I vividly remember the first time I performed a random vibration (PSD) analysis. I was at my first job, with a couple of years of dynamics analysis under my belt and, one day, I was dragged into a meeting where panicked project engineers were showing graphs of field failures to management. An unacceptable number of products were breaking due to high cycle fatigue during operation over rough terrain. If this problem was not corrected quickly, the costs to the company would be astronomical. An outside consultant was brought in and suggested that random vibration was the cause of the problem.
At the time, nobody in our organization had any experience with how to tackle these problems. I was given the FE analysis responsibility for this project. At the time, I felt honored, thinking it was because my management had confidence in me to get this critical problem solved. Looking back, I realize that it was because nobody else wanted to get anywhere near this potentially careerending disaster, so they gave it to “the kid”.
After many long days setting up the analysis, and long nights reading Crandall and Mark (Ref. 1) so I could understand what it was that I was doing during the long days, the analysis was complete. We identified the problem, and the fix was relatively straightforward. Problem solved, bullet dodged, but I had no real confidence that I understood random vibrations.
A review of the project, once we had time to breathe, identified a host of things we had done wrong. Not surprising, since we were learning everything on the fly and had to produce results very quickly.
This “trial by fire” scenario is not the way I would suggest that anybody take on a type of analysis that they are not familiar with, but sometimes outside pressures trump proper protocol. For this project, the thing that scared me the most was that I had no sanity check on my PSD results – I was relying completely on the software to produce the correct results without any kind of hand calculation. The natural frequencies and mode shapes compared reasonably well with experimental data, so I had passed one important hurdle from the validation and verification perspective. After that, due to my own ignorance, I was pretty much flying blind. If you've read my posts about validation, you know this is a good recipe for disaster – in this case, we just got lucky.
During my many subsequent years working with companies doing PSD analysis and teaching classes to engineers looking to do random vibration analysis, I realized that many don’t have a good way to check the validity of the PSD solution. With input in units like /Hz and output in terms of RMS 1σ stress, it’s difficult to get any intuitive feel for what the results should be.
The good news is that there are some very simple hand calculations that can be performed, which produce excellent approximations to PSD solutions for most problems. The one I use most often can be found on pages 11 and 12 in the Shock and Vibration Handbook (Ref. 2).
Let’s look at a simple example. Suppose we have a simplysupported 10” long, by 0.1” high, by 1” deep steel beam, excited by a constant 0.1/Hz base excitation in the “height” direction from 20 to 3000 Hz. We will assume a damping ratio of 2%.
The first natural frequency of this beam is 92.05 Hz. From the nature of excitation and the geometry of this simple case, we know that the first mode will be dominant. In a more general case, it may not be so obvious and a review of the response PSD function would be required in order to assess the most dominant frequency or identify multiple modes that play a big role in the overall response.
The massnormalized mode shape in Figure 1 shows that the displacement at the center of the beam at this frequency is 52.363. The participation factor for this mode in the direction of excitation is found from the modal analysis output as 0.02436.
Figure 1 – Mass Normalized Mode Shape

We now have all the information needed for our calculation:
Inserting these values into our equation yields a 1 (RMS) displacement response at the beam center of approximately .028 inches. This would be the value to compare to the results of a random vibration analysis at this same location.
There are other variants of this equation for a force or pressure excitation. The equations assume that one mode is dominant in the response and that participation factor and mode shape data are available for this mode. This method can be extended to include more than one mode by assuming the modes are decoupled and using a square root of the sum of the squares approach to sum individual mode contributions.
This simple check has served me well over the years to quickly validate PSD results. It would be great to hear about how others ensure that their random vibration solutions pass muster.
References
1. Crandall, S.H., and Mark, W.D. Random Vibration in Mechanical Systems. New York, Academic Press Inc., 1973.
2. Harris, C.M., ed., Shock and Vibration Handbook, 3rd Ed., New York, McGrawHill , 1988.