Browsing by Subject "Vibration -- Mathematical models"
Now showing 1 - 3 of 3
Results Per Page
Sort Options
Item Numerical approximation of the spectrum of a nonselfadjoint operator governing the vibrations of a nonhomogeneous damped string(Texas Tech University, 2004-08) Busse, Theresa NicoleIn my dissertation, I present the results of numerical investigations of the spectrum of the damped hyperbolic equation involving damping terms, both in the equation and in the boundary conditions. This equation describes the vibrations of a realistic string having distributed (Kevin-Voight) damping and smart material inclusions (self-straining actuators) in the string. The action of the actuators is modeled through specific boundary conditions that involve two independent parameters reflecting the strengths of smart materials. The dissertation has two distinct parts: the first part is devoted to the physical background of vibrational motion, and the second part is devoted to the formulation of a specific problem, presentation, and discussion of the main findings of research. In my dissertation I consider a non standard Strum-Liouville problem. The problem is nonstandard due to two factors, the model incorporates two energy decay mechanisms, i.e., the energy of a string dissipates through the internal friction (Kevin- Voight damping) and through the end points of the string. Many researchers have studied the Strum-Liouville problem for the string equation in the absence of any damping. My problem involves two different types of damping, which makes the problem much more difficult. My particular goals were to investigate the distribution of the eigenfrequencies for the string (in mathematical language, the distribution of the eigenvalues of some linear operator), to analyze dependence on the damping coefficient, and finally to support a well known conjecture in the mathematical community about the multiple eigenfrequencies. In the course of my work, an unexpected discovery was made. I already mentioned, my main goal originally was to support a well known idea concerning the behavior and properties of purely imaginary eigenvalues for the aforementioned Strum-Liouville problem. 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; these points form a set symmetric with respect to the imaginary axes. It has been assumed tin the mathematical community that with a change in the damping of small steps, the two eigenvalues closest to the imaginary axes would move toward each other, and finally they merge into one double eigenvalue. A minor change in the damping instantly breaks the double eigenvalues into two different simple purely imaginary eigenvalues. A goal of this thesis was to support that idea numerically. Unexpectedly, it was observed that the actual behavior of the two eigenvalues closest to the imaginary axes is totally different from the prediction. Namely, when I change the damping by a very small step (10"^^), 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, those two eigenvalues totally 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 axes. When I continue changes in the damping, the pair of eigenvalues moves along the imaginary axes. So, my calculations show that contrary to the widespread opinion, the eigenvalues never merge to create a multiple one. This interesting and important behavior had not previously been observed. It should be emphasized that in my research I have changed the values of the coefficient that stands for the first order derivative in time of the unknown function {Ut). In the dissertation of R. Plant II [9], the dependence of multiple eigenvalues upon changing the density of the string, i.e., the coefficient standing before the highest derivative in time {Uu)- The fact that R. Plant obtained similar results, validates the importance of the discovery made in this thesis. The results here create more unsolved questions, and opens a rich and exciting area for future research in this area.Item Numerical investigation of a damped wave equation with distributed and boundary energy dissipation(Texas Tech University, 2004-08) Plant, Robert Earl,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.Item Response of a nonlinear two-degree-of-freedom system to a horizontal harmonic excitation(Texas Tech University, 1985-12) Li, WenlungAn elastic structure containing a fluid subjected to a horizontal sinusoidal excitation is investigated. The system is found to include cubic nonlinearities. The system response is determined by using the multiple scales asymptotic approximation method. The method predicts that primary resonances may occur when the excitation frequency, Ω is close to either the first mode natural frequency, ω1, or the second mode natural frequency, ω2. The system behavior under the fourth order internal resonance condition (ω2 ≈ 3ω1) is predicted. The system response under conditions of primary resonances (Ω ≈ω1 and Ω≈ω2), together with internal resonance is also considered. Other features, such as amplitude jump phenomenon and chaotic-like response have been observed. Two possible responses have been found when Ω is near ω2 = unlmodal response and autoparametric interaction response. The boundaries of these two motions are defined in the excitation amplitude - frequency plane. Moreover, the so called "static attractor" is also observed.