Until very recently, the wrong belief that given forces in mechanics cannot depend on acceleration was a wide-spread opinion. As a consequence, little effort, if any, has been made to develop a formal mechanical theory for such forces.At the present time, a class of new problems involving the study of mechanical systems acted on by forces that depend on acceleration and higher derivatives can be considered on the basis of traditional statements of mechanical problems (of stabilization, stability, optimal control, etc.).The current paper is primarily devoted to equilibrium stabilization using accelerationdependent forces. For definiteness, the analysis is concerned with the tilting railway car, which is a mechanical system where the use of such forces is indispensable. In spite of the specificity of this system, the analysis allows one to get an idea of the generalities of the use of acceleration-dependent forces in mechanics and shows that they can be used in practice to good advantage.
In his book “A Treatise on Analytical Dynamics,” Pars asserted that acceleration-dependent forces are inconsistent with one of the fundamental principles of mechanics, namely, with the superposition principle, thus spreading among mechanical scientists the idea that such forces are not admissible in mechanics. This article demonstrates that given forces that depend on acceleration or higher derivatives are admissible in mechanics and shows that this assertion in Pars’s book is fallacious and the only condition for the applicability of such forces is the equation of motion possessing a unique solution.
In many cases, mechanical systems include elements that differ greatly in inertia characteristics. It seems to be quite natural for a researcher who has to deal with such a system to have the desire to neglect its comparatively small inertia characteristics by putting them equal to zero. On such a simplification, the researcher has to do with a mechanical system that, along with 'massed' bodies (all inertia characteristics of which are distinct from zero), also includes 'massless' bodies (some inertia characteristics of which are zero). An important feature of systems of massed and massless bodies is that they may turn out to be singular. The analysis of systems of this type, in comparison with regular ones, involves some additional problems, whose solution, despite currently available methods of study of singular and singularly perturbed equations, may present a considerable challenge.In the current paper, the notion of the 'rank of a system of massed and massless bodies' is introduced, and an approach to the solution of the above problems based on this notion is proposed. This approach makes it possible to write generalized Lagrange equations for singular mechanical systems, to identify conditions for going from singularly perturbed equations of motion to singular ones to be correct, to identify conditions for the unique existence of the solution of the resulting singular equations of motion and to write them in normal form.
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