# Peak Response for Random Vibration Single-degree-of-freedom System

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Please see the following post for an important revision:  Peak Response to Random Vibration with Probability of Exceedance

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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 ) ]

where

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.

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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

– Tom Irvine

## 5 thoughts on “Peak Response for Random Vibration”

1. We found that most vibration controllers allowed for exceedances in 3-sigma acceleration peaks, despite the use of 3-sigma clipping (on the control). We typically measured in the range of 4-sigma responses, sometimes more, sometimes less. Either way, peak structural or part stress may fatigue significantly due to these higher peak stresses, despite their infrequent occurrence.

2. Scot McNeill on said:

sigma*sqrt [ 2 ln (fn T ) ] is actually the formula the most probable extreme value for a narrow-band, Gaussian process. This is slightly lower than the expected (average) extreme, which is sigma*( sqrt [2ln(fnT)] + 0.5772/sqrt [2ln(fnT)] ). The nuber 0.5772 is Euler’s constant (gamma, not e). If the process is not narrow-banded, replace fn with fz, where fz is the average, mean-value up-crossing frequency.

All these formulas come from the Gumbel distribution, which is the extreme-value distribution for a Gaussian process with mild bandwidth restrictions.

3. senthil on said:

Tom
How do you add a static stress value to a random vibration and calculate fatigue damage from the resultant stress? Imagine a plate with a static installation because of the plate mounting pads are out of flatness and goes through a random vibration. How do you combine two stresses? Thanks

Senthil V