Mechanical Systems, Classical Models potx

[...]... reproduce the objects under study in a simplified form, constituting thus – as all models – a certain idealization of the reality MECHANICAL SYSTEMS, CLASSICAL MODELS 4 The types of ideal models mentioned above can be considered as limit cases In fact, there exist ideal models that have common features to both types of models which have been described; they contain systems of notions and axioms which... order and their structure From the very beginning, the models can be divided in two great classes: technical (material) models and ideal (imaginary) models; this classification is made considering the construction of the models as well as the possibilities by which the objects to be studied may be reproduced Newtonian model of mechanics 3 The technical models are created by man, but they do exist objectively,... introduced are basic notions; these ones cannot be defined with the aid of other notions, and they must be described by their various properties, obtaining thus the considered mathematical models 1 MECHANICAL SYSTEMS, CLASSICAL MODELS 2 1.1.1 Notion of system Notion of mathematical model In the study of a science of nature, more and more the connection between the modern methods of the theory of systems and... where they fill a certain position with respect to other objects of the material world; they have also a certain extent One can thus say that the space 10 MECHANICAL SYSTEMS, CLASSICAL MODELS represents a support of matter If, after a series of mechanical phenomena (for instance, after a succession of motions), material bodies maintain their properties of extent and relative situation with respect... mentioned above does not mean that we know how the point P is moving along this trajectory; to state accurately this motion we must define the velocity and the acceleration of the point MECHANICAL SYSTEMS, CLASSICAL MODELS 14 The mechanical nature of the phenomenon of motion imposes certain conditions to the vector function (1.1.16) So, it must be continuous (the trajectory is continuous, the point cannot... mathematical study of the mechanical systems 1 Mechanics, science of nature Mechanical systems All the phenomena of nature are characterized by motion; this is an essential property of matter, having infinity of aspects For instance, the motion can be mechanical, physical, chemical and biological; there correspond various sciences of nature Among them, mechanics studies the objective laws of the mechanical motion... by its physical nature, we have to do with analogic systems E.g., electrical models may reproduce processes analogous to those encountered in mechanics, qualitatively different, but described by similar equations These models, as others of the same kind, take part in the class of mathematical models One can construct such models, for instance, to study the torsion of a cylindrical straight bar of arbitrary... motion) The intuitiveness of these models is put into evidence first of all by the fact that the models themselves, formed by elements sensorial perceptible (plates, levers, tubes, fluids, vortices etc.), are intuitive, and – on the other hand – by the fact that they are intuitive images of the objects themselves Sometimes, these models are fixed in the form of schemata The models of second order are systems... fluids), under the denomination of mechanical systems Mathematics plays a very important rôle in the study of a science of nature Mechanics has been the first science of nature which was strongly mathematicized, by considering various mathematical models, which have been proved to correspond to the surrounding reality To mechanical systems there correspond thus mathematical models of organized matter, having... exists a relation of equality if they are characterized by the same elements (direction, modulus, point of application, support) or the same canonical co-ordinates For instance, for the free MECHANICAL SYSTEMS, CLASSICAL MODELS 6 vectors V (Vi ) and W (Wi ) we can write V = W, Vi = Wi , i = 1, 2, 3 ; (1.1.3) for the sliding vectors we must have also the same support, while for the bound vectors the same . model. These models reproduce the objects under study in a simplified form, constituting thus – as all models – a certain idealization of the reality. MECHANICAL SYSTEMS, CLASSICAL MODELS 4. described by their various properties, obtaining thus the considered mathematical models. 1 MECHANICAL SYSTEMS, CLASSICAL MODELS 2 1.1.1 Notion of system. Notion of mathematical model In the study. 0-387-23195-1 SINGULAR PERTURBATION THEORY Robin S. Johnson, ISBN 0-387-23200-1 Mechanical Systems, Classical Models article MechanicsVolume I: P by Petre P. Teodorescu Faculty of Mathematics,