A quick note regarding vibration and noise units

Just a quick note regarding expressions of vibration and acoustical data. Every now and then we come upon a vexing problem related to full expressions of the units of a measurement (or criterion).

I'm not talking about gross errors, like confusion of "inches-vs-centimeters" or "pounds-vs-newtons". Instead, I'm referring to some of the other, more subtle parts of the expression, like scaling and bandwidth.

Take a look at the plots below; this is from a vibration measurement in a university electron microscopy suite. Note that all of the data shown in this blog post are completely identical; however, they are expressed in different terms. I've re-cast this same singular spectrum in different terms so that you can see how much it matters to have a full expression of the units we're talking about.

To start, we surely won't confuse big-picture terms, like the difference between acceleration, velocity, and displacement. Which one you work with doesn't matter much, but we'd better be sure we understand the difference between them:

Data above are from a single measurement, expressed in acceleration, velocity, and displacement. Obviously, these units are all different, so the curves look different, despite the fact that each spectrum relates exactly the same information.

Also, we probably won't get fooled by the physical units, like inches-vs-meters. At the very least, we have a good "order-of-magnitude" sense of where things ought to land; if you have an electron microscopy suite with vibrations in double-digits, then they'd better be micro-inches/sec (rather than micro-meters/sec), or else you're in trouble:

The same data from before, now expressed using two different sets of units of velocity. Obviously, if you have a criterion like "0.8um/sec" then you'd better compare against the curve expressed in um/sec rather than the one in uin/sec. But are we finished? Do we have a complete expression of the "units" yet?

We're not quite finished, though. Just because we've agreed on terms (velocity) and physical units (micro-meters/sec), we still have some work to do. We never said what the measurement bandwidth should be. We've been showing narrowband data, but what if the criterion is expressed in some other bandwidth? Maybe it's not even a constant bandwidth, but rather a proportional bandwidth, like (commonly-used) 1/3 octave bands:

This is still all the same data, only we are now showing it in narrowband (1Hz bandwidth) as well as in 1/3 octave bands. Note that widths of the 1/3 octave bands scale with frequency as f*0.23, so at low frequencies (below 4Hz) the 1/3 octave band is actually smaller than 1Hz.

OK, so now we have terms (velocity), physical units (micro-m/sec), and bandwidth (let's choose 1/3 octave band). But we're still not quite finished: we still need to say what signal scaling we're using. You might have seen this referred to using phrases like "RMS" or "Peak-to-Peak":

Again, these are the same data as above, but now we've chosen the 1/3 octave band velocity in micro-m/sec. But if we're supposed to compare against a criterion, which scaling do we use? There's a big difference between the RMS, zero-to-peak, and peak-to-peak values. 

If I told you that the limit was 0.8um/sec, then would you say that this room passes the test? As you might surmise, you can't answer that question if all I told you that the limit was 0.8um/sec. You need to know exactly what I mean by "0.8um/sec". I know it sounds funny, but just plain micro-meters-per-second is not a complete expression. You have to tell me whether we're talking 0.8um/sec RMS; or zero-to-peak; or peak-to-peak. You'll also have to tell me what bandwidth you want: PSD? 1/3 Octave Band? Narrowband, with some specific bandwidth? 

If you were to tell me that you need to meet 0.8um/sec RMS in 1/3 octave bands, then we can plot the data appropriately and make some intelligent statements:

Since we've been given a full expression of the criterion (0.8um/sec RMS in 1/3 octave bands, which happens to be IEST's VC-G curve), we can plot the data with those units and overlay the criterion. This room passes the test, but without a full expression, we couldn't say one way or the other.

We see this kind of problem all the time. Most notably, we see people comparing narrowband measurement data against a 1/3 octave band criterion like those in the VC curves. This is just plain wrong, because the measurement and criterion are literally expressed in different units. The VC-G criterion isn't simply "0.8um/sec"; instead, it is actually "0.8um/sec RMS in 1/3 octave bands from 1 to 80Hz". 

This is important, and it matters a lot!