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**Step 1 – Swept Sine Testing**

The simplest test specification to analyze is that of a *Swept Sine* test. Figure 9 illustrates the *Test Profile* of a swept-sine test. The profile is an acceleration-verses-frequency spectrum, normally presented in log-log format. The acceleration amplitude is normally expressed in peak “**g**” units, where 1**g** =9.807 m/s2 = 386.4 inch/sec2. The frequency,** f**, is normally expressed in Hz (Hertz meaning cycle/second), though circular frequency, **ω**, in radian/second (1/s) is sometimes used. (Note **ω**=2π** ****f** = 6.2832 **f**.)

The test profile is (normally) composed of a series of straight-line segments. Note the frequency,** f**, and acceleration,** g**, of each point defining the end of a line segment. Calculate the corresponding peak velocity and peak-to-peak displacement for each endpoint using the following equations:

Retain the largest result from equation 1) as the *Peak Velocity*. Retain the largest result from equation 2) as the *PTP Displacement*. These actions are summarized for the Test Profile of Figure 9 in the following table:

**Step 1 – Random Testing**

Random testing involves generating and controlling a random signal with a specifically shaped spectrum and Gaussian amplitude statistics. The signal is defined by a spectral test profile called a *Power Spectral Density* (PSD) which is always measured and it exhibits an amplitude histogram or *Probability Distribution Function* (PDF) which can also be measured. Figure 10 illustrates these two complimentary views of a random acceleration.

Like a swept-sine test, a *Random* test has an acceleration spectrum as the *Test Profile*. However, the vertical units of the random spectrum are very different; they are in units of **(g _{RMS})^{2}/Hz** conventionally abbreviated to

**g**. Such a “squared amplitude” spectrum is called a

^{2}/Hz*Power Spectral Density*or PSD. The area under a PSD is the signal’s “power” or mean-square value. Hence the area under a

**g**acceleration is the square of the signal’s root-mean-square (RMS) value,

^{2}/Hz**g**. The “per Hertz” part of the

_{rms}**g**unit label refers to the resolution

^{2}/Hz*Noise Bandwidth*of the instrument that measured the PSD.

A modern vibration controller measures a PSD by using *Fast Fourier Transform* (FFT) processing. The FFT works upon sequentially gathered blocks of **N** successive signal amplitude samples separated equally in time by **Δt** seconds. It mathematically converts these into an **N/2** point spectrum with spectral amplitudes measured every **Δf = 1/NΔt Hertz**. A “power spectrum” is produced by squaring the spectral amplitudes and ensemble averaging many squared spectra. The averaged *Power Spectrum* becomes a *Power Spectral Density* when the amplitudes are divided by the *Noise Bandwidth*. The noise bandwidth of such a measurement is **kΔf**, where **k** is determined by the “window” or weighting applied to each acquired block of **N** time samples. A typical value of **k** is 1.5 when the (very common) Hanning window is applied. A window functions (such as Hanning) is necessary to suppress spectral distortions resulting from the asynchronism between the acceleration signal and the analyzer.

Because the random testing signal exhibits “bell-shaped” *Gaussian* amplitude statistics, the peak value, **g _{peak}**, can be precisely estimated from,

**g**. The PDF is an amplitude histogram (counts or occurrence versus signal value) normalized to enclose a unit area. In vibration work it is frequently useful to present a PDF with its amplitude on a log scale as in Figure 11.

_{rms}
The horizontal (signal amplitude) axis can be scaled to multiples of **σ**, the *standard deviation* synonymous with **g _{rms}** when the signal is of

*zero mean value*. Figure 11 illustrates such scaling and shows the fraction of area subtended by ±1, 2, 3, 4, 5 and 6

**σ**amplitude bands. Since the total are under the PDF is unity, these areas represent the probability of the signal being within the ± bounds at any time. Figure 11 shows that a Gaussian acceleration signal of RMS value,

**g**, will have an instantaneous peak value within ±3

_{rms}**g**99.73% of the time. In other words, if the random signal has an RMS value of

_{rms}**g**, 3

_{rms}**g**is a very good estimate of the peak acceleration.

_{rms}