Mechanics of non-holonomic systems

Abstract

Further volumes of this series can be found on our homepage: springer.com Slivker, V.I. Elements, 2007 ISBN 978-3-540-44718-4 Elsoufiev, S.A. Geomechanics, 2007 ISBN 978-3-540-37052-9 Awrejcewicz, J., Krysko, V.A., Krysko, A.V. Plates and Shells, 2007 ISBN 978-3-540-37261-8 Wittbrodt, E., Adamiec-Wojcik, I., Wojciech, S. Systems, 2006 ISBN 3-540-32351-1 Aleynikov, S.M. Geotechnics, 2007 ISBN 3-540-25138-3 Skubov, D.Y., Khodzhaev, K.S. Non-Linear Electromechanics, 2007 ISBN 3-540-25139 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Mechanics of Structural Strength Analysis in Thermo-Dynamics of Dynamics of Flexible Multibody Spatial Contact Problems inThe use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, BerlinPrinted on acid-free paper 9 8 7 6 5 4 3 2 1 springer. AnnotationA general approach to the generation of equations of motion of holonomic and nonholonomic systems with the constraints of any order is proposed. The system of equations of motion in generalized coordinates is regarded as 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 law of motion is given by the equations of constraints and in another, for ideal constraints, it is described by the vector equation not involving reactions of constraints. In the whole space the law of motion involves the Lagrange 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 the Lagrange multipliers to holonomic systems permits us to construct two new methods for determining the normal frequencies and normal forms of oscillations of elastic systems and also to propose a special form of equations of motion for system of rigid bodies. The nonholonomic constraints, the order of which is greater than two, are regarded as program constraints such that their validity is provided by the existence of generalized control forces, which are determined as the functions of time. The closed system of different...

Trajectories of a Ball Moving Inside a Spherical Cavity Using First Integrals of The Governing Nonlinear System

“…(2) are formulated as linear functions of velocities 153q j ; this has been shown to be satisfactory concerning the problems considered. For more details and generalization; see [4] or [48].…”

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

“…Taking this into account, the necessary dynamic conditions for realizing the motion of the system in accordance with restrictions (1) and (2) [9,10], based on the Coulomb friction laws, are that intensities of the interaction forces between points A and B of the system and the horizontal plane of motion should not exceed the corresponding Coulomb forces of sliding friction.…”

10.5937/fmet1403199r = Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

“…1. ), taken completely from [10], it is necessary first to introduce two Cartesian reference coordinate systems: the stationary coordinate system Oξηζ , whose coordinate plane Oξη coincides with the horizontal plane of vehicle motion, and a movable coordinate system Axyz , which is rigidly attached to the vehicle body, so that the coordinate plane Axy coincides with the plane Oξη . The axis of the movable coordinate system Ax is defined by the direction of a normal to the axis of the vehicle rear axle and point C, where C Ax  .…”

“…Having this in mind, the necessary dynamic conditions for realizing such motion [10], and based on the Coulomb laws of sliding friction, are Based on above considerations, it can be inferred (see Fig. 5.)…”

“…This paper, using the example of a nonholonomic mechanical system with limited reactions of constraints, presents the procedure of creating the differential equations of motion where both reactions of nonholonomic constraints and control forces figure, based on the general theorems of dynamics [9]. Applying the general theorems of dynamics, one obtains, on one hand, more suitable relations in determining the reactions of constraints and control forces, while, on the other hand, physical interpretation of the multipliers of constraints is unnecessary as in the case of applying some other equations of analytical mechanics that refer to nonholonomic mechanical systems [10]. This paper also provides the procedure for solving brachistochronic motion of a nonholonomic mechanical system in a plane at the steady value of mechanical energy during motion, when initial and end positions are specified [14].…”