Some Nonlinear Sine Sweep Vibration Test Data

Certain equipment must be designed and tested to withstand external vibration excitation.  This is common in the military, naval, aerospace and other industries.

The equipment is typically mounted on a shaker table and subjected to base excitation.  The input may be random vibration if the field environment is likewise.  In other cases, random vibration is used to verify the integrity of parts and workmanship separately from the maximum expected field environment.

The random vibration is typically specified as a power spectral density (PSD).  Note that the workmanship screen and field level can be enveloped by a single PSD. A goal is to verify that the equipment operates properly before, during and after the random vibration test.

A more thorough test is to perform a sine sweep test before and after the random vibration test.   A response accelerometer is mounted on the test article, in addition to the control accelerometer at the base input location.   The objective is to determine whether any natural frequencies have shifted, or any other changes have occurred, as a result of the random test.  Such changes could indicated loosened fasteners, crack formation or other defects.

A case history is given next.  The data was sent to me by a colleague.  I have requested further information on the equipment and will post a photo or diagram later if permission is granted.


Figure 1.

Figure 2.

A rocket engine assembly was subjected to a sine sweep test in conjunction with a random test.  A resonant response occurred when the excitation frequency was swept through 85 to 86 Hz as shown in Figure 1.  The equipment response would have had a similar frequency content to the input if it had been a well-behaved, linear, single-degree-of-freedom system.  The response Fourier transform for the corresponding duration did have a spectral peak at 85.45 Hz matching the sweeping input frequency as shown in Figure 2.

(Note that this is an approximation because the Fourier transform is taken over a short duration and represents an average, whereas the input frequency has instantaneous change.)

But the response also showed integer harmonics with the highest peak at 683.6 Hz, which was 8x the fundamental frequency.

Please let me know if you have observed similar effects or have other insights.  Hopefully, I can post more details later…

Sine Sweep Time History Data

Thank you,
Tom Irvine

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My colleague Albert Turk sent me a reply, paraphrased as follows:

I suspect a component with a resonance at the input frequency that is excited to the point of metal-to-metal impact. I have seen data from repetitive impact machines (HASS) and also from gunfire (50 cps) that had these integer multiples.

If so, the sinusoidal excitation has turned the assembly into a repetitive impact machine near 85 Hz. It would be interesting to see if there is a sine input amplitude threshold below which this suddenly goes away.

And Steve Zeise wrote:

I have observed this phenomenon and tracked it down to loose joints introducing impacts into the system.

Note that joints can slip under shock & vibration loads.

“Loss of clearance” of “loss of sway space” may be appropriate, related terms to describe the problem shown in the data.  Further investigation is needed.

Peak Response to Random Vibration with Probability of Exceedance

An important parameter in random vibration analysis is the peak response, which can be the maximum relative displacement, velocity, acceleration, stress or strain. The peak response can then be compared with the threshold for yielding, ultimate failure, etc. The peak is also important for fatigue analysis, particularly for materials with higher exponents. A common approach is to consider that the peak response is 3σ, where 1σ is the standard deviation. But higher responses often occur > 3σ.

This paper presents a method for estimating the peak response for a desire probability of exceedance.  peak_response_random.pdf

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The equation is included in: Vibrationdata Matlab Signal Analysis Package

The function can be accessed via:

vibrationdata > Miscellaneous Function I > SDOF Response: Sine, Random & Miles > SDOF Response: Peak Sigma for Random Base Input > Risk of Overshoot, XRS, URS

This function may also be used for MDOF systems if the positive slope zero-crossing rate is used in place of the natural frequency.

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See also:

European Cooperation for Space Standardization, Mechanical Shock Design and Verification Handbook, ECSS-E-HB-32-25A

Equivalent Static Loads for Random Vibration

– Tom Irvine

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

Webinar 41 – PSD Special Topics

1. Band-Splitting
2. Time-Level Equivalence
3. PSD Synthesis using Sine Series

PowerPoint Slides:  webinar_41_psd_topics.pptx

Audio/Visual File:

NESC Academy PSD Special Topics – Recommend viewing in Firefox with Sliverlight Plugin

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Matlab script: Vibrationdata Signal Analysis Package

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Here is a brief guideline paper from Martin-Marietta:  bandsplit.pdf

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Thank you,
Tom Irvine

Swept Sine-on-Random Environments

Launch vehicles may encounter a variety of mixed sine and random vibration environments during powered flight. The random vibration is typically driven by turbulent boundary layers, shock waves, and other aerodynamic flow effects. The sine vibration may be due to a thrust oscillation for the case of a solid motor. Furthermore, the thrust oscillation frequency and amplitude may each vary with time. Both the sine and random environments may thus be nonstationary.

Avionics components must be designed and tested to withstand the composite vibration environment. A single power spectral density specification which envelops the complete environment is usually desired for simplicity. Furthermore, the power spectral density specification is assumed to have a corresponding time history which is both stationary and Gaussian.

Traditional specification derivation methods involve assuming piecewise stationary flight data and making a maximum envelope from the piecewise segments. A more discerning method is to use the fatigue damage spectrum method to derive a stationary power spectral density which yields the equivalent fatigue damage of the composite nonstationary flight data. This paper demonstrates this fatigue damage enveloping method.   swept_sine_fds.pdf

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See also:

Matlab script: Vibrationdata Signal Analysis Package

Optimized PSD Envelope for Nonstationary Vibration

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– Tom Irvine

Using Random Vibration Testing to Cover Shock Requirements

Aerospace and military components must be designed and tested to withstand shock and vibration environments.

Some of this testing occurs as qualification, whereby a sample component is tested to levels much higher than those which it would otherwise encounter in the field. This is done to verify the design.

Now consider a launch vehicle component which must withstand random vibration and pyrotechnic shock. The random vibration specification is in the form of a power spectral density (PSD). The shock requirement is a shock response spectrum (SRS).

Pyrotechnic-type SRS tests are often more difficult to control and thus more expensive than shaker table PSD tests. Furthermore, some lower and even mid-level SRS specifications may not have the true damage potential to justify shock testing.

The fatigue damage spectrum (FDS) can be used to further determine whether the PSD specification covers the SRS requirement.  If so, then shock testing can be omitted in some cases.

Here is a paper: random_cover_shock.pdf

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See also:

Shock Severity Limits for Electronic Components

Rainflow Cycle Counting

Fatigue Damage Spectrum

Matlab Mex – fds_main script

SDOF Response to an acceleration PSD Base Input – VRS script with FDS option

Matlab script: Vibrationdata Signal Analysis Package  – SRS damped sine time history synthesis function

Direct Fatigue Damage Spectrum Calculation for a Shock Response Spectrum

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– Tom Irvine

Webinar Unit 4 Random Vibration

Here are the materials for random vibration:

PowerPoint File: webinar_random_vibration.ppt

Audio/Visual File: NESC Academy Random Vibration
– Recommend viewing in Firefox with Sliverlight plugin

Matlab script: Vibrationdata Signal Analysis Package

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Python version PowerPoint File:  webinar_4_Random_python.ppt

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See also:


Random Vibration & Integration of the Normal Distribution Curve

Rayleigh Distribution


Vibrationdata Webinars

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– Tom Irvine

Response of an SDOF System to an Applied Force PSD or Acoustic SPL


This set of scripts calculates the response of an SDOF system to an applied force PSD.

A transfer function is derived. Then its square is applied to the force PSD to calculate the response, using point-by-point multiplication and integration.

Important note:  The following scripts in this post require the supporting functions which are in the above zip file.

And here is a script that calculates the vibration response spectrum for an applied force:  vrs_force.m

It also has an option for a fatigue damage spectrum.

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Here is a script which calculates the SDOF response to an external SPL.  This  assumes that the sound pressure can be converted to an equivalent force for a fully correlated, uniform pressure distribution across the top surface of the mass with no flanking paths:  sdof_spl.m

And here is the full vibration response spectrum for the SPL:  vrs_spl.m

Both of these scripts require these additional supporting functions:


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Here are related papers.

The Steady-state Response of Single-degree-of-freedom System to a Harmonic Excitation Force:  sforce.pdf

An Introduction to Frequency Response Functions:  frf.pdf 

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See also:

Modal Transient Analysis Numerical Engine, Applied Force

Vibrationdata Structural Dynamics – Classical Methods

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– Tom Irvine