# 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. 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. 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