دینامیک سیستم‌های غیرهولونومیک

TABLE OF CONTENTS PREFACE.............................................................. iii INTRODUCTION ......................................................... 1 CHAPTER I. KINEMATICS OF NONHOLONOMIC SYSTEMS ........................ 3 1. Holonomic and nonholonomic discrete mechanical systems..............3 2. Configuration space ................................................7 3. Virtual displacements. The number of degrees of freedom ...........12 4. Phase space .......................................................15 5. Kinematics of the rolling of one surface on another............... 17 6. Kinematic integrating mechanisms...................................22 7. Holonomy criteria for a system with linear kinematic constraints.. 29 CHAPTER II. STUDY OF THE MOTIONS OF NONHOLONOMIC SYSTEMS ON THE BASIS OF THE GENERAL LAWS OF DYNAMICS. CLASSICAL PROBLEMS OF RIGID BODIES ROLLING ON A SURFACE........................ 44 1. General laws of dynamics. Generalization of the angular momentum integral.................................................... 45 2. Rolling of a disc and a torus on a horizontal plane................55 3. The Bobylev-iukovskii problem of a rolling sphere containing a gyroscope...........................................................64 4. Caplygin's problem of nonholonomic motion on a plane...............71 5. Rolling of a sphere on an absolutely rough surface................ 76 CHAPTER III. ANALYTIC DYNAMICS OF NONHOLONOMIC SYSTEMS............... 87 1. Principle of virtual displacements and the d' Alembert-Lagrange equations ........................................................... 87 2. Equations of motion of nonholonomic systems with Lagrangian multipliers. The reactions of ideal nonholonomic constraints..........92 3. Nonholonomic Caplygin systems. Caplygin's equations. Vorenec' s equations................................................ 100 4. Equations of Volterra and Maggi ..................................114 5. Equations of motion in quasi-coordinates......................... 120 6. Transpositional relations in the analytic mechanics of nonholonomic systems..............................................................135 7. Canonical form of the equations of motion of nonholonomic systems............................................................. 143 8. Appell's equations................................................147 9. Impulsive motion of nonholonomic systems..........................159 10. Variational principles in the mechanics of nonholonomic systems..............................................................175 11. First integrals of the equations of motion of nonholonomic systems..............................................................187 12. Theory of Caplygin' s reducing multiplier (last multiplier) .....199 CHAPTER IV. VALIDITY OF THE MATHEMATICAL MODELS IN THE MECHANICS OF NONHOLONOMIC SYSTEMS................................212 0. Introduction..................................................... 213 1. Errors arising from neglect of the finite size of the region of contact of rolling bodies .......................................... 214 2. The Appell-Hamel example of a system with a nonlinear nonholonomic constraint ..........................................................223 3. On the realizability of a nonholonomic constraint by forces of anisotropic viscous friction......................................233 CHAPTER V. SMALL OSCILLATIONS AND THE STABILITY OF NONHOLONOMIC SYSTEMS..............................................................238 1. General results from the theory of small oscillations and the theory of stability..................................................238 2. Stability and small oscillations of nonholonomic systems near equilibrium states..............................................261 3. Stability of steady motions of holonomic and nonholonomic systems..............................................................294 CHAPTER VI. DYNAMICS OF NONHOLONOMIC SYSTEMS AND TECHNICAL PROBLEMS OF THE DIRECTIONAL STABILITY OF ROLLING SYSTEMS ........... 308 1. Theory of rolling of an elastic pneumatic tire. Equations of motion of vehicles with pneumatically tired wheels...................308 2. Stability of a bicycle and a motorcycle ..........................330 3. Shimmy of the nose wheel ofa three-wheel aircraft landing gear ...374 4. Shimmy of the front suspension of an automobile...................393 5. Directional stability of an automobile .......................... 408 6. Traveling stability of pairs of railroad wheels and trucks....... 419 CHAPTER VII. DYNAMICS OF NONHOLONOMIC SYSTEMS AND THE GENERAL THEORY OF ELECTRICAL MACHINES .............................. 426 1. Maxwell's equations. The concept of a state in electrodynamics....426 2. Derivation of the equations of electrodynamics from a variational principle........................................................... 430 3. Quasisteady approximation ........................................435 4. Discrete description of electromagnetic processes in the quasi-steady approximation.......................................... 439 5. Electrodynamics of slowly moving media. The ponderomotive forces ............................................................. 442 6. Lagrange-Maxwell equations for electromechanical systems......... 447 7. Extension of the Lagrange-Maxwell equations to electro- mechanical systems with currents that do not flow in closed loops.............................................................. 455 8. Models of electrical machines described · by the Lagrange -Maxwell equations...................................................463 9. Examples of electromechanical systems with nonholonomic constraints realized by sliding contacts.............................470 10. Gaponov's equations of motion of nonholonomic electrical systems............................................................. 478 BIBLIOGRAPHY ........................................................495