Advanced theory of mechanisms and machines

Foundations of Engineering Mechanics M.Z Kolovsky, A.N Evgrafov, Yu A Semenov, A V Slousch Advanced Theory of Mechanisms and Machines Tai ngay!!! Ban co the xoa dong chu nay!!! Springer-Verlag Berlin Heidelberg GmbH Engineering ONLINE LIBRARY 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 Babitsky, DSc Loughborough University Department of Mechanical Engineering LEII 3TU Loughborough, Leicestershire United Kingdom J Wittenhurg Karlsruhe (TH) Institut fiirTechnische Mechanik KaiserstraBe 12 D-76128 Karlsruhe I Germany Universităt 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 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; NewYork; 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 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 configuration 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 investigation 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 distinguish 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 Structure of Machines and Mechanisms Machines and Their Role in Modem Production Structure of a Machine and its Functional Parts Mechanisms Links and Kinematic Pairs Kinematic Chains and Structural Groups Generation of Mechanisms 1.5 Mechanisms with Excessive Constraints and Redundant Degrees of Movability 1.6 Planar Mechanisms 1.7 Mechanisms with Variable Structure Strucural Transformations of Mechanisms 1.8 Examples of Structural Analysis of Mechanisms 1.9 Problems 1.1 1.2 l.3 1.4 Geometric Analysis of Mechanisms 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Problems of Geometric Analysis Geometric Analysis of Open Kinematic Chains Derivation of Equations of Geometric Analysis for Closed Kinematic Chains Solution to the Equations of Geometric Analysis The Inverse Problem of Geometric Analysis Special Features of Geometric Analysis of Mechanisms with Higher Kinematic Pairs Problems Kinematic and Parametric AnalYSis of Mechanisms 3.1 3.2 3.3 3.4 3.5 3.6 Kinematic Analysis of Planar Mechanisms Kinematic Analysis of Spatial Mechanisms Kinematic Analysis of a Mechanism with a Higher Pair Kinematics of Mechanisms with Linear Position Functions Parametric Analysis of Mechanisms Problems 1 10 17 19 24 27 33 41 41 44 52 58 66 70 72 79 79 85 90 93 103 108 VIII Contents Determination of Forces Acting in Mechanisms 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Geometric Conditions for Transmission of Forces by Mechanisms Determination of Forces Acting in Mechanisms by the Graph-Analytic Method and the Method of Opening Kinematic Chains Application of Equilibrium Equations of a Mechanism to its Kinematic and Parametric Analysis General Formulation of the Force Analysis Problem Equations of Kinetostatics Determination of the Resultant Vector and ofthe Resultant Moment ofInertia Forces of Links Solution of the Equations of Kinetostatics Application of the General Equation of Dynamics for Force Analysis of Mechanisms Force Analysis of Mechanisms with Higher Kinematic Pairs Problems Friction in Mechanisms 5.1 5.2 5.3 5.4 Friction in Kinematic Pairs Models of Kinematic Pairs with Friction Force Analysis of Mechanisms with Friction Problems Equations of Motion for a Mechanism with Rigid Links 6.1 6.2 6.3 6.4 Lagrange's Equations of the Second Kind for a Mechanism with a Single Degree of Movability Lagrange's Equations of the Second Kind for Mechanisms with Several Degrees of Movability An Example for Derivation of the Equations of Motion of a Mechanism Problems Dynamic Characteristics of Mechanisms with Rigid Links 7.1 7.2 7.3 7.4 Internal Vibration Activity of a Mechanism Methods of Reduction of Perturbation Moments External Vibration Activity of Mechanisms and Machines External Vibration Activity of a Rotating Rotor and of a Rotor Machine , , 121 121 128 133 138 143 147 152 157 158 175 175 178 185 194 211 211 216 219 224 235 235 237 239 242 Contents 7.5 7.6 7.7 7.8 Balancing of Rotors Vibration Activity ofa Planar Mechanism Loss of Energy due to Friction in a Cyclic Mechanism Problems Dynamics of Cycle Machines with Rigid Links 8.1 8.2 8.3 Mechanical Characteristics of Engines Equations of Motion of a Machine State of Motion Determination of the Average Angular Velocity of a Steady-State Motion for a Cycle Machine 8.4 Determination of Dynamic Errors and of Dynamic Loads in a Steady-State Motion 8.5 Influence ofthe Engine Dynamic Characteristic on Steady-State Motions 8.6 Starting Acceleration of a Machine 8.7 BrakingofaMachine 8.8 Problems Dynamics of Mechanisms with Elastic Links 9.1 9.2 Mechanisms with Elastic Links and Their Dynamic Models Reduction of Stiffuess Inlet and Outlet Stiffuess and Flexibility of a Mechanism 9.3 Reduced Stiffuess and Reduced Flexibility of a Mechanism with Several Degrees of Movability 9.4 Determination of Reduced Flexibilities with the Help of Equilibrium Equations of a Rigid Mechanism 9.5 Some Problems of Kinematic Analysis of Elastic Mechanisms 9.6 Dynamic Problems of Elastic Mechanisms 9.7 Free and Forced Vibration of Elastic Mechanisms 9.8 Problems 10 Vibration of Machines with Elastic Transmission Mechanisms 10.1 Dissipative Forces in Deformable Elements 10.2 Reduced Stiffuess and Reduced Damping Coefficient 10.3 Steady-State Motion of a Machine with an Ideal Engine Elastic Resonance 10.4 Influence of the Static Characteristic of an Engine on a Steady-State Motion IX 245 247 252 254 269 269 276 278 280 286 289 294 295 301 301 305 308 311 313 315 318 321 327 327 330 332 339 X Contents 10.5 Transients in an Elastic Machine 10.6 Problems 11 Vibration of a Machine on an Elastic Base Vibration Isolation of Machines 11.1 Vibration of the Body of a Machine Mounted on an Elastic Base 11.2 Vibration of a Machine in the Resonance Zone Sommerfeld Effect 11.3 Vibration Isolation of Machines 11.4 Problems 12 Elements of Dynamics of Machines with Program Control 12.1 Basic Principles of Construction of Machines with Program Control 12.2 Determination of Program Control Sources of Dynamic Errors 12.3 Closed Feedback Control Systems 12.4 Effectiveness and Stability ofa Closed System 12.5 Problems 342 349 361 361 364 367 369 371 371 373 378 380 383 References 387 Index 389 382 12 Elements of Dynamics of Machines with Program Control equation (12.43) the condition of Hurwitz must be fulfilled which in the present case takes the form TTMrs Denoting /(1"S-1 = a, I(lrs- = p, -II( < TM (1 +rs -I) 1(1 (12.44) we bring the condition (12.44) to the form Ta < 1+ p Fig 12.4 represents the areas of stability of the system in the plane of the parameters a, p corresponding to different values of T The larger the engine time constant T, the smaller is the domain of admissible values a, p, and, hence, the smaller gain coefficients I( and 1(1 p Stability area Or-~~~~ - a -1 Fig 12.4 Area of stability of the closed system Thus, an increase in the gain coefficients of a feedback system can lead to instability of the closed system A negative feedback, which by the principle of action should induce a diminution of the dynamic error, in reality, turns out to be the cause of its unlimited increase Without recourse to a detailed description of all processes occuring in a closed system, let us only point out that instability is, in essence, caused by the engine inertia, a characteristic of which is the engine time constant T This inertia leads to a phase shift of the engine vibration moment with respect to that vibration component of the transient which must be damped by the moment As a result, the engine moment provided by the feedback signal becomes an excitation moment instead of a damping moment The larger the value of T, the stronger this curcumstance manifests itself It must be noted that several other elements of the control system possess inertia, as well E.g., the signal at the controller output !1u is related to the dynamic 12.5 Problems 383 error If/' through a more complex function than the one described by expression (12.32) In first approximation the dynamic processes carried out in a controller are described by an equation of the kind (12.45) where 1:c is the controller time constant Usually the "delay" at the controller is small ( 1:c « 1:,1:M ), so that it can be neglected for small gain coefficients How- ever, with the increase in K and K\ the influence of the small time constant 1:c on the stability of the system becomes essential In general, the larger the gain coefficients of a feedback loop are, the more precise the dynamic model of the system must be In particular, this concerns the analysis of the elasticity of the links of a mechanical system This analysis appears necessary in systems for motion control of precision machines, in which the program motions must be carried out with high accuracy 12.5 Problems 12.1 A vertical column rotating about its axis and with a gripper attached to it is considered as automatic operator The moment of inertia with respect to the rotation axis of the column including the gripper and the payload is Jc = 1.5 kg m The system of program control must ensure the rotation of the column from the initial position cp(O) = to the end position cp(T) = CPT =, for the positioning time T law of the column angular acceleration is 2, ("\ Cpp = CPT -2 smut, T = s The required variation ("\ 2, u =- T The column is set in motion by a direct-current electric motor with separate excitation whose dynamic characteristic 1:Q+Q=ru-s{j) has the following parameters: 1: = 0.01 s, r = 0.232 Nm/V, s = 0.486 Nms Here, u is the control voltage and (j) is the angular velocity of the motor shaft The moment of inertia of the motor rotor together with the transmission mechanism is J r = 0.01 kg m ; the transmission ratio is i = 50, so that {j)=iq, 384 12 Elements of Dynamics of Machines with Program Control Formulate the program variation law of the control voltage and find the dynamic errors during its execution Solution: 1) The program control, as a rule, is defined on the basis of the ideal engine characteristic, i.e up t =~OJp, OJ p =if fPp(t)dt = irp; {1-cosOt} o Hence, u = si rpr {1-cosOt} p r T 2) The motion equation of the column is solved together with the equation of the engine dynamic characteristic Here, J = J p + J c /;2 = 0.0106 kg m After the usual transformations we obtain where rM = J I s = 0.0218 s The general solution of this equation is OJ Here, D and p = De-nt cos{k1t + p}+ m are arbitrary constants and n=1/2r=50s- l , k=~l/rrM =67.7s- l , OJ kl is a particular solution, =~k2_n2 =45.7s- l We note that the first term is rapidly damping: Within 0.1 s the amplitude of vibrations diminishes by the factor 148 We seek the second term in the form = A + Bcos{Ot - 8} Substituting this expression into the equation and solving the resulting system we find m A = i rpr , B = T sin = cos = k irpr IT ~(k2 _02 Y+4n202 ~(k2 2nO _0 Y ~(k2 T =0.137, +4n 202 k _0 _0 = 0.9991 rpr , Y +4n 202 = 0.991 12.5 Problems 385 Thus, the angular velocity of the column after the transient is m = - 'Pr [1- 0.999Icos{Ot T 0)] and the dynamic velocity error is Iif = i 'Pr T [cosOt - 0.9991cos{Ot - 0)] = i 'Pr T (0.009cosOt - 0.137sin Ot) The dynamic error of the angle is determined by the expression 'I' = i 'Pr [sinO! - 0.9991sin{0! - m 0)] = i 'Pr m (0.009sin O! + 0.137 cosOt) 12.2 The column of the machine described in the foregoing problem is accelerated; the required program acceleration is ijJ P = & = 0.1 S -2 Find the dynamic position error, applying a program control and employing a control system with position feedback with a gain coefficient K = 50 V in the feedback loop Find the maximal admissible value of this coefficient Solution: 1) We define the program control on the basis of the engine ideal characteristic: s s .s u =-q =1-'P =1-&t PrP r P r' where, the parameters of the engine dynamic characteristic are those of the previous problem: s = 0.486 Nms, r = 0.232 Nm/V, T = 0.01 s, while the mechanical time constant is TM = 0.0218 s The motion equation of the controlled system is: 2) In the case of program control we have u = up TTMCi + TMq +q = i&t We determine only the forced motion since the free motion vanishes rapidly: q = i&{t - TM)' Moreover, the dynamic velocity error is lifO = -iu M = -0.109 s-l, -iuMt while the dynamic angle error is11'° = 3) In the case of a feedback control system we have u = up + K{q P- q) with q p = i&t /2 We obtain 386 12 Elements of Dynamics of Machines with Program Control Assuming again that the free motion is rapidly damping, we determine only ST Mis /(rK) The the forced component of the solution q = ict /2 - dynamic position error is If/ = -STMis/{nc}-;::; -0.0457 -;: ; 2.62° We note that the calculated dynamic position errors are those of the engine shaft The errors of the column position are i = 50 times less 4) The limit value of the gain coefficient in the feedback loop is determined by the inequality K