Nonsmooth impact mechanics. Models, dynamics and control.

*(English)*Zbl 0861.73001
Lecture Notes in Control and Information Sciences. 220. Berlin: Springer. xvi, 404 p. (1996).

The book deals with very important phenomena in engineering and other fields of applied science, namely dynamic phenomena with impacts. In order to illustrate the importance of such problems in engineering, let us mention clattering oscillations in automobile gear boxes which sometimes occur for wheels which are not shifted to transmit a tourque. These oscillations basically are very high frequency impact oscillations due to the clearances depending on the teeth-meshing of the gear wheels. To understand and to prevent such oscillations, it is essential to have an effective mathematical theory available to treat impact problems in mechanics. To develope such a theory is the main aim of this book which focusses on the mathematical aspects in a very competent and comprehensive way.

Basically, the physical impact problem can be transformed into the mathematical problem for a dynamical system with unilateral constraints. The unilateral constraints depending on the position variables are treated in the book. Especially, the control of systems with unilateral constraints is addressed. If the bodies are considered to be rigid as it is in general, then it is well known that certain indetermined problems may arise for which the mathematical question of well-posedness, that is, such properties of solutions as existence, uniqueness and continuous dependence on initial data and parameters, will be very important. However, it should be noted that at several points in the book also the influences of flexibility of the structures are discussed. Further questions of global motions of multibody systems with impacts, the construction of proper Poincaré mappings resulting in time-discrete nonlinear dynamical systems, and stability problems are treated in detail.

The presentation is excellent in combining rigorous mathematics with a great number of examples ranging from simple mechanical systems to robotic systems allowing the reader to understand the basic concepts. Abundant literature citations are given in the text and in the list of references comprising more than 500 citations and allowing the reader to enter easily more specialized fields.

This book can be strongly recommended to engineers wishing to better understand the mathematical background of impacting systems, and to applied mathematicians who will still find a great number of open mathematical questions to be attacked in the future.

Basically, the physical impact problem can be transformed into the mathematical problem for a dynamical system with unilateral constraints. The unilateral constraints depending on the position variables are treated in the book. Especially, the control of systems with unilateral constraints is addressed. If the bodies are considered to be rigid as it is in general, then it is well known that certain indetermined problems may arise for which the mathematical question of well-posedness, that is, such properties of solutions as existence, uniqueness and continuous dependence on initial data and parameters, will be very important. However, it should be noted that at several points in the book also the influences of flexibility of the structures are discussed. Further questions of global motions of multibody systems with impacts, the construction of proper Poincaré mappings resulting in time-discrete nonlinear dynamical systems, and stability problems are treated in detail.

The presentation is excellent in combining rigorous mathematics with a great number of examples ranging from simple mechanical systems to robotic systems allowing the reader to understand the basic concepts. Abundant literature citations are given in the text and in the list of references comprising more than 500 citations and allowing the reader to enter easily more specialized fields.

This book can be strongly recommended to engineers wishing to better understand the mathematical background of impacting systems, and to applied mathematicians who will still find a great number of open mathematical questions to be attacked in the future.

Reviewer: H.Troger (Wien)

##### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

74A55 | Theories of friction (tribology) |

74M15 | Contact in solid mechanics |

34H05 | Control problems involving ordinary differential equations |

70K99 | Nonlinear dynamics in mechanics |