trick question on beams
trick question on beams
(OP)
If I take a beam and support it two different ways:
free/free
or
clamped/clamped
Which one will have the higher first resonant frequency?
Which one will have the higher second resonant frequency?
Be careful. It's not a trick in the wording but it's a tricky question.
free/free
or
clamped/clamped
Which one will have the higher first resonant frequency?
Which one will have the higher second resonant frequency?
Be careful. It's not a trick in the wording but it's a tricky question.
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RE: trick question on beams
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Eng-tips forums: The best place on the web for engineering discussions.
RE: trick question on beams
I guess I need to reset my clocks, could it be April first is here already.
RE: trick question on beams
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RE: trick question on beams
I have been puzzled by this same problem before.
I would say free/free has higher frequency than clamped. Looking at mode shapes will reveal the answer. This goes against common sense as, in general, adding constraints increases natural frequency.
In clamped case first bending mode shape is full lenght, L.
In free case first bending mode shape crosses at points 0.25 and 0.75 of the total lenght, L. As a result, shorter L will increase frequency.
If I remember correct, clamped one has higher second frequency. Guessing here. I never convinced myself thoroughly by setting up the stiffness matrix and doing the whole thing the long way.
suviuuno
RE: trick question on beams
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: trick question on beams
RE: trick question on beams
Are you sure this is always the case? I agree, generally this is the case, but I surprised myself with this "odd ball" a year ago. Cantilever case is different from beam supported at both ends -> therefore not relevant.
suviuuno
RE: trick question on beams
RE: trick question on beams
Now I have to agree with Greg there are size 0-frequency modes for the free/free case which don't exist for the clamped/clamped. Those would be linear translation at constant velocity along the three axes or rotation at constant velocity about the three axes.
I understand the experimment with gradual increasing stiffness of the springs. I have to ponder a little bit about what light it sheds on this thread.
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RE: trick question on beams
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: trick question on beams
Instead of the cantilever beam, I could take my fixed/fixed beam and build some flexibility into the end supports. That flexibility would allow some displacement from 0 position (with a resulting spring force in the restoring direction) and some angle from horizontal (with a resulting moment in the restoring direction).
Now imaigine I can vary that flexibility. If I make it infinitely stiff I have my original fixed/fixed. If I make it infinitely flexible, I have my original free/free. I can vary it smoothly anywhere in between.
Now start with the free/free and increase flexibility just a little bit. Now I have the rigid body mode with lowest frequency very close to 0... in fact as low as I want (can achieve any value by varying that stiffness). Of note, the frequency is much lower than first resonance of the original fixed/fixed system. As I gradually increase the stiffness, that original resonance frequency that started at 0 (free/free beam) gradually moves all the way up to the first resonance of the fixed/fixed.
So in this case gradually adding the constraint(by increasing stiffness) increased the resonant frequency from 0 (free/free) to the fixed/fixed first resonant frequency.
It seems to answer the contradiction that seemed apparent when initially looking at the problem. Thx Greg.
The purpose of the time-domain / impact discussion was to prove that 0hz was in fact a mode of the free/free system? I agree with that. To me a mode is the motion that the body will keep (neglecting damping) when the forcing function is removed. It is clear the free/free beam can continue to move linearly or rotationally at constant velocity (0hz.)
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