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Re: Lamb wave dispersion equations

Posted by: Michael Conry (PID_1251), E-mail: michael.conry@ucd.ie, on December 02, 2003 at 17:43 :

In Reply to: Lamb wave dispersion equations posted by : Protopappas Vasilis , E-mail: me00642@cc.uoi.gr, on December 02, 2003 at 16:42 :

----------- Start Original Message -----------
: Hello,
: Could anyone suggest a way to solve the Lamb wave dispersion equations? I
: want to plot the dispersion curves for a number of symmetric and
: antisymmetric modes, the equations, however, need a numerical evaluation.
: Thank you in advance,
: Vasilis Protopappas
------------ End Original Message ------------

Plotting the curves can be tricky. The easiest way is as follows:

Divide the frequency/phase-velocity (f/c) space up into small increments.

At each point within this space calculate the value of the
characteristic equation F(f,c)

Then do a contour plot of the resulting numbers, including just a
single contour (at F(f,c)=0)

This is easy to do with Matlab, and I have also done it with Gnuplot and
Python.

This is not an efficient way to plot the curves (it requires a lot of
calculations) but even for a multilayered plate I found it took a pretty
short length of time to prepare a nice dispersion plot.

Doing the job in a more rigorous way requires either that you use curve
following algorithms to plot the curves (e.g. find the solutions for a
particular value of phase velocity, then incrementally reduce the phase
velocity and find solutions at each subsequent point). A difficulty in
this case can be correct handling of the crossovers of the curves.

Best regards,
Michael

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----------- Start Original Message ----------- : : Hello, : : Could anyone suggest a way to solve the Lamb wave dispersion equations? I : : want to plot the dispersion curves for a number of symmetric and : : antisymmetric modes, the equations, however, need a numerical evaluation. : : Thank you in advance, : : Vasilis Protopappas : Plotting the curves can be tricky. The easiest way is as follows: : Divide the frequency/phase-velocity (f/c) space up into small increments. : : At each point within this space calculate the value of the : characteristic equation F(f,c) : Then do a contour plot of the resulting numbers, including just a : single contour (at F(f,c)=0) : This is easy to do with Matlab, and I have also done it with Gnuplot and : Python. : This is not an efficient way to plot the curves (it requires a lot of : calculations) but even for a multilayered plate I found it took a pretty : short length of time to prepare a nice dispersion plot. : Doing the job in a more rigorous way requires either that you use curve : following algorithms to plot the curves (e.g. find the solutions for a : particular value of phase velocity, then incrementally reduce the phase : velocity and find solutions at each subsequent point). A difficulty in : this case can be correct handling of the crossovers of the curves. : Best regards, : Michael ------------ End Original Message ------------

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