# Mechanics of non-holonomic systems: A New Class of control systems (Google eBook)

Springer Science & Business Media, May 27, 2009 - Technology & Engineering - 332 pages

A general approach to the derivation of equations of motion of as holonomic, as nonholonomic systems with the constraints of any order is suggested. The system of equations of motion in the generalized coordinates is regarded as a one vector relation, represented in a space tangential to a manifold of all possible positions of system at given instant. The tangential space is partitioned by the equations of constraints into two orthogonal subspaces. In one of them for the constraints up to the second order, the motion low is given by the equations of constraints and in the other one for ideal constraints, it is described by the vector equation without reactions of connections. In the whole space the motion low involves Lagrangian multipliers. It is shown that for the holonomic and nonholonomic constraints up to the second order, these multipliers can be found as the function of time, positions of system, and its velocities. The application of Lagrangian multipliers for holonomic systems permits us to construct a new method for determining the eigenfrequencies and eigenforms of oscillations of elastic systems and also to suggest a special form of equations for describing the system of motion of rigid bodies. The nonholonomic constraints, the order of which is greater than two, are regarded as programming constraints such that their validity is provided due to the existence of generalized control forces, which are determined as the functions of time. The closed system of differential equations, which makes it possible to find as these control forces, as the generalized Lagrange coordinates, is compound. The theory suggested is illustrated by the examples of a spacecraft motion. The book is primarily addressed to specialists in analytic mechanics.

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### Contents

 HOLONOMIC SYSTEMS 1 2 Lagranges equations of the ﬁrst and second kinds 4 3 The DAlembert Lagrange principle 12 4 Longitudinal accelerated motion of a car as an example of motion of a holonomic system with a nonretaining constraint 15 NONHOLONOMIC SYSTEMS 25 2 Equations of motion of nonholonomic systems Maggis equations 28 3 The generation of the most usual forms of equations of motion of nonholonomic systems from Maggis equations 38 4 The examples of applications of different kinds equations of nonholonomic mechanics 45
 6 Transformation of the frequency equation to a dimensionless form and determination of minimal number of parameters governing a natural freque... 173 7 A special form of equations of the dynamics of system of rigid bodies 178 8 The application of special form of equations of dynamics to the study of certain problems of robotics 181 9 Application of the generalized Gaussian principle to the problem of suppression of mechanical systems oscillations 183 EQUATIONS OF MOTION IN QUASICOORDINATES 193 2 The PoincareChetaevRumyantsev approach to the generation of equations of motion of nonholonomic systems 201 3 The approach of J Papastavridis to the generation of equations of motion of nonholonomic systems 207 THE METHOD OF CURVILINEAR COORDINATES 213

 5 The Suslov Jourdain principle 66 6 The definitions of virtual displacements by Chetaev 74 LINEAR TRANSFORMATION OF FORCES 77 2 Theorem on the forces providing the satisfaction of holonomic constraints 83 3 An example of the application of theorem on the forces providing the satisfaction of holonomic constraints 88 4 Chetaevs postulates and the theorem on the forces providing the satisfaction of nonholonomic constraints 92 5 An example of the application of theorem on forces providing the satisfaction of nonholonomic constraints 97 6 Linear transformation of forces and Gaussian principle 100 APPLICATION OF A TANGENT SPACE TO THE STUDY OF CONSTRAINED MOTION 105 2 The connection of differential variational principles of mechanics 109 3 Geometric interpretation of linear and nonlinear nonholonomic constraints Generalized Gaussian principle 113 4 The representation of equations of motion following from generalized Gaussian principle in Maggis form 119 5 The representation of equations of motion following from generalized Gaussian principle in Appells form 121 THE MIXED PROBLEM OF DYNAMICS NEW CLASS OF CONTROL PROBLEMS 125 2 A generation of a closed system of differential equations in generalized coordinates and the generalized control forces 128 3 The mixed problem of dynamics and Gaussian principle 131 4 The motion of spacecraft with modulo constant acceleration in Earths gravitational field 137 5 The satellite maneuver alternative to the Homann elliptic motion 144 APPLICATION OF THE LAGRANGE MULTIPLIERS TO THE CONSTRUCTION OF THREE NEW METHODS FOR THE STUDY OF MECHANI... 149 1 Some remarks on the Lagrange multipliers 150 2 Generalized Lagrangian coordinates of elastic body 152 3 The application of Lagranges equations of the first kind to the study of normal oscillations of mechanical systems with distributed parameters 154 4 Lateral vibration of a beam with immovable supports 160 5 The application of Lagranges equations of the ﬁrst kind to the determination of normal frequencies and oscillation modes of system of bars 165
 2 The relation between a reciprocal basis and gradients of scalar functions 215 3 Covariant and contravariant components of vector 216 4 Covariant and contravariant components of velocity vector 217 5 Christoffel symbols 218 6 Covariant and contravariant components of acceleration vector The Lagrange operator 220 7 The case of cylindrical system of coordinates 222 8 Covariant components of acceleration vector for nonstationary basis 225 9 Covariant components of a derivative of vector 227 STABILITY AND BIFURCATION OF STEADY MOTIONS OF NONHOLONOMIC SYSTEMS 229 THE CONSTRUCTION OF APPROXIMATE SOLUTIONS FOR EQUATIONS OF NONLINEAR OSCILLATIONS WITH THE USAGE OF THE G... 235 THE MOTION OF NONHOLONOMIC SYSTEM WITH OUT REACTIONS OF NONHOLONOMIC CONSTRAINTS 239 2 Free motion of the Chaplygin sledge 240 3 The possibility of free motion of nonholonomic system under active forces 243 THE TURNING MOVEMENT OF A CAR AS A NONHOLONOMIC PROBLEM WITH NONRETAINING CONSTRAINTS 245 2 The turning movement of a car with retaining bilateral constraints 246 3 The turning movement of a reardrive car with nonretaining constraints 249 4 Equations of motion of a turning frontdrive car with nonretaining constraints 256 5 Calculation of motion of a certain car 259 6 Reasonable choice of quasivelocities 261 CONSIDERATION OF REACTION FORCES OF HOLONOMIC CONSTRAINTS AS GENERALIZED COORDINATES IN APPROXIMATE DETE... 263 THE DUFFING EQUATION AND STRANGE ATTRACTOR 281 References 287 INDEX 327 Copyright