Foundations of Engineering Mechanics M.Z. Kolovsky, A.N. Evgrafov, Yu. A. Semenov, A. V. Slousch Advanced Theory of Mechanisms and Machines Springer-Verlag Berlin Heidelberg GmbH ONLINE LIBRARY Engineering http://www.springer.de/engine/ M.Z. Kolovsky, A.N. Evgrafov, Yu. A. Semenov, A. V. Slousch Advanced Theory of Mechanisms and Machines Translated by L. Lilov With 250 Figures , Springer Series Editors: V. 1. Babitsky, DSc J. Wittenhurg Loughborough University Universităt Karlsruhe (TH) Department of Mechanical Engineering Institut fiirTechnische Mechanik LEII 3TU Loughborough, Leicestershire KaiserstraBe 12 United Kingdom D-76128 Karlsruhe I Germany Authors: M.Z. Kolovsky A.N. Evgrafov Yu. A. Semenov A. V. Slousch State Technical University St. Petersburg Kondratievsky 56-24 195197 St. Petersburg Russia Translator: Prof. Dr. 1. Lilov ul. Rajko Jinzifov 1606 Sofia Bulgaria Cataloging-in publication data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Advanced theory of mechanisms and machines / M.Z. Kolovsky. Translated by L. Lilov Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Foundations of engineering mechanics) ISBN 978-3-642-53672-4 ISBN 978-3-540-46516-4 (eBook) DOI 10.1007/978-3-540-46516-4 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, re citation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereofis 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-Verlag. Violations are liable for prosecution act under German Copyright Law. © Springer-Verlag Berlin Heidelberg 2000 Originally published by Springer-Verlag Berlin Heidelberg New York in 2000 Softcover reprint ofthe hardcover Ist edition 2000 The 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. Typesetting: Camera-ready copy from authors Cover-Design: de'blik, Berlin Printed on acid-free paper SPIN 10728537 62/3020 5432 1 O Preface This book is based on a lecture course delivered by the authors over a period of many years to the students in mechanics at the St. Petersburg State Technical University (the former Leningrad Polytechnic Institute). The material differs from numerous traditional text books on Theory of Machines and Mechanisms through a more profound elaboration of the methods of structural, geometric, kinematic and dynamic analysis of mechanisms and machines, consisting in both the development of well-known methods and the creation of new ones that take into account the needs of modem machine building and the potential of modem computers. The structural analysis of mechanisms is based on a new definition of structural group which makes it possible to consider closed structures that cannot be reduced to linkages of Assur groups. The methods of geometric analysis are adapted to the analysis of planar and spatial mechanisms with closed structure and several degrees of movability. Considerable attention is devoted to the problems of con- figuration multiplicity of a mechanism with given input coordinates as well as to the problems of distinguishing and removing singular positions, which is of great importance for the design of robot systems. These problems are also reflected in the description of the methods of kinematic analysis employed for the investi- gation of both open ("tree"-type) structures and closed mechanisms. The methods of dynamic analysis were subject to the greatest extent of modification and development. In this connection, special attention is given to the choice of dynamic models of machines and mechanisms, and to the evaluation of their dynamic characteristics: internal and external vibration activity as well as frictional forces and energy losses due to friction at kinematic pairs. The dynamic analysis of machine assemblies is based on both models of "rigid" mechanism and models that take into account the elasticity of links and kinematic pairs. Different engine characteristics are considered in the investigation of the dynamics of machine assemblies. Special attention is given to the dynamics of machines with feedback systems for motion control. The limited volume of the text book did not allow the authors to include some traditional topics (the investigation of geometry of gearings, cam mechanisms, the parametric synthesis). The authors assume that these topics are presented to a satisfactory extent in the available text books. The text book sets a large number of problems. Some of them are solved in details, the rest have only answers. The authors believe that the solution of the problems is necessary for the full understanding of the course. In order to successfully master the material in the text book, the reader should possess a certain level of knowledge in the field of mathematics and theoretical mechanics. On the whole, the required level corresponds to the common progams taught in higher technical educational institutions. VI Preface The text book has been written by a team of authors and it is difficult to distin- guish the participation of anyone of them. The authors would like to note that the successful preparation of this new course was fostered with the great help of the lecturers of the Chair of Theory of Machines and Mechanisms CSt. Petersburg State Technical University) and, most of all, with the continual support of Prof. G.A. Smirnov who was for many years the head of this chair. As it is known, the work on a text book is not finished with its publication. Coming out of press only signifies the beginning of this work. The authors will be genuinely grateful to the readers for any critical remarks on the material presented in this text book and for any suggestions for its improvement. Authors M.Z. Kolovsky A.N. Evgrafov J.A. Semenov A.V. Slousch Contents 1 Structure of Machines and Mechanisms 1 1.1 Machines and Their Role in Modem Production 1 1.2 Structure of a Machine and its Functional Parts .................... . 2 l.3 Mechanisms. Links and Kinematic Pairs ............................. . 3 1.4 Kinematic Chains and Structural Groups. Generation of Mechanisms ............................................. . 10 1.5 Mechanisms with Excessive Constraints and Redundant Degrees of Movability .................................... . 17 1.6 Planar Mechanisms .................................................... .. 19 1.7 Mechanisms with Variable Structure. Strucural Transformations of Mechanisms .......................... . 24 1.8 Examples of Structural Analysis of Mechanisms .................. .. 27 1.9 Problems .................................................................. . 33 2 Geometric Analysis of Mechanisms 41 2.1 Problems of Geometric Analysis ............................ ............. 41 2.2 Geometric Analysis of Open Kinematic Chains ...................... 44 2.3 Derivation of Equations of Geometric Analysis for Closed Kinematic Chains ................................................ 52 2.4 Solution to the Equations of Geometric Analysis ..................... 58 2.5 The Inverse Problem of Geometric Analysis .......................... 66 2.6 Special Features of Geometric Analysis of Mechanisms with Higher Kinematic Pairs ............................. 70 2.7 Problems ... ... ... ... ......... ......... ......... ...................... ...... 72 3 Kinematic and Parametric AnalYSis of Mechanisms 79 3.1 Kinematic Analysis of Planar Mechanisms . .. . . . . .. .. . .. . . .. . . . . .. ... 79 3.2 Kinematic Analysis of Spatial Mechanisms ........................... 85 3.3 Kinematic Analysis of a Mechanism with a Higher Pair . . . . . . . . . ... 90 3.4 Kinematics of Mechanisms with Linear Position Functions . .. .. .. 93 3.5 Parametric Analysis of Mechanisms ... ...... ... ...... ... ......... ... 103 3.6 Problems ... ............ ......... ...... ......... .................. ...... .... 108 VIII Contents 4 Determination of Forces Acting in Mechanisms 121 4.1 Geometric Conditions for Transmission of Forces by Mechanisms ... 121 4.2 Determination of Forces Acting in Mechanisms by the Graph-Analytic Method and the Method of Opening Kinematic Chains................................................. 128 4.3 Application of Equilibrium Equations of a Mechanism to its Kinematic and Parametric Analysis .................. .................. 133 4.4 General Formulation of the Force Analysis Problem .. .............. 138 4.5 Equations of Kinetostatics. Determination of the Resultant Vector and ofthe Resultant Moment ofInertia Forces of Links.... 143 4.6 Solution of the Equations of Kinetostatics ............ ...... ........... 147 4.7 Application of the General Equation of Dynamics for Force Analysis of Mechanisms ...... ...... ............ ...... ............ 152 4.8 Force Analysis of Mechanisms with Higher Kinematic Pairs ...... 157 4.9 Problems ............ .................................... .................. 158 5 Friction in Mechanisms 175 5.1 Friction in Kinematic Pairs .............................................. 175 5.2 Models of Kinematic Pairs with Friction .......................... ..... 178 5.3 Force Analysis of Mechanisms with Friction .......... ............... 185 5.4 Problems .................................................................... 194 6 Equations of Motion for a Mechanism with Rigid Links 211 6.1 Lagrange's Equations of the Second Kind for a Mechanism with a Single Degree of Movability .......... ...... ..... 211 6.2 Lagrange's Equations of the Second Kind for Mechanisms with Several Degrees of Movability . .. .. . . .. . . . . . . .. .. 216 6.3 An Example for Derivation of the Equations of Motion of a Mechanism ....... .. .. .. .. .. .. .. .. .. .. .. ...... 219 6.4 Problems ................................................................... 224 7 Dynamic Characteristics of Mechanisms with Rigid Links 235 7.1 Internal Vibration Activity of a Mechanism ......................... 235 7.2 Methods of Reduction of Perturbation Moments ... ......... ... .... 237 7.3 External Vibration Activity of Mechanisms and Machines 239 7.4 External Vibration Activity of a Rotating Rotor and of a Rotor Machine ........... , .................... , ... .......... ... ... ... ..... 242 Contents IX 7.5 Balancing of Rotors ..................... ...................... ........ ... 245 7.6 Vibration Activity ofa Planar Mechanism ... ...... ... ......... ...... 247 7.7 Loss of Energy due to Friction in a Cyclic Mechanism . .. .. ...... ... 252 7.8 Problems ... ......... ... ...... ... ...... ... ............ ... ...... ....... ... ... 254 8 Dynamics of Cycle Machines with Rigid Links 269 8.1 Mechanical Characteristics of Engines ............ ... ... ... ........ .... 269 8.2 Equations of Motion of a Machine. State of Motion ................. 276 8.3 Determination of the Average Angular Velocity of a Steady-State Motion for a Cycle Machine ............................. 278 8.4 Determination of Dynamic Errors and of Dynamic Loads in a Steady-State Motion . .. . . . .. . . .. ... .. . . .. . .. .. . . .. . .. .. . .. . . .. . . ... . . .. ... 280 8.5 Influence ofthe Engine Dynamic Characteristic on Steady-State Motions ........................ ...... ...... ... ... ...... ... ... 286 8.6 Starting Acceleration of a Machine ................................. .... 289 8.7 BrakingofaMachine .................................................... 294 8.8 Problems ................................................................... 295 9 Dynamics of Mechanisms with Elastic Links 301 9.1 Mechanisms with Elastic Links and Their Dynamic Models 301 9.2 Reduction of Stiffuess. Inlet and Outlet Stiffuess and Flexibility of a Mechanism...... ...... ......... ... ...... ... ... ........ ... 305 9.3 Reduced Stiffuess and Reduced Flexibility of a Mechanism with Several Degrees of Movability .......................................... 308 9.4 Determination of Reduced Flexibilities with the Help of Equilibrium Equations of a Rigid Mechanism . .. .. . . .. .. . . .... . . . .... 311 9.5 Some Problems of Kinematic Analysis of Elastic Mechanisms ... 313 9.6 Dynamic Problems of Elastic Mechanisms ....................... ..... 315 9.7 Free and Forced Vibration of Elastic Mechanisms ........... ..... .... 318 9.8 Problems ........ .... ......... ... ...... ...... ... ...... ... ... ....... ... ... ... 321 10 Vibration of Machines with Elastic Transmission Mechanisms 327 10.1 Dissipative Forces in Deformable Elements .................. ...... ... 327 10.2 Reduced Stiffuess and Reduced Damping Coefficient ............ 330 10.3 Steady-State Motion of a Machine with an Ideal Engine. Elastic Resonance ........................................................ 332 10.4 Influence of the Static Characteristic of an Engine on a Steady-State Motion ................................................... ... 339 X Contents 10.5 Transients in an Elastic Machine .......................... ............. 342 10.6 Problems ... ...... ... ... ... ... ...... ... ... ... ...... ... ... ... ... ... ... ... .... 349 11 Vibration of a Machine on an Elastic Base. Vibration Isolation of Machines 361 11.1 Vibration of the Body of a Machine Mounted on an Elastic Base ................................................................ 361 11.2 Vibration of a Machine in the Resonance Zone. Sommerfeld Effect........ ...... ... ... ...... ... ...... ... ... ... ... ... ... .... 364 11.3 Vibration Isolation of Machines ...... ... ... ...... ... ...... ... ...... .... 367 11.4 Problems ................................................................... 369 12 Elements of Dynamics of Machines with Program Control 371 12.1 Basic Principles of Construction of Machines with Program Control... ... ... ... ... ... ... ... ... ... ... ....... 371 12.2 Determination of Program Control. Sources of Dynamic Errors... 373 12.3 Closed Feedback Control Systems ................. ................. .... 378 12.4 Effectiveness and Stability ofa Closed System ...... ... ... .......... 380 12.5 Problems ................................................................... 383 References 387 Index 389