Dynamics of nonholonomic systems

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