source=bl&
ots=ID063AMux9&sig=PpOW0RvrTeobHMot4bHXpps6rbg&hl=en&
ei=thZKS53ZGY320gSk3cXpAQ&sa=X&oi=book_result&ct=result&resnum=4&
ved=0CBIQ6AEwAzge#v=onepage&q=&f=false

desertfox

 

[TERIO]

I think I see what you are getting at. It would take some distance from the connection between the larger and smaller shaft for the torque to distribute itself over the entire cross section of the larger part (see attached sketch). For long shafts this end effect would be negligible but since your section is short compared to its diameter this could be important. Off the top of my head I don't have a good idea of how to estimate this impact.

 

[ortabe]

TERIO, that's what I mean. Your sketch illustrates it nicely.

 

[dhengr]

ortabe:
RE: your 10JAN 7:32 post; that’s called pulling something out of your butt, but it may not be too bad a guess.
RE: your 10JAN 12:36 post, 1st para., last sentence; I think you read that right and have good memory.

To get your brain and thinking in gear; think of stress concentrations at shaft transitions, more generous radius means lower stresses; the point being, how do the stresses want to flow and if you pinch them down too much you have high stresses (concentrations). And you have that problem to consider in your shaft from the stress standpoint. But, assume no radius from 300mm to 900mm for our discussion, and still ask the question, how do the stresses and strains act; they do not turn 90° and go up to 900mm and turn 90° again and go down the length of the shaft. Rather, they will tend to make some nice gradual transition to a larger radius, and if the middle length is long enough, they will attain your full calc’d. angular stiffness k=JG/l, or corresponding stresses ? and ? for some distance. In this transition length they will be influenced by the material outside the transition zone. I would spend pages and pages and hours too, of Theory of Elasticity clacs. and verbiage to try to prove this. From the stress and stiffness standpoint, I dare-say, you could run a line from the 300mm dia. to the 400mm dia. and have a fair approx. for the stiffness, given the lengths.

Most of you have probably read my standard harangue about FEA (access to a FEA program) does not an engineer make. But, here it seems to me, you have a good application for FEA biting you in the butt and instead you pulled 450mm out of your butt; which I think might still be a pretty good first guess, given the 300mm length. Model this shaft and apply a unit torque or a unit rotation and develop your own stiffness; much better than any guess. For at least a dia. on either side of the transitions you want a fairly tight mesh; I guess that means the whole shaft, given the dia’s. and lengths.

Remember, your stiffness (k=JG/l) is an idealized stiffness which only gives correct results at points somewhat removed from transitions and points of torque application, etc. (Saint-Venant’s Principle, et.al.) An old rule of thumb is shaft dia. or beam depth, plus. Thus, it’s probably somewhat nebulous given the dimensions of your shaft. But, you should be able develop a more meaningful stiffness and stress distribution with good FEA modeling and proper interpretation of the results. You must still pay attention to stress concentrations, and total system balancing. You must include the total mass distribution in your dynamic analysis.

For all this free advice, I want a copy of your work and results, for my own edification. I am awaiting the results. Good Luck, prove me right, please.

 

[ortabe]

> that's called pulling something out of your butt
Absolutely.

> you should be able develop a more meaningful stiffness
> and stress distribution with good akward modeling and
> proper interpretation of the results.
I was hoping to steer clear of FEA for one because I currently don't have access to Ansys or the like but lacking a more classical approach FEA seems the way to go.

> For all this free advice, I want a copy of your work and
> results, for my own edification. I am awaiting the results.
If I manage to develop a rule of thumb that I can validate with FEA I'll be sure to let you know.
Thank you for your insights!

And just for fun I'll add the awkward first mode shape of the above rotor without any correction to k.

http://img36.imageshack.us/img36/5392/testms1.png

 

[gwolf2]

dhengr,

Please stop posting these ludicrous long monologues. They stray far too wide of the point and are just you mouthing off without solving the poster's problem.

"For all this free advice, I want a copy of your work and results, for my own edification. I am awaiting the results."

REALLY!

ishvaaag gave a perfectly good answer on post #1!

Annoyed,

Gwolf2

 

[TERIO]

In the book "Advanced Strength of Materials" by J.P. Den Hartog there is a solution (turns out to be graphical in the end) for shafts with variable diameter starting on page 38 using a stress function. There is a good sketch on page 44.

http://books.google.com/books?ei=a8...t&q=Round Shafts of Variable Diameter&f=false

 

[ortabe]

Thanks TERIO. That looks very usefull to my problem.