Natural Frequency of a Simply Supported Beam

[_MechEng95]

I have been trying to find some formula to follow to calculate the frequency of a simply supported beam for some time and there are so many suggestions I am not sure what to follow.

I have a beam with 2 point loads, equidistant from the beam supports and, a uniform distributed load across the length of the beam.

I looked at Roark's Formulas for Stress and Strain, that gave me a suggestion of

https://gyazo.com/ca0ea05de9c4d33e8bde0f0fa3634116

However, I searched the forum and found suggestions for using:
(n[sup]2[/sup]π/2L[sup]2[/sup])(√EI/m)

Any clarification on the matter would be much appreciated. I have to compare hand calculations to results found in ANSYS. Neither of these formulas give me similar results to ANSYS but that is most likely a fault on my end with the programme.

Any help is appreciated!

 

[GregLocock]

The two equations you have found are dimesnionally incompatible. Obviously the dimension should be T-1.

Why do you think Roark is wrong?

Cheers

Greg Locock


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[JStephen]

A pure "load" wouldn't affect the natural frequency of the beam.
If that load consists of a mass resting on the beam, then the mass will affect the natural frequency.
If that load is applied by some connecting member, then the stiffness of that connecting member will also affect the natural frequency.
That's not my normal line of work, but seems like I remember there being a Rayleigh/Ritz method for approximating the combined effects of various masses like that. I'd have to go hunt it up in my vibrations textbook.

 

[_MechEng95]

The only dimension should be T-1? I don't follow.

I don't think roark is wrong, there are just a lot of opinions on which formula to use.

A MIT spreadsheet for beam frequency I found suggested I use the following equation:

https://gyazo.com/dda4d2cc79a8376b17f3d00490483006

where k = npi

So roark should be used for this issue?

 

[Denial]

The three formulae you have located are in fact the same (for the fundamental mode).
PROVIDED:
» You note the distinction between frequency in cycles per second and frequency in radians per second.
» You note the distinction between mass and mass per unit length.
» You use a "consistent" set of units and stick rigidly to it.
» You note the distinction between mass and force/weight.
[sub][ ] [ ][/sub] (If the two point LOADS have significant MASS associated with them then the formulae — all three of them — no longer apply.)

 

[_MechEng95]

Ok thank you, so for the use of roarks formula, the frequency is given in Hz right?

 

[ChadV]

The start of each table in Roark's includes a description of the notation. If you look just above the page you snapshotted the formula from, you'll see that f is in cycles per second (Hz).

As others have noted, the formulas you've found are the same, with differences only in the notation and units. Make sure you're very careful about consistent units, distinction between mass and force, etc... I can speak from experience of having torn my hair out for an hour trying to match a natural frequency from a software last week before realizing that N, mm, and MPa weren't consistent with kg, meaning my hand-calc was off by a factor of sqrt(1000).

The best advice I can give is to write the formula out by hand, including units, to make sure your final answer is in the units you think it is.

 

[ThomasH]

Hi

On a fundamental level the natural frequency for a beam is a function of the beams mass and its stiffness. The loading is not a factor. Self-weight may cause load effects due to gravity but that will not effect the natural frequency.

For a simply supported beam with uniform stiffness and mass the natural frequency for the first mode is: pi/2 * sqrt ((E*I) / (m*L^4)), use a consistent set of units ant the result will be in Hz. In the SI system load is measured in terms of Newton while mass is in terms of kg.

If you Google it there are sources that have a confusing use of mass and load. My approach is that load has direction while mass doesn't. Also, in space you are wheight-less, not mass-less.

Hope it helps

Thomas

 

[JStephen]

I was looking through the vibrations textbook, and find several different approaches depending on the details.
If the "loads" are in fact lumped masses at those points, and if the beam mass can be neglected compared to those masses, then you have a 2-degree of freedom system which can be analyzed as such.
The Rayleigh method can be used to treat a beam as a series of lumped masses, and I assume adjusting two of those lumped masses can account for the extra loads in this case.
If the loads are fairly negligible compared to the beam weight, just use the uniform beam equation and add in the extra loads.

 

[WARose]

[blue]MechEng95[/blue]

I have to compare hand calculations to results found in ANSYS. Neither of these formulas give me similar results to ANSYS....

That isn't surprising. Assuming you have the beam meshed well, you will be getting results for multiple modes in multiple directions. (That is also assuming your "beam" has different moment of inertias about it's axis. I.e. a "strong"/"weak" axis.) You might get a majority of mass participating in a lower mode......but to get to (say) 90%, it may take much higher modes. (Depending on the stiffness.)

Comparing it to formulas in Roark's are misleading because you have no idea what that number represents (in terms of mass participating).

 

[GregLocock]

Looks like I was wrong on the difference between the two equations. The answer has to have dimensions of T-1 as it is a frequency.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[ThomasH]

Hi

Regarding the difference between hand calculation and ANSYS. I often work with vibration analysis in FEM-software (Nastran). For situations when I can compare hand calculation with FEM the results for natural frequencies are usually more or less spot on.

Also, in my experience, the most common misstake with this type of analysis is the use of inconsistent units. Yoyu can find examples of consistent unit sets here:

https://femci.gsfc.nasa.gov/units/index.html

Good Luck

Thomas