Natural Frequency Question? 1

For a simply supported beam, I can calculate the natural frequency of the composite section using the equation below. To properly use the equation, I assume that I should use the beam's weak axis & channel's strong axis moment of inertia since the vibration was observed laterally. Also, the weight would only be the steel without the suspended load below since it is not vibrating vertically. Is this correct?

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I am not familiar enough with that (particular) equation to say.....but the equations I've used (out of Roark's) for this sort of thing....I typically include all weights for each particular direction of vibration. (Just like if you did it via FEA....speaking of that....)

By the way, that equation likely just gives the [first] fundamental frequency. If you find that is lower than the operating frequency of the fans, you may want to do a dynamic/frequency analysis via FEA software to capture the higher modes of the system.

I am still puzzled how the vibration from the cooling tower fans could be transmitted through the pipes and up the hangers in order to excite the beam vibration.

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You'd be amazed how vibration can get around [in] a structure.

This equation does the fundamental mode or first mode. The second mode is 2^2 or 4 times that and the third mode is 3^2 or 9 times and so on.

So, you would include the pipe weight w/water in calculating the frequency in both directions. Correct?

The equation is based on a uniform load. It is from AISC Design Guide 11.

The second mode is 2^2 or 4 times that and the third mode is 3^2 or 9 times and so on.

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I don't think that is true. (Especially for a multi-degree of freedom system. Which this unquestionably is.)

So, you would include the pipe weight w/water in calculating the frequency in both directions. Correct?

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Correct.

That equation will give you the natural frequency of the beam under UDL, but you have several point loads. Now if the point loads are small relative to the weight of the beam, it might give you a decent approximation to validate computer results. I don't think you can assume the pipes aren't vibrating at all, there certainly has to be some participation if there is load in the hangers. In your position, I would put together a simple beam model in software, include enough nodes that you get a deflected shape that matches the actual shape being observed, and vary masses until you find a configuration that would be excited by your forcing frequency. Then you can decide to either conservatively include the masses (lower Fn), ignore the masses (higher Fn), and then figure what fix keeps you away from the forcing frequency.

Good point canwesteng. Those equations are built around specific loadings.

Aren't your hangers the most flexible part of the system?
A stiffer beam won't help if the pipe is allowed to 'float'.
Sounds like the hangers need to be examined.
Just be careful, the forces will not change.
So making it stiffer will increase the stress someplace.

= = = = = = = = = = = = = = = = = = = =
P.E. Metallurgy, consulting work welcomed

SteveGregory said:

...When you push on the top flange laterally near the middle, it starts vibrating laterally.

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What is “it”? The beam, pipe, or both? Can you tell if they’re moving laterally together?

Equation 3-1 should give a good approximation of the natural frequency for vertical motion. Include the beam, pipe, and fluid contents in plf or klf. This equation isn't sensitive to whether there are several point loads vs w. Note that this equation is used for girders with point loads. Even with two point loads, it can be used without adjustment.

If the pipe is hanging 45 in. below the beam, then it’s hard to see how stiffening the beam’s EIy would help. There is probably a mode that looks like a pendulum with the pipe moving transverse to the beam. To compute the lateral natural frequency of the beam, I would only include the beam weight because I don't think the beam can drag the pipe back and forth as it vibrates -- at least not significantly.

Similarly, if the pipe tries to move transversely, I don't think it can get much help from the beam. Does this pipe have any lateral bracing along the 35 ft span? If not, then that seems like a long span for a 6 in. to 8 in. pipe. Is there any chance something is repeatedly bending it horizontally over this long span?

You could try this: Build a model of this beam. Add the hangers below. Add the pipe plus weight of its contents. See what the model comes up with for the first few natural frequencies and mode shapes. If you have RISA-3D, SAP2000, or most other programs, this should be a quick exercise. You can use Equation 3-1 to verify some of the frequencies to make sure your analysis is correct.

All that typed, I’m wondering if something else is going on.

Do you know for sure that vibration is the problem?
Where the pipe leaks, is it just leaking at a gasket, are nuts backing off, is it cracking? A more elastic gasket might help.
If you have adequate load capacity, you can adjust frequencies of the pipe and beam somewhat by adding weight to them.
If you're getting higher-mode vibration, you can possibly shift supports around and cut it down.
Would it be feasible to add an intermediate brace or guy wires?

Thanks for the comments. The building is only 4 years old!

The center-line of the beam is about 9" to 9.5" from the face of the cooling tower. The pipe is 45" clear and directly below the beam. It runs parallel with the beam and it is not braced. The end of the pipe turns 90 degrees up vertical for a short distance below the beam and then turns 90 degrees from the beam horizontal into the side of the cooling tower. The leak is at a flange where the water goes into the cooling tower. The picture shows the pipe, 2 hangers and the beam above.

If it is an 8" pipe with insulation and water, I am estimating it's weight at 52 plf. The beam weighs 65 plf. The equation I used was derived from Roark. Roark has a constant Kn that gives the modes 1-5. Mode frequencies for n = 2 through 5 can be found by multiplying the fundamental frequency by n^2.

If you estimate the natural frequencies, what would compare them to? Say you get 2 Hz. Is that OK?

Is there any chance that liquid movement is such that lateral forces are repeatedly being induced? With the long unbraced length, that seems like the best candidate to me.

I am waiting on the fan rpm values to see if they hit near any of the modes. I got 2.2 hz for the bare beam weak axis fundamental frequency and 3.8 hz with adding a C12x25 cap channel. Those numbers are based on using the pipe weight for the lateral vibration. Without the pipe weight, those numbers are 2.95 Hz and 4.78 Hz. Your earlier post suggested not using the pipe weight. WARose thought I should use it.

I could see the vibration from the fans being transmitted through the pipe enough to excite the beam into a lateral oscillation. I could do that just standing next to it and pushing on it with my hands. It is a long unbraced beam and no way to brace it.

The opinion in your first post made sense "I would only include the beam weight because I don't think the beam can drag the pipe back and forth as it vibrates -- at least not significantly." You can see the difference it makes for the beam.

It may not make enough difference for the pipe leak and I already told our client that I am not a piping expert and this may not fix your pipe problem. I don't know if there is some type of flexible connection that would be appropriate. Maybe that big guy that used to do the Flexseal commercials could fix it...

SteveGregory said:

...The second mode is 2^2 or 4 times that and the third mode is 3^2 or 9 times and so on. ...

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Backing up a ways in the thread, this is correct. I'm looking right at it in a vibrations textbook.

SteveGregory said:

...I am waiting on the fan rpm values to see if they hit near any of the modes. ...

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I expect the fan operating frequency to be much higher than your 3 - 5 Hz estimates, so I guess you're considering the third mode, which would be between 27 Hz and 45 Hz. That's quite a range, so it's tough to ensure that the operating frequency doesn't match this natural frequency. I think I'd build a little model like I described before. That might allow you to pin down a decent estimate that can be used for frequency tuning.

I'm pretty skeptical about higher mode vibrations causing this problem because higher frequency vibrations will have such a small displacement amplitude.

I'm imagining something like a pendulum mode with larger amplitude, lower frequency vibrations wrenching the connection back and forth. Maybe wind or fluid flow could cause this. I'm not sure.

Another idea: The vertical support is statically indeterminate because the pipe can go a long ways toward supporting itself over a 35 ft span. Maybe there is a significant reaction at the pipe flange connection that is leaking.

Interesting problem! I know "interesting" might not be how you'd describe it, being in the soup. LOL

If you push laterally on the beam flange, you see the vibration, right? Have you considered a torsional mode of vibration? Just curious. Strong axis and weak axis are pretty easy to figure. Torsional vibration is definitely more tricky.

I always push the mechanical engineers to use flex connections whenever practical on exterior stuff like this. Piping systems grow and shrink and have various dynamic loads and induced vibrations. A lot of times leaks are caused by an overly restrained pipe "structure" that causes nozzle connections to behave as anchor points in the system. Flex connections help both with relieving those induced reactions at the nozzle and with damping out equipment vibrations. A pipe stress engineer could take a look at the system as a whole and identify or rule out any restraint problems in the layout.

To tackle the beam issue, I would try to have someone measure the beam vibration amplitude in the field rather than trying to analytically determine it. If it is determined that the beam is being excited by the fan vibrations and oscillating vertically enough to yank on the pipe hanger, one possible solution is to remove or relocate that hanger closest to the nozzle. It's such a short span that relocating would probably not help much. If the pipe stress engineer says you cannot just remove that hanger altogether, maybe a spring can can be incorporated into the hanger to absorb the vibrations in the rod. Again that would be likely need to be a pipe stress engineer's design.

Here are some excerpts from MSS-SP-58 for reference. The table is for nominal max support spans of horizontal pipe.

Thanks Bones for your posts. I think I will suggest a Piping Engineer. I would guess the supports are about 10 feet apart now. Spring Isolaters in the hangers might help the beam.

Would you believe the building is only 4 years old looking at the photo?

Yea looks like there is some galvanic corrosion going on.

271828 said:

Interesting problem! I know "interesting" might not be how you'd describe it, being in the soup. LOL

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I find these type of problems really fun and interesting as well. I used to work in the same firm as a mechanical engineer who was basically a living legend in the piping and vibration world. He made it fun and was truly a pipe whisperer... he would walk a plant and literally touch the tip of a ball point pen to a spot on a pipe run, feeling the vibration through the pen for a few seconds, then say something like "hmm, we are gonna need to put a such-and-such spring can here, and shift that sliding support over about 18 inches." He would design the whole project in the field in his head without running a single calc, just pure intuition and feel. Unfortunately I never got to pick his brain since I was in the structural department, but I learned a lot just following him around the plant a couple times, taking down his declarations on a clipboard.

bones206 said:

I find these type of problems really fun and interesting as well. I used to work in the same firm as a mechanical engineer who was basically a living legend in the piping and vibration world. He made it fun and was truly a pipe whisperer... he would walk a plant and literally touch the tip of a ball point pen to a spot on a pipe run, feeling the vibration through the pen for a few seconds, then say something like "hmm, we are gonna need to put a such-and-such spring can here, and shift that sliding support over about 18 inches." He would design the whole project in the field in his head without running a single calc, just pure intuition and feel. Unfortunately I never got to pick his brain since I was in the structural department, but I learned a lot just following him around the plant a couple times, taking down his declarations on a clipboard.

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Great story. Some people are awesome!