Tractable Models of Solid Mechanics Formulation, Analysis and Interp
This book describes significant tractable models used in solid mechanics - classical models used in modern mechanics as well as new ones. The models are selected to illustrate the main ideas which allow scientists to describe complicated effects in a simp
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Discrete Finite Systems
An asymptotic approach is rather efficient when dealing with the theory of oscillations, since one often can figure out a number of relatively simple limit cases which can be efficiently treated and completely understood. Such an approach allows the deepest possible simplification but preserves the most significant features of dynamical behavior. At the same time, one can use the expansions by small parameters characterizing the deviation of the system from the tractable limit case. The mathematician supposes usually that the small parameters are already known and that construction and substantiation of the asymptotic expansions is the only problem. Contrary to this, for physicists and engineers the choice of the appropriate limit cases and corresponding small parameters is the most important step. Systems with a relatively small number of degrees of freedom offer convenient frameworks for the explanation of the basic concepts and ideas related to the simplification and formulation of the tractable models. Thus, in this part of the book we are going to discuss the discrete models of a dynamical system – from one to few degrees of freedom. The step-by-step approach will reveal essential new features arising from each stage of complication of the discussed models.
2.1 Linear Oscillators Beyond any doubt, a linear oscillator and a system of coupled linear oscillators are the most popular and the best understood models of discrete dynamical systems. If parameters of the oscillators are time-independent, then a general analytic solution can be provided in the form of exponents (combined with polynomials in resonant cases). This theory is widely known and can be found in standard textbooks (see, e.g. Den Hartog, 1956; Meirovitch, 2000), so there is no reason to repeat it here. Instead, we will concentrate on asymptotic aspects of behavior of individual and coupled linear oscillators, keeping in mind possible generalizations for less common systems. L.I. Manevitch, O.V. Gendelman, Tractable Models of Solid Mechanics, Foundations of Engineering Mechanics, DOI 10.1007/978-3-642-15372-3_2, C Springer-Verlag Berlin Heidelberg 2011
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2 Discrete Finite Systems
2.1.1 Linear Conservative Oscillator The system consists of a point-like mass m and a linear spring (with stiffness c) connecting the mass with an immobile point (Fig. 2.1) is one of the most simple and important models of mechanics and physics. A pendulum with small amplitudes (without friction) and oscillatory contour in radiophysics (without resistance) are well known realizations of this model. This mathematical model is described by the well known differential equation m
d2 U + cU = 0 dt2
(2.1)
(U is the displacement of point-like mass with respect to its equilibrium state, m is the mass and c – the rigidity of the elastic spring). The initial conditions in certain dU (0). initial instant have to be supplied as U 0 =U(0) and V 0 = dt√ 0 In dimensionless variables τ = ω0 t, u = U/U , ω0 = c/m one obtains d2 u +u=0 dτ 2
(2.2)
d
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