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We will also need the equivalent equations for segments of the velocity and displacement PSDs. These may be obtained by synthetic integration of equation 1. Recall A is of dimension g2/Hz where g is understood to be an RMS measurement. We require a velocity power spectrum, V, of IPS2/Hz amplitude and a displacement power spectrum, D, of inch2/Hz dimension where both IPS and inch are understood to be RMS units. The required segment equations are:
The areas under each segment are obtained by integrating equations 1, 2 and 3 with respect to frequency, f, over the interval from f1 to f2. This results in the mean-square values:

Adding the mean-square for all segments in the PSD yields the total mean-square values of acceleration, velocity and displacement. Taking the square root of each mean-square provides the RMS value. Multiplying the RMS values for g2, IPS2 and inch2 by 3 provides the peak acceleration, velocity and displacement. Multiply the displacement result by 2 to obtain a peak-to-peak value. Let’s consider an example.

Note that some specifications will specify the slope of rising or falling segments. The slope is normally presented in either dB/octave (decibels per frequency doubling) or dB/decade (dB change over a ten-to-one frequency span). This allows m to be evaluated as:

Figure 14 illustrates a typical Random Test Profile. The frequency, f, and g2/Hz of the four points defining this three-segment spectrum were placed in an Excel® spreadsheet. The mean-square values of g2, IPS2 and inch2 for each segment were computed using equations 4, 5 and 6. The results for all segments were summed and the required conversion to peak and peak-to-peak values made. These calculations are summarized in Figure 15.

Figure 14: A Random Test Profile is a g2/Hz amplitude Power Spectral Density function.

Figure 15: Calculating the span and peak acceleration, velocity and displacement for the test of Figure 11.