In the plot above, taken from space craft system engineering book, chapter 2.

This is the plot showing the random vibration power spectrum of Ariane 4. The unit is $g^2/Hz$, I understand how the unit came by reading the Wikipedia page of spectral density. But I am unable to comprehend the physical meaning of this graph.

As far as I understand it, this is the power spectrum of the acceleration signal experienced by any payload inside the fairing. How does a payload engineer go about from looking at this graph to testing the payload for such vibration? I mean do they reverse calculate signal required to feed in vibration table having same spectral density? How is this graph practically used and understood?

Edit: I drew question mark in the plot, because I don’t get what acceleration squared per unit frequency means physically.


Let's start with the text in the middle of the plot: "r.m.s acceleration 7.3 g"

That means that, typically and without paying attention to frequency, you'll be seeing a vibrational acceleration around 7.3 g. You need to design for that.

Now, what if your device responds differently at different frequencies? I.e. it has a beam inside it that resonates at some frequency? Or the bonds in its chips pretty much ignore low frequencies, but are very sensitive at very high frequencies? That's where the chart comes in.

From about (by eye) 150Hz to 700Hz, the vibration is described by that flat line. Throughout that region, the amount of vibration is constant at about .004 $g^2/Hz$. What does that unit mean? Let's try an example: If your device was sensitive from 200 to 250 Hz, a bandwidth of 50 Hz, the vibration is $(50Hz \times 0.004 g^2/Hz) = 0.2 g^2$. Since that $g^2$, you then take the square root (the R in RMS) to get an RMS acceleration of 0.44 g.

At higher and lower frequencies, the vibration falls off as you get away from the core frequencies. The "+6db/oct" and "-3db/oct" are synonymous with "rising like $f^2$" and "falling like $1/f$" respectively because "oct" is short for octave, a factor of two in frequency, and +6db and -3db are factors of 4 and 1/2 respectively. (Why do engineers do that? Long story...)

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The choice of units is more a historical/maths thing than the physical meaning being very clear.

In power spectral densities, the 'power' is defined as the square of the signal. This is useful to engineers because it make the maths work, but it make sense to think of this as sort of like a power, as in typical SHM conditions this is strongly related to power (thought of as force * displacement, but force is proportional to displacement, hence signal squared). There is of course a constant missed out here, the relation between force and displacement, but its often easier to deal with the constant separately. Mainly because this is the measured data and its the easiest to work with. As (you would hope) the vibrations in a spacecrafts payloads are elastic, acceleration and displacement are also proportional for the individual frequencies (ish/some caveats) and acceleration is easier to measure the same thing happens again.

So you could use this interpretation again, but its starting to become a little meaningless.

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