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

327 | |

### Common terms and phrases

acceleration applied center of mass Chaplygin Chaplygin's equations Chapter Chetaev coefficients computed considered control force corresponding curvilinear coordinates deﬁned deﬁnition diﬀerent diﬀerential equations disk dynamics elastic equa equations of constraints equations of motion example ﬁnd ﬁrst kind ﬁxed form of equations formulas friction force functions Gaussian principle given HerojiOHOMHBix holonomic constraints holonomic systems ideal constraints integrals introduced kinetic energy L-space Lagrange multipliers Lagrange’s equations Lagrangian coordinates linear Maggi's equations Maggi’s mechanical system MexaHHKa motion of nonholonomic nonholonomic mechanics nonholonomic systems nonlinear nonholonomic constraints obtain oscillations parameters plane problem quantities quasivelocities relations representation point represented rigid body satisﬁed second kind solution steady motions straints system of equations take the form Taking into account tangent space Tejia theorem tions type of motion values variables vector velocity virtual displacements wheels yH-Ta Λκ