Ασυμπτωτική ανάλυση ταλαντωτικών συστημάτων με χρήση της μεθόδου CSP

Abstract

In spite of the rapid and significant increase in computing power of current processors and parallel computational architectures, the numerical solution of multi-scale problems (chemical kinetics, biological modeling, atmospheric prediction, control, electronics, etc.) remains a daunting task, mostly because the time scales of interest are usually much slower than the fastest time scales characterizing this class of systems. The mathematical models which attempt at reproducing this phenomenology, are usually classified as being ‘‘stiff’’. Stiffness can indeed be associated with a spread in the magnitude of (i) the real, negative part of the eigenvalues of the system, and/or (ii) their imaginary part; the first category being associated with the presence of dissipative processes, such as viscous dissipation and/or chemical reactions, and the latter to nonlinear convective transport or nonlinear oscillatory behavior Specifically, the thesis is formatted as follows. In the first chapter, we will define the singular perturbation method will in parallel we will present a short description of the CSP method and its fundamental features, as they are mandatory for the understanding of the present work. Then, we will refer and try to prove the basic points of the modified CSP method, as they were first stated by S.H.Lam and D.A.Goussis. Finally, in two different chapters, we will study two specific examples, both analytically and numerically. The first problem consists of three masses which can move horizontally. The masses are connected with a linear and a non-linear spring, as well as with two dampers. The coupled motion of the three masses in combination with the slight difference in their frequencies characterize the problem as "stiff". The second problem, consists of a rod which can move on the vertical axis. At the end of the rod, lies a pendulum. The coupled motion of the pendulum and the rod describes a stiff system, with the fraction of their natural frequencies as the small parameter. These two mechanical systems are two simple approaches of soft/stiff systems.