Nonstationary Vibration Enveloping Method Comparison

There is a need to derive a power spectral density (PSD) envelope for nonstationary acceleration time histories, including launch vehicle data, so that components can be designed and tested accordingly.

Three methods are considered in the following paper using an actual flight accelerometer record.

The first method divides the accelerometer data into segments which are idealized as “piecewise stationary” in terms of their respective PSDs. A maximum envelope is then drawn for the superposition of segment PSDs. This method initially requires no assumptions about the response characteristics of the test item, but vibration response spectra may used for peak clipping as shown in the example.

The following two methods apply the time history as a base input to a single-degree-of-freedom system with variable natural frequency and amplification factors. The response of each system is then calculated. Upper and lower estimates of the amplification factor can be used to cover uncertainty.

The first of this pair is the energy response spectrum (ERS), which gives energy/mass vs. natural frequency, as calculated from the relative response parameters.

The final method is the fatigue damage spectrum (FDS), which gives a Miners-type relative fatigue damage index vs. natural frequency based on the response and an assumed fatigue exponent, or upper and lower estimates of the exponent.

The enveloping for each of the response spectra methods is then justified using a comparison of candidate PSD spectra with the measured time history spectra. The PSD envelope can be optimized by choosing the one with the least overall level which still envelops the accelerometer data spectra, or which minimizes the response spectra error.

This paper presents the results of the three methods for an actual flight accelerometer record. Guidelines are given for the application of each method to nonstationary data. The method can be extended to other scenarios, including transportation vibration.

Paper:  enveloping_comparison.pdf

Slides:  Irvine_IEST_2016.pptx

The Matlab scripts for the enveloping methods are included in  Vibrationdata GUI package

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See also:

Rainflow Cycle Counting

Energy Response Spectrum

Dirlik Rainflow Counting Method from Response PSD

Fatigue Damage Spectrum, Frequency Domain

Optimized PSD for Nonstationary Vibration Environments

– Tom Irvine

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

Fatigue Damage including Mean Stress

Options for including mean stress have been added to the Vibrationdata GUI package’s Fatigue Toolbox, for both stress time histories and PSDs.

Four methods are available:


Matlab script: Vibrationdata Signal Analysis Package

Here are some charts from Iowa State University: Fatigue Mean Stress

– Tom Irvine

NASGRO Coefficients

The following scripts calculate the A, B, C, P coefficients to model a set of SN curves with varying R values:

sn_curve_fit_R.m is the main script.

The remaining scripts are supporting functions.

The equations are given in: sn_curvefit_equation.pdf

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– Tom Irvine

Webinar 37 – Acoustic Fatigue

PowerPoint Slides:  webinar_37_acoustic_fatigue.pptx

Audio/Visual File:

NESC Academy Acoustic Fatigue – Recommend viewing in Firefox with Sliverlight Plugin

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Rainflow Fatigue Posts

Acoustic Fatigue of a Plate

Acoustic Power Spectra

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Matlab script: Vibrationdata Signal Analysis Package

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See also: Vibrationdata Webinars

Thank you,

Tom Irvine

Low Risk Parts for Spaceflight

NASA-STD-5019, Fracture Control Requirement for Spaceflight Hardware (excerpt) Low-Risk Part

This section addresses parts that can be classified non-fracture critical because of large structural margins and other considerations that make failure from a pre-existing flaw extremely unlikely.

a. For a part to be classified low risk, it shall be constructed from a commercially available material procured to an aerospace standard or equivalent.

b. Aluminum parts shall not be loaded in the short transverse direction if this dimension is greater than 7.62 cm (3 in).

c. A part whose failure directly results in a catastrophic hazard shall be excluded from being classified low risk, except when the total (unconcentrated) stresses in the part at limit load are less than 30 percent of the ultimate strength for the material used and requirements (1) through (3) and either (4) or (5) are met.

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Thus fatigue and fracture analyses are not required for parts with peak stress less than 0.3*ultimate.

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The peak stress can also be compared to the endurance limit, but some materials do not have identified endurance limits.  Here is a rule-of-thumb for these cases from NASA-HDBK-5010.

Perform endurance limit analysis to show the maximum stress does not exceed the endurance limit or

Smax < Ftu/( 4{1-0.5 R} )


Smax is the local concentrated stress
Ftu is the tensile ultimate stress
R is the ratio of minimum stress to maximum stress in a fatigue cycle

Note that R=-1 for fully reversed stress with zero mean stress.  For this case:  Smax < Ftu/6

This formula has some limitations and needs further research.

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– Tom Irvine

SDOF Response to Sine or Sine Sweep Base Input, Rainflow


Rainflow fatigue cycles can be easily calculated for a single-degree-of-freedom subjected to a sine or sine sweep base input.  The reason is that each pair of consecutive positive and negative response peaks forms a half-cycle.

The relative fatigue damage can then be calculated from the rainflow cycles.

Here are Matlab scripts for performing the rainflow and damage calculations.

rainflow_sine.m is for the case where the natural frequency is known.

rainflow_sine_fds.m gives the fatigue damage spectrum for a family of natural frequencies.

The remaining scripts are supporting functions.

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See also:

Rainflow Cycle Counting


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– Tom Irvine

Acoustic Fatigue of a Plate

Here is a paper showing how fatigue damage can be calculated from a stress response PSD for a plate excited by an acoustic pressure field:  acoustic_fatigue_plate.pdf

The calculation method is given at:
Fatigue Damage for a Stress Response PSD

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The Matlab scripts for calculating the plate responses are included in the vibroacoustics section of the Vibrationdata GUI package, available at:   Vibrationdata Signal Analysis Package

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See also:

Steady-State Response of a Rectangular Plate Simply-Supported on All Sides to a Uniform Pressure:  ss_plate_uniform_pressure.pdf

Steady-State Vibration Response of a Plate Fixed on All Sides Subjected to a Uniform Pressure: fixed_plate_uniform_pressure.pdf

Aircraft Fuselage Fluctuating Pressure

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– Tom Irvine

Fatigue Damage for a Stress Response PSD

Here is a paper.

Estimating Fatigue Damage from Stress Power Spectral Density Functions: estimate_fatigue_psd.pdf

Note that the duration for the example in the paper was 600 sec.

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This following Matlab program calculates the cumulative rainflow fatigue damage for an input stress PSD using the following wideband methods:

1. Wirsching & Light
2. Ortiz & Chen
3. Lutes & Larsen, Single-Moment
4. Benasciutti & Tovo, alpha 0.75
5. Dirlik
6. Zhao & Baker


Random Vibrations: Theory and Practice (Dover Books on Physics)

The stress PSD and the fatigue strength coefficient must have consistent stress units.

The input PSD must have two columns: freq(Hz) & stress(unit^2/Hz)

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Main scripts:

Vibrationdata Signal Analysis Package

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The following values are “For Reference Only.”

m = fatigue exponent
A = fatigue strength coefficient

Aluminum 6061-T6 with zero mean stress

A=9.7724e+17 (ksi^9.25)
A=5.5757e+25 (MPa^9.25)

Butt-welded Steel Joints

A=1.255e+11 (ksi^3.5)
A=1.080e+14 (MPa^3.5)

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See also:

Rainflow Fatigue

Mrsnik, Janko Slavic, Boltezar, Frequency-domain methods for a vibration-fatigue-life estimation – Application to real data:  mrsnik_article_vib_fatigue.pdf

Spectral Moments Notes

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– Tom Irvine