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.

The first step is to calculate the relative acceleration in the time domain.  Then integrations are performed to calculate the relative velocity and the relative displacement.  Highpass filtering and trend removal are optional steps in this process.

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 > Integrate or Differentiate >  Relative Displacement from Two Acceleration Time Histories

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

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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

Low Frequency Structural Loads for Secondary Structures & Components

Method 1

Secondary & Component loads can be specified in the form of a Mass-Acceleration curve as described in:

NASA-STD-5002, Paragraphs, 5.3.2, 5.4.

The Mass-Acceleration curve can be derived from measured flight or test data, or analytical loads. Care must be taken to avoid “double dipping,” if two or more sources are used.

This method effectively assumes that the component’s fundamental frequency is much higher than the excitation frequencies. The resulting analysis becomes a rigid-body analysis.

Method 2

Component loads can be expressed as a Vibration Response Spectrum (VRS).

This assumes that the component responds as a single-degree-of-freedom (SDOF) system subjected to base acceleration.

The Y-axis for this function is peak response acceleration. The X-axis is natural frequency (Hz). The amplification factor Q must also be noted. A family of curves for various Q factors can be included in a single plot.

The VRS curve(s) can be derived from measured flight or test data, or analytical loads. Again, care must be taken to avoid “double dipping,” if two or more sources are used.

The resulting VRS can then be used for design purposes by picking off the peak response acceleration value for a given natural frequency and Q.

A similar VRS can be derived for relative displacement. This is important for cases where clearance, sway space, alignment, or isolator deflection are concerns.

My colleagues and I used the VRS method at my previous workplace, Orbital Sciences Corporation, Chandler, AZ.

I have posted references for this method at:  VRS Link

Method 3

A measured or synthesized acceleration time history can be used to base drive a finite element model of the component, as a modal transient analysis.

The time history can be derived from measured flight or test data, or analytical loads. Again, care must be taken to avoid “double dipping,” if two or more sources are used.

I also occasionally used this method at Orbital Sciences Corporation.

Shaker Table Testing

Avionics components are typically tested over the frequency domain 20 to 2000 Hz for random vibration.

Components may also need to withstand transportation vibration below 20 Hz.

See SMC-TR-06-11, sections 3.24 – 3.26

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See also:

Launch Vehicle Coupled Loads Analysis (CLA) Upper Frequencies

Equivalent Static Loads for Random Vibration

Mass Acceleration Curves

Effective Modal Mass

JPL D-5882 Trubert

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– Tom Irvine

Optimized PSD Envelope for Nonstationary Vibration


There is a need to derive a PSD envelope for nonstationary acceleration time histories, including launch vehicle data, which may be similar to that shown in the above figure.

A PSD can be derived using rainflow fatigue cycle counting along with a Miners-type relative fatigue damage index.  The enveloping is then justified using a comparison of fatigue damage spectra between the candidate PSD and the measured time history.

The derivation process can be performed in a trial-and-error manner in order to obtain the PSD with the least overall GRMS level which still envelops the flight data in terms of fatigue damage spectra.  The Dirlik method can be used to calculate the fatigue damage spectrum of each candidate PSD in the frequency domain, instead of using the longer, time domain synthesis approach.

Furthermore, this can be done for a number of Q and fatigue exponent permutations for the case where these values are unknown.  This adds conservatism to the final PSD envelope.

Again, the goal is to derive the minimum PSD which envelopes the measured data in terms of fatigue.  The PSD’s duration is selected by the user.  It may or may not be the same as that of the measured data.  The method will scale the PSD to compensate for either a shorter or longer duration.

Statistical uncertainty factors or safety margins can then be added as a separate, post-processing step.

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Here is a C++ program for applying this acceleration PSD derivation method for a user-supplied base input time history.



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Here is a similar C++ program for deriving a force PSD for an applied force time history.



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Here is a similar C++ program for deriving an acoustic SPL for an acoustic pressure time history.  The  applied and derived pressure fields are assumed to be uniform and fully correlated.



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Here is a Matlab MEX script set for an acceleration PSD:

Instructions: Go to the Matlab Command Window.


>>mex -setup

The C++ source code is compiled with Matlab as:

>>mex rainflow_mex.cpp

Then run the script:


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Here is a corresponding paper:  optimize_psd_fds.pdf 

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See also:

Rainflow Fatigue Cycle Counting

Fatigue Damage Spectrum, Time Domain

Dirlik Rainflow Counting Method from Response PSD

Here is a previous method which performs fatigue comparison calculations strictly in the time domain for a single PSD candidate.  It does not have automatic optimization capabilities, however.

Enveloping Nonstationary Random Vibration Data

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– Tom Irvine

The Great Spacecraft Base Input Vibration Test Debate


Engineers prepare the MESSENGER spacecraft for a vibration test at The Johns Hopkins University Applied Physics Laboratory, Laurel, Md., where the Mercury-bound NASA spacecraft was designed and built.

* * *


Flight configured spacecraft are subjected to base input vibration tests for certain programs.  The spacecraft are often one-of-a-kind, so the vibration test is effectively a proto-qualification test covering both design and workmanship verification.

The tests may be sinusoidal or random.   The sine vibration is typically at low frequencies, below 100 Hz.

My colleagues are divided on whether these spacecraft system-level tests are prudent and effective.   The following is a brief summary of key points.

* * *

Arguments Against Spacecraft Vibration Testing

The following assertions are made by A.M. Kabe and E. Perl from The Aerospace Corporation.

Vibration tables cannot replicate the impedance of the launch vehicle interface, nor the interaction that occurs between the launch vehicle and spacecraft when they are a coupled system; hence, the modes of vibration will not be the same as in flight.

Only translational motions are applied at the base, one axis at a time, whereas during flight, the launch vehicle/spacecraft system will vibrate simultaneously in all six degrees of freedom at each mass point and at each interface point between the launch vehicle and spacecraft.

The total acceleration load during powered flight also depends on the spacecraft rigid-body acceleration which a shaker cannot replicate.

Derivation of a “base input” environment from a few accelerometer locations at the launch vehicle/spacecraft interface will generally lead to an over prediction of the motions at the interface, since local deformations are mapped on the assumption that the interface acts as a rigid plane.

The use of (response)/(base motion) ratios to extract damping, a common practice, is not a valid approach for multi-degree-of-freedom systems it fails to account for the mode participation factor.

The test article may not include the actual spacecraft launch vehicle adapter or the propellant mass in the tanks because of safety and contamination concerns.

The test requirement forces the spacecraft organization to design its system to not only survive the launch environment, but also to survive an artificial test that more often than not produces overly conservative loads in many parts of the structure while not adequately testing others.

The test can pose unnecessary risk of damaging flight hardware late in the program.

Note that A.M. Kabe advocates acoustic reverberant chamber testing of spacecraft as a workmanship screen, as an alternative to base shake testing.  He also favors shaker table vibration testing on a component or subsystem level.

* * *

Arguments For Spacecraft Vibration Testing

The following justification points are made by NASA engineers Daniel Kaufman, Scott Gordon, Steve Hendricks and Dennis Kern.

Essentially all current launch vehicle organizations (Delta, Atlas, Taurus, Pegasus, Ariane, HII, Proton, Long March, Falcon, etc.) specify and require or strongly recommend a spacecraft sine or random vibration test.

Note that this point needs further investigation.  Some launch vehicle providers may specify optional sine vibration levels depending on the coupled-loads analysis (CLA). 

Testing is also required by NASA documents, such as NASA-STD-7002A (2004) & GSFC-STD-7000 (2005).

Insurance companies require vibration tests on all commercial communications satellites.

Some test facilities have the capability to perform simultaneous multi-axis vibration testing, as needed.

The vibration test provides qualification for tertiary/ancillary hardware that would not otherwise be tested.  This includes:  Cable harnesses, bellows, connectors, actuators, plumbing lines, wave guides, brackets, dampers, shades and shields, articulation/deployment mechanisms, shunt heaters, louvers, purge equipment, hinges and restraints, blankets/supports.

The test provides an opportunity to determine the structural linearity in the operational vibration range of response.   Note that linearity is a typical CLA assumption.

Force limiting reasonably accounts for the interaction with the base motion, and has been effectively employed in spacecraft vibration testing, thus reducing the potential for an over-test at the spacecraft’s natural frequencies in the test configuration.  The force limiting takes into account the CLA response levels.

Force gauges under the spacecraft provide a very accurate method of measuring and limiting to the CLA loads during the vibration test for mid to high apparent mass modes.

Numerous case histories have shown that vibration testing is effecting for uncovering design or workmanship flaws which would have otherwise caused mission degradation or failure.

As an aside, NASA/GSFC typically uses sine vibration testing, whereas JPL tends to use random.

A few examples from sine testing at GSFC are:

  • TRMM:  During Observatory sine testing, found that the NASDA supplied PAF clamp band had insufficient tension and gapped during the test.  As a result, the clamp band tension was increased for flight.
  • GOES had a workmanship problem involving a missing or loose bolt which caused structural failure of a mission-critical antenna. It was detected during the lateral sine test.
  • NOAA-K experienced IMU saturation during sine sweep testing.  Because the spacecraft IMU provides guidance information for the Titan II launch vehicle during ascent, IMU saturation during launch would have resulted in a mission failure.  Changes were made and launch vehicle restraints were implemented to resolve the problem, including wind restrictions at launch and a commanded first stage shutdown vs. fuel depletion.
  • TDRS-H: During the sine vibration test, the first two modes for the Space Ground Link antenna (SGL) were lower than predicted by the model.  The first mode dropped from 15 Hz to 11 Hz and the second mode dropped from 33 Hz to 25 Hz.  It turned out that the mathematical model of this “simple” antenna was wrong and therefore the Verification Loads Cycle had to be rerun.

A few examples from random vibration testing at JPL are:

  • Cassini: Experienced an RTG electrical short to its spacecraft mount in system random vibration test. Significant degradation in spacecraft electrical power could have resulted. Spacecraft mount was redesigned.
  • CloudSat: Cloud Profiling Radar waveguide failure in spacecraft random vibration test due to apparent poor workmanship of adhesive bonding.  Possible loss of science data averted.
  • MER 1: Fundamental modes of the Rover in spacecraft random vibration test were 20% greater than predicted in all three axes. (Fixed base modal test had been performed on Rover, Lander, and Cruise Stage separately; FE models were then combined. Estimated stiffness of Lander attachment to Rover was too low.) FE model was updated just in time for the verification CLA cycle. Vibration test also revealed improper torque of bolts on some tanks in low level runs. Bolts were properly torqued and test completed successfully.
  • MSL Rover: experienced several motor encoder screws backed out of at least one of the Rover actuators during Rover random vibration test.  The actuators are used throughout Rover and the issue was unlikely to have otherwise been found before launch, which could have been a serious threat to the mission.

JPL prefers random vibration because it easier to control, particularly with respect to force limiting.

Note that JPL tested the SMAP spacecraft to the following workmanship PSD:

20 Hz to 250 Hz, 0.01 G^2/Hz

The overall level was 1.5 GRMS.  The duration was one minute.  The vibration was applied in the vertical axis and in one “45 degree” lateral axis.

Sine Sweep Control

Sine sweep vibration is more difficult to control than random especially for the case of lightly-damped modes.  The sweep rate, compression factor, and tracking filter must be selected with great care.

INVAP experienced control issues during sine vibration testing of the ARSAT-1 structural test model.  But note that dynamic simulator were used for the test, which can have much less damping than the actual flight hardware.

* * *

All things considered, I favor spacecraft shaker table vibration testing.  

Note that a spacecraft may also be tested in an acoustic reverberant chamber.  Typical acoustic test specifications extend over the frequency domain up to 10 KHz.  See also NASA-STD-7001A.  

The acoustic test serves a different purpose than the base shake test, although there could be some overlap in terms of workmanship screening.

The acoustic test represents the airborne acoustic environment inside the payload fairing, particularly for the liftoff event.   In contrast, the shaker table test represents structural-borne energy transmitted from the launch vehicle to the base of the spacecraft during powered flight.  This energy could come from pogo, thrust oscillation or a main-engine cutoff (MECO) event.

* * *

Other spacecraft tests include static proof testing and modal surveys.  Static proof testing is particularly important for composite materials.

* * *


What is your experience in vibe testing spacecraft that normally would be fueled with Hydrazine for launch? Do you vibe with empty tanks, fill them with DI water, or do something else? There are obvious challenges with each case from not being flight representative, to having to bake out the water prior to fueling. Can you please let me know what your experience is.

Answer from a NASA colleague:

It would be very rare to have to test wet. Pre-test analysis is used to check whether test objectives can be met dry.

Answer from another NASA colleague:

Typically we would either test with dry tanks or with DI water. We have also used mass simulators attached externally to the tank structure to simulate the mass-loading of the propellant during vibration testing. I know some folks use isopropyl alcohol (IPA) in their tanks as a propellent simulator. Less dangerous than testing with a live propellant but this can still be a problem as the flammability of the IPA brings a number of additional safety considerations into the mix during the test.

We do sine vibration testing on our spacecraft as a final dynamic verification that everything will perform as expected after being exposed to the low-frequency launch environment. We perform analysis to compare the response of a wet vs dry spacecraft to make the determination if the mass of the propellant has a significant effect on the responses in critical areas. If based on the analysis results, we don’t think we can achieve the goals of the vibration test with dry tanks, then we would push to do the test with a propellant simulator (typically DI water). Most tanks tend to have a pretty direct load path to the spacecraft-launch vehicle interface so the mass loading of the tank doesn’t usually drive the spacecraft responses and you can make the case based on analysis that a dry test will meet the test goals.

Recently, we had a spacecraft that had tank modes at 35 Hz when filled that drove the response of hardware mounted to a spacecraft deck. When the tanks were empty you didn’t get that interaction. We worked with the propellant group to run the test with DI water in the tanks. The prop folks didn’t like it at first but after looking into it, the drying process wasn’t a significant schedule hit and could be performed in parallel with other spacecraft activities.

Bottom line is we don’t have a single fixed approach to testing wet vs dry. We look at the goals of the test and the risks of not testing in a flight like configuration vs the impact to the verification flow to make the decision.

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See also:

NASA-HDBK-7008, section 7.1.1.

Spacecraft Vibration Testing: Benefits and Potential Issues
Ali R. Kolaini, Walter Tsuha and Juan Fernandez

Comparison of Direct Acoustic Impingement of Avionics
Equipment to Vibration Base Shake Response  13_Tyler_Scogin.pdf

Substituting a Shaker Vibration Test for a Reverberant Acoustic Test

Derivation of the Sinusoidal Sweep Vibration Test Levels for the Cosmic Background Explorer COBE.pdf

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– Tom Irvine

Integrating Accelerometer Time History Signals

The scripts in integrate an acceleration time history to a velocity time history and/or a velocity time history to a displacement time history.

The main script is: integrate_th.m

The remaining scripts are supporting functions.

Integrating acceleration to velocity typically causes a spurious offset in the velocity signal, which in turn causes a “ski slope” effect in the resulting displacement signal.

So options are included for fading, trend removal, and mean filtering.  These options must be used with “engineering judgment.”

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Here is a script for differentiating a time history  differ.m

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These scripts in correct an acceleration time history so that its corresponding velocity and displacement time histories each oscillate
about the zero baseline.

The main script is: acceleration_correction.m

The remaining scripts are supporting functions.

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– Tom Irvine

Seismic Peak Ground Acceleration

The Iwate-Miyagi Nairiku earthquake struck northeast Honshu, Japan, on 14 June 2008.

This earthquake had a moment magnitude Mw 6.9 according to the USGS.

The peak ground acceleration (PGA) had a maximum vector sum (3 component) value of 4278 cm/sec^2 (4.36 G).

This is the highest ever recorded PGA, although other quakes have had higher moment magnitudes.  The Richter and moment magnitudes are a measure of the total energy released by a quake.

The PGA is measured at a point.  It depends on soil conditions, distance from the hypocenter, and other factors.


Masumi Yamada et al (July/August 2010). “Spatially Dense Velocity Structure Exploration in the Source Region of the Iwate-Miyagi Nairiku Earthquake”. Seismological Research Letters v. 81; no. 4;. Seismological Society of America. pp. 597–604. Retrieved 21 March 2011.

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Tohoku, Japan Earthquake 2011

The 2011 earthquake off the Pacific coast of Tōhoku was a magnitude 9.0 (Mw) undersea megathrust earthquake off the coast of Japan that occurred at 14:46 JST (05:46 UTC) on Friday 11 March 2011.

The largest peak ground acceleration (PGA) of 2.7 G was recorded in the North-South direction at Miyagi prefecture – MYG04 station.

Reference 1

Reference 2

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The highest PGA for earthquakes in the USA was 1.7 G for the 1994 Northridge, California quake, which had a 6.7 moment magnitude.

Reference:  Lin, Rong-Gong; Allen, Sam (26 February 2011). “New Zealand quake raises questions about L.A. buildings.” Los Angeles Times (Tribune). Retrieved 27 February 2011.

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The peak ground velocity (PGV) has a better correlation with structural damage according to some sources.

The largest recorded ground velocity from the 1994 Northridge earthquake, made at the Rinaldi Receiving station, reached 183 cm/sec (72 in/sec).

Reference:  USGS ShakeMap

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Further information is given at:  Vibrationdata Earthquake Engineering Page

– by Tom Irvine

The Great Amplitude Format Debate

Shock and vibration test specifications for avionics and military equipment have almost always been specified in terms of acceleration. The main reason is that acceleration can easily be measured by accelerometers.

Velocity sensors are also available but are less common.

Gaberson, Chamblers et al, claim that pseudo velocity bests represents the damage potential of a shock or vibration event.  Pseudo velocity is the relative displacement multiplied by the natural frequency (rad/sec).  This assertion has merit.  I have written a paper on this subject at: sv_velocity.pdf

On the other hand, Steinberg gives empirical formulas for the fatigue potential for both shock and vibration for circuit boards in terms of relative displacement.  The formulas are given in Steinberg’s Book.

The shock response spectrum can be plotted in tripartite format, showing each of the three amplitude metrics as a function of natural frequency.  A good engineering practice is to review all three response parameters in this format for thoroughness.

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There are two pseudo velocity metrics.

Let omega be the natural frequency in (rad/sec)

The pseudo velocity shock spectrum (PVSS) is calculated by multiplying the relative displacement SRS value by omega.

The acceleration pseudo velocity shock spectrum (APVSS) is obtained by dividing each acceleration SRS value by omega.

Dr. Howard Gaberson has generated some examples which show that the PVSS and APVSS are nearly equal except at very low natural frequencies where the APVSS tends to be higher.

In addition, the true relative velocity can be calculated using the method in:  ramp_invariant_base.pdf

The two pseudo velocity metrics and the true relative velocity metric can be used somewhat loosely and interchangeably in regard to damage potential estimation.  Further experiments and research are needed to refine these concepts.

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See also:

Dr. Howard Gaberson’s Papers

SRS Tripartite

Stress-Velocity Relationship

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– Tom Irvine

FAQ: Can Shock or Vibration be used to Cover Acceleration?


Neither shock nor vibration response should be used to cover rigid-body acceleration because material limits depend on the strain rate and on the duration of the load.

Furthermore, static deflection shapes differ from dynamic mode shapes.

Material Stress Limits

The following is an excerpt from Reference 1 with some minor editing:

A material can sometimes sustain an important dynamic load without damage, whereas the same load, statically, would lead to plastic deformation or to failure.  Many materials subjected to short duration loads have ultimate strengths higher than those observed when they are static.

Hopkinson noted that copper and steel wire can withstand stresses that are higher than their static elastic limit and are well beyond the static ultimate limit without separating proportionality between the stresses and the strains.  This is provided that the length of time during which the stress exceeds the yield stress is of the order of 1 millisecond or less.

From tests carried out on steel (annealed steel with a low percentage of carbon) it was noted that the initiation of plastic deformation requires a definite time when stresses greater than the yield stress are applied.  It was observed that this time can vary between 5 milliseconds (under a stress of approximately 352 MPa) and 6 seconds with approximately 255 MPa; with the static yield stress being equal to 214 MPa).  Other tests carried out on five other materials showed that this delay exists only for materials for which the curve of static stress deformation presents a definite yield stress, and the plastic deformation then occurs for the load period.

The equivalent units are as follows

Table 1.  Annealed Steel Test Results






5 msec for plastic deformation onset



6 sec for plastic deformation onset



Static Yield Stress



Dynamic Strength

Reference 2 notes:

As far as steels and other metals are concerned, those with lower yield strength are usually more ductile than higher strength materials.  That is, high yield strength materials tend to be brittle.  Ductile (lower yield strength) materials are better able to withstand rapid dynamic loading than brittle (high yield strength) materials.  Interestingly, during repeated dynamic loadings low yield strength ductile materials tend to increase their yield strength, whereas high yield strength brittle materials tend to fracture and shatter under rapid loading.

Reference 2 includes the following table where the data was obtained for uniaxial testing using an impact method.

Dynamic Strengthening of Materials


Static Strength


Dynamic Strength (psi)

Impact Speed


2024 Al (annealed)




Magnesium Alloy




Annealed Copper




302 Stainless Steel




SAE 4140 Steel




SAE 4130 Steel








Shock vs. Acceleration

The following paragraph is taken from Reference 3.

Acceleration loads are expressed in terms of load factors which, although dimensionless, are usually labeled as “g” loads. Shock environments (methods 516.5 and 517) are also expressed in “g” terms. This sometimes leads to the mistaken assumption that acceleration requirements can be satisfied by shock tests or vice versa. Shock is a rapid motion that excites dynamic (resonant) response of the materiel but with very little overall deflection (stress). Shock test criteria and test methods cannot be substituted for acceleration criteria and test methods or vice versa.


1.  C. Lalanne, Sinusoidal Vibration (Mechanical Vibration and Shock), Taylor & Francis, New York, 1999.

2.  R. Huston and H. Josephs, Practical Stress Analysis in
Engineering Design, Dekker, CRC Press, 2008.  See Table 13.1.

3.  MIL-STD-810F, Method 513.5, Section 1.3.3 Acceleration versus shock

Further information is given at:  Vibrationdata Acceleration Page

– by Tom Irvine