Natural frequency and mode shapes of continuos systems

[mina1363]

Hi,
Is there anyone who has a good understanding of vibration of continuos systems. I need to calculate the natural frequencies of a 1D bar governed by the wave equations: d^u/dt^2 = sqrt(E/rho)*d^u/dx^2 . I have an idea of how to obtain the natural frequencies using fixed or free boundary conditions. I am not sure what I need to do in cases where there is a force at the end of the bar (say Edu/dx = F/area)
I am really desperate and would appreciate any help!!

 

[GregLocock]

"Is there anyone who has a good understanding of vibration of continuos systems."

I dare say there are a few.

An axial force will have no effect on the 1D wave equation, because E and rho are unaffected. If E and rho were affected (ie you have a gas) then just use the new values.

Cheers

Greg Locock


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[mina1363]

Thanks Greg,
The reason I am confused is that I have seem cases where there was a mass at the end of a 1D bar and that has been taken into account when working out the natural frequencies. For instance case 4 on page 11 of the attached file. Another case I have come across is where you have a fluid piston attached at the end of a spring, and something similar have been done to work out the couple frequency of the spring-fluid piston system.
I really need to understand why they have done this.

 

[GregLocock]

Well, those cases aren't static axial forces, so your initial post was misleading in the extreme.

Yes, adding a mass at the end of a bar will affect both the mode shape and the frequency. For simple cases you might be able to solve it analytically, or else you can model it. An old method for looking at complex systems is called the theory of receptances, I'm not sure if I'd actually recommend it never having used it since uni.

Rayleigh Ritz is another approach.

Cheers

Greg Locock


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[electricpete]

Is there anyone who has a good understanding of vibration of continuos systems .

Greg certainly knows a few things about vibration including continuous systems. More than anyone else I have ever met. We are lucky to have him on the forum.

Side note not important to the op after it was clarified: A static axial force (no mass) could tend to increase natural frequency of a beam slightly.

My simplistic view of it: In addition to PE from bending there is another PE from tension/compression similar to that of a string as shown.
thread384-312306

But it tends to be a small effect as illustrated for the particular case of pinned/pinned beam with assumed half-sin modeshape:
PEbending/ PEtension= (pi^2*E*Id) / (T*L^2)

Using circular/rod geometry and steel material properties and relatively high loading 30ksi it simplifies to:
PEtension/PEbending ~ 0.005*pi* (L/D)^2

It is negligible for most purposes, but becomes larger as L/D increases. This matches our intuition that increasing L/D toward a very very long thin rod will make it act more like a stretched string and less like a beam.


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(2B)+(2B)' ?

 

[electricpete]

Ha. "Minor" typographical correction:

But it tends to be a small effect as illustrated for the particular case of pinned/pinned beam with assumed half-sin modeshape:
PEbending/ PEtension= (pi^2*E*Id) / (T*L^2)

Using circular/rod geometry and steel material properties and relatively high loading 30ksi it simplifies to:
PEbending/ PEtension ~ 0.005*pi* (L/D)^2

It is negligible for most purposes, but becomes larger as L/D increases. This matches our intuition that increasing L/D toward a very very long thin rod will make it act more like a stretched string and less like a beam.

Should have been

typo[/s] CORRECTED]
But it tends to be a small effect as illustrated for the particular case of pinned/pinned beam with assumed half-sin modeshape:
PEbending/ PEtension PEtension/ PEbending = (pi^2*E*Id) / (T*L^2)

Using circular/rod geometry and steel material properties and relatively high loading 30ksi it simplifies to:
PEbending/ PEtension PEtension/ PEbending~ 0.005*pi* (L/D)^2

It is negligible for most purposes, but becomes larger as L/D increases. This matches our intuition that increasing L/D toward a very very long thin rod will make it act more like a stretched string and less like a beam.

=====================================
(2B)+(2B)' ?

 

[mina1363]

Thank you both. I hope I have not insulted anyone by asking if anyone has a good understanding of continuos system, I was not very familiar with the forum.

With regards to my question, from your replies I am concluding that an added mass does effect the natural frequencies and mode shapes but an external force only does so insignificantly.

I have written a code to numerically calculate the displacements of a 1D bar fixed at one end and at the other end I have set the stress term Edu/dx to an arbitrary sinusoidal force over area. I was trying to make sense of the my results by looking at the natural frequencies and mode shapes of the continuos system. To do so, can I follow a similar procedure to the one shown in the example in the file I attached previously since in a sense the force I am applying is being applied internally as oppose to an external force applied at the boundary using a separate term. Please let me know if I am not being clear.

Thanks again.

 

[electricpete]

My input was addressing effect static/dc axial force upon lateral natural frequency of a beam, which doesn't apply to your question.

I'm not understanding your question:

at the other end I have set the stress term Edu/dx to an arbitrary sinusoidal force over area

What is this supposed to represent physically?

=====================================
(2B)+(2B)' ?

 

[mina1363]

I am trying to excite the bar using some sinusoidal function at that free boundary. I am not quite sure if what I have done is correct.

 

[electricpete]

So you are looking at simple 1-d axial beam, free at one end?

Rao's Mechanical Vibrations 3'rd edition has analysis in section 8.3 resulting in analytical solutions in Table 8.7. Also, as an approximation, the first mode frequency can be calculated by lumping an equivalent mass at the end of a massless bar (with same E)… where the equivalent mass is approx M/3


=====================================
(2B)+(2B)' ?

 

[mina1363]

Thank you. Unfortunately, I don't have access to this book.

I am not sure what the book suggests but I need to use the methods described in the attachment to arrive at the natural frequencies and mode shapes of a 1D continuos bar governed by the wave equation. The point I am not sure about is whether the way I have excited the bar at the boundary is makes sense and if this excitation should be included when working out the natural f's and mode shapes.

 

[electricpete]

It is starting to sound like homework, so I'm going to limit my comments.

From a quick skim, your attachment seems to contain means for analytical solution of that problem, but not for numerical/finite element solution. You'll need to read your attachment closer for analytical solution or find another reference for numerical solution.


=====================================
(2B)+(2B)' ?

 

[mina1363]

I am solving the equations numerically but I want to find the mode shapes and natural frequencies analytically (according to the paper attached) to validate my numerical results. Like I said, the thing I am not sure about is the way I have excited the bar. I am not that young to do homework and I do not need anyone to do the work for me. I understand the analytical and numerical methods. My question is about that particular boundary. I would appreciate it if you commented on the boundary term.

 

[GregLocock]

The method of excitation of the bar has basically no effect on frequency or mode shape (OK if you try and excite a mode at a node it won't work, but that is an odd case and I suspect you'd still get a good analytical solution).

So I don't really understand your problem.

The BC at the fixed end is x=0 xdot=0 xdotdot=0 and at the other end, with the mass, the mass is free to move axially.

I'd excite the system using a force F applied to the mass, and then draw a FBD for the mass to derive the force applied to the bar (basically F-m*xdotdot).

A book that has solutions for all manner of systems like this is

http://www.amazon.com/Formulas-Natural-Frequency-Mode-Shape/dp/1575241846

and is pretty much essential for calibrating homegrown dynamic models, unless you have more time than money.



Cheers

Greg Locock


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