Numerical investigation of a damped wave equation with distributed and boundary energy dissipation

Date

2004-08

Journal Title

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Volume Title

Publisher

Texas Tech University

Abstract

In this thesis, I study numerically the distribution of the eigenvalues of a specific matrix differential operator, which governs the vibrations of a string having damping of two types: distributed (or Kevin-Voight) damping and the end-point damping. I investigate a corresponding Sturm-Liouville problem that is not standard because of the damping terms both in the equation and in the boundary conditions. The boundary damping reflects a contemporary approach to the modeling of the action of smart material inclusions along the string.

My thesis has two distinct parts: the first part is devoted to the physical background of vibrational motion, and the second is devoted to the formulation of my specific problem as well as the presentation and discussion of the main findings of my research. My original main goal was to support a well-known idea concerning the behavior and properties of purely imaginary eigenvalues for the aforementioned Sturm-Liouville problem. It should be emphasized that since I have a string with energy dissipation, the corresponding spectrum is a countable set of complex points. These points geometrically converge to some horizontal asymptote and form a set that is symmetric with respect to the imaginary axis. It has been assumed in the mathematical community that when one changes the density with small steps, the two symmetric eigenvalues closest to the imaginary axis move toward each other, finally merging into one double eigenvalue on the axis. A minor change in the density instantaneously breaks the double eigenvalue into two different simple purely imaginary ones.

So my goal was to support that idea numerically. Unexpectedly, I have observed that the actual behavior of the symmetric pair of eigenvalues closest to the imaginary axis is totally different from the prediction. Namely, when I change the density by a very small step (10 -14), I find that for the first dozen steps, the eigenvalues behave as expected: they slowly move in such a way that the distance between them decreases. However, at some moment immediately following, those two eigenvalues drastically change their behavior and begin moving apart in opposite directions. The movement continues along two branches of a hyperbola-like curve until the moment when both eigenvalues reach the imaginary axis. As the density continues to change, the new pair of purely imaginary eigenvalues moves along the imaginary axis. In conclusion, my calculations show that contrary to the widespread opinion, the eigenvalues never merge to create a multiple eigenvalue. It is a very interesting and important discovery.

In my study, I have changed the value of the coefficient standing before the highest order derivative in the original hyperbolic differential equation (uu). It is the most influential coefficient, so the accuracy of calculations must be extremely high. I wish to add here that similar- in spirit- numerical simulations are presented in the thesis of T. Busse [1]. She also observed the nonexistence of multiple eigenvalues by changing the distributed damping coefficient. In addition, I have done numerical simulations related to the possible appearance of multiple eigenvalues when one changes, with small steps, the boundary parameter (i.e., the "strength" of the smart material inclusions along the string). Though this set of results is not included in the present manuscript, it is interesting to note that that I again obtain the hyperbola-like curves, which suggests that there are no multiple eigenvalues as well.

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Keywords

Energy dissipation -- Mathematical models, Wave equation -- Mathematical models, Vibration -- Mathematical models, Eigenvalues, Damping (Mechanics) -- Mathematical models

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