Quote of the Day

The most important 6 inches on the battlefield is between your ears.

## Introduction

I was asked last week to write a vibration test plan for a mobile electronic product. I am used to writing vibration test plans that follow canned procedures in standards like MIL-STD-810F or SAE J1455, but this case is different because the customer has specified a non‑standard random vibration acceleration profile, which is also called a Power Spectral Density (PSD). I need to determine the RMS g level for this profile. This post shows how I go about this calculation. I am not going to showing the customer's vibration PSD because it is proprietary. Instead, I will use a well‑known US Navy vibration PSD as a computation example (Figure 1).

In the case of vibration testing, the term PSD is a misnomer. Vibration testing is normally based on an Acceleration Spectral Density (ASD) function expressed in the form of a graph or table. However, the term PSD is commonly used because the analysis methods are identical to those used with PSDs in digital signal processing, where the term PSD really does refer to a power spectrum.

## Background

### Objective

For my random vibration test problem, I was given a normalized ASD shape (no absolute levels) and the customer's desired RMS g (acceleration) level. I must integrate the ASD to determine the RMS g level. Given the integral, I can set the absolute levels to obtain the required RMS g level. This effort is complicated slightly by the fact that the ASD is usually specified in terms of piecewise linear segments on a log-log graph. What is the function that I must integrate? This post will show you how to integrate an ASD to determine the RMS g level.

### ASD Example

Figure 1 shows a US Navy example that has been used for decades to test electronic gear, which developed by Willis Willoughby, who developed much of the US Navy's quality program. These piecewise linear approximations are important because vibration test equipment (Figure 2) is usually programmed using piecewise linear approximations to ASDs. The approximation process is actually quite interesting, and NASA presents a good example of the process here.

Figure 2 shows a typical vibration table.

## Analysis

### Derivation

A line segment on a log-log graph does not represent a line segment on a linear scale. My integration needs to be performed with the function on a linear scale. So I need to convert that log-log line segment into its linear scale form. Figure 3 shows how I did that conversion.

The yellow highlighted areas of Figure 3 shows how to compute the slope and intercept of the line segment on a log-log plot. The green highlighted area shows how to convert a point on a log-log plot to a linear scale.

Now that I have my vibration function, the integration operation is straightforward. I will be using Mathcad's built-in integration function, but many tools support similar capabilities (e.g., R has its trapz function, which I use regularly).

### Example

To compute the RMS g level, I must integrate the ASD curve for the frequencies of interest, in this case, 10 Hz to 2000 Hz. Figure 4 shows my work for the US Navy example of Figure 1. I computed 6 g_{RMS,} which I highlight in yellow in Figure 4. The US Navy lists this profile as 6.06 g_{RMS,} which is close agreement.

I should mention that there are other ways to compute RMS g level. Take a look at this website and this website.

I've attempted, without success to use Excel to calculate the Grms of the PSD curve by calculating the area of the 3 parts of the curve, similar to the NAVMAT example you show above. It's clearly more complicated than I first assumed. Tom Irvine attempts to provide tutorial instruction, but without detailed examples of the calculation.

My slope calculations are simple enough, but the "area" calculation is eluding me, even for the portion of the curve where the slope is zero. Any support is greatly appreciated.