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# random vibration analysis2

## random vibration analysis

(OP)

I am looking for empirical/handy formulae or equations that can be used to calculate the resulting displacement and forces of simple strucures (plates, beams, etc.) that are subjected to random vibration. The input random vibration load is presented in the form of power-spectral density (G**2/Hz) versus frequency(hz). Amplification factor/transmissibility factor need to be considered.

### RE: random vibration analysis

Basically, the square root of the sum of the squares at varying frequencies is taken. See details below.

This information comes from Chapter 13, Random Vibrations, of "Theory of Vibration with Applications", 3rd Ed., by Thomson.

Define S(f) as the power spectral density function
Define H(f) as the frequency response function for the structure. Each degree of freedom will have its own H(f), and it can be determined by shaker testing at varying frequncies.

Next, pick several discrete frequencies, equally spaced over the spectrum range. Calculate the mean square response at each frequency. It is defined as
a^2 = S(f)*¦H(f)¦^2*df, where df is the frequency increment. Sum up all of the a^2 values, and take the square root. This will give the acceleration response of the structure. For a multiple degree of freedom system, each DOF will have its own results.

It is possible to calculate the probability that a given excitation will be exceeded. See Thomson for details.

### RE: random vibration analysis

Based on my understanding of your question, check out text book by Blevins, Roark and the Miles equation.

### RE: random vibration analysis

http://www.vibrationdata.com/tutorials.htm

Specifically, refer to the vibration response spectrum tutorial, vrs.pdf.  Also, refer to psdinteg.pdf.

Please let me know if you have any questions after reading these tutorials.

Sincerely, Tom Irvine
Email: tomirvine@aol.com

### RE: random vibration analysis

Look out for Mile's equation.  Terry Scharton of JPL has written several papers on the subject in conjunction with force-limited testing.  Mile's gives the correct ratio of response acceleration-to-base acceleration, but overlooks the fact that a valley in the base acceleration spectra will always occur at the resonance frequency of the mounted system, such that the effective "Q" at this frequency is more normally one or two.  I've heard Terry put it this way - Miles equation only works in textbooks and on shakers.  Remember that the natural frequency of a mounted part is never a natural frequency of the assembly.  The Mile's concept of a finite spring-mass system on an infinite base is easy to understand, mathematically correct, but misleading.  If you do take the math one step further and apply a finite force to an infinite mass on an infetismal spring, with a finite spring-mass (mass m) undamped system attached.  The result is that at resonance the response of the finite mass to the finite force follows - guess what - F=ma. The response of the infinite mass m is not F/M, but zero.  Ponder them apples.

### RE: random vibration analysis

I am also looking for the same thing as "rama", including the Miles equation but I can't find anything! Those books and papers are impossible to find...
Have you already have anything on the subject?

### RE: random vibration analysis

The Blevins book was available through ASNT until recently (so it probably still is).

When lloking for technical references, I've found Amazon.com to be at times extremely useful

### RE: random vibration analysis

RMS Response = sqrt(pi/2 * fn/zeta * PSD), Schiff D. - "Dynamic Analysis and Failure Modes of Simple Structures"

### RE: random vibration analysis

hi rama..
i c that ur working on random vibration analysis of beams...
how do u apply the random loads on the beam...i am using MSC/NSATRAN v70.7 should i use a large mass ..and what should be th order of the mass..
do we get the out put as PSD ,Mean and mean square response for the load applied ....what other data is equired to specified for th analysis
gautam

### RE: random vibration analysis

Tom wrote:

By the way... there is A SMALL FEE!

I have found your advice on the forum to be very helpful, and I'm sure I speak for all in saying that you are a valued contributor.  But in the future it might be helpful if you could warn us of the fee so that the 99% of us who don't routinely pay for information on the internet will not waste time following your links.

### RE: random vibration analysis

I think a simple formula is out of the question for a multi-modal continuous structure (except for a few very specific cases for which there are analytical solutions). The problem also depends on the frequencies you are interested in. At high frequencies where there are many modes in the structure, and asymptotic statistical solution may be possible (I don't have the reference for these at the moment but if this is applicable then please contact me). At lower frequencies, (as I think has already been pointed out) you need to have knowledge of the system's frequency response function. As you do not have any phase information in the input signal (it its a power spectum) you would only require the amplitude of the FRF. You can approximate the FRF (Ewins, "Modal Testing: Theory and Practice) if you have knowledge of the mode shapes, natural frequencies and dampings of the structure (these can be found in Blevins, "Formulas for Natural Frequency and Modeshape, a VERY difficult book to get hold of unless you have access to a university library).

You do not say if your random loading is distributed or applied at a point. I would be tempted to use an FE model if it is the former.

Regards

Michael

### RE: random vibration analysis

If you can easily find the eigenmodes of the system
then you can use classical modal analysis in conjunction
with random vibration analysis. The response power spectra
can be related to the input excitation cross-spectra by
way of the system spatial modes and transfer function.
Check out a paper by Dowell(1995) on Asymptotic Modal
Analysis for some details...the paper will give most
of the important relations and will give an easily calculated result in the asymptotic limit.

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