Optimized PSD Envelope for Multiple Accelerometer Time Histories

Prerequisite Reference Papers

David O. Smallwood, An Improved Recursive Formula for Calculating Shock Response Spectra, Shock and Vibration Bulletin, No. 51, May 1981.  DS_SRS1.pdf

Rainflow Counting Tutorial

Fatigue Damage Spectrum, Time Domain

Fatigue Damage Spectrum

Dirlik Method for PSDs

Optimized PSD FDS Nonstationary 

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Main Paper

Consider a component mounted on a structure where the base input is measured by an adjacent accelerometer on the structure. An envelope power spectral density (PSD) is needed so that component design and test levels can be derived, with the appropriate added statistical uncertainty margin.

Assume that the base input has been measured over a series of accelerometer time histories. This could be the case for an automobile driven at different speeds over different road conditions, for example.

The envelope PSD can be derived using fatigue damage spectra as shown in:  FDS_PSD_multiple.pdf

The C++ programs are:


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Here is an alternate program that allows for repetition for a given time history file.  This is useful, for example, if a short time duration was measured to represent a longer service duration.


Now assume that there are three measured acceleration time histories where the repetition number is 10, 50 and 100, respectively.

The input file format would be:

time_history_1.txt 10
time_history_2.txt 50
time_history_3.txt 100

Substitute your own file names and multipliers accordingly.

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– Tom Irvine

Relative Displacement from Two Accelerometer Time Histories

Assume that input and response acceleration PSDs have been measured.  The corresponding displacement PSDs can be readily calculated.

But the relative displacement cannot be calculated accurately because the phase angles are discarded in the PSD calculation.

The best method is to perform the relative displacement calculation in the time domain. The relative displacement PSD can then be calculated if desired.

A function for calculating the relative displacement from two accelerometer time histories is given in:

Matlab script: Vibrationdata Signal Analysis Package

The function can be accessed via:

>> vibrationdata > Time History > Relative Displacement from Two Signals

Typically, a highpass filter from 5 to 10 Hz is needed.

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– Tom Irvine

Some Vibration Testing Q & A

Here are some questions from students that I recently received for shaker table testing of equipment.

  1. How are vibration standards developed and how do they relate to the operational environment?

Standards may have different purposes.  Some of the common ones are:

  1. Envelope the actual field or service vibration environment
  2. Screen for manufacturing and workmanship defects
  3. Combined envelope & screen
  4. Identify natural frequencies and damping


The envelope levels are ideally based on measured accelerometer data from actual service environments.  But the background data might not be readily available.  The levels become a sort of tribal folklore.

Some standards specify generic envelope levels, but also have caveats that the users should make their own accelerometer measurements for their specific applications and develop levels accordingly.

Screen Levels

NAVMAT P9492 is a seminal reference with a generic workmanship vibration level.  But there is an emphasis on users developing their own levels through trial-and-error.

For further information, please visit:



Natural Frequency Identification

The natural frequency identification tests are flexible and can be performed either with random or sine sweep base input.  Also the input levels can be varied for a linearity study.

Another option is modal testing with an applied force input.

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  1. What are there two options of dwell and random and when is it suitable or not to use either one?

Both sine dwell and random can be used for either enveloping an operational environment or for a workmanship screen.  Ultimately any sort of testing comes down to experience and engineering judgment.  Contractual requirements may also be a driver.

A sine dwell would be especially appropriate if the service environment was sinusoidal.

Sine dwell may also refer to exciting a test item at its natural frequency with a sinusoidal input, for a resonant response.

Likewise, a random test would be advisable if the service environment was random.

Some environments may also be sine-on-random, such as helicopter vibration.

See also:



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  1. Can we obtain transient time domain data from a PSD? Is there a method for doing this?

We can synthesis a time history to satisfy a PSD.  The time history will typically be stationary and Gaussian, but other distributions are possible.  This is essentially what a vibration control computer does for a closed-loop shaker test.  We can also do this for analysis purposes.




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  1. When creating a PSD profile for a shaker table test, are there specific methods that can be followed for reducing the number of breakpoints and keeping the Grms in the test the same as in the measured data?

A PSD test specification usually consists of a few straight line segments in log-log format.  The number of breakpoints can be reduced, but this will tend to increase the overall GRMS level.  The GRMS increase is usually considered acceptable as part of the added uncertainty margin.

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  1. For torsional acceleration should we use a torsional accelerometer (expensive) or can we obtain the torsional acceleration from the translational accelerometers?

You should be able to calculate the torsional acceleration from two closely-spaced translational accelerometers, by subtracting one signal from the other.  You would need to assume that the difference was due to pure rotation.  This requires engineering judgment.

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  1. How do we determine the dependencies between the different PSDs-PSDx and PSDy?

You really need to measure the accelerometer data in all three axes.  There is also a cross-PSD function that shows the correlation as a function of frequency between two PSDs.   The cross-PSD functions are usually only a concern for simultaneous multi-axis testing.

In aerospace avionics testing, a common level which envelops all three axes is typically derived, for the environment enveloping.  Then some statistical uncertainty margin is added.  Then the component is tested to the same level in each of three orthogonal axes.

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  1. PSDs at a different axis: Say we obtain the PSD in a given axis. Is there a way we can obtain the PSD is some other axis? Are there transformation equations?

No methods or equations are available.   You need to take measurements in all three axes.

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  1. We are hoping to gain a better understanding of testing methodologies (comparing sine sweeps and random vibration methods).

There are peak and fatigue damage spectra comparison methods for this very purpose.   These functions relate the peak response or fatigue damage as a function of natural frequency.







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  1. How do the Normal distribution and Rayleigh distributions of random vibration differ?

The PSD test specifications have an implicit assumption that the base input random instantaneous values will have a normal distribution.  Now consider a lightly-damped single-degree-of-freedom oscillator subjected to the base input.  Reduce the response time history to a subset of its local positive and negative peaks.  Use absolute values of each.  Plot those as a histogram.  The resulting response histogram will have a Rayleigh distribution.   Rayleigh distributions are used for fatigue calculations and for the estimation of the absolute maximum peak response.



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  1. How can we compare different PSDs? For example we have a Standard PSD profile and can we compare it to our PSD from measured data? How can we determine if excursions from our PSD over the Standard PSD are significant?

Assume that the test item behaves as an SDOF system.  The vibration response spectrum (VRS) may then be used, which gives the response as a function of natural frequency and the amplification factor Q.



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  1. What is the relationship between an accelerated life testing input level and the time duration such that input level and the time can be adjusted to get the equivalent accelerated life testing.

There are some approaches, each of which requires assumptions.  The key assumption is the inverse slope of the S-N fatigue curve.  Also, linearity must be assumed.

See Time-Level Equivalence in:


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Course Slides

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– Tom Irvine

Acoustics, Seismic & Mechanical Vibration Waves

Here are some papers…

Notes on Mechanical Shock & Bending Waves

Waves in Structures

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Functions for calculating wave speeds are given in:

Matlab script: Vibrationdata Signal Analysis Package

The functions can be accessed via

>> vibrationdata > Miscellaneous > Acoustics & Vibroacoustics > Structural Wave Speeds


>> vibrationdata > Miscellaneous > Acoustics & Vibroacoustics > Acoustics > Speed of Sound

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Wave Propagation Videos

Group Speed less than Phase Speed

Group Speed equal to Phase Speed

Group Speed greater than Phase Speed

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– Tom Irvine

Transverse Vibration of a Rotating Beam via the Finite Element Method


A function for calculating the natural frequencies and mode shapes for an elastic beam undergoing rotation is given in:

Matlab script: Vibrationdata Signal Analysis Package

The function can be accessed via:

>> vibrationdata > Miscellaneous > Structural Dynamics > Beam Bending > Rotating Beam, FEA

An option is included for calculating the response to a uniform force/length.

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See also:  Beam_FEM_rotating.pdf

– Tom Irvine

HALT/HASS for Product Reliability


Highly Accelerated Life Testing (HALT) is a process for ruggedization of preproduction products.  It is also referred to as AST (Accelerated Stress Testing).

Highly Accelerated Stress Screening (HASS) is the production screen for products once they have been characterized in HALT.

HALT and HASS each use very high rate of change temperature chambers which are combined with multi-axis pneumatic vibration systems.

Slides: HALT/HASS for Product Reliability: HALT_HASS.pptx

See also: Environmental Stress Screening (ESS) for Product Reliability:  ESS.pptx

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A function for synthesizing a rough approximation of a HALT/HASS acceleration time history (repetitive shock) is included in:

Matlab script: Vibrationdata Signal Analysis Package

The function can be accessed via:

>> vibrationdata > Miscellaneous > generate > HALT/HASS simulation

The Matlab script also has a function for thermal & vibration screen strengths

>> vibrationdata > Miscellaneous > ESS Screening Strengths

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United States Army, Belvoir Research, Development & Engineering Center, ESS Guide

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– Tom Irvine

Beam Supported by End Springs

A function for calculating the natural frequencies and mode shapes for an elastic beam supported by end springs is included in:

Matlab script: Vibrationdata Signal Analysis Package

The function can be accessed via:

>> vibrationdata > Miscellaneous > Structural Dynamics > Beam Bending > Beam with End Springs, FEA

Options for calculating the beam response to an applied force will be added soon.

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See also:  Beam Bending Finite Element Analysis

– Tom Irvine

Normal Tolerance Factors

Maximum predicted environment (MPE) levels can be derived for acoustic, shock, and vibration levels where flight data is available.  The enveloping for multiple sets of data can be performed using tolerance factors as described in the following papers.

Normal Tolerance Factors for Upper Tolerance Limits: normal_tolerance_factors.pdf

Uncertainty Margins in Aerospace Vibroacoustic Levels: uncertainty.pdf

A function for calculating tolerance factors is given in:

Matlab script: Vibrationdata Signal Analysis Package

>> vibrationdata > Miscellaneous > Statistical Distributions & Tolerance Factors > Tolerance Factors

– Tom Irvine

Electronic Circuit Board Design Reliability Lessons Learned

Here are some papers:

Electronic Circuit Board Design Reliability Lessons Learned: electronics_design_reliability.pdf

Effect of Conformal Coating on Circuit Board Vibration Response:  conformal_coat.pdf

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Steinberg’s Vibration Analysis for Electronic Equipment describes the methods for designing electronic equipment that must work with a high degree of reliability in severe shock and vibration environments. This text provides practical methods for analyzing circuit boards and other equipment. Topics include sinusoidal vibration, random vibration, fatigue, classical shock, and shock response spectrum.

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See also:  Vibrationdata Steinberg

– Tom Irvine

SDOF System Response to Initial Velocity & Displacement


I have added a function for calculating the free vibration response of an SDOF system to initial conditions to the Vibrationdata GUI package.

Matlab script: Vibrationdata Signal Analysis Package

>> vibrationdata  > Miscellaneous > Structural Dynamics > Spring-Mass Systems > SDOF Free Vibration Response to Initial Conditions

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Reference: free.pdf

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– Tom Irvine