dtanner2013
Mechanical
I have cantilever beam in which I found the modes 1 and 2 by way of FEA (see attachment). How would I go about verifying this by hand?
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Cantilever Beam Vibration
- Thread starter dtanner2013
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spongebob007
Military
Seems like something you should be able to find in any textbook on vibrations.
electricpete
Electrical
Spongebob - did you look at the attachment.
It is not a simple classical cantilever beam.
PERHAPS we could simplify it for order-of-magnitude hand calc by imagining that the center section is flipped past the end to create more of a textbook beam which has roughly twice the length and half the width (of all 3 sections) or original problem.... but it would be very much an open question how close that would be.
=====================================
(2B)+(2B)' ?
It is not a simple classical cantilever beam.
PERHAPS we could simplify it for order-of-magnitude hand calc by imagining that the center section is flipped past the end to create more of a textbook beam which has roughly twice the length and half the width (of all 3 sections) or original problem.... but it would be very much an open question how close that would be.
=====================================
(2B)+(2B)' ?
electricpete
Electrical
And even if it's close, it would only be so for the first mode.
=====================================
(2B)+(2B)' ?
=====================================
(2B)+(2B)' ?
spongebob007
Military
I didn't look at the attachment. I guess it's not your average beam. I would say just by looking at the plot that the mode shapes seem about reasonable. What is the need to verify by hand? If you could calculate by hand there really is no need for FEA. If you are confident that you are using the program correctly and that you built the model correctly you should have reasonable faith in the results. Having a lot of experience with FEA helps. I don't veryify a lot of the FEA work I do with hand calculations. The point of FEA is that it allows for analysis of complex structures that can't be analyzed by hand. When I first learned to do FEA in college it was by making models of structures that had textbook solutions. If nothing else, this gives one confidence in the results the program gives. If you are modeling something that is within the limitiations of what FEA can do and you model the structure correctly and make sure the loads and boundary conditions are applied correctly, your results will be correct. The experience part comes in to play as far as having a feel for whether or not your results make sense. I have seen FEA results that seem to defy the laws of physics. On closer inspection it has always turned out that there was an error in the model.
GregLocock
Automotive
Rayleigh Ritz is the usual approach for complex systems.
However that system looks like even a moderately skilled analyst could work out the first mode exactly from first principles, assuming constant thickness and that the torsional strength of the bit joining the two is high. I'd guess it would take me a couple of hours.
The first mode is 'almost' a classical beam mode with a 50% reduction in root stiffness.
Another approach would be theory of receptances.
An approach for the second mode would be to work out the mass and moment of inertia of the central tongues, and apply that to the tip of a cantilever, and work out the mode from that. It isn't perfect, it should be close.
Incidentally a front view of each mode shape would be far more informative.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376
However that system looks like even a moderately skilled analyst could work out the first mode exactly from first principles, assuming constant thickness and that the torsional strength of the bit joining the two is high. I'd guess it would take me a couple of hours.
The first mode is 'almost' a classical beam mode with a 50% reduction in root stiffness.
Another approach would be theory of receptances.
An approach for the second mode would be to work out the mass and moment of inertia of the central tongues, and apply that to the tip of a cantilever, and work out the mode from that. It isn't perfect, it should be close.
Incidentally a front view of each mode shape would be far more informative.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376
spongebob007
Military
"I'd guess it would take me a couple of hours."
Which leads me back to my original question? Why? Looking at the image of the structure I figure I could build this FEA model and have many frequencies and mode shapes in less than five minutes. As engineers we often need to balance accuracy against budget and schedule. I would certainly question spending 5 minutes to do an FEA model and then 2 hours of hand calculations to verify it. On the flip side, I would question doing an FEA model of something I could easily figure out within five minutes using a hand calculation. In my mind at least, FEA is not for solving the easily solvable, but for coming up with approximate numerical solutions to highly complex problems that can't be solved easily (or at all) by other means.
In situations similar to this there are a couple of things I might do. Easiest thing you could do is perform a mesh refinement study. Run the model again and again with smaller and smaller elements. If your model is adequately meshed you will see little change in your results between successive iterations. In some cases I might say that I know my results should be between two extremes, so I run the extreme cases to come up with lower and upper bounds on my problem. My results should fall somewhere between the bounds. Another idea along these lines would be to come up with a very simplified and easily solvable model that will give you an idea if you are in the ballpark with your results.
Which leads me back to my original question? Why? Looking at the image of the structure I figure I could build this FEA model and have many frequencies and mode shapes in less than five minutes. As engineers we often need to balance accuracy against budget and schedule. I would certainly question spending 5 minutes to do an FEA model and then 2 hours of hand calculations to verify it. On the flip side, I would question doing an FEA model of something I could easily figure out within five minutes using a hand calculation. In my mind at least, FEA is not for solving the easily solvable, but for coming up with approximate numerical solutions to highly complex problems that can't be solved easily (or at all) by other means.
In situations similar to this there are a couple of things I might do. Easiest thing you could do is perform a mesh refinement study. Run the model again and again with smaller and smaller elements. If your model is adequately meshed you will see little change in your results between successive iterations. In some cases I might say that I know my results should be between two extremes, so I run the extreme cases to come up with lower and upper bounds on my problem. My results should fall somewhere between the bounds. Another idea along these lines would be to come up with a very simplified and easily solvable model that will give you an idea if you are in the ballpark with your results.
GregLocock
Automotive
Why would I spend 2 hours verifying a model that runs in 5 minutes? Because if that model was to be used as the basis for many models, or for a very expensive decision, then the 2 hours spent checking it is worthwhile.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376
GregLocock
Automotive
I forgot to add, "or safety critical".
If the consequences of failure are virtually zero then I wouldn't check my work either. Build and break is a very traditional form of design and is quick, informative and fun in the right arena. It also allows you to develop solutions in cases where the optimum is not obvious.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376
If the consequences of failure are virtually zero then I wouldn't check my work either. Build and break is a very traditional form of design and is quick, informative and fun in the right arena. It also allows you to develop solutions in cases where the optimum is not obvious.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376
amanuensis
Automotive
The shape of this beam is quite interesting.
We can see that the two resonance frequencies are close together - ((3415-3240)/3415)*100 is the relative difference in percent-
This fact can be explained by the theory. Indeed, resonance frequencies only increase with thickness of the beam. Width has no influence on the resonance frequency.
We can think this structure as two beams with the same length, the same thickness and almost the same boundary conditions.
The only difference is the width which has actually no influence.
So the shape of the two modes and the value of the resonance frequencies are quite coherent from a physical point of view.
We can see that the two resonance frequencies are close together - ((3415-3240)/3415)*100 is the relative difference in percent-
This fact can be explained by the theory. Indeed, resonance frequencies only increase with thickness of the beam. Width has no influence on the resonance frequency.
We can think this structure as two beams with the same length, the same thickness and almost the same boundary conditions.
The only difference is the width which has actually no influence.
So the shape of the two modes and the value of the resonance frequencies are quite coherent from a physical point of view.
electricpete
Electrical
I can believe that beam resonant frequency doesn't depend on width (depends on thickness,length, and material properties).
I can't follow the rest of your thought process.
=====================================
(2B)+(2B)' ?
I can't follow the rest of your thought process.
=====================================
(2B)+(2B)' ?
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