This book considers methods of approximate analysis of mechanical, elecยญ tromechanical, and other systems described by ordinary differential equaยญ tions. Modern mathematical modeling of sophisticated mechanical systems consists of several stages: first, construction of a mechanical model, and then writing appropriate equations and their analytical or numerical exยญ amination. Usually, this procedure is repeated several times. Even if an initial model correctly reflects the main properties of a phenomenon, it deยญ scribes, as a rule, many unnecessary details that make equations of motion too complicated. As experience and experimental data are accumulated, the researcher considers simpler models and simplifies the equations. Thus some terms are discarded, the order of the equations is lowered, and so on. This process requires time, experimentation, and the researcher's intuยญ ition. A good example of such a semi-experimental way of simplifying is a gyroscopic precession equation. Formal mathematical proofs of its admisยญ sibility appeared some several decades after its successful introduction in engineering calculations. Applied mathematics now has at its disposal many methods of approxiยญ mate analysis of differential equations. Application of these methods could shorten and formalize the procedure of simplifying the equations and, thus, of constructing approximate motion models. Wide application of the methods into practice is hindered by the folยญ lowing. 1. Descriptions of various approximate methods are scattered over the mathematical literature. The researcher, as a rule, does not know what method is most suitable for a specific case. 2
CONTENT
I Dimensional analysis and small parameters -- 1 Dimensional analysis -- 2 Introduction of small parameters -- II Regularly perturbed systems. Expansions of solutions -- 3 The Poincarรฉ theorem. The algorithm of expansion -- 4 Applications of the Poincarรฉ theorem -- 5 Poincarรฉ - Lyapunov method -- 5.1 Algorithm of the method -- 5.2 Examples. Nonisochronism of nonlinear system oscillations -- III Decomposition of motion in systems with fast phase -- 6 Method of averaging in systems with a single fast phase -- 7 Applications of the method of averaging -- 8 Method of harmonic linearization -- 9 Method of averaging in systems with several fast phases -- 10 Averaging in systems without explicit periodicities -- IV Decomposition of motion in systems with boundary layer -- 11 Tikhonov theorem -- 12 Application of the Tikhonov theorem -- 13 Asymptotic expansion of solutions for systems with a boundary layer -- V Decomposition of motion in systems with discontinuous characteristics -- 14 Definition of a solution in discontinuity points -- 15 Examples -- VI Correctness of limit models -- 16 Limit model of holonomic constraint (absolutely rigid body) -- 17 Limit model of kinematic constraints -- 18 Limit model of servoconstraint -- 19 Precession and nutation models in gyro theory -- 20 Mathematical model of a โ{128}{156}man โ{128}{148} artificial-kidneyโ{128}{157} system -- 21 Approximate models of an aircraft motion -- 22 Automobile motion decomposition -- References -- Author Index
