Finding the natural frequency of a pipe with two 90 degree bends

[shreddinglicks]

I'm trying to find the natural frequency of a pipe that is roughly z shaped (two 90 degree bends). Is it possible to break this pipe up into three straight pieces and model each piece as a beam experiencing transverse vibration?

For example, modeling the end segments as cantilevers and the middle segment as a pin-pin. Upon getting the natural frequencies of each segment, summing them to get the natural frequency of the entire pipe?

 

[Ivan Nikiforov]

Hello! It seems to me that first of all it is necessary to determine the plane in which fluctuations occur. In addition, it is necessary to determine the attachment point (support) relative to which fluctuations occur. For example, if a pipe is fixed at the end, then it will have one natural oscillation frequency. if the pipe is fixed to the center of a long section, then it will have a different natural oscillation frequency. That is, the fulcrum is very important. If your pipe has a fulcrum in the center of a long section, then the calculation can be performed based on the following assumption. Short sections of pipe can be equivalently represented as weights of equal mass mounted at the ends of a long section.

 

[shreddinglicks]

But what about a pipe of this shape being broken up into three pieces.

Two cantilevers and one pin-pin. Is it possible to sum the individual natural frequencies to get the frequency of the entire pipe? Does plane matter if the natural frequency is scalar?

 

[Ivan Nikiforov]

It seems to me that the natural frequencies simply cannot be added together, since this is a single system and each of its elements affects the conjugate element. This system needs to be brought to a standard model, for which there is a mathematical apparatus. The plane of oscillation matters. Does the system oscillate parallel or perpendicular to the screen plane?

 

[Ivan Nikiforov]

It seems to me that your system can be brought to such a model. Regardless of the plane in which the oscillations occur, the frequency of natural oscillations is determined by the stiffness of the short sections and the mass of the long (central) section. The central section does not deform during vibrations, so it represents only a mass.

 

[jrmichler]

For example, modeling the end segments as cantilevers and the middle segment as a pin-pin. Upon getting the natural frequencies of each segment, summing them to get the natural frequency of the entire pipe?

No. This never works. Starting with some simple assumptions: You have a pipe with normal pipe elbows, the elbows are as stiff as the pipe, the pipe is in a single plane, and the ends are truly fixed. In that case, the model would not have any concentrated masses because the entire pipe would bend during vibration. The middle section has bending stiffness, and the stiffness of the elbows makes it flex with the end pieces. The calculation is then the calculation for straight length of pipe that is the same length as the total length of the three pieces.
Real world factors will cause errors. Elbows have more mass than an equal length of pipe, and different bending stiffness. Fixed ends are never truly fixed. Vibration modes out of the plane of the pipe flex the center piece in torsion, where the effective stiffness will be larger, so the first natural frequency will be higher than the first natural frequency in the plane of the pipe.

Any calculation will give an answer with a high probability of significant error, but will give you a rough idea. If you need better than that, you need to measure the natural frequencies.

 

[DaveE]

Is it possible to break this pipe up into three straight pieces and model each piece as a beam experiencing transverse vibration?

Of course that analysis is possible. But it won't give you the correct answers.

 

[Baluncore]

Break the structure into three sections. Study each as a separate module.

The two cantilevers will have different natural frequencies in vertical movement. Each will have a restoring force, and a virtual end mass, that can be modelled as a pendulum.

The mid-section will have only vertical, that is axial movement, so with a higher longitudinal speed of sound, will appear as a lump of pendulum mass, that is attached to the two cantilevers.

For an estimate of the vertical mid-section, oscillating in the vertical direction, combine the restoring force of the two cantilever tubes, and combine the mass of all three. That will make a single pendulum, with one natural frequency. That mode of oscillation will have a frequency, that is below the mean natural frequency, of the two independent cantilevers.

 

[DaveE]

The mid-section will have only vertical, that is axial movement

Isn't there also a "twisting mode", rotation about the longitudinal axis of the whole structure with radial motion of the center section? What about a "shear wave" sort of motion where the center section is translated horizontally? It seems to me that whatever direction you hit the center section with a hammer, there will be a natural oscillatory mode excited.

I guess I'm confused about "the natural frequency". Frequency of which mode? Is it just the lowest frequency that counts?

 

[Baluncore]

Isn't there also a "twisting mode", rotation about the longitudinal axis of the whole structure with radial motion of the center section? What about a "shear wave" sort of motion where the center section is translated horizontally?

Yes, there are many modes that can be excited. Since the end sections can oscillate as catenary pendulums, the complexity will come from the ways in which the mid-section might couple those two separate resonators. I was looking for a mode that could be computed accurately, and that could make a high Q, test case.

There are a number of modes where a harmonic relationship, (ratio of integers), between the two catenary pendulums might be excited. Those could be coupled through the mid-section, also as a harmonic resonator. That could be a particularly difficult model to analyse, so might come as quite a surprise, especially with a node somewhere on the mid-section.

The frequency of oscillation will also depend on the unspecified mass of the pipe contents.

When I first read the thread title, I imagined an organ pipe, with the complexity of elbows causing partial internal reflections.

 

[shreddinglicks]

View attachment 356613
It seems to me that your system can be brought to such a model. Regardless of the plane in which the oscillations occur, the frequency of natural oscillations is determined by the stiffness of the short sections and the mass of the long (central) section. The central section does not deform during vibrations, so it represents only a mass.

Interesting. I have FEA results for this problem. I wanted to see if there was a way to get the solution using a pencil and paper.

I'll try what you have here and see what I get.

 

[shreddinglicks]

No. This never works. Starting with some simple assumptions: You have a pipe with normal pipe elbows, the elbows are as stiff as the pipe, the pipe is in a single plane, and the ends are truly fixed. In that case, the model would not have any concentrated masses because the entire pipe would bend during vibration. The middle section has bending stiffness, and the stiffness of the elbows makes it flex with the end pieces. The calculation is then the calculation for straight length of pipe that is the same length as the total length of the three pieces.
Real world factors will cause errors. Elbows have more mass than an equal length of pipe, and different bending stiffness. Fixed ends are never truly fixed. Vibration modes out of the plane of the pipe flex the center piece in torsion, where the effective stiffness will be larger, so the first natural frequency will be higher than the first natural frequency in the plane of the pipe.

Any calculation will give an answer with a high probability of significant error, but will give you a rough idea. If you need better than that, you need to measure the natural frequencies.

I'll try that approach.

 

[shreddinglicks]

Of course that analysis is possible. But it won't give you the correct answers.

I have FEA results for this. I'm trying to see if this problem can be solved by hand with reasonable accuracy.

 

[shreddinglicks]

Thanks for the responses. I'll respond back with my attempts.