Analysis Methods Of Dual Variables Interval And Its Applications

Great deals of new subjects are coming forward and many fields are intersecting are contemporary characteristics of science. There exist analogue relationships among structural mechanics, theory of elasticity, electric circuit analysis and control theory, etc. The analogue relationships can be founded in the general theory system, dual variables system. In dual variables system, many disciplines have common foundations of mathematics, their analysis and solution have the same principles and methods, and their physical systems can be described by two kinds of variables with which the dynamic characteristics can be revealed profoundly.Electrical networks theory studies two kinds of variables, voltage and current in electrical system with matrix analysis methods, and contains a series of concepts and methods. Based on analogy method, dual variables system's two kinds of variables can be simulated as voltage and current of electrical system, so the concepts, the analysis and solving methods of electrical networks theory can be applied in dual variables system. Thus, study on dual variables system can be developed greatly, and the application scope of dual variables system can be extended to discrete and continuous systems, conservative and nonconservative systems, structural boundary problems and time course problems, etc.This dissertation presents analysis methods of dual variables interval by the combination of electrical networks theory and dual variables system, and includes the typical problems and its analysis and solving methods for the discrete, continuous dual variables interval and its chain dual system. The thesis elaborates the applications of analysis methods of dual variables interval in electrical circuit system, elastic bar structure, elastic beam structure, electromechanical system, plane elastic wave, modern control theory, etc. The main contents in this dissertation are as follows:1. By analogy with electrical networks theory, the typical problems and its analysis and solving methods for dual variables interval and its chain dual system are proposed. This part provides different forms of parameter equation of dual variables interval and the relationships of conversion among them, and gives T parameter method and Z(Y) parameter method for nodal variables of chain system. This part presents the unilateral equivalent quantities of chain system node and its recursive calculation method, and studies the relationships between nodal response and nodal excitation of chain system, and so forth.2. The typical problems and its analysis and solving methods for continuous dual variables system are proposed. In this part, there are three primary forms of solution of dual equation summed up:T(T') parameter form solution, eigenvector expansion form solution, modal expansion form solution. For one-dimensional continuous dual variables interval, finite element equation and the shape functions, boundary integral equation and the fundamental solutions are deduced by solutions of dual equation. The recursive calculation method for relational matrix P(x) and equivalent action g(x) of continuous dual variables system are provided. The general methods on solving the boundary value problems (or depending on a parameter) of dual variables interval are given.3. The applications of analysis methods of dual variables interval in elastic bar and elastic beam (including static and dynamic analysis of Timoshenko beam, Euler beam) structures are elaborated. In this part, the T (T1) parameter form solution, eigenvector expansion form solution, modal expansion form solution of elastic bar and beam are given. The finite element equation and its shape functions, the boundary integral equation and its fundamental solutions of bar and beam are established, and the relationships of them can be revealed profoundly. The model analyses of bar and beam are achieved by dual equation's modal expansion method and boundary value problem depending on a parameter. The eigenvector expansion methods of dual equation are well suitable for analyses of reflection and transmission of elastic waves in bar and beam. Segmented bar and beam structures can be analyzed and solved as chain dual system. Based on analogy method, the dual equations of damping bar and lossy transmission line are established and the comparative studies between them are illustrated. Considered as chain dual systems, mechanical vibration model, mechanical transmission system and DC servomotor system are analyzed and solved with analysis methods of dual variables interval.4. The applications of analysis methods of dual variables interval in elastic wave problems are expounded. In dual variables system, there are general analysis and solving methods for the normal incidence problems of elastic plane P-and S-waves, the oblique incidence problems of elastic plane SH-and P-SV waves, uniform wave and inhomogeneous wave problems. The reflection and transmission of elastic waves are analyzed and solved based on the eigenvector expansion form solution of dual equation. Stoneley wave, Rayleigh wave and LOVE wave are studied, and the corresponding surface wave's equations are derived based on boundary value problems of dual variables interval depending on a parameter. The modal problems of elastic wave guide are studied, and the frequency equations are deduced with the modal expansion form solution of dual equation. Layered elastic media structures are analyzed and solved as chain dual system with analysis methods of dual variables interval. The wave impedance problems of elastic layered media are accounted, and the recursive calculation methods of wave impedances, reflection coefficient and transmission coefficient of P-SV waves in elastic layered media are given.5. The applications of analysis methods of dual variables interval in linear quadratic optimal control and Kalman filtering are accounted. LQ optimal control and Kalman filtering problems can be described uniformly by Hamiltonian dual equations. For LQ optimal control and Kalman filtering problems, considering the canonical equations of continuous systems as dual equations, or considering the canonical equations of discrete systems as H parameter equations and converting them to T (T1) parameter equations, state feedback matrix and equivalent action of LQ optimal control, state variance matrix and mean vector of Kalman filtering are solved by the recursive calculation method for equivalent quantities of dual variables system.This dissertation provides the general analysis and solving methods for multidisciplinary problems with the new systematic methodology and new ways of thinking, and reveals the intrinsic relationships among many subjects, which are beneficial to promoting the intersection among different disciplines. The practical applications in thesis show that analysis methods of dual variables interval are well suitable for analysis and solving methods of multidisciplinary problems. The results obtained are identical with those solved by traditional methods, the analyses are clear, the computational methods are concise, and the physical meanings are distinct, so the essences of the problems are well revealed. All of those indicate the effectiveness and universal applicability of analysis methods of dual variables interval for multidisciplinary fields.