Rotating beam stress calculation

[IKBrunel]

I'm struggling go to find the formula to calculate the stress & strain developed along a beam that is rotating around one end (simple rotor blade). Roarks give a simplified calc for max stress/strain in a pinned beam, I need to know the stress as it develops from the tip inwards.

Does anyone know this formula or where I should look?

Details:
- Rectangular section beam
- Beam aligned radially (axis intersects rotational axis)
- ideally will include inside rad as well as outside rad (length)


Many thanks

Matt

 

[drawoh]

IKBrunel,

Is this like the rotor blade of a helicopter?

The beam is solidly supported at one end, with a non-constant, distributed load. This may be in a handbook somewhere. You can work everything out by double integration.

It does not sound like your pinning is in a direction that affects stress and strain.

JHG

 

[BobM3]

How simple can you afford to be? In addition to the obvious radial force there will be tangential forces, aerodynamic forces (lift), natural frequencies and probably other forces that would affect the stress and strain.

 

[btrueblood]

Pretty simple integral problem.

The stress should vary from zero at the outer tip, to the value from Roark's at the hub. The stress should vary as the square of radial position.

 

[rb1957]

assuming you're just whirling this beam about a point, the endload in the beam (total length L) at a station x is the mass of the beam between x and the tip * centipedal acceleration.

as other posters have mentioned, the real situation could be more complicated if there are aerodynamic forces acting on the beam.

 

[zekeman]

The centripetal force on any section of the beam at a distance r ( from CL) for a beam of ro length (outer edge ro from the centerline of rotation )is:
the integral of rho* w^2*a(x)*b(x)*x*dx with limits from r to ro where
w=angular velocity, radians/sec
a(x) width of beam at pos x
b(x)= thickness at x
rho = mass density, e.g., slugs/in^3
In your rectangular problem a and b are constant so the integration is very simply
rho*w^2*a*b*x^2/2 between limits of r and ro
invoking limits this becomes:
rho*w^2*a*b(ro^2-r^2)/2
since crossesctional area is a*b,
the stress is the above expression absent a*b

 

[drawoh]

zekeman,

A slug is 1ft/lb.sec[sup]2[/sup]. Don't mix it with inches. If you want to use English units, use m=w/g, where w is in pounds, and g equals 32.2ft/sec[sup]2[/sup], or 386in/sec[sup]2[/sup].

JHG

 

[zekeman]

Yes,you are right; I am out by a factor of 12. The integration term, a*b*dx*rho is in slugs, but the x should have been in feet , thus the factor of 12. So divide my result by 12 for the answer

 

[rb1957]

just to fuss this, aren't the dimensions of slugs lb*sec^2/ft (so that when you multiply by acceleratoion (ft/sec^2) you get lb

 

[IKBrunel]

Thanks to all for the responses.

Thanks for the formula, Zekeman, I'm new at my company and on this problem so I'm kicking myself a bit for not seeing how to derive it. I've unneccessarily and pedantically checked it with Quickfield ... it checks fine.

My problem involves turbo rotor blades spinning at 1KHz / 60,000rpm. The blades are a circular array of radial thin beams, with the 'paddles' angled up to 45°.
Example image:

http://uweb.txstate.edu/~wg06/manuals/ModulabPVD/turborotstat.gif

We have FEA results show some concerning stress concentrations from root radiuses and twist (induced by centripetal imbalance). My question was there to help me build a simple rule-of-thumb model to help me develop the blades without running FEA for each iteration. I can come back to FEA when I've got a design based on basic principles.

My start point was to look at extension stress (as discussed). My next quest is to model the twist that happens as the blade flattens at speed. That's much harder, I expect to have to use beam torsion formula using a moment that develops towards the hub.

Does anyone know of good sources/books that would help with rotor blade design?


Many thanks

 

[rb1957]

with your more detailed explanation i think we're missing something.

we've considered inertia effects only (with cavets about this limitation).

a "turbo-blade disc" is bound to have significant aero. loads as well. but as you say, maybe you're only interested in some aspects of the design (how much the blades will grow under load?) rather than the detialled stress analysis of the blades. but these aero. loads have a significant effect on the blade twist. i'd expect to see the blades twisted along their length, so as to keep a constant angle of attack with the local airflow (the vector sum of the rotational speed of the disc and the incoming airflow). a high angle of attack is probably appropraite for the outer portion of the blade, but will caused stalled airflow on the inner portions, and this is not good.

 

[IKBrunel]

rB1957,

Thanks for the comment, I should have said, it's for a vacuum pump so the rotor isn't started until the system has been pumped down such that aero lift is negigible. That said, we still consider abuse states where a fault or misuse results in atmospheric air being dumped into the pump chamber. This results in a high axial load and I would like to consider this as an embellishment to the calcs. we generally design and test for that rather than analyse but it's a much discussed subject. It can lead to a very high energy catastrophic failure (0.5Kg rotor at 60krpm stops in a few seconds!)

Many thanks for the interest in the problem.

 

[rb1957]

seems a bit odd to me that a rotor and stator being used where there isn't much airflow ... isn't the point of a stator to remove swirl ?

 

[drawoh]

just to fuss this, aren't the dimensions of slugs lb*sec^2/ft (so that when you multiply by acceleratoion (ft/sec^2) you get lb

rb1957,

You are right. So much for quickly scanning Wikipedia.

JHG

 

[rb1957]

ah yes ... wiki, the font of all knowledge
(sorry, but an intern here uses wiki instead of text books, geez)

 

[btrueblood]

Rb1957,

not that odd:

http://en.wikipedia.org/wiki/Turbomolecular_pump

 

[rb1957]

fair enough, but all this "molecules impacting the blades" should implies some sort of lateral forces.