Fractional Analysis: Methods of Motion Decomposition (1997)

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 Poincare theorem. The algorithm of expansion.- 4 Applications of the Poincare theorem.- 4.1 Stokes' problem.- 4.2 Secular terms.- 4.3 Systematic drifts of a gyro in gimbals. Method of successive approximations.- 5 Poincare - 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.