| Abstract: | The primary objectives of this study are formulation and solution validation of mathematical models for hybrid (discrete and distributed) coordinate dynamical systems. In the first part of this dissertation, an explicit generalization of the classical Lagrange's equations (for discrete coordinate dynamical systems) is presented; this formulation applies to a large family of multibody hybrid discrete/distributed parameter systems. The coupled system of ordinary and partial differential equations follows directly from spatial and time differentiation of various Lagrangian functionals, whereas the boundary conditions are directly established from another explicit set of symbolic variational equations. The work-energy rate principle is introduced to efficiently obtain the nonconservative virtual work terms. The specific governing equations and boundary conditions for five example hybrid systems are obtained from the resulting general equations, as illustrations. In the second part of this dissertation, an inverse dynamics method is introduced for constructing exact special case solutions for hybrid coordinate ordinary/partial systems of differential equations (hybrid ODE/PDE systems). The solution is constructed such that it lies near a given approximate numerical solution, and therefore the special case solutions can be generated in a versatile and physically meaningful fashion and can serve as a benchmark problem to validate approximate solution methods. A multi-variable orthogonal function expansion method and computer symbol manipulation are successfully used for this process. Since the exact special case algebraic solutions can be evaluated anywhere in space and time, this approach is ideally suited to providing a true exact motion and the corresponding forces for studying the convergence errors in a family of approximate solutions. This methodology makes it possible for one to rigorously determine exact solution errors for a significant class of hybrid differential equation systems for which the initial value problem is not, in general, exactly solvable. Finally, a simple method is suggested to provide a physically meaningful performance index for the Linear Quadratic Regulator (LQR) problem for flexible space structure models which have a rigid body mode. A numerical procedure is introduced to solve a time-variant LQR problem with inequality control constraints. The method of particular solutions and an associated quasi-linearization method are used in this procedure. |
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