Fractional Analysis : Methods of Motion Decomposition

This book considers methods of approximate analysis of mechanical, elec­ tromechanical, and other systems described by ordinary differential equa­ 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 ex­ amination. Usually, this procedure is repeated several times. Even if an initial model correctly reflects the main properties of a phenomenon, it de­ 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 intu­ ition. A good example of such a semi-experimental way of simplifying is a gyroscopic precession equation. Formal mathematical proofs of its admis­ sibility appeared some several decades after its successful introduction in engineering calculations. Applied mathematics now has at its disposal many methods of approxi­ 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 fol­ 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.

I Dimensional analysis and small parameters.- 1 Dimensional analysis.- 1.1 The main concepts of dimensional analysis.- 1.2 Transformations in dimensional analysis.- 2 Introduction of small parameters.- 2.1 Normalization of equations of motion.- 2.2 Variants of small parameter introduction.- 2.3 Regular and singular perturbations with respect to the small parameter.- 2.4 Two types of power series expansion with respect to a small parameter.- 2.5 Redundancy in methods of approximation.- II Regularly perturbed systems. Expansions of solutions.- 3 The Poincaré theorem. The algorithm of expansion.- 4 Applications of the Poincaré theorem.- 4.1 Stokes’ problem.- 4.2 Secular terms.- 4.3 Systematic drifts of a gyro in gimbals. Method of successive approximations.- 5 Poincaré - 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.- 6.1 Krylov - Bogolyubov equations in standard form.- 6.2 Algorithm of asymptotic expansion.- 6.3 Approximation accuracy.- 6.4 Averaging over trajectories of the generating system.- 6.5 Variants of averaging methods.- 7 Applications of the method of averaging.- 7.1 Free oscillations with friction of various types.- 7.2 Free oscillations of a tube generator.- 8 Method of harmonic linearization.- 8.1 Foundations of the method.- 8.2 Examples.- 9 Method of averaging in systems with several fast phases.- 9.1 Averaged equations of the first approximation.- 9.2 Resonances in multifrequency systems.- 9.3 Averaging algorithm in the case of resonance.- 9.4 Pendulum resonance oscillations.- 9.5 Resonant oscillations with friction.- 10 Averaging in systems without explicit periodicities.- 10.1 Volosov averaging scheme.- 10.2 Separation of characteristic motions of an oscillator with high friction.- IV Decomposition of motion in systems with boundary layer.- 11 Tikhonov theorem.- 11.1 Introductory considerations.- 11.2 Tikhonov theorem.- 11.3 Decomposition of motion on an infinite time interval.- 12 Application of the Tikhonov theorem.- 12.1 Quasistatic motions of mechanical systems.- 12.2 The method of “frozen coefficients”.- 12.3 The limit model for a double pendulum of high stiffness.- 12.4 Relaxation oscillations of the valve generator.- 13 Asymptotic expansion of solutions for systems with a boundary layer.- 13.1 Algorithm of expansion.- 13.2 Asymptotic expansions for the Stokes problem.- 13.3 Asymptotic expansions on the problem of pendulum motion in a medium of high viscosity.- 13.4 Decomposition of motions of a railway car in magnetic suspension.- V Decomposition of motion in systems with discontinuous characteristics.- 14 Definition of a solution in discontinuity points.- 15 Examples.- 15.1 Relay control of angular motion of spacecraft. Sliding mode.- 15.2 Disc rolling motion with Coulomb friction.- 15.3 Relaxation oscillations of the Froude pendulum.- VI Correctness of limit models.- 16 Limit model of holonomic constraint (absolutely rigid body).- 16.1 Conditions for correctness of the model in statically definable and indefinable cases.- 16.2 Examples.- 17 Limit model of kinematic constraints.- 17.1 Conditions of model correctness in kinematically definable and indefinable cases.- 17.2 Change of kinematic constraints for rolling of a braked wheel.- 17.3 Kinematic indefinability in a rolling rail car problem.- 18 Limit model of servoconstraint.- 18.1 Conditions of servoconstraint realizability.- 18.2 Realization of servoconstraints, defining the manipulator extremity motion.- 19 Precession and nutation models in gyro theory.- 19.1 Correctness conditions for an extended precession model.- 19.2 Precession model for a gyrotachometer.- 19.3 Precession model of a three-axis force gyrostabilizer.- 19.4 Two-step method for stability approval of the nutation model for a three-axis gyrostabilizer.- 20 Mathematical model of a “man ― artificial-kidney” system.- 21 Approximate models of an aircraft motion.- 21.1 Models of zeroth approximation with respect to small parameters.- 21.2 Refined models of motion.- 22 Automobile motion decomposition.- 22.1 Structure of automobile motion partial models.- 22.2 Mathematical model of rolling for a deformed wheel. Are nonlinear nonholonomic constraints possible?.- References.- Author Index.