Mode Acceleration and Inertia Relief for a Semi-definite Structural Dynamics System


Aircraft and launch vehicles behave as unconstrained systems in flight, with six rigid-body modes. These vehicles may be “trimmed” using aerodynamic control surfaces and thrust vector control to prevent rotation about the vehicle center-of-gravity (CG).

There is a need to calculate the vehicle’s displacement response to wind, gusts, buffeting and other external forces. This process requires separating the rigid-body response from the elastic response. The elastic response is the relative displacement referenced to the CG displacement. The stress and strain can then be calculated from the elastic displacement response. The method is carried out by inertia relief, where rigid-body motion is constrained by applying an inertial acceleration that is opposite to the acceleration resulting from the external forces.

Here is a paper, which is a work-in-progress inertia_relief.pdf

– Tom Irvine

Fatigue Analysis Webinars


This is a work-in-progress…

I am creating a series of webinars with Matlab exercises for fatigue analysis

Matlab script: Vibrationdata Signal Analysis Package

Here are the slides:

Unit 1  Fatigue Introduction

Unit 2  Fracture

Unit 3  Sine Vibration

Unit 4  Random Vibration

Unit 5  Rainflow Cycle Counting, Time Domain

Unit 6  Sine Sweep Vibration

Unit 7   Synthesizing a Time History to Satisfy a PSD Specification

Unit 8  Drop & Classical Shock  & Video Half-Sine SRS Animation

Unit 9  Seismic & Pyrotechnic Shock & Video Delta 4 Shock Events

Unit 10  SRS Synthesis

Unit 11  Vibration Response Spectrum

Unit 12  Rainflow Fatigue, Spectral Methods, Fatigue Damage Spectrum

Unit 13  Modifying Spectral Fatigue Methods for S-N Curves with MIL-HDBK-5J Coefficients

Unit 14a  Enveloping Nonstationary Vibration via Fatigue Damage Spectra

Unit 14b  Enveloping Nonstationary Vibration, Batch Mode for Multiple Inputs

Unit 15  Using Fatigue to Compare Sine and Random Environments

Unit 16  Sine-on-random Conversion to a PSD via Fatigue Damage Spectra

Unit 17  Non-Gaussian Random Fatigue and Peak Response

Unit 18   Acoustic Fatigue

Unit 19  Shock Fatigue

Unit 20  Fatigue Damage including Mean Stress

Unit 21  Electronic Circuit Board Fatigue, Part 1

Unit 22  Electronic Circuit Board Fatigue, Part 2

Unit 23  Time-Level Equivalence

Unit 24  Multiaxis Fatigue, Constant Amplitude Loading

Unit 25  Multiaxis Fatigue, Stress Ratio Methods

Unit 26  Multiaxis Fatigue, Variable Amplitude Loading

Unit 27  Airbus Fatigue Manual

More later…

– Tom Irvine

Waterfall SRS, 1940 El Centro Quake



The top figure is the time history from the El Centro earthquake, North-South horizontal component.  The second is the corresponding Waterfall FFT with 4 second segments and with 50% overlap.

The Waterfall FFT is calculated by first taking the complete response time history for each natural frequency of interest.  Then the time history for each is divided into segments.  Finally, the shock response spectrum (SRS) is taken for each natural frequency and for each segment, by taking the peak positive and peak negative responses.

The Waterfall SRS function is given in:

Matlab script: Vibrationdata Signal Analysis Package

>> vibrationdata > Time History > Shock Response Spectrum, Various > Waterfall FFT

See also:  El Centro Earthquake

– Tom Irvine

Tom’s Video & Animation Files

Another work-in-progress…

Step 1: Download a video file.
Step 2: Play using VLC media player.

* * * * *

Launch Vehicles

Delta 4 Heavy Launch Vehicle Shock Events

Pegasus Launch Vehicle

Linear Shaped Charge Test

* * * * *


Chinook Ground Resonance

* * * * *

Fixed Wing Aircraft

MD80 Tail Failure

Boeing 747 Wind Tunnel

C-5 Tail Wind Tunnel

Twin Commanche

* * * * *

Automotive & Transportation

Triple Trailer Oscillation

* * * * *

Shock & Vibration Testing

Generator Seismic Shaker Test

* * * * *

Fluid Systems

Pool Slosh 1

Pool Slosh 2

* * * * *


Half-Sine SRS Animation

Tom’s Conference Papers & Slide Index

I am trying to collect all my presentations. This is a work-in-progress…

Thank you,
Tom Irvine

* * * * * *

NAFEMS World Congress 2017

Introduction to Vibration

Spectral Functions

Random Vibration

Vibration Fatigue

Shock 1 & 2


* * * * * *

Aerospace Spacecraft & Launch Vehicle Dynamic Environments Conference

2017, Statistical Energy Analysis Software & Training Materials, Part 2

2016, Statistical Energy Analysis Software & Training Materials

2015, Seismic Analysis and Testing of Launch Vehicles and Equipment using Historical Strong Motion Data Scaled to Satisfy Shock Response Spectra Specifications

2014, Optimized PSD Envelope for Nonstationary Vibration

2013, Extending Steinberg’s Fatigue Analysis of Electronics Equipment to a Full Relative Displacement vs. Cycles Curve

2012,  Keynote, Dynamics Engineering: A Call to Serve  

2012, An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

2011, The NASA Engineering & Safety Center (NESC) Shock & Vibration Training Program

* * * * * *

European Space Agency


ESA Pyrotechnic Shock Distance & Joint Attenuation via Wave Propagation Analysis

ESA Shock Analysis of Launch Vehicle Equipment using Historical Accelerometer Records to Satisfy Shock Response Spectra Specifications 

2016, European Conference on Spacecraft Structures Materials and Environmental Testing

Modifying Spectral Fatigue Methods for S-N Curves with MIL-HDBK-5J Coefficients

* * * * * *

Various Vibration & Fatigue Conferences

VAL2015, A review of spectral methods for variable amplitude fatigue prediction and new results

VAL 2015, Using a Random Vibration Test Specification to Cover a Shock Requirement via a Pseudo Velocity Fatigue Damage Spectrum

ICoEV 2015, International Conference on Engineering Vibration, Derivation of Equivalent Power Spectral Density Specifications for Swept Sine-on-Random Environments via Fatigue Damage Spectra

MOVIC & RASD 2016, Multiaxis Fatigue Method for Nonstationary Vibration

* * * * * *

Shock and Vibration Exchange (formerly SAVIAC)

* * * * * *

Institute of Environmental Sciences and Technology (IEST)

IESTECH 2016, Nonstationary Vibration Enveloping Method Comparison

* * * * * *

Earthquake Engineering Conferences

16th WCEE, Seismic Analysis and Testing of Equipment using Historical Strong Motion Data Scaled to Satisfy Shock Response Spectra Specifications

* * * * * *


2003, A Time Domain, Curve-Fitting Method for Accelerometer Data Analysis

2003, Practical Application of the Rayleigh-Ritz Method to Verify Launch Vehicle Bending Modes

* * * * * *


Sine Sweep vs. Random for Pre and Post Testing


Certain equipment must be designed and tested to withstand vibration.  This is common in the automotive, aerospace, military and other industries.    The equipment is typically mounted to a shaker table and then subjected to a base input random or sine sweep vibration test.   The random vibration is usually in the form of a power spectral density (PSD).

The sine sweep or random test level may represent a maximum expected field environmental, a parts and workmanship screen, or an envelope of both.   The level may also include a statistical uncertainty margin or a safety factor.

A common practice is to perform low-level sine sweep test before and after the full-level test in order to measure the transmissibility ratio and identify natural frequencies and damping ratios.   There must be at least one base input control accelerometer and one reference accelerometer for this test, where the reference accelerometer is mounted somewhere on the test item.   The before and after transmissibility curves are then compared to assess whether any of the response peaks have shifted in frequency or magnitude.   Any shift may indicate that some fasteners have loosened or some other change has occurred.  If so, further investigation is needed.  Ideally, two curves are identical such that no further evaluation is required.

Sine sweep is the traditional vibration test for the pre and post tests.  The purpose of this paper is to determine whether random vibration can be substituted for sine sweep, via an example.  This could be done for time saving.  Also, random vibration is easier to control than sine sweep.

A difference between sine sweep and random is that all modes are excited all the time for stationary broadband random.  There is only one excitation frequency at a given time in sine sweep vibration, and each mode will be excited individually if the modal frequencies are well-separated.  In addition, the random vibration used for shaker testing typically has a bell-shaped histogram curve, whereas sine sweep vibration with constant amplitude has a bathtub-shaped histogram.

Both sine sweep and random should give the same transmissibility results for a linear system per textbook theory, but there are some practical concerns for implementation of each.  The numerical example results will show that random vibration is adequate, although sine sweep remains the best choice because it can give finer resolution.

An example is given in:  sine_sweep_random_pre&post_test.pdf

See also:

Webinar Unit 3 Sine Sweep Vibration 

Beam Bending Natural Frequencies & Mode Shapes

– Tom Irvine

Multi-axis Shock & Vibration Testing

Equipment must be designed and tested to withstand shock and vibration.  Ideally, all equipment would be tested on a shaker table with six-degree-of-freedom control (three translations and three rotations).  Such tables and control systems exist but are very expensive.  Furthermore, any multi-axis testing requires careful consideration of phase angles between the six degrees.

Another option is to test equipment on a triaxial table where the three translations are controlled, and the three rotational degrees are constrained to zero motion.  Testing on a biaxial table is yet another choice.

The most common test method, however, remains testing in each of three orthogonal axes, one axis at a time, on a single-axis shaker.  This is simplest and least expensive method.

The question arises “Should the acceleration level be increased for the case of single-axis testing?”

There is a tacit understanding that aerospace and military equipment test levels already have a sufficient uncertainty margin or safety factor so that the levels can be used without further increase.  In other words, the specifications are already intended for single-axis testing.  In many cases, a uniform level is used in each axis which is the maximum envelope of the maximum expected levels in the three axes plus some margin.

* * *

The standards which address testing equipment for earthquakes take a different approach. The following descriptions are taken from five common standards.

Only KTA 2201.4 gives a scaling formula.  This is also the only standard from the five samples which may be freely downloaded.

* * *

IEEE 344-2013  Standard for Seismic Qualification of Equipment for Nuclear Power Generating Stations

8.6.6 Multiaxis tests

Seismic ground motion occurs simultaneously in all directions in a random fashion. However, for test purposes, single-axis, biaxial, and triaxial tests are allowed. If single-axis or biaxial tests are used to simulate the 3D environment, they should be applied in a conservative manner to account for the absence of input motion in the other orthogonal direction(s). One factor to be considered is the 3D characteristics of the input motion. Other factors are the dynamic characteristics of the equipment, flexible or rigid, and the
degree of spatial cross-coupling response. Single and biaxial tests should be applied to produce adequate levels of excitation to equipment where cross coupling is significant and yet minimize the level of overtesting where the cross coupling is not significant.

* * *

KTA 2201.4   Design of Nuclear Power Plants against Seismic Events, Part 4: Components

This document may be freely downloaded: link

See paragraphs

5.3.3 Excitation Axes Simultaneity of excitation directions

Simultaneous three-axis testing is preferred. But single-axis testing can be substituted by testing in each of three axes sequentially.

The standard shows, for example, that the uniform single-axis level should be the “square root of the sum of the squares” of the three orthogonal installation site levels.

* * *

IEC 980 Recommended practices for seismic qualification of electrical equipment of the safety system for nuclear generating stations

6.2.9 Qualification test method General

As is well known, seismic excitation occurs simultaneously in all directions in a random way. According to this point of view, the test input motion should consist of three mutually independent waveforms applied simultaneously along the three orthogonal axes of the equipment.

However, taking into account that three axial testing installations are rare and that triaxial testing is desirable when significant coupling exists simultaneously between the two preferred horizontal axis of the specimens, biaxial testing with multifrequency independent input motion in the horizontal and vertical direction is an acceptable test.

Tests shall be performed according to 6.3.2 and, in terms of total duration and fatigue induced, are intended to become conservative.

In some cases, single axis tests with multiple, or single frequency excitation are also acceptable methods of test if properly justified considering the effect of coupling between axes.

* * *

Telcordia GR-63-CORE

Assumes single-axis testing.  The base input time history is specified in the standard.

* * *

IEEE 693-2005 – IEEE Recommended Practice for Seismic Design of Substations

paragraph 4.9

The shaker table shall be biaxial with triaxial preferred.

* * *

See also:

Seismic Test & Analysis Webinars

Hypersphere SRS


– Tom Irvine

Webinar Index

Here is a listing of the webinars and related materials.

Matlab script: Vibrationdata Signal Analysis Package

1. Natural Frequencies

2. Sine Vibration

3. Sine Sweep Vibration

4. Random Vibration

5. Fourier transforms

6. Leakage Error, Hanning Window

7. FFTs

8. Waterfall FFT

9. White Noise FFT

10. Sample Rate & Aliasing

11. Power Spectral Density

12. Power Spectral Density Functions of Measured Data

13. SDOF Response to Power Spectral Density Base Input

14. Synthesizing a Time History to Satisfy a PSD Specification

15. SDOF Response to Base Input in the Frequency Domain

16. Vibration Response Spectrum

17. SDOF Response to Applied Force

18. Force Vibration Response Spectrum

19. Digital Filtering

20. Digital Filtering, Part 2

21. Integration & Differentiation of Time Histories

22. Integration and Differentiation of Time Histories & Spectral Functions

23. Classical Shock Pulse

24. Seismic Shock

25. Pyrotechnic Shock

26. Pyrotechnic Shock, part 2

27. SRS Synthesis

28. Multi-degree-of-freedom SRS

29. Stress-Velocity Relationship

30. Rectangular Plate Shock & Vibration

31. Rectangular & Circular Plate Shock & Vibration

32. Electronic Circuit Board Fatigue

33. Rainflow Fatigue

34. Rainflow Fatigue for Continuous Beams

35. Using Fatigue to Compare Sine and Random Environments

36. Non-Gaussian Random Fatigue and Peak Response

37. Acoustic Fatigue

38. Electronic Circuit Board Fatigue Part 2

39. Sine-on-Random Vibration

40. Shock Fatigue

41. PSD Special Topics

42. Shock Special Topics

43. Two-degree-of-freedom System, Two-stage Isolation

44. Sine Filtering

45. Two-degree-of-freedom System with Rotation and Translation

46. Two-degree-of-freedom System with Multi-point Enforced Motion

47. Shock Response Spectrum Synthesis, Special Topics

Seismic Test & Analysis Webinars

Structural Dynamics Webinars

Fatigue Webinars

Circuit Board Shock & Vibration Analysis

HALT/HASS for Product Reliability

More later. . .

– Tom Irvine

Nonlinear Modeling of Bolted Interfaces & Joints

Mechanical joints may have nonlinear damping and stiffness, due to.frictional slipping between the connected members, etc.  I am enclosing modeling advice from a colleague and related links.

– Tom Irvine

* * *

If the nonlinearity stays fairly weak then there are a few options to do a worst case analysis:

1.) For broadband loads the linear model with low-level damping is usually a very conservative model for the system at high amplitude (in our cases we frequently see damping increase by a factor of 3-4). Of course if you have a sharp harmonic then you need to consider the downward shift in frequency, but that is usually small and you’re not likely to design something to have a strong excitation frequency just barely below resonance.

2.) To improve fidelity, these uncoupled 1DOF oscillator models with power-law dissipation (i.e. log(damping) vs log(displacement amplitude) = linear) can do a very good job of capturing how the damping changes with amplitude. It APPEARS (no guarantees with nonlinearity) that one can obtain a “worst case” analysis by using a linear model with damping near the maximum damping expected at that amplitude. However, a word of caution: the frequency shift smears the resonance peak, so one cannot necessarily assume that the damping measured by a half-power method will be accurate! We use a Hilbert transform or some other time domain technique to estimate damping at a time instant (and therefore at a certain amplitude). We have also used step-sine tests (in a paper for this year’s IMAC) with a phase condition to find the resonance frequency and damping with good success. In any event, an approach such as this will be less over-conservative than a linear model based on low-amplitude response and if one is careful it is probably possible to still make sure it is conservative.

3.) If there is nonlinearity then one should also think whether any nonlinear phenomena might come into play: super-harmonic resonance (exciting a mode at omega by applying a force at (1/2)*omega, (1/3)*omega, etc…; modal coupling (modes are excited that shouldn’t be based on linear theory), chaos, etc…

Dr. Matt Allen, University of Wisconsin-Madison

Dynamics of Bolted Interfaces

Numerical Study Iwan Model

Nonlinear Characterization of a Bolted Structure

* * *

See also:

Some Nonlinear Sine Sweep Vibration Test Data 

Tall Building Natural Frequencies and Damping