Calulating G_{rms} (RootMeanSquare Acceleration)
It is very easy to describe the G_{rms} (rootmeansquare acceleration, sometimes written as GRMS or Grms or grms or g_{rms}) value as just the square root of the area under the ASD vs. frequency curve, which it is. But to physically interpret this value we need to look at G_{rms} a different way. The easiest way to think of the G_{rms} is to first look at the mean square acceleration.
Meansquare acceleration is the average of the square of the acceleration over time. That is, if you were to look at a time history of an accelerometer trace and were to square this time history and then determine the average value for this squared acceleration over the length of the time history, that would be the mean square acceleration. Using the mean square value keeps everything positive.
The G_{rms} is the rootmeansquare acceleration (or rms acceleration), which is just the square root of the mean square acceleration determined above.
If the accelerometer time history is a pure sinusoid with zero mean value, e.g., a steadystate vibration, the rms acceleration would be .707 times the peak value of the sinusoidal acceleration (if just a plain average were used, then the average would be zero). If the accelerometer time history is a stationary Gaussian random time history, the rms acceleration (also called the 1 sigma acceleration) would be related to the statistical properties of the acceleration time history (you may have to refresh your probability and statistics knowledge for this):
 68.3% of the time, the acceleration time history would have peaks that would not exceed the +/ 1 sigma accelerations.
 95.4% of the time, the acceleration time history would have peaks that would not exceed the +/ 2 sigma accelerations.
 99.7% of the time, the acceleration time history would have peaks that would not exceed the +/ 3 sigma accelerations.
There is no theoretical maximum value for the Gaussian random variable; however, we typically design to 3 sigma since it would only be theoretically exceeded 0.3% of the time. In addition, from a practical point of view, we know that it would be physically impossible to achieve unreasonably high sigma values.
Below is presented the method to calculating the rootmeansquare acceleration (G_{rms}) response from a random vibration ASD curve.
Typical random vibration response curve:
G_{rms} values are determined by the square root of the area under a ASD vs. frequency response curve. The Acceleration Spectral Density values are in g^{2}/Hz and the frequencies are in Hz.
The figure above shows a bandwidth of 10 Hz, which will be used as an example for calculating G_{rms}.
In order to calculate the G_{rms} value for the entire curve, sum up all the areas (A_{1} + A_{2} + A_{3} + ... + A_{n} = A) and take the square root of the sum.
NOTE: 3dB is a factor of 2 for ASD curves (g^{2}/Hz) while 6dB is a factor of 2 for G_{rms} values. For example, reducing a peak ASD value of 12g^{2}/Hz by 3dB would give you 6g^{2}/Hz; reducing a value of 12G_{rms} 3dB results in a value of 9G_{rms} and reducing it 6dB results in a value of 6G_{rms}. This tends to be confusing for people new to random vibration.
An Excel 97 spreadsheet, grms.xls, written by Bob Coladonato that calculates all these values is available for downloading. The only input values necessary are frequencies and their respective ASD levels.
Microsoft Excel 97 Spreadsheet for G_{rms} calculations. (Hold your shift key down while clicking on the link to save the file to your hard drive.)
An Excel 95 version, grms95.xls, is also available.
Thanks to Bob Coladonato and Bill Case, both now retired from Goddard, and Jaap Wijker of University of Technology Delft in the Netherlands for their assistance with this page.
Ryan Simmons August 1997
