dielwa-elba: Dispersion of Elastic Waves in Elliptical Bars using Fraser's collocation method

A python module to compute the dispersion of elastic waves in elliptical bars using the collocation method according to the following article of Fraser:

Fraser, W. B. (1969). Dispersion of elastic waves in elliptical bars. Journal of Sound and Vibration, 10(2), 247‑260. https://doi.org/10.1016/0022-460X(69)90199-0

Also contains a module to compute the dispersion of longitudinal elastic waves in round bars (Pochhammer-Chree equation).

Dispersion curves visualization

Two methods are available to visualize the dispersion curves:

1. compute the values of the characteristic function on the whole (K,C) domain of interest. Dispersion curves are then visualized with a contour plot (with level 0 being the solutions) or a sign plot (most changes of sign being the solutions).
2. numerical solving of the characteristic equation with a prediction-correction algorithm. Prediction uses basic polynomial extrapolation, correction uses Regula Falsi method.

Warnings

The second option is only possible if the starting of the curve is known, ie. only for the first branch of the longitudinal mode. Higher branches are harder to catch and follow automatically. It is not possible, for now, to follow the first branch of the other modes (torsion and flexion).

The first method requires some car in the interpretation of the diagrams. Depending on the order of approximation (ie number of collocation points), either the real part or the imaginary part of the characteristic function is of interest.

Contents

This repository contains the following Python and Fortran files:

• ellipticReferenceSolutions.py is the module used to plot the dispersion curves of Fraser's article;
• round_bar_pochhammer_chree.py contains the mother class to handle characteristic equations (it is applied here to the dispersion of longitudinal waves in round bars, with Pochhammer-Chree equation);
• elliptical_bar_fraser.py contains the class used to handle Fraser's approximate equations.
• fraser_matrix.f90 computes Fraser's characteristic matrix. It is used to speed up calculations (10x approximate speed-up compared to pure Python version)
• special_functions.f90 contains Bessel functions, used in the previous file.

Installation

1. Extract all the files in the same folder;
2. Compile special_functions.f90:

• gfortran -c special_functions.f90 -fPIC

3. Compile fraser_matrix.f90 with f2py to make the fraser_matrix module available within Python:

• f2py -c -I. special_functions.o -m fraser_matrix fraser_matrix.f90

4. The Python files should now be usable !

Documentation and usage

Sorry, this is a small project, read the docstrings for the documentation.

For examples of how to use the modules, see the script part at the end of each module. It serves as both examples and tests of the methods.

The names of the methods should be explicit enough to guess what they do. I hope the names of the attributes are not so obscure.

Testing

Well, you can use ellipticReferenceSolutions.py to plot Fraser's curves and compare them with the dispersion curves computed with elliptical_bar_fraser.py.