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

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. 1. Babitsky, DSc

Loughborough University

Department of Mechanical Engineering

LEII 3TU Loughborough, Leicestershire

United Kingdom

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

J. Wittenhurg

Universităt Karlsruhe (TH)

Institut fiirTechnische Mechanik

KaiserstraBe 12

D-76128 Karlsruhe I Germany

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

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© Springer-Verlag Berlin Heidelberg 2000

Originally published by Springer-Verlag Berlin Heidelberg New York in 2000

Softcover reprint ofthe hardcover Ist edition 2000

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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 Machines and Their Role in Modem Production

1.2 Structure of a Machine and its Functional Parts .................... .

l.3

1.4

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 .................................................................. .

2 Geometric Analysis of Mechanisms

1

1

2

3

10

17

19

24

27

33

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

1 Structure of Machines and Mechanisms

1.1

Machines and Their Role in Modern Production

Modem industrial production is reduced in the end to the execution of a great

number of diver~e working processes. Most processes are associated with

treatment and transformation of initial raw materials into half- or fully fmished

products; such working processes are referred to as technological. Technological

processes involve transportation of materials to the place of utilization as well as

energy processes, i.e. generation and transformation of energy in forms most

convenient for the respective proccess. Also, in/ormation processes, i.e.

transmission and transformation of information are of great importance in modem

production, ensuring execution of operations associated with control and

organization of production.

The accomplishment of many working processes requires realization of certain

mechanical motions. For instance, material processing on a lathe requires shifting

the blank and the instrument; transportation of raw materials and of finished

products is reduced to mechanical shifting; transformation of heat energy into

electric energy requires rotations steam turbines and generators, and so on. The

execution of working processes is also associated with the application of/orces to

materials in process in order to balance the weight of transported objects. A

person is able to realize directly mechanical motions which allow him to carry out

certain working processes manually. In modem production however the

overwhelming majority of working processes associated with the realization of

mechanical motions is carried out by machines.

We will call machine (or machine aggregate) a system designed to realize

mechanical motions and force actions related to the execution of one or another

working process. Machines are divided into technological, transport, energy￾converting and information machines depending on the kind of working process.

In industrial production, in addition to machines, various apparatuses are used

which are not directly associated with mechanical motion but with chemical,

thermal and other processes or with transmission and transformation of

information. Sometimes some of them are called machines, as well (e.g.,

electronic computing devices); however, the term "machine" will be used, in this

course, only in the indicated sense.

M. Z. Kolovsky et al., Advanced Theory of Mechanisms and Machines

© Springer-Verlag Berlin Heidelberg 2000

2 1 Structure of Machines and Mechansims

1.2

Structure of a Machine and its Functional Parts

Modem machines are, as a rule, complex systems consisting of several sub￾systems. These subsystems are referred to as functional parts of machines. To the

functional parts of a machine belong the engines, the mechanical system and the

motion control system. The functional diagram of a very simple one-engine

machine is represented in Fig. 1.1. E stands for the engine, MS for the

mechanical system, PCS for the program control system, FCS for the feedback

control system and WP for the working process performed by the machine.

Q .. P

uP,...&\.. u ... ~

... ,. ,.. PCS ... ~ ,. E ,.. MS x( WP

,4, q' l1u x' n

.J

FCS ......

..J

....

Fig. 1.1. Functional diagram of a one-engine machine

The completion of mechanical motions in a machine is always accompanied by

transformation of some kind of energy into mechanical work. The engine is that

part of the machine where such transformation actually takes place. Electric,

thermal, hydraulic, pneumatic engines can be distinguished depending on the kind

of the transformed energy. An input engine parameter u controls the energy

transformation process. For electrical engines such a control parameter is the

electrical voltage (for direct current engines) or the alternate current frequency;

for internal-combustion engines control is achieved through change of the fuel

quantity entering the combustion chamber; and so on. Each engine has an output

link. This is a rigid body performing rotational (rotary engine) or reciprocating

motion (reciprocating engine). The output engine coordinate is the generalized

coordinate q determining the position of the link. The generalized driving force

Q is generated in the engine acting on another functional part - the mechanical

system connected with the engine; an equal and oppositely directed force -Q acts

on the output engine link in accordance with Newton's third law.

The mechanical system transforms the simplest motions created by engines into

complex motions of the machine working organs, ensuring execution of working

processes. Henceforth, such motions will be referred to as machine program

motions. Output engine links are usually the inputs of a mechanical system.

Therefore, the number of system inputs is equal to the number of engines. This

number is referred to as number of degrees of movability of a machine. Fig. 1.2

shows the functional diagram of a machine with m degrees of movability. The

1.3 Mechanisms. Links and Kinematic Pairs 3

input parameters of its mechanical system are the coordinates ql> ... ,qm of the

output engine links and the output parameters are the coordinates xI"",xn of the

machine working organs.

U.1 r-- .. ~

E1 , ~

~ ... P2

---- q' ..:

1 ~

r-- Q2 Pn

U2 -.: .:

,,:; E2 ~ ~

PCS ~ .. MS WP

---- q' x..1

2 ,

x 2

Qm .:

- ~

Um ..:.: .,;,: Em ~ xn ~ .. .;.

---- q' ,

U ~~ m

.. FCS ~

~

"'

Fig. 1.2. Functional diagram of a multi-engine machine

The execution of a working process causes workloads, i.e. active forces

Ps (s=l, ... ,n) acting on machine working organs. Mechanical systems of

machines are in tum divided into simpler subsystems called mechanisms.

Systems for motion control are important functional parts of modem machines.

Systems of program control produce program control signals up prescribing

program motions of machines. Perturbation factors which will be considered in

detail below cause errors, i.e. deviations of actual motions from the program

motions. The correction of motion is achieved through a feedback system. It

receives information about errors in positions, velocities or accelerations and

forms correcting controls !t.u which diminish these errors.

1.3

Mechanisms. Links and Kinematic Pairs

A connected system of bodies ensuring transmission and transformation of mech￾anical motions is called a mechanism. The bodies constituting a mechanism are

referred to as links. Most often, the links of a mechanism are rigid bodies but

mechanisms with liquid or elastic links exist, as well.

4 1 Structure of Machines and Mechansims

z {cp= ~ A

~

BJ. x

a) b)

B

.JL

\A

y

x

Fig. 1.3. Kinematic pairs of movability one: a) revolute, b) prismatic, c) screw

The constructive elements connecting links and imposing constraints on their

motion are referred to as kinematic pairs. In mechanisms with links that are rigid

bodies the kinematic pairs are realized in the form of cylindric (Fig. l.3a) or

spheric (Fig. I.Sa) joints, sliders and guides (Fig. l.3b), screw couplings

(Fig. l.3c), contacting cylindric or planar surfaces (Fig. 1.6) and a lot of other

constructive elements. Henceforth, only kinematic pairs constituted by rigid links

will be considered.

Different physical models corresponding to different degrees of idealization of

mechanism properties are used in the study of mechanisms. The choice of one or

B

~ cP A A

~2<;

a) b)

Fig. 1.4. Kinematic pairs of movability two: a) cylindric, b) spheric pair with a pin

1.4 Kinematic Chains and Structural Groups. Generation of Mechanisms 5

a)

Fig. 1.5. Kinematic pairs of movability three: a) spheric, b) planar contact pair

another model depends primarily on the investigation goals and on what

information about mechanism behaviour is needed in the analysis process. At

different stages of a machine construction and investigation one and the same

mechanism is described by different physical models. In the study of mechanism

structure and kinematics one of the simplest physical models, referred to as a

mechanism with rigid links, is usually used. The transition from a real mechanism

to this model is based on the following assumptions:

1. All links and elements of kinematic pairs are considered nondeformable and

rigid links are considered to be perfectly rigid bodies.

2. It is assumed that in a motion process no violation of the constraints imposed

by kinematic pairs takes place and that these constraints themselves are

holonomic, stationary and bilateral.

Henceforth, to make it short, mechanisms with rigid links will be referred to as

rigid mechanisms, and the physical model of a machine consisting of only rigid

mechanisms will be referred to as a rigid machine.

Like every physical model of a real system, the rigid mechanism model has limi￾tations. For the solution of a large number of problems of statics and, particularly,

of dynamics of mechanisms, one must use more complex models, taking into

account deformations of links and of elements of kinematic pairs. Henceforth, such

B

~

a) b)

Fig. 1.6. Kinematic pairs of movability four and five: a) cylinder-plane, b) sphere-plane