3 Variational Foundations of Modern Structural Dynamics 23 Mathematical formulations and procedures based on variational principles continue to influence developments both in structural dynamics and other branches of applied physics. The complementary viewpoint introduced by Castigliano in 1873 was generalized for dynamics by Toupin [23]. Toupin’s variational principle was found to be quite useful in fluid–structure interaction studies [24]. In addition, variational formulations that are direct consequences of Galerkin’s method have been introduced by Biot [25] for heat transfer and MacNeal et al. [26] for electromagnetics. 3.2 Economy in Nature and Basic Variational Formulations Philosophers of antiquity, well before the advent of modern science initiated by Newton [1], believed that nature operated in accordance with a rule of economy as noted by Aristotle (d. 312BC), “Nature follows the easiest path that requires the least amount of effort”. In the medieval era, William of Ockham (1347) suggested an economic principle for human reasoning with his famous saying, “It is futile to employ many principles when it is possible to employ fewer” (popularly known as Ockham’s razor). It is fascinating that Newton’s second law may be used as a postulate to deduce variational formulations of mechanics [2–4] (as theorems). Moreover, when the variational principle due to Hamilton [3], is postulated, Newton’s second law follows as a theorem. The postulate-theorem interrelationship is demonstrated below for a mechanical system composed of a single particle restricted to motion along one direction. Newton’s second law [1] states the postulate, F D d dt m du dt (3.1) D’Alembert [2] claimed that the system follows an “economy” in variational terms that F d dt m du dt •u D0 (3.2) where “•u” is defined as a variation off the true path of “u”. Hamilton [3] added to D’Alembert’s principle by claiming that the following time integral is true: t2 Z t1 F d dt m du dt •u dt D0 (3.3) After rearranging terms and integrating by parts, the above integral is expressed as t2 Z t1 • 1 2 m du dt 2! CF •u ! dt- m du dt •uˇ ˇ ˇ ˇ t2 t1 D0 (3.4) This variational integral is more compactly expressed as t2 Z t1 .•TC•W/ dt D0subjected to the end conditions m du dt •uˇ ˇ ˇ ˇ t2 t1 D0 (3.5) T D 1 2 m du dt 2 is the system’s kinetic energy and •WDF•u is the system’s virtual work. In addition, in order to curtail mathematical imposition of “Calvinist” predestination doctrine, we avoid imposition of the initial and final time constraints by extending t2 to infinity and enforce two initial (state) conditions, namely that u(0) and Pu.0/ are specified. The variational integral defined in Eq. (3.5) is known as Hamilton’s principle. A more familiar form of Hamilton’s principle results from dividing the total applied force into two components, namely (a) a conservative potential energy (U) based force, and (b) a non-conservative component, which are expressed as
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