# Mechanics of non-holonomic systems

## A New Class of control systems

• Gives deeper insight into theory and applications of Analytical Mechanics

Book

Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)

1. Front Matter
Pages I-XXXII
2. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 1-24
3. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 25-76
4. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 77-104
5. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 105-124
6. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 125-148
7. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 149-192
8. Shervani Kh. Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda
Pages 193-212
9. Back Matter
Pages 213-329

### Introduction

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.

### Keywords

Non-holonomic systems control derivation of equations of motion mechanics programming vector relation

#### Authors and affiliations

1. 1.Academy of Sciences of the Chechen RepublicGroznyRussia 364906
2. 2.Dept. Mathematics & MechanicsSt. Petersburg State UniversitySt. PetersburgRussia 198504

### Bibliographic information

• Book Title Mechanics of non-holonomic systems
• Book Subtitle A New Class of control systems
• Authors Sh.Kh Soltakhanov
Mikhail Yushkov
S. Zegzhda
• Series Title Foundations of Engineering Mechanics
• DOI https://doi.org/10.1007/978-3-540-85847-8
• Copyright Information Springer Berlin Heidelberg 2009
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages Engineering Engineering (R0)
• Hardcover ISBN 978-3-540-85846-1
• Softcover ISBN 978-3-642-09938-0
• eBook ISBN 978-3-540-85847-8
• Series ISSN 1612-1384
• Series E-ISSN 1860-6237
• Edition Number 1
• Number of Pages XXXII, 332
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
Industry Sectors
Automotive
Chemical Manufacturing
Biotechnology
Consumer Packaged Goods
Energy, Utilities & Environment
Aerospace
Oil, Gas & Geosciences
Engineering

## Reviews

From the reviews:

“‘This monograph can be useful for English scientists. It will help them to get acquainted with a rather great number of works by Russian scientists.’ … the book gives an exhaustive account of many theoretical results and applications on non-holonomic mechanics, which often had little circulation in the literature; in this respect, the historical overview of the subject, with an emphasis on the contribution by the Russian school, is interesting. … Many applications are discussed in detail using different approaches … .” (Carlo Morosi, Mathematical Reviews, Issue 2011 j)