In-plane Rectangular Plate Vibration, FEA


I have written a function for calculating the natural frequencies and modes shapes of a rectangular plate undergoing in-plane vibration via the finite element method.

The function is included in the software posted at in the Matlab GUI package.

The function can be accessed via:

vibrationdata > Structural Dynamics > Plates, In-plane

I will post a corresponding paper later…

– Tom Irvine

Kinetic & Strain Energy Formulas

Kinetic & strain energy formulas can be used as a basis for Rayleigh-Ritz or finite element analysis of vibrating structures.

Equations for common structures and loading conditions are given in strain_energy.pdf

The cases include:

Uniform Bar under a Constant Axial Load
Bar under a General Axial Load
Beam in Bending
Circular Rod in Torsion
Beam or Rod in Shear
Rectangular Plate in Bending
Rectangular Plate in Bending & Shear, Stress-Strain Formulation
Circular Plate in Bending
Rectangular Plate, In-plane Displacement
Axial Spring
Ring, In-Plane, Extensional Displacement

See also: Continuum Mechanics

– Tom Irvine

Vibroacoustics/Statistical Energy Analysis


Like Robert Frost, I have miles to go before I sleep, and all because I took the road less traveled…

This is a long-term work-in-progress…

I am including SEA in my Matlab GUI package.

The functions can be accessed via:

vibrationdata > Miscellaneous Functions I > Acoustics & Vibroacoustics

I am using textbooks by Lyon, DeJong, Wijker, Norton & Karczub, along with AutoSEA and VAPEPS manuals.

Please feel free to recommend additions or corrections.

More later…


Compendium of Modal Densities for Structures, NASA-CR-1773

Cremer, Heckl, Ungar, Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies

Tom Irvine

Payload Fairing Foam Blankets

A spacecraft at launch is subjected to a harsh acoustic and vibration environment resulting from the passage of acoustic energy, created during the liftoff of a launch vehicle, through the vehicle’s payload fairing. In order to ensure the mission success of the spacecraft it is often necessary to reduce the resulting internal acoustic sound pressure levels through the usage of acoustic attenuation systems. Melamine foam, lining the interior walls of the payload fairing, is often utilized as the main component of such a system.

Here are some NASA reference papers:

29th_ATS_Absorption_Paper_23September2015_Final_as submitted to ATS


TM-2014-218350 Noise Con 2014 on Melamine Foam Acoustic Testing

TM-2014-218127 ATS version of NEMFAT

– Tom Irvine

Adding Uncertainty Margin

One of my colleagues asked whether uncertainty margin should be added to the input or the response, particularly for a nonlinear system.  I recommend adding to the response.  He asked that I write a paper using an acceleration base input which he supplied.  The results are given in the following paper for a spring-mass system with nonlinear stiffness.


Note that waiting until the end of the analysis to add margin is the method used by Sam DiMaggio at SpaceX.

On the other hand, NASA’s Isam Yunis recommends applying the uncertainly factor to the input.

Isam and I both agree that deriving environments is ultimately a matter of engineering judgment.

See also:  Normal Tolerance Factors

– 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

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

* * *

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

NASA Employment Opportunity

From a colleague…

I just wanted to let you know that we would have a GS-13 posting out next week seeking for an experienced strength/fracture engineer to fill a vacant position in my branch here at LSP.  I would appreciate very much if you can pass this info to folks that may be interested or know somebody that may be interested in this positon and willing to work at KSC, Florida. Specifically, please forward this email to Ivatury Raju since I don’t know him personally but have seen his email correspondences may times through you and other folks at NESC.  If somebody is looking for additional details on the job description, please have that person contact me via email or phone (321-222-8935).  The posting has been cleared to be posted NASA-wide next week (Oct 26-30)  Thanks and have a good time in San Diego next week.


Energy Response Spectrum

My colleagues at Sandia National Laboratories have presented some conference papers recently on energy response spectrum.

Here is a paper that I wrote on this topic:  energy_response_spectrum.pdf

Also, I have added the energy response spectrum as an option for acceleration time histories to the Vibrationdata GUI package.

See also:
Shock Response Spectrum
David O. Smallwood Papers
Temporal Moments

More later …

Tom Irvine

Employment Opportunity

Here is a job opening in Huntsville, Alabama for helicopter air frame finite element analysis:

I spoke with the CEO, Michele Kochoff Platt.  This is an excellent opportunity.  So please forward this link to any of your colleagues who are seeking employment.  Note that that U.S. citizenship is required.

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 

* * *

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:


* * *

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.

* * *

– Tom Irvine