Determine bending moment from acceleration measurements

[BevanSmith]

I am trying to determine bending moments from acceleration measurements. I can integrate the acceleration signal twice to get deflection, but then am unsure how to obtain the bending moment from that deflection data.

Deflection theory works from the bending moment to get the deflection but I am trying to get the bending moment from the deflection.

Any assistance would be appreciated.

 

[electricpete]

First a simplistic suggested approach, then the caveats.

The simplistic approach. Assume that an unknown force F or a moment M is applied to the shaft at some location based on your knowledge of the system and your suspicion of the vibration cause. Use that force or moment to determine a deflection shape to within a scalar constant. Use measured amplitude to determine the scalar constant. Moment is E*I*dy^2/dx^2.

Now the caveats.
First – you are measuring acceleration I am guessing it is on bearing housing. Relationship between bearing housing displacement and shaft displacement is unknown.

Second – is it really fair to model the vibration as originating with a force or moment at one location. Forcing functions such as misalignment are much more complicated.

I’m sure there are more caveats, but those two come to mind.


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[electricpete]

Maybe third caveat would be the question: how well do you know the stiffness of your bearings / supports. Those should be an input into your model.

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[electricpete]

There there are dynamic effects. For one thing it affects the way you look at those support stiffnesses (they should be dynamic stiffnesses).

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[GregLocock]

What you need is the fundamental beam equation.

From 3 adjacent deflections work out the radius of curvature of the beam (schoolboy geometry)

Then

M/I=E/R

I bet you thought it was harder than that!



Cheers

Greg Locock


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[electricpete]

Greg - So you don't care where along the shaft the force is applied or how stiff are the supports?

I would think in that case the bending stresses you calculate will represent a minimum value.... the actual are unknown and can be much higher than calculated and you have no way to bound them without a better model.

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[GregLocock]

No, in order to bend a beam of properties EI the moment required is always EI/R

This M/I=E/R=s/y is the fundamental beam equation, everything else is derived from it (for elastic beam theory).

Cheers

Greg Locock


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[electricpete]

If you know the profile of displacement along the entire beam, then you are certainly in business to calculate moment and bending stress. But we are talking about only a handful of displacement (from acceleration) measurements... I assumed on each bearing housings, and you mentioned a number 3. Three displacements is not enough to characterize the entire shape of a continuous beam subject to unknown reaction forces (unknown because you imply support stiffness is irrelevant) and unknown applied force at an unknown location.


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[GregLocock]

I've reread the OP's post he doesn't mention how many data points he has. OK, he'll need 3 between each support or applied load, ideally.

Cheers

Greg Locock


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[electricpete]

We can place an accelerometer on the bearing housing to get an estimate of the shaft movement at the location of the bearing. Maybe you work with different instrumentation than I, but in my experience there is no practical way to place an accelerometer on a rotating shaft to measure shaft acceleration between bearings.

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[electricpete]

Actually, the original poster said nothing about shaft... so I may be way off base. My apologies.

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[FeX32]

Why would you calculate a moment from an accel. signal?
Could that not be misleading, such a signal could indicate oscillation without any applied moment. And any static moment would not be picked up.
Why not perform a dynamic characterization.


Fe

 

[GregLocock]

the stresses in the beam still exist. Otherwise you couldn't break stuff without applying external loads.

Cheers

Greg Locock


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[drawoh]

What you need is the fundamental beam equation.

From 3 adjacent deflections work out the radius of curvature of the beam (schoolboy geometry)

...

That only works if you have a constant moment going through the beam. If this is a cantilever beam, he needs to work out the bending using McCauley's method or by looking it up in a handbook. Going from deflection to moment is then fairly easy.

JHG

 

[electricpete]

I'm with Greg on that. The Euler Bernoulli beam approximation is:
M = E*I*dy^2/dx^2
SigmaBendingMax = c*M / I where c is distance neutral axis to extreme fiber
It doesn't matter what applied or reaction forces or moments created the bending, we can solve that maximum bending stress from the deflection shape. (Assume shear stress is negligible... beam is generally long and thin etc).

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[electricpete]

... and again that would require fairly finely spaced measurements. If the data is not so finely spaced, or there is excess error in it, it could be beneficial to have an idea of the functional form of the expected displacement solution and use the data to calibrate the coefficients of the solution.

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