Welcome to Vibrationdata
Random Vibration & Power Spectral Density Page

Dear Guest,
Writing tutorials and maintaining a website are expensive endeavors. I need your help to continue this effort. Your subscription will allow me to continue this site and to add new materials.

The software and tutorials require a username and password which are given by subscription. The subscription instructions are given below.
The subscription entitles the user to download all of the items on this page, as well as other Vibrationdata pages.

Thank you for your support.
Sincerely, Tom Irvine
Email: tom@vibrationdata.com

 An Introduction to Random Vibration. A random vibration signal is analyzed in terms of its histogram, probability density function, and descriptive statistics.
 Power Spectral Density Units [ G^2 / Hz ]. The bandpass filtering method is used to demonstrate a power spectral density calculation.
Integration of the Power Spectral Density Function:
The overall GRMS value is obtained by integration, for log-log format.
Acceleration, Velocity, and Displacement Power Spectral Density Functions. Calculation methods are presented to convert any of these function to the other two functions.
A Method for Power Spectral Density Synthesis. The method begins with a white noise time history. The Fourier transform is then manipulated to obtain the desired power spectral density functions.
PSD synthesis example using step-by-step software program instructions.
Equivalent Static Load for Random Vibration. The static load is taken from the three-sigma vibration response.
Power Spectral Density Calculation via Matlab:
Guest tutorial submitted by Bob Light.
 Estimating the Transmissibility Q for Random Vibration

dB/octave Calculations for Log-Log Plots


Piersol, The Analysis of Nonstationary Vibration Data

Further materials for nonstationary vibration is given at:  nonstationary page


Tolerance Factor Tutorials

Uncertainty Margins in Aerospace Vibroacoustic Levels

This paper also includes recommendations for establishing maximum expected flight levels, as well as acceptance and qualification test levels.


Normal Tolerance Factors for Upper Tolerance Limits

Matlab Script: k_factor.m


C. Link, An Equation for One-Sided Tolerance Limits for Normal Distributions


P95/50 Rule -- Theory and Application 

I. Yunis & D. Ludwiczak, On the Use of 3dB Qualification Margin for Structural Parts on ELV



 Sine-on-Random Vibration

Single-degree-of-freedom System Response to Sine-on-Random Vibration

A. Gokce & M. Baker, Sine on Random Analysis: Alternatives and Challenges


Andrew Brown & D.S. McGhee, NASA Marshall Space Flight Center
NASA/TP-2003-212257, Statistical Evaluation and Improvement of Methods for Combining Random and Harmonic Loads,
Statistical Comparison and Improvement of Methods for Combining Random and Harmonic Loads (non-copyright AIAA paper)

 Sine, Random and Shock Vibration Equivalence 


On the use of a Narrowband Random Power Spectral Density to Cover a Pure Sine Vibration Environment via Rainflow Fatigue


 Sine and Random Vibration Equivalent Damage
Sine and Random Vibration Equivalence
Extracting Sine Tones from a Power Spectral Density extract_sine_tones.pdf
 Fackler, SVM-9, Equivalence Techniques for Vibration Testing
 A Comparison of Sine and Random Vibration via Tustin's Demonstration Device
Shock and Random Vibration Equivalence

 Vibraton Response Spectrum

 Introduction to the Vibration Response Spectrum
 Derivation of Miles Equation
 Simmons, Miles Equation
 Enveloping Data via the Vibration Response Spectrum

Deriving a Random Vibration Maximum Expected Level with Consideration for Kurtosis



Optimizing a Circuit Board Natural Frequency with Respect to Random Vibration.

The optimization is achieved using the vibration response spectrum.
The Response of a Single-Degree-of-Freedom System to Excitation from Random Surface Irregularities. (The input function may be a s patial power spectral density).
Methods for Converting a Power Spectral Density to a Shock Response Spectrum

 Random Vibration Stress & Fatigue

 Random Vibration Fatigue:
Includes Miner's Cumulative Damage Index.
Time-Scaling Equivalence Methods for Random Vibration Testing
A Fatigue Equivalence Method for Enveloping Nonstationary Random Vibration
 Vibration Fatigue Criteria for Electrical Components

Sandia Report, An Efficient Method for Calculating RMS von Mises Stress in a Random Vibration Environment


Effect of Conformal Coating on Circuit Board Vibration Response

 Random Vibration Testing

Random Vibration Band-Splitting, Martin Marietta
Time-Scaling Equivalence Methods for Random Vibration Testing
Bastien, Random Vibration (Stress Screening) of Printed Wiring Assemblies

 Statistical Distributions

 The Rayleigh Distribution
 Integration of the Normal Distribution Curve
 Additional information regarding random vibration is given in Course Materials

 Software Description



This program accepts an acceleration, velocity, or displacement power spectral density function. Given one of these functions, the program then integrates or differentiates as appropriate to find the other two functions. The program also calculates the overall level of each of the three functions.
Synthesis of a time history to satisfy a power spectral density using white noise
 Windows version
DOS version:

 DOS version:
Synthesis of a time history to satisfy a power spectral density using a sine series
psd_sine.exe psd_sine.cpp
Synthesis of a time history to satisfy a Fourier magnitude using a sine series
fourier_mag_sine.exe fourier_mag_sine.cpp
Synthesis of a time history to satisfy a sine-on-random specification
Power spectral density of a time history signal. This program uses the FFT method. Single PSD file output.
Power spectral density of a time history signal. This program uses the FFT method. Various output files.
Acceleration PSD from a velocity time history, such as data from a laser vibrometer.
Calculate a PSD via successive bandpass filtering.
 Energy Spectral Density
Calculations involving slopes in dB/octave. Can be used for both power spectral densities and shock response spectra.


 Vibration response spectrum of a single-degree-of-freedom system subjected to base excitation.
Conversion of a vibration response spectrum to a power spectral density.
Response of an individual single-degree-of-freedom system to base excitation PSD.
Response of an individual single-degree-of-freedom system to an applied force PSD.
Optimum envelope of a power spectral density function using the vibration response spectrum method.

Derive a Random Vibration Maximum Expected Level with Consideration for Kurtosis using 3-sigma VRS



Derive a Random Vibration Maximum Expected Level with Consideration for Kurtosis using the n-sigma VRS method,
where n=sqrt[ 2 ln (fnT) ]



Maximum envelope of a PSD or SRS function.
Generation of a "white noise" time history.
Circuit board fatigue via Steinberg's formula. 


Logarithmic interpolation of a frequency domain function.
Response PSD calculation from a power transmissibility and input PSD.
Linear interpolation of a time history.
The peak distribution of a time history in terms of a probability density function. The peak distribution of a narrowband process may follow a Rayleigh distribution.
 Histogram of a time history.
 Zero crossings of a time history.
This program multiplies the Fourier transform of a base input function by the transfer function of a single-degree-of-freedom system.

Convert a PSD with constant bandwidth to octave, one-third octave, or one-sixth octave format.



This program transform a PSD from one resolution to another. The input PSD must have a constant spectral bandwidth.
 Average PSD from individual PSD records.
 PSD sum from individual PSD records.
psd_add_set.exe psd_add_set.cpp
Maximum PSD from individual PSD records.
This program calculates the P95/50 level of a number of PSD functions.

The P95/50 level is the 95th percentile value with a 50 percent confidence.
Calculate magnitude versus frequency from PSD function.

Convert a displacement PSD in dB format to a velocity RMS spectrum



Sine and Random Vibration Equivalent Damage
Peak SDOF response from the Rayleigh distribution
Probability calculator for the following distributions:

1. Normal
2. Binomial
3. Poisson
4. Rayleigh


Unit conversion program.
DLL files:


Please recommend this site by clicking on the Google +1 Button.

Other Vibrationdata Pages:    HomeMatlab random vibration | Tutorials