This page contains links to computer programs for calculating natural frequencies and modes of some continuous systems using exact methods and approximate methods such as the Rayleigh-Ritz approach, and a general purpose program that searches the natural frequencies of a standard eigenvalue matrix equation of the form [k]{x}-lamda[M]{x}={0} where lamda is the square of the natural frequency. All resources are freely available for educational or research purposes, provided their use is acknowledged in any publications for which the results were generated using the programs. Please e-mail me if you are interested in a program that uses Lagrangian multiplier method to calculate natural frequencies of continuous systems with constraints. I would appreciate feedback from users. The programs currently available were developed for specific research and teaching requirements by the author (yes me, just pretending to be polite:-), using Macromedia Authorware, and the programs available through this page may be run on Windows environment (will run on Windows 95 or later versions including NT). If you wish to share any of your own electronic resources with the research community freely, please contact me to arrange them to be placed on this site or to put links to the sites where they are currently available. More interactive programs in vibration and Mechanics of Materials are available, but due to lack of server disk space (some programs are very large) and to protect copyrights of potential publications, they are not available on www. However, if you need any specific program for research or teaching, please feel free to e-mail me with your request, and if a program is readily available, arrangements could be made for you to use it for a specified period. I am also happy to share my lecture notes.

Interactive Programs Currently Available

Asymptotic.exe in zipped archive asymptotic.zip

Down load asymptot.zip

This program may be used to calculate the natural frequencies of uniform beams carrying a mass and partially restrained by an elastic spring. The spring stiffness and the magnitude of the attached mass and its inertia may be varied and their effect on the frequencies and modes may be studied. The centre of mass may be placed at an eccentricity with respect to the point where it is attached to the beam. Currently only two boundary conditions are available. One is for a beam that is pinned (simply supported) at one end and the other is a beam clamped at one end. However it is possible to fully constrain the translation at any given point in the beam by using a spring with very large positive or negative stiffness or by using a large mass. The rotation could be prevented by using a large inertia.

This is useful in the Rayleigh-Ritz analysis where the functions to be used for the vibratory displacements must not violate any geometric constraints. Research has shown that by replacing all rigid supports or connections with flexible supports or connections that have very high stiffness (thus practically, rigid) the natural frequencies of constrained systems can been obtained using a Rayligh-Ritz procedure, but without any restrictions on the choice of displacement functions. This makes the Rayleigh-Ritz approach quite versatile, but the only problem is one cannot be sure if the stiffness values used in the modelling are sufficiently large in order to fully constrain the displacements (and/or rotations) at the rigid supports and connections. However, recently it has been shown that the maximum error in such modelling may be found by using a combination of large positive and negative values for ths stiffness. By the way have you visited the electronic discussion forum of the Journal of Sound and Vibration, JSV+ ?

Now here is the latest news on asymptotic modelling: It is possible to use large positive and negative artificial mass (or inertia) to model a cosntraint! In fact using a combination of psoitive and negative artificial mass appears to be the best way to asymptotically handle constraints. If you are interested in learning more about this click here for the media release on a paper published in the Physical Sciences Proceedings of the London Royal Society

Galerkin's Method for Postbuckling and Vibration Analysis of Simply Supported Slightly Curved Rectangular Plates.

Galerkins.zip

An interactive program to calculate the static out-of-plane displacement of initially curved, simply supported reactangular plates subject to biaxial in-plane loading. The program also calculates the natural frequencies about the statically deformed configuration. The static analysis is the solution to the cubic equations obtained by applying Galerkin's method to von Karman's non-linear equilibrium equation. In some cases, three real roots for the displacements are possible, and where this is the case, the program may be used to find all three roots, includingthe snap-through buckling configuration. The natural frequencies are obtained by solving the eigen value problem arising from applying Galerkin's method to the linear shell vibration equations, about the statically deformed shape.

Asymptotic modelling with virtual and artificial inertia.

virtuali.zip

An interactive program to calculate the natural frequencies of a masless elastic beam carrying a virtual mass and an optional artificial mass to study the asymptotic modelling in linear stress nalysis. For clarification, contact the author.

It is available now. Please click the link on the left labelled newtonian.zip, if you want a program to calculate natural frequencies and modes of some beams and bars. Unzip and run on windows.

Gaussian.a4r in zipped archive gaussian.zip

You also need the runtime file in zipped form, available from the Interactive Programs Page