Random Vibration Max Displacement Calculation

[kaiserman]

I have a question in regards to random vibration profiles. What is the equation that vibration controllers use to calculate the maximum displacement of certain random vibration profiles? Displacement calculations are straight forward for sinusoidal vibration, however, displacement calculations regarding random vibration are not as obvious.

Currently, we dump a random vibration field profile into our vibration controller and the system calculates a maximum displacement, but how? We also have a random vibration calculator to aid in obtaining the maximum displacement. We need this to determine if we can perform certain tests on our existing equipment.

It would be convienent to have an equation to use without having to program in a full profile.

Thanks,
Kaiserman

 

[tomirvine]

The equations are beyond the scope of hand calculations.

There are two calculation required.

1. Convert the acceleration power spectral density to a displacement power spectral density.

2. Calculate the area under the displacement power spectral density curve. The square root of the area is the displacement in RMS.

The units must be carefully considered.

I have a software program that will do these calculations. Please send me an Email if you would like the program.

Tom Irvine
Email: tomirvine@aol.com

http://www.vibrationdata.com

 

[GregLocock]

I'm a bit surprised that you can get a /maximum/ displacement for truly random inputs with a Gaussian distribution. What is the ten millionth standard deviation signal? twenty millionth?

Most of the vibration table controllers I've seen use a crest factor, typically 5, which means that the height of the highest peak is limited to 5 times the RMS value.

Cheers

Greg Locock

 

[Tobalcane]

Kaiserman,

Tomirvine, correct me if I’m wrong.

When I want to get a ballpark figure for displacement at frequency and G^2/Hz out, I would use the following equations (using a G^2/Hz vs Hz plot with PSD input and PSD output(response) profile):

Transmissibility Q = sqrt(PSDout/PSDin)

Gout=sqrt((pi/2)*PSDin*Fn*Q)

Where Fn=frequency at max G^2/Hz of interest

Thus

Displacement=(g*Gout)/(Fn*2*pi)^2

Where g=gravity

I’m not sure if your program uses this methodology, but at least you have some equations to work with.

Hope this helps…
Tobalcane

 

[Tobalcane]

Kaiserman,

A second look at your question, are you asking for the displacement of the armature (vibration head) or where the response accelerometer sits? What I posted was equations for response accelerometer and where it sits.

Tobalcane,

 

[MikeyP]

Some comments, one practical and one not so...

If you are trying to reduce the peak levels of the input signal, you can do this by "crest factor optimisation". The common way to do this is by defining a spectrum of the desired amplitude but with random phase (a "random phase multisine"). This spectrum is then inverse Fourier transformed into the time domain, where the time signal is clipped to the desired crest factor. The signal is then transformed back to the frequency domain where the phase components at each spectral line are kept as they are, but the spectral amplitudes are changed back to the original values. After several iterations of this process you are left with a signal which has the desired frequency vs amplitude profile but which has a probability density function (pdf) - and hence crest factor - that tends to that of a sine wave (ie minimum at zero and maximum at the maximum amplitudes).

The issue of predicting maximum response levels due to random excitation is a hot topic in the applied mathematics world (I should point out that I am not an applied mathematician except in the sense that all engineers are applied mathematicians!). To the maths boffs, these problems are known as Stochastic Partial Differential Equations (SPDEs). For more info try searching for "SPDE", "Ito fomula" and "FPK equation" on Google. Essentially the FPK equation converts the dynamic equation of motion of system into a PDE where the unknown variable is the pdf of the response. This process is so straightforward that even I can follow it. The problem is that the resulting PDE is usually insoluable. Last I heard solutions had been found for about 6 very specific cases! Of course, if you can solve it, then you can get all sorts of useful probabilistic information about the system, eg. mean time to failure, fatigue estimates, the probability of the system exceeding a certain displacement in a specified time etc.

And now it's time for my pills.

M

--
Dr Michael F Platten

 

[tomirvine]

There are two displacements of concern.

One is the base input displacement, which is the displacement of the shaker head. It is also the displacement of the mounting fixture assuming that the fixture is rigid. The original question was in regard to this displacement. I offered a program that calculates the RMS displacement of the shaker head. Note that 1 sigma equals RMS assuming zero mean. The exact peak displacement is unknown. It can only be estimated in terms of probability, assuming a normal distribution. In practice, the peak displacement might be 3 or 4 sigma.

The other displacement is the response displacement of the test item. A previous reply gave Miles equation for this calculation. This approach assumes single-degree-of-freedom behavor. It also assumes that the input PSD is flat within plus or minus one octave of the natural frequency, as a rule-of-thumb.

Tom Irvine

http://www.vibrationdata.com