trick question on beams 1

[electricpete]

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.

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[electricpete]

I should point out that for purposes of this question, the beam follows the Euler Bernoulli model.

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[hacksaw]

I guess I need to reset my clocks, could it be April first is here already.

 

[electricpete]

Not exactly. What's your answer?

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[suviuuno]

Hi,

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

 

[GregLocock]

Adding constraints does increase the frequencies of the system. The free-free beam has six 0 Hz modes. If you lime, consider a cantilever with an adjustable spring at the attachment to the wall. Dial in a very soft rate, you'll get rigid body behaviour for 6 modes, and then a flexural mode. Now start to stiffen the spring, the rigd nody modes will increase in frequency until at some point the first mode will be the flexural mode instead of rigid body.





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[hacksaw]

They are identical...

 

[suviuuno]

Quote by Greg: "Adding constraints does increase the frequencies of the system."
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

 

[EnglishMuffin]

According to my copy of Blevins, Hacksaw is correct (ie identical natural frequencies), but the mode shapes appear to be somewhat different.

 

[electricpete]

Yes, Englishmuffin and hacksaw are correct that the 1st, 2nd, 4rd etc non-zero mode frequencies are the same for the free/free and clamped/clamped. That was the part I was surprised about and prompted me to pose the question. I thought like suvionno that adding contstraints should increase the natural frequency.

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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[GregLocock]

The reason that works is that you need to step back from a frequency domain perspective, and think about the time domain. What is the response of the system to an impact? The modes that we see are the way that the system distributes that energy. One reason I think this way is the theory of receptances, which is a rather theoretical way of looking at what happens when two MDOF systems are joined. I've not found much practical use for it but it certainly inspires some of my more hand-waving arguments.



Cheers

Greg Locock

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[electricpete]

I think I have figured out how Greg's cantilever with variable spring relates to the question about the surprising result that the first non-zero resonant frequency of the clamped/clamped is the same as the free/free.

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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