After a detailed discussion of the variation theorem for upper bound calculation of eigenvalues, many standard procedures for determining lower bounds to eigenvalues are presented with chemical applications in mind. A new lower bound method, arbitrary choice without trunctation is presented and tested on the helium atom. This method is attractive because it does not require knowledge of the eigenvalues or eigenvectors of the base problem. In application, however, it is shown that the method is disappointing for two reasons: 1) the method does not guarantee improved bounds as calculational effort is increased; and 2) the method requires some a priori information which, in general, may not be available. A possible direction for future work is pointed out in the end. An extension of a lower bound method by Calogero and Marchioro has been developed and is presented in appendix G along with comments on the effective field method in appendix H for Virginia Tech access only. / Ph. D. / To avoid copyright infringements, access to these three appendices (G, H, and I) has been permanently limited to the Virginia Tech campus. In the case that Virginia Tech places these appendices freely on the internet, Virginia Tech is solely responsible for copyright violations.

Thesis. 1975. Ph.D.--Massachusetts Institute of Technology. Dept. of Meteorology. / Vita. / Bibliography: leaves 168-170. / Ph.D.

The energy of a graph began with German physicist, Erich H¨uckel’s 1931 paper, Quantenttheoretische Beitr¨age zum Benzolproblem. His work developed a method for computing the binding energy of the π-electrons for a certain class of organic molecules. The vertices of the graph represented the carbon atoms while the single edge between each pair of distinct vertices represented the hydrogen bonds between the carbon atoms. In turn, the chemical graphs were represented by an n × n matrix used in solving Schr¨odinger’s eigenvalue/eigenvector equation. The sum of the absolute values of these graph eigenvalues represented the total π-electron energy. The criteria for constructing these chemical graphs and the chemical interpretations of all the quantities involved made up the H¨uckel Molecular Orbital theory or HMO theory. In this paper, we will show how the chemical interpretation of H¨uckel’s graph energy evolved to a mathematical interpretation of graph energy that Ivan Gutman provided for us in his famous 1978 definition of the energy of a graph. Next, we will present Charles Coulson’s 1940 theorem that expresses the energy of a graph as a contour integral and prove some of its corollaries. These corollaries allow us to order the energies of acyclic and bipartite graphs by the coefficients of their characteristic polynomial. Following Coulson’s theorem and its corollaries we will look at McClelland’s first theorem on the bounds for the energy of a graph. In the corollaries that follow McClelland’s 1971 theorem, we will prove the corollaries that show a direct variation between the energy of a graph and the number of its vertices and edges. Finally, we will see how this relationship led to Gutman’s conjecture that the complete graph on n vertices has maximal energy. Although this was disproved by Chris Godsil in 1981, we will provide an independent counterexample with the help of the software, Maple 13

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New modal synthesis methods are developed for calculating the exact eigenproperties of structures divided into two substructures. Unlike the conventional mode synthesis techniques, here the synthesis of modes is carried out by solving a second eigenvalue problem by nontraditional means: the eigenvalues of the combined structure are obtained by solving simple characteristic equations which,in the proposed approach, can be defined in closed-form. These equations can be easily solved by a simple Newton-Raphson iterative scheme, especially when good initial estimates of the roots are available. Herein, explicit expressions are provided to calculate these initial values; these expressions are obtained via a second order matrix perturbation analysis of the algebraic eigenvalue problems. Once the eigenvalues are known, the eigenvectors can be calculated with closed-form expressions without solving any simultaneous equations. Several variants of the methods are developed to consider various damping cases which can be encountered in practice. Furthermore, for each damping case, two parallel approaches are developed that allow one to utilize two different types of modes - free and fixed attachment modes - of one of the substructures to be synthesized with the free attachment modes of the other structure. The eigenproperties of the combined system, once calculated, can be utilized in a mode superposition approach for the calculation of the system response for any forcing function . For seismic analysis of combined primary and secondary structures, such as a main structure supporting piping or other auxiliary system, the combined structure is divided into two substructures. The eigenproperties of these two substructures are then synthesized to obtain the eigenproperties of the combined system. To obtain response for seismic design motions defined in terms of ground response spectra, response spectrum approaches are presented which utilize the eigenproperties of the combined system. Closed-form formulas are presented to obtain any displacement-related response quantity of interest for the seismic input defined in terms of ground response spectra. The seismic response calculated by the proposed procedure accounts for the dynamic interaction effects between the primary and secondary structures and the nonclassically damped characteristics of the combined system in an analytically exact way. Numerical results showing the performance of the modal synthesis methods are presented. The applicability of the response spectrum approaches is verified by numerical simulation studies. The approach is applicable to light as well as heavy secondary structures equally effectively. It can also be used for seismic analysis of tertiary systems in industrial units. The potential application of the approach for efficient and accurate vibration analysis of aerospace and aircraft structures should also be of interest. / Ph. D.

A theoretical investigation of internal flow subjected to an axial variation of the external convection coefficient is presented. Since the variable boundary condition parameter causes the problem to become nonseparable, conventional techniques do not apply. Instead, the Green's function technique is used to convert the governing partial differential equations into a singular Volterra integral equation for the temperature of the fluid at the wall. The integral equation is resolved numerically by the trapezoid rule with the aid of a singularity subtraction procedure. The solution methodology is developed in terms of a fully turbulent flow which is shown to contain fully laminar and slug flow as special cases. Before examining the results generated by numerical solution of the integral equation, a thorough study is made of each of the building blocks required in the solution procedure. A comparison of the respective dimensionless velocity profiles and dimensionless total diffusivities for each of the flow models is presented. Next, an analysis of the eigenvalue problem for each flow model is presented with consideration given to the normalized eigenfunctions and the eigenvalues themselves. Finally, the singular nature of the Green's function is examined showing the effect of the parameters Ho, Re and Pr. The technique is applied to study the heat transfer from a finned tube. A parameter study is presented to examine the effects of the external finning and the flow model. The effect of external finning is examined through specific variations of the external convection coefficient, while the flow model is selected through the velocity profile and eddy diffusivity. In examining turbulent flow, the effects of the parameters, Re and Pr, are considered. / M.S.

The objective of this research work was to develop a design tool to analyze and design undamped beam and rotor systems in two dimensions. Systems modeled in two dimensions, such as beams with different moments of inertia, could produce varying responses in the each direction of motion. A coupling between the vertical and horizontal motions also exists in rotor systems mounted of fluid film bearings. The computer program called 2DBEAM has been developed to model and provide analyses of such systems in two dimensions. The tool has been based on an existing design package, BEAM9, which in its present state provides the response of beams and rotors in one plane of motion. The 2DBEAM program has the capability of performing the static response, free vibration, forced dynamic response, and frequency response analyses of a system. The Transfer Matrix Method has been used in the development of the software and an explanation of the method is included in this thesis. Mathematical problems and solutions encountered while developing 2DBEAM are also documented in this study. The code has been tested against analytical and published solutions for the types of analysis mentioned above and on coupled and uncoupled system models. / Master of Science