Please see the following post for an important revision: Peak Response to Random Vibration with Probability of Exceedance
Consider a single-degree-of-freedom system subjected to a broadband random vibration base input, where the input is stationary and has a normal distribution. (The following calculation also works for an applied force.)
The system’s response will be narrowband random, with amplification at the system’s natural frequency. The response will also have a normal distribution.
The typical assumption is that the peak response will be “3-sigma.”
Note that 1-sigma is equivalent to the GRMS value assuming a zero mean.
Higher sigma values can actually occur depending on the duration and natural frequency.
The highest expected response peak is estimated as:
sqrt [ 2 ln (fn T ) ] + 0.5772 / sqrt [ 2 ln (fn T ) ]
sqrt is the square root
ln is the natural log
fn is the natural frequency (Hz)
T is the duration(sec)
This formula is derived from the Rayleigh distribution as shown in:
Equivalent Static Loads for Random Vibration: eqstatic.pdf
A Matlab script for this calculation is given at: peak_response_random.m
As an example, consider a system with a natural frequency of 200 Hz subjected to stationary random vibration over a duration of 60 seconds. The maximum expected peak is 4.3-sigma for this case.
See also: Peak Response for Multiple Random Vibration Events peak_response_multiple_events.pdf
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The method can be extended to a multi-degree-of-freedom-system as shown in: mdof_peak.pdf
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Matlab script: Vibrationdata Signal Analysis Package
See also: Rayleigh Distribution
– Tom Irvine