kinematic design of mechanisms in a computer aided …

KINEMATIC DESIGN OF MECHANISMS IN A COMPUTER AIDED DESIGN

ENVIRONMENT

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

ERALP DEMIR

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

IN MECHANICAL ENGINEERING

MAY 2005

Approval of the Graduate School of Natural and Applied Sciences

_______________________

Prof. Dr. Canan ÖZGEN Director

I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science. _______________________

Prof. Dr. Kemal İDER Head of Department

This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. _______________________

Prof. Dr. Eres SÖYLEMEZ Supervisor

Examining Committee Members Prof. Dr. Eres SÖYLEMEZ ________________________

Prof. Dr. Kemal ÖZGÖREN ________________________ Prof. Dr. Reşit SOYLU ________________________ Asst. Prof. Dr. Ergin TÖNÜK ________________________ Prof. Dr. Şakir BOR ________________________

ii

I hereby declare all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Eralp Demir

iii

ABSTRACT

KINEMATIC DESIGN OF MECHANISMS IN A COMPUTER AIDED DESIGN ENVIRONMENT

Demir, Eralp

M.S., Department of Mechanical Engineering

Supervisor: Prof. Dr. Eres Söylemez

MAY 2005, 79 Pages

CADSYN (Computer Aided Design SYNthesis) is a visual, interactive computer

program working under Computer Aided Design (CAD) enviroment, which

accomplishes the synthesis and analysis of planar four-bar mechanisms. The

synthesis tasks are motion generation, path generation and function generation.

During synthesis, the dyadic approach is utilized which introduces vector pairs and

complex number algebra to model the motion. The possible solutions can be limited

for link dimensions, the center circle point curves within a certain region,

transmission angle characteristics, branch and order defects. The designed

mechanism can be analyzed for velocity, acceleration and transmission angle and any

of the data can be exported to Excel® for further analysis.

The software is designed to provide the user maximum feasible number of solutions.

In four multiply separated position synthesis, if there is flexibility in the value(s) of

one or any number of input parameter(s), designer can obtain different Burmester

curves by changing those parameter(s). Designer can also simulate the kinematics of

the mechanism by using drawing functions that are available from the CAD

iv

enviroment at any time. Drawing parts in the design plane can be attached to any link

of the mechanism and can be simulated throughout the motion as part of the link it is

attached. As a whole, this computer program is designed to satisfy the needs of

mechanism designers while working in CAD enviroment.

Keywords: Mechanism Synthesis, CAD, Burmester Theory, Multiply Separated

Positions, CADSYN.

v

ÖZ

BİLGİSAYAR DESTEKLİ TASARIM ORTAMINDA MEKANİZMA SENTEZİ

Demir, Eralp

Yüksek Lisans, Makina Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Eres Söylemez

MAYIS 2005, 79 Sayfa

CADSYN (Computer Aided Design SYNthesis), düzlemsel dört çubuk

mekanizmaların sentez ve analizini bilgisayar destekli tasarım ortamında

gerçekleştiren görsel, etkileşimli bir bilgisayar programıdır. Ele alınan sentez

yöntemleri olarak konum, yörünge ve fonksiyon sentezi seçilmiştir. Sentez

çözümleri için hareketin kompleks sayı modellemesini gerektiren vektör çifti

yaklaşımı kullanılmıştır. Olası çözümler arasından istenilen özelliğe sahip

mekanizmaları bulmak için program uzuv boyutlarına, sonucu sınırlandıran alan

seçimine, bağlama açısı değerine, uzuvların sıra ile elde edilip edilmemesine, ve

mekanizmanın konumlararası mafsallar sökülmeden geçmesine göre yapılan

sınırlamalara uygun çözümleri kullanıcıya sunabilmektedir. Sentez işlemi sonunda

elde edilen mekanizma; konum, hız ve ivme gibi kinematik değerler için analiz

edilebilmekte ve farklı analizler için Excel®’e bir kütük şeklinde gönderebilmektedir.

Yazılım kullanıcıya mümkün olduğunca fazla çözümü sunacak şekilde tasarlanmıştır.

Dört konum sentezinde ise, eğer bir ya da birkaç giriş parametre değerlerinde

esneklik varsa tasarımcı elindeki çözüm kümesini yeni çözümler ekleyerek

büyütebilir. Tasarımcı mekanizma hareketini veya çalışma ortamını, çizim

vi

programının komutlarını kullanarak oluşturabilir. Mekanizmanın çalışma

düzlemindeki herhangi bir eleman, herhangi bir uzuva bağlanarak, o uzuvla birlikte

hareket ettirilebilir. Sonuç olarak bu program mekanizma tasarımcısının bilgisayar

destekli tasarıma olan ihtiyacına cevap vermek için hazırlanmıştır.

Anahtar kelimeler: Mekanizma Sentezi, BDT, Burmester Teorisi, Sonlu ve/veya

Sonsuz Yakın Konumlar, CADSYN.

vii

ACKNOWLEDGEMENTS

I would like to thank to:

Prof. Dr. Eres Söylemez, for supervising me for four years and sharing his

experience on mechanism design,

Prof. Dr. Kemal Özgören, for his invaluable efforts on correcting this thesis and his

help on forming the motion theory for the animations of the mechanism,

My mother, Tülay Demir, struggling to provide me a working atmosphere at home

and supporting me all through my life,

My father, Necdet Demir, the best engineer I have seen, directing and helping me

whenever I need and showing me how to be a good engineer and an ideal father,

My sister, Pelin Demir, the best thing in my life, being a part of me,

… and everyone whose studies I used.

viii

TABLE OF CONTENTS

ABSTRACT .............................................................................................................. iv

ÖZ ............................................................................................................................. vi

ACKNOWLEDGEMENTS .................................................................................... viii

TABLE OF CONTENTS .......................................................................................... ix

LIST OF FIGURES ................................................................................................. xii

LIST OF TABLES .................................................................................................. xiv

CHAPTER 1. INTRODUCTION .............................................................................................. 1

1.1 .General ......................................................................................................... 1

1.2 Literature Survey ......................................................................................... 4

1.3 Scope of This Study ..................................................................................... 6

2. THEORY AND FORMULATION ..................................................................... 7

2.1 General ......................................................................................................... 7

2.2 Multiply Separated Position Synthesis ..................................................... 11

2.2.1 Dyadic Approach ............................................................................. 12

2.3 Synthesis for Motion Generation ............................................................... 15

2.3.1 Synthesis for Three Multiply Separated Positions ........................... 16

2.4.2 Synthesis for Four Multiply Separated Positions ............................. 17

2.4 Synthesis for Path Generation .................................................................... 20

2.5 Synthesis for Function Generation.............................................................. 22

2.6 Special Cases for Motion Generation .........................................................23

2.6.1 Special Cases for P-P-P ................................................................... 23

ix

2.6.2 Special Cases for PP-P-P ................................................................. 23

2.6.3 Special Case for P-P-P-P ................................................................. 24

2.7 The Branch and Circuit Defects .................................................................. 26

2.8 Order Problem ............................................................................................. 28

2.9 Synthesis with Different Positions .............................................................. 29

2.10 Slider-Crank Design (Ball’s Point) ........................................................... 31

2.10.1 Motion Generation with Slider Crank ............................................ 31

3. CADSYN MECHANISM DESIGN SOFTWARE ........................................... 35

3.1 General ........................................................................................................ 35

3.2 Visual Basic Programming Environment under AutoCAD® ...................... 36

3.3 The Code structure of CADSYN ............................................................... 37

3.4 Mechanism Design with CADSYN ........................................................... 38

3.5 Synthesis Types........................................................................................... 49

4. TEST CASES .................................................................................................... 50

4.1 Test Case 1 ................................................................................................. 50

4.1.1 The Design Constraints .................................................................... 50

4.1.2 The Synthesis Procedure .................................................................. 51

4.2 Test Case 2 ................................................................................................. 53



4.3 Test Case 3 ................................................................................................. 55



4.4 Test Case 4 ................................................................................................. 59



4.5 Test Case 5 ................................................................................................. 62


x


4.6 Test Case 6 ................................................................................................. 64



4.7 Test Case 7 .................................................................................................. 67



4.8 Test Case 8 .................................................................................................. 69



4.9 Test Case 9 .................................................................................................. 72



5. DISCUSSION AND CONCLUSION ............................................................... 75

5.1 Summary and Discussion ..................................................................... 75

5.2 Future Recommendations .................................................................... 77

REFERENCES ......................................................................................................... 78

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

xi

LIST OF FIGURES

FIGURES

2.1 Absolute motion of the moving frame with respect to the fixed frame .............. 8

2.2 The fixed and moving centrode curves ............................................................. 10

2.3 One dyad of the four-bar mechanism ................................................................ 12

2.4 The dyad vector in path generation synthesis formulation ............................... 20

2.5 Construction to use motion generation calculations in path generation ........... 21

2.6 Output link motion in function generation ........................................................ 22

2.7 The trailer conditions in four-bar mechanisms ................................................. 26

2.8 Allowable regions for a trailer .......................................................................... 27

2.9 The rocker conditions in four-bar mechanisms ................................................. 27

2.10 The immediate effect on solution curves for manipulated inputs ................... 30

2.1 The slider crank design parameters ................................................................... 31

3.1 Form to input selection method ........................................................................ 39

3.2 Form to input positions ...................................................................................... 39

3.3 Form to input infinitesimal position vector components .................................. 40

3.4 Form to enter any limitation ............................................................................... 40

3.5 Form to change position properties .................................................................... 41

3.6 Form to input required data for three positions ................................................. 41

3.7 Additional options for four positions ................................................................ 42

3.8 Selection form to construct mechanisms............................................................ 42

3.9 Link repository used to store link data .............................................................. 44

3.10 The mechanism data stored ............................................................................. 44

3.11 Filter for used to differentiate solutions with order and branch problem ....... 45

3.12 Problems that can be transferred to Excel® for further analysis ..................... 45

3.13 Analysis form for simple kinematic entities ................................................... 46

xii

3.14 Animation form used for mechanism analysis ................................................ 47

3.15 Block addition form ....................................................................................... 48

3.16 Form used warn the user that the CAD functions are active ........................... 48

4.1 The design domain of test case 1 ...................................................................... 51

4.2 The synthesized mechanism for test case 1 ...................................................... 52

4.3 The synthesized mechanism properties of test case 1 ....................................... 53


4.5 The desired path of load of test case 3 ............................................................... 56


4.7 The analysis window for the mechanism of test case 3 .................................... 58

4.8 The working plane of the mechanism of test case 4 ......................................... 59

4.9 The designed meat cutting mechanism ............................................................. 61

4.10 Epicyclic gear train of test case 5 .................................................................... 62

4.11 The synthesized mechanism for test case 5 ..................................................... 64

4.12 The design domain of test case 6 .................................................................... 65



4.15 The design domain of test case 8 .................................................................... 70


4.17 The synthesized mechanism and Burmester curves for test case 9 ................. 74

xiii

LIST OF TABLES

TABLES

2.1 Number of solutions for multiply separated positions ...................................... 14

2.2 The specified parameters for motion generation ............................................... 15

2.3 Parameters for motion generation ..................................................................... 16

2.4 Explicit terms used in the dyad loop equation for three positions .................... 17

2.5 Coefficients of the dyad loop equations for four multiply separated positions

motion generation ............................................................................................. 18

2.6 Special Cases for P-P-P ..................................................................................... 23

2.7 Special Cases for PP-P-P .................................................................................. 24

2.8 Special Cases for P-P-P-P ................................................................................. 24

2.9 Coefficients of loop equations for three positions ............................................ 33

2.10 Coefficients of loop equation for four positions ............................................... 33

xiv

CHAPTER 1

INTRODUCTION 1.1 GENERAL

The very first step in machine design is kinematic synthesis of mechanisms that is

followed by analysis. Analysis and synthesis tasks should be done together to obtain

an acceptable optimum design. Analysis can be in many forms like dynamic analysis,

kinematic analysis, force analysis and finite element analysis which consider

different matters of the design. However synthesis task, deals with two main

problems; to determine the type (Type Synthesis) and to find the dimensions

(Dimension Synthesis) of the mechanism which suits best to the desired motion

charactersitics [3,4].

There are several approaches for the dimensional synthesis of mechanisms. Usually

two of them find general use: Prescribed Position Synthesis and Optimization

Synthesis. Optimization Synthesis looks for the solution with the least error between

the realized output and the desired output. On the other hand, Prescribed Position

Synthesis looks for the mechanism dimensions satisfying the desired finite number of

positions exactly at the prescribed positions.

There are mainly three types of multiply separated position synthesis methods which

are: Motion Generation, Path Generation with Prescribed Timing and Function

Generation. In motion generation, the mechanism has to move a body through

several positions, so the location and orientation of a body are the input parameters.

In path generation with prescribed timing, the mechanism has to move a body

through several locations corresponding to a connecting link rotation. In function

generation input link rotations, are correlated with output link rotations to simulate a

defined function. The solution method does not vary considerably among these

synthesis procedures as the only changing parameters are the inputs [2].

A point on the moving plane which satisfies these conditions and draws a circular arc

on the fixed plane is called the Circle Point, and the corresponding center of this

circular arc is called the Center Point. The main aim of prescribed position synthesis

is to find the center points and circle points for a given case in order to define a

linkage by connecting these two points.

In kinematic synthesis of mechanisms, intuition and experience of the designer play a

major role compared to other design stages [12]. However, just like in every

engineering problem, synthesis problems require the solution of mathematical and/or

geometrical systems as well. Even though calculation procedure can be carried out in

many programs easily, without a user interface the synthesis task becomes a

cumbersome and time consuming problem. Computer programs with user interface

not only take over the duty of solving the mathematics and/or geometry of the

problem from the designer but also help the user visualize the design. At the end, the

designer will have to use his intuition and experience for the selection of the most

suitable mechanism out of the possible combinations.

As a whole the necessity of using computer programs for synthesis become abundant

in our time with the arising development in computer technology. The computer

programs are capable of reaching the best solutions with user interaction at every

design stage both in analysis and synthesis.

The aim of this study is to construct a computer program for synthesis of planar

mechanisms of slider crank and four-bar. The program is named as CADSYN which

is a combination of the terms Computer Aided Design (CAD) and Synthesis. The

software is capable of directing the user to feasible solutions. CADSYN can conduct

2

analysis of some basic properties of the synthesized mechanism such as kinematic

entities like positions, velocities etc. In addition, CADSYN can export the results to

Microsoft Excel® for further analysis.

CADSYN works under CAD software which has its visual interface already built.

This opportunity saves a considerable programming effort and time. Besides that, the

user will be able to work in a reliable and a familiar environment with additional

utilities of the CAD program.

The software CADSYN is written in Visual Basic language and supported by the

program; AutoCAD®. Therefore the designer has the opportunity to access the same

utilities of the CAD software during synthesis. This will present the user to

synthesize mechanisms through constructing its own working environment.

CADSYN allows the user to work with technical drawings, imported pictures and

self-drawn drawings so he/she can observe his/her design working in its real

environment. Therefore it will determine how the links of the mechanism interferes

with its working plane which provides the user to see whether the links coincide with

some other parts. The shapes of the links can be easily determined by using the

drawing commands of the CAD program. This will eliminate the necessity of redoing

or checking the design to find the appropriate shape of links or bodies which requires

plotting the mechanism several times.

Dynamic analysis and three dimensional visualization of the designed mechanism are

beyond the scope of this thesis.

3

1.2 LITERATURE SURVEY

The theory behind this study is based on Precision Point Synthesis which was first

solved by Burmester [3, 4, 8] graphically. Freudenstein [6] contributed to this theory

by formulating the problem analytically. Erdman and Sandor [3-4] introduced the

dyadic approach which is easy to implement to numerical solutions. The theory

mentioned, forms the base of the many of the software packages on the market.

Some of the software on the other hand, uses optimization routines to generate the

desired motion or path.

In the Mechanical Engineering Department of Middle East Technical University two

previous works were done by Polat[11] and Sezen[13]. Polat has generalized the

dyad vector approach for finitely and infinitesimally separated positions and he

prepared the computer program MECSYN for synthesis of planar four bar and slider

crank mechanisms in Fortran® programming language. The program is capable of

outputting center and circle point locations for desired motion characteristics

however it has no means of interacting with the user. Sezen using Delphi® and

Pascal®, prepared QuadLink that is a user friendly software with graphical

capabilities. However, in his work there is no interaction with CAD software and

design tools to realize the design enviroment are limited.

Polat in his study discusses the synthesis for slider crank mechanisms which is not

much different than the synthesis procedures followed for a four-bar. The design

only differs from four-bar synthesis in the way that, after user selects the appropriate

dyad, the Ball’s Point locations are evaluated as the circle point whose center of

curvature is at infinity and presented to the user to be selected as slider points.

The solutions found by using Burmester theory might result incorrect mechanisms.

This is either because the mechanism satisfies the precision positions at two different

configurations (Branches) or because the positions are not in sequence. Filemon [5]

and Waldron [17, 18] have shown the regions of the Burmester curve which result in

a practical mechanism. Prentis [12] introduced an analytical method to eliminate

4

defective solutions from the solution set, proposing 20th order analytical function,

however the solution is rather complicated and time consuming even for computers.

Polat also reflected the theory of Filemon to crank rocker type of mechanisms.

Holte [7] indicates that a change in positions may change the solution curve

completely. Mlinar [9] carried out a sensitivity analysis to see the rate of change of

solution curves with a change of orientation of the input positions. He has reached

the conclusion that a characteristic point at the solution set is determinant on how

much the solution is affected. Mlinar and Holte together signify the importance of a

topic that is not discussed before by Polat and Sezen that the solution curves may

change drastically by changing an input incrementally.

Many computer programs are written for kinematic synthesis of mechanisms. In

addition to the literature presented Ulushan [16] and Sezen, WATT® [20] is a good

reference for synthesis of mechanisms. The software interacts with other common

programs like Excel® and AutoCad® that the user can import or export data. The

software uses optimization techniques to find solutions which are very different than

precision point synthesis solution and it can be used to design mechanisms for path

generation only, however with its user-friendly design atmosphere, WATT® can be

considered as a good implementation example.

5

1.3 SCOPE OF THIS STUDY

The objective of this study is to develop an interactive software package for the

kinematic synthesis and analysis of planar four bar and slider crank mechanisms with

prismatic and revolute joints. As was mentioned before the fundamental theories

were developed more than a century ago and they did not change significantly. But

with the increasing necessities and increase in computation facilities; new solution,

design and analysis methodologies and techniques have been developed with

improved graphical utilities. By the integration of these techniques with the extended

power of new generation computers, plenty of software packages were and are still

being developed by many groups. At this point, specialization, in other words, event

specific software development, ease of usage, low cost software packages and quick

integration and compatibility with other CAD/CAM software packages become the

major goal. For this reason, AutoCAD 2004® has been chosen for the software

development environment since it is very common software for 2-D and 3-D design

and drafting.

The software designed is named as CADSYN. It possesses an interactive user-

friendly environment with CAD functions applicable at any time. The software

works in AutoCad2004® as a Visual Basic® application but it can run under any

CAD program supported by AutoDesk®.

CADSYN can take into account the approximate position inputs to increase the

number of feasible solutions during design for four multiple position case. This

allows greater flexibility in the type of design problems that can be solved. The

selection is left totally to the user by showing the restrictions to get a working

mechanism and it is believed that the user can decide better on the characteristics of

the mechanism, other than kinematics.

6

CHAPTER 2

THEORY AND FORMULATION

2.1 GENERAL

The theory and formulation for the synthesis of mechanisms with prescribed position

synthesis by dyadic approach is applied in this study [3, 4, 8, 6]. The synthesis

method used is the same as presented in the studies of Polat [11] and Sezen [14]. The

theory will be briefly summarized in the following sections. For details please refer

to these two studies.

In planar motion, every moving body defines a moving plane. If the motion of this

plane is measured with respect to a fixed plane for which all kinematic properties of

which are taken as zero at all times, this motion is called absolute motion. If the

motion of a moving plane is considered relative to another moving plane, then this

motion is called relative motion [2, 11, 13].

In order to define the motion in an independent and unique way, the canonical

representation is used. When coordinate frames are defined on the fixed and

moving planes, the position of a moving plane relative to the fixed plane is defined

by the time dependent quantities and )(ta→

)(tφ as seen in Figure (2.1). Let A be a

point selected on the moving plane with its coordinates A(X,Y), A(x,y) and position

vectors and relative to the fixed and moving frames respectively. If one lets: )(tZ→ →

z

ibaa +=→

(2.1)

7

then the coordinate transformation for point A can be handled by [1]:

)sin()cos( φφ yxaX −+= (2.2.a)

)cos()sin( φφ yxbY ++= (2.2.b)

Figure 2.1 Absolute motion of the moving plane with respect to the fixed frame

From Figure (2.1), the vector can be expressed in terms of vector as: →

Z→

z

φiezaZ→→→

+= (2.3)

In the analysis of motion of mechanisms, usually we are only interested in the

geometry of motion. Thus, without loss of generality, once the assumption of

0≠dtdφ is made, one can take t=φ or 1=

dtdφ . In order to clarify this expression,

an example from the four-bar mechanism can be given: The angular velocity of the

crank of a four-bar has nothing to do with the shape of coupler curve. Hence one can

take the derivatives with respect to φ instead of t. Letting κ be any variable, then

8

φκκ

dd

=/ and defining n

n

n dad

φ=a and n

n

n dbd

φ=b (where n=1,2,3), if the derivatives

of Equations (2.2.a), (2.2.b) and (2.3) are taken with respect to φ :

) (2.4.a) cos()sin(1/ φφ yxaX −−=

)sin()cos(1/ φφ yxbY −+= (2.4.b)

φieziaZ→→→

+= 1

/

(2.5)

can be obtained. Since Equations (2.4.a), (2.4.b) and (2.5) are valid for all points on

the moving plane, one can search for a point P, for which, at an instant:

0)cos()sin(1/ =−−= φφ PPP yxaX (2.6.a)

0)sin()cos(1/ =−+= φφ PPP yxbY (2.6.b)

01/ =+=

→→→ φiPP eziaZ (2.7)

using Cramer’s rule and solving for xp , yp and : →

pz)cos()sin( 11 φφ baxP −= (2.8.a)

)cos()sin( 11 φφ aby P += (2.8.b)

φiP eaiz −

→→

= 1 (2.9)

substituting these values in global coordinates expressions:

1baXP −= (2.10.a)

1abYP += (2.10.b)

→→→

+= 1aiaZP (2.11)

This point P is called the instantaneous center or the pole, and it exists in every

plane motion where the angular velocity of the moving plane is not zero.

The differential of a curve s defined by its Cartesian coordinates is:

22 dydxds +=

9

The differentials of the instantaneous center on the fixed and moving frames are:

On the fixed frame: 221

221

222 )()()()()( abbad

dYd

dXds PP ++−=+=φφ

(2.12.a)

On the moving frame: 221

221

222 )()()()()( abbaddy

ddxds PP ++−=+=

φφ (2.12.b)

Equations (2.12.a) and (2.12.b) imply that the two rates of change of the pole are the

same. This leads to a valuable expression [2]: The curve attached to the moving

frame, which is drawn on the moving plane all through the motion by the

instantaneous center (pole) and characterized by the formula of , is called the

moving centrode. The curve attached to the fixed plane, which is drawn on the fixed

plane by the instantaneous center (pole) and characterized by the formula of , is

called the fixed centrode. The motion of a moving plane can be uniquely determined

by the pure rolling of the moving centrode curve attached to the moving frame over

the fixed centrode curve attached to the fixed frame and is shown in Figure (2.2).

This is the canonical representation of plane motion and it is unique and

independent of every observer.

Pz→

PZ→

Figure 2.2 The fixed and moving centrode curves

10

2.2 MULTIPLY SEPARATED POSITION SYNTHESIS

In this type of synthesis method the mechanism passes exactly from the prescribed

precision points but the motion in between these input positions is uncontrollable.

The designer has to solve the loop closure equations for several variables to get the

mechanism dimensions. The main advantage of the prescribed positions technique is

that the number of solution is infinite.

In multiply separated position synthesis, both finite and infinitesimal displacements

may exist between the prescribed positions. Tesar introduced a notation in which the

finitely separated positions are shown with a hyphen placed in between the positions

(i.e. for two finitely separated positions P-P) and infinitesimally separated positions

are shown with nothing in between the positions. For example, P-PP shows three

multiply separated positions in which the infinitesimally separated position belongs

to the second finitely separated position and it is the first order derivative of the

prescribed position vector.

The synthesis problem is actually to find the some parameters by assigning values to

the remaining parameters other than input parameters. These parameters change with

respect to the type of the design problem. In motion generation the location and

orientation changes (α, δ) of the coupler where the moving body is attached are the

inputs and crank rotations (β) corresponding to that motion are to be found to

construct mechanism. In path generation the location changes of the coupler (δ) and

crank rotations (β) are the inputs and coupler rotations (α) corresponding to that

motion have to be found. In function generation the rotation of crank and output links

(φ, ϕ) are the inputs and coupler rotations (α) corresponding to that motion have to

be evaluated to construct mechanism (For finite position case).

11

2.2.1 Dyadic Approach

In dyadic approach, the mechanism to be synthesized is divided into vector pairs

called dyads, each of which carries out the motion independently through the

prescribed positions [3, 4]. Then these dyads can be combined to form the whole

mechanism. In the case of a four-bar mechanism, there are two dyads which must be

synthesized independently and must then be combined.

The prescribed position inputs can be either finitely separated or infinitesimally

separated. In this thesis “P-P” represents finitely, “PP” represents infinitesimally

separated positions. In CADSYN dyadic approach is used to obtain solution which is

the easiest way to implement. In a four bar mechanism there are two dyads which is

formed by two individual vector pairs.

In Figure (2.3) one dyad of the four–bar mechanism is shown. The fixed and moving

coordinate systems are taken coincident at the first prescribed position, at point P1.

The first position is defined by the vector pair and , and the j→

W→

Z th position (where

j=2,3,4) is defined by the modified forms of the same vectors which are multiplied

by rotational modifiers. The rotation of the moving plane from first position to jth

position is αj and the corresponding crank rotation is βj. The position of point Pj is

moved in amount from initial position Pj

→

δ 1, which is also called the dyad vector.

Figure 2.3 One dyad of the four-bar mechanism

The dyad equation can be written as;

12

0=−−++→→→→→ βi

eWieZδZW jjj

α where j = 1, 2, 3 … (2.13)

or in standart form:

jδieZi

eW jj→→→

=−+− )1()1( αβ where j = 1, 2, 3 … (2.14)

Instead of using vector , it has better physical meaning to use the fixed pivot

(center point) coordinate vector

→

W→

R and the moving pivot (circle point) coordinate

vector - . Hence, substituting the following relation into Equation (2.14) : →

Z

→→→

−−= WZR (2.15)

the dyad loop closure equation for the finitely separated positions becomes:

jδi

eieZi

eR jjj→→→

=−+− )()1(ββ α where j = 1, 2, 3 … (2.16)

For infinitesimally separated positions, the derivative of above equation is taken with

respect to the angular displacement. For instance, for motion generation synthesis the

derivatives are taken with respect to the rotation of the moving (coupler) plane, α.

The values of α and β are evaluated at the finitely separated position at which the

infinitesimally separated position occurs. Hence, the loop closure equation for

infinitesimally separated positions becomes:

j

j

j

jj

δddee

ddZe

ddR iii

ααββαα

β

ββ

β αααα ==

==

→→→

=−+− )()()1( α (2.17)

Tesar[15, 16] introduced a form which combines the equations for the finitely and

infinitesimally separated positions. The indices used in this equation are as follows:

j: index of the finitely separated position.

13

k: index of the infinitesimally separated position corresponding to a finitely

separated position.

l: index of the (total) multiply separated position.

j

k

k

j

jk

k

j

k

k

δddee

ddZe

ddR iii

ααββαα

β

ββ

β αααα ==

==

→→→

=−+− )()()1( α (2.18)

A more compact and useful form of this equation can be written as [1]:

lllll δbaZbσR =−+−→→

)()( (2.19)

where:

jασαα

ll

== αie

dda k

k

jβσβ

βα

ll

== ie

ddb k

k

j→

=→

→

=δσδ

δα

ll k

k

ddδ (2.20)

)0,01,0(0 =≠=== lll σσσ korkk (2.21)

Depending on the synthesis type, (i.e. motion, function and path generation) the

unknowns and the precribed position data can change. However the number of

unknowns, number of free choices and number of solutions are the same for the same

number of prescribed positions and are independent of the type of synthesis. These

relations are given in Table (2.1).

Table 2.1 Number of solutions for multiply separated positions

Number of Positions

Number of Scalar

Equations

Number of Scalar

Unknowns

Number of Free

Choices

Number of Solutions

TWO 2 5 3 ∞3

THREE 4 6 2 ∞2

FOUR 6 7 1 ∞ FIVE 8 8 0 Finite

14

2.3 SYNTHESIS FOR MOTION GENERATION

In motion generation the input for synthesis is the coupler link motion that can be

expressed by both translation (δ) and rotation (α) for finitely separated positions. For

infinitesimally separated position the translation is represented with ( )k

jk

dd

αα

→

δ . In

this study only three and four positions will be taken under consideration. The

parameters to be specified for different cases of input positions are presented in

Table 2.2.

Table 2.2 The specified parameters for motion generation

Multiply Separated

Positions

No Case

The Parameters to be Specified:

P-P-P α 2 , 2δr

, α 3 , 3δr

PP-P α 2 , 2δr

, 0=ααdδdr

THR

EE

PPP 0=ααdδdr

, 02

2

=ααdδdr

P-P-P-P α 2 , 2δr

, α 3 , 3δr

, α 4 , 4δr

PP-P-P α 2 , 2δr

, α 3 , 3δr

, 0=ααdδdr

PPP-P α 2 , 2δr

, 0=ααdδdr

02

2

=ααdδdr

PP-PP α 2 , 2δr

, 0=ααd

δdr

, 2ααα =d

δdr

FOU

R

PPPP 0=ααdδdr

,02

2

=ααdδdr

,03

3

=ααdδdr

The unknown parameters to be selected for different type and number of positions,

are shown in Table 2.3. From this table the free parameters and corresponding input

parameters for motion generation can be seen.

15

Table 2.3 Parameters for motion generation

Number of

Positions

Number of Scalar Equations

The Specified Parameters

The Parameters To Be Determined (and number of scalar

unknowns)

The Arbitrarily Picked Parameters

(number of free choices)

THREE 4 a 2 , a 3 , 2δ

r,

3δr

b 2 , b 3 , R

r, Zr

( 6 )

b 2 , b 3( 2 )

FOUR 6 a 2, a 3, a 4, 2δ

r,

3δr

, 4δr

b 2 , b 3 , b 4 , R

r, Zr

( 7 )

b 2 or b 3 or b 4( 1 )

2.3.1 Synthesis for Three Multiply Separated Positions

In the loop equation (Eq’n 2.19), there exist six scalar unknowns for four scalar

independent equations in three position synthesis (Refer to Table 2.4). In order to

solve the equations, therefore, two inputs must be specified for three multiply

separated position case. Designer can specify these two free parameters in three

ways:

- b2 and b3 specification; i.e. crank or output link rotation vectors

- Center point specification; fixed pivot location at initial position

- Circle point specification; moving pivot location at initial position

Besides these three specifications, in order have a greater number of solutions, b2 (or

b3) can be entered and the other input b3 (or b2) in that case, is (i.e. for finite

position case, rotated 0o to 360o) can be varied over a range. Then the circle and

center point curves which are both circles are plotted.

The center and circle point vectors can be solved from loop equations as;

2332322233

2233

babaσ)ba(σ)b (a)b (a)b (a

−++−+−

−−−=

→ 32 δδR (2.22)

16

2332322233

2233

babaσ)ba(σ)b (a)σ (b)σ (b

−++−+−

−−−=

→ 32 δδZ (2.23)

Table 2.4 Explicit terms used in the dyad loop equation for three positions

CASE σ 2 σ 3 a 2 a 3 b 2 b 3

2δ 3δ Selected Parameters

P-P-P 1 1 e iα2 e iα3 e iβ2 e iβ3

2

→

δ 3

→

δ 2β and 3β

PP-P 0 1 i e iα2 iβ•

e iβ2αd

dδr

2

→

δ•

β and 3β

PPP 0 0 i -1 i β•

ββ i2 •••

+− αd

dδr

2αdd2δr

•

β and ••

β

Table 2.4 shows the variation of input and output parameters for different types of

position inputs.

2.3.2 Synthesis for Four Multiply Separated Positions

In this case a single parameter is varied over a range to get the solutions like b2 or b3

or b4 for finitely and b for infinitesimally separated solutions. The resulting points

define a special curve called Burmester curve. The solution of loop equation for

finite position case, will yield [11] equation 2.24 that have coefficient vectors D1, D2

and D3. These coefficients vary with the type of the position inputs.

043321 =++ ββ ii eDeDD (2.24)

Different cases of inputs (P-P-P-P, PPP-P, PP-PP, PP-P-P, PPPP) are given in Table

2.5. The other cases; which are P-PPP,P-PP-P, P-P-PP are treated same as the case

PPP-P, PP-P-P and PP-P-P respectively. [13]

17

2.4 SYNTHESIS FOR PATH GENERATION

Path generation is the synthesis of mechanisms for the generation of specified closed

curves possessing certain desirable characteristics, or the guiding of points through a

number of prescribed positions [11, 13]. The utilization of coupler curves in design

includes applications in single dwell mechanisms, straight line motion mechanisms,

special indexing devices, double dwell linkages and applications associated with

agricultural, textile and food processing machinery.

This time, crank rotation (β) is correlated with the position change of the part (δ) that

is going to be fixed to coupler of the four-bar as shown is Figure 2.4. In this case, the

loop equation (Eq’n. 2.19) remains the same where bj is known rather than aj.

Therefore only the unknown parameters are reversed during path generation in an

algebraic linear equation set when compared to motion generation. Indeed the

problem can be treated as a motion generation problem and if bj values entered as if

they were aj parameters in motion generation calculation routines, the center and

circle pivots are found. The center points found this way are the same as in path

generation case, only circle points are different which could also be found by a

simple trick for path generation.

Figure 2.4 The dyad vector in path generation synthesis formulation

The rotations of a parallelogram constructed to a dyad are shown in Fig.2.5. It can be

observed that the link constructed parallel to the coupler link, rotates the same

amount as the crank (β) link whereas the coupler link for motion generation case

20

rotates as much as the crank link that is similarly constructed for path generation

case. Therefore the circle point corresponding to the path generation case can be

found by a simple function used to evaluate corresponding location by using the

same calculation procedures for the motion generation case (Eq’n 2.25). Hence the

circle point for path generation will simply be;

iZ R s e α→ →

⋅= + ⋅ (2.25)

Figure 2.5 Construction to use motion generation calculations for path generation

21

2.5 SYNTHESIS FOR FUNCTION GENERATION

In function generation the output angle (ψ) of a mechanism is correlated with input

crank angle (β) (For finite position case). Higher order synthesis is required if low

structural error is desired in the vicinity of the precision points. A minimization

calculation is necessary to evaluate the input positions with corresponding output

positions with the least error. Chebychev [14] spacing is recommended for this

purpose.

The synthesis method for function generation is a little different than the methods

discussed in motion generation and function generation. First of all, the output link at

any length having its fixed and moving pivot at any location is chosen arbitrarily

since the ratio of link lengths can be changed. Then the output rotation angle (ψ) can

be used to estimate the coupler position change (δ). Then the problem turns into

“Path Generation with Prescribed Timing” in which an input angle is correlated with

the position change of coupler.

Figure 2.6 Output link motion in function generation

22

2.6 SPECIAL CASES

The synthesis procedures fail in some special cases of prescribed positions while

motion generation in which the circle and/or center point curve degenerate into a line

or a circle. These cases are discussed and proved in detail in the studies of Polat and

Sezen [11, 13]. Only the outputs will be tabulated in the following sections. These

special cases are treated in a different subroutine in CADSYN to avoid complexity.

2.6.1 Special Case for P-P-P

In Table 2.6 special cases corresponding to three multiply separated positions are

given. The degenerated forms of center & circle points are indicated in the table.

Table 2.6 The special cases for case P-P-P

Condition Center point Curve

Circle Point Curve

→

→

3

2

δ

δ is real At infinity

(β2 = β3 = 0)

Every point

(β2 = β3 = 0) α2 = α3 = 0

→

→

3

2

δ

δ is not real Every point (β2 and β3 have fixed values)

Every point (β2 and β3 have fixed values)

α2 = 0, α3 ≠ 0 β2 is fixed, β3 in [0, 2π] Line Line

α2 ≠ 0, α3 = 0 β2 is fixed, β3 in [0, 2π] Line Line

α2 =α3 ≠ 0, β2 is fixed, β3 in [0, 2π] Line Line

2.6.2 Special Case for PP-P-P

While the three finite positions are parallel to each other, special cases appear. The

special cases for four multiple positions with one infinitesimal input are given in

Table 2.7.

23

Table 2.7 Special cases for case PP-P-P [1]

Condition Center point Curve Circle Point Curve

→

→

2

3

δ

δ

is real

At infinity →→→ −

−=− 32)1( 2

δβ

δβ

ieiZ

i

α2 =0 α3 =0

→

→

2

3

δ

δ

is not real

Line

)1()()1(

2

2 32

−−−−

=

→→

•→→

β

β δβδi

i

eiiieR

Line

)1()1(

2

2 32

−+−−

=−

→→

•→→

β

β δβδi

i

eiieZ

2.6.3 Special Case for P-P-P-P

While three or four positions are parallel to each other following special cases that

are shown in Table 2.8 happen.

Table 2.8 Special cases for case P-P-P-P

Condition Center point Curve Circle Point Curve

µδ

δ=→

→

2

3 At infinity Line α2 = α3 = 0, α4 ≠ 0

µδ

δ≠→

→

2

3 Circle Circle

µδ

δ=→

→

2


µδ

δ≠→

→

2

4 Circle Circle

µδ

δ=→

→

3


µδ

δ≠→

→

3

4 Circle Circle

24

Table 2.8 (Continued)

νδ

δµ

δ

δ== →

→

→

→

2

3

2

4 , At infinity Line α2= α3 = α4 ≠ 0

νδ

δµ

δ

δ== →

→

→

→

2

3

2

4 , Circle Circle

µδ

δδ=

−→

→→

2

43 Line Two lines α2 = 0, α3 = α4

µδ

δδ≠

−→

→→

2

43 Obtained by the general procedure

Obtained by the general procedure

µδ

δδ=

−→

→→

3


µδ

δδ≠

−→

→→

3



µδ

δδ=

−→

→→

4


µδ

δδ≠

−→

→→

4



25

2.7 BRANCH AND CIRCUIT DEFECTS

The solutions for mechanism synthesis may lead to mathematically correct but

infeasible linkages that are associated with the configuration of the mechanism. The

positions belong to different configurations (Branches) in Branch problem.

A circuit defect occurs when the input link has two disjointed ranges of motion and

not all of the prescribed positions belong to the same input link range in rocker-crank

and double-rocker mechanisms [1,5,17,18].

Trailer: A crank which has limited rotation about its moving pivot (Both

cranks of double-crank and rocker of crank-rocker)

Rocker: A crank which has limited rotation about its fixed joint (Rocker of

crank-rocker)

Using Figure 2.7, the following inequalities (Ineq. 2.26) can be geometrically

derived.

Figure 2.7 The trailer conditions in four-bar mechanisms

0 < µ1 < π

- α2 + β2 < µ1 < π- α2 + β2

- α3 + β3 < µ1 < π- α3 + β3 (2.26)

- α4 + β4 < µ1 < π- α4 + β4

26

With the inequalities in 2.26, an interval for m can be found (θ1 < µ1 < θ2). In order

to find a defect free, working mechanism; the driving link or the other link of the

four-bar mechanism must be selected in the Filemon [5] region limited by θ1 and θ2

as shown by Fig. 2.8.

Figure 2.8 Allowable regions for a trailer

Similiarly from the geometry of the four-bar mechanism of Figure (2.9), Polat[11]

derived the relations in Ineq. 2.27 for rocker condition:

Figure 2.9 The rocker conditions in four-bar mechanisms

0 < ψ1 < π

- β2 < ψ1 < π- β2

- β3 < ψ1 < π- β3 (2.27)

- β4 < ψ1 < π- β4

27

If there is an interval for ψ1, then, naming the interval boundaries as θ1 and θ2, where

θ1 <ψ1< θ2, two regions can be formed. In order to avoid the branch problem, the

other center point A0 has to be selected in the appropriate region.

As similar fashion, for crank rocker type mechanisms, the rocker condition is applied

to limit the center points rather than limiting circle points. Hence a selection option

should be supplied the user to indicate whether he/she desires a crank rocker or a

different type of mechanism.

2.8 ORDER PROBLEM

The order problem is simply the positions that are not passed in advance as desired

with the rotation of driving link in the same direction. The driving crank rotations

(β) are the implication of order problem.

In crank rocker mechanisms; the crank link indicates the order problem [12]. Hence

crank link must satisfy;

If 0>dtdα then

⎪⎪⎩

⎪⎪⎨

⎧

><

<>

•

•

32

32

0

0

βββ

βββ

andor

and

In double-rocker mechanisms, both of the rocker links rotate without changing their

direction and following the same order of the prescribed positions. Taking the case

P-P-P-P as an example, both of the rockers must satisfy the following inequalities:

4320 βββ <<< or πβββπ 2234 <<<< (2.28)

In double-crank mechanisms, the two cranks rotate without changing their direction,

hence following the same order through the prescribed positions. Taking the case P-

P-P-P as an example, the following conditions must be satisfied:

28

432 βββ << or 234 βββ << (2.29)

The order problem can be solved following the same method for different

combinations of positions.

2.9 SYNTHESIS WITH DIFFERENT POSITIONS

In CADSYN, by changing the type and/or value of input parameters like (i.e. “β2

fixed at value … and rotate β3 to get solutions”) crank rotation (β) or center or circle

point location, different solutions can be obtained. These solutions can be combined

to yield different mechanisms. Therefore, in addition to present as many solutions to

the user, different preferences might be utilized for selecting two dyads for three

position input. Actually that is the reason for using three positions for synthesis.

If an additional position of the part is predicted, this position can be used in four

position synthesis to get solutions. This saves the user from decisions that have to be

made for three position synthesis and the addition of the position makes the solution

more reliable in the sense that the mechanism will be passing through four positions

rather than three. But the problem in this case is that the designer will be having only

one Burmester solution curve which has a specific trend rather than different curves

in three position case. From problem to problem it changes but a position can be

added to three position inputs in some problems to make the problem a four position

synthesis problem. The entered position input can be changed and/or a different

position can be added to have many Burmester curves each having a different trend.

29

Figure 2.10 The immediate effect on solution curves for manipulated inputs

To illustrate this fact, horizontal location of the third position is changed and a

different input position is obtained (only δ2y changes). The new position is drawn on

the screen immediately which is shown in Figure 2.10 as a screen shot. After

pressing “Apply” button visible on the form, the new Burmester curves are plotted

with a different color. By pressing the button “Done” after the new curve is plotted,

the new solution will be added to the solution set for dyad selection process.

The addition of positions can be applied to a four precision position problem as well.

If the user is not sure about the location of a position then he/she can change that

input and add it to the solution set or change it and recalculate Burmester curves.

This provides the flexibility of solving the problem with different inputs to the user

which may be useful different kind of problems. Especially it should be mentioned

that the solution can totally change by slightly changing input parameters (i.e. like α

or δ in some special locations).

30

2.10 SLIDER-CRANK DESIGN

For slider crank synthesis, the design methodology is the same as four-bar except one

of the center point of the four-bar is at infinity forming the special case of a four-bar.

2.10.1 Motion Generation with Slider-Crank

In precision point synthesis the crank rotation becomes infinitesimal for a center

point at infinity shown in Fig. 2.11, the center point vector becomes infinite in length

which also means the circle point for a slider-crank travels in a linear path. The

solution for multiple position case (P-P-P-P) is examined below.

Figure 2.11 The slider crank design parameters

The loop equation can be written as;

jij jZ s Z e αδ

→ → → →

+ = + ⋅ (2.30)

Rearranging eq’n. 2.30 we get;

31

( 1)jij jZ e α δ

→ →

⋅ − = − s→

2

(2.31)

Replacing the position data to this equation;

22( 1)iZ e α δ

→ →

⋅ − = − s→

(2.33)

33 3( 1)iZ e α δ

→ →

⋅ − = − s→

(2.34)

Knowing that the slider displacements are multiples of each other;

3 2s s λ= ⋅ (2.32)

By specifying λ, we can solve for - Z→

to get center point locations. Z→

, whose locus is

a circle, can be determined as below;

2

32

2

1

1 11

i

i

Zee

α

α

δ

δ λ

λ

→

→→

−

− =−−

(2.33)

For infinitesimal positions the analysis is very similar, detailed derivations are

proved by Polat [11]. The cases for three positions and their results are presented in

the Table 2.9. The solution is a circle which takes the form of “Inflection Circle” for

three infinitesimally separated position input (PPP) case. Inflection circle is the locus

of points for a moving body on which the moving points have radius of curvature of

infinite length. That means all points selected on the inflection circle will follow a

straight path.

32

Table 2.9 Coefficients of loop equations for three positions

CASE λ SOLUTION & TYPE Result is a “Circle” locus

P-P-

P λ is varied

3 2

3 2

1 ( 1i iZe eα α )

δ λ δλ

− + ⋅− =

− − ⋅ −

Result is a “Circle” locus which is called “Point Tangent Locus”

PP-P

λ is varied

3

3 2

(1 )iZi e α

δ λ δλ

→ →→ − ⋅

− =⋅ + −

Result is a “Circle” locus which is called “Inflection Circle”

PPP

λ is varied

3 2

1Z

iδ λ δ

λ

→ →→ − ⋅

− =⋅ +

In four positions the solution will yield a single point which is called Ball’s Point.

The results are obtained by solving the general equations 2.22 & 2.23 for different

position inputs. The solution is done for the first displacement coefficient λ1. It could

be done for λ2 or λ3 also but the result doesn’t change as there is only one solution.

The results are presented in Table 2.10.

Table 2.10 Coefficients of loop equations for four positions

CASE Centre Pt. Location (Z) 1Re( ) Im( ) Im( ) Re( )Re( ) Im( ) Re( ) Im( )

V T V TU V V U

λ ⋅ − ⋅=

⋅ − ⋅

P-P-

P-P

3 2

3 2

1 ( 1i iZe eα α )

δ λ δλ

→ − + ⋅− =

− − ⋅ −

3 4

4 2

32

4 3

2 4

3 2

( 1) (

( 1) (

( 1) (

i i

i i

ii

T e e

U e e

V e e

α α

α α

αα

δ δ

δ δ

δ δ

1)

1)

1)

= ⋅ − − ⋅ −

= ⋅ − − ⋅ −

= ⋅ − − ⋅ −

33

Table 2.10 (Continued)

PP-P

-P

3

2 3

(1 )iZi e α

δ λ δλ

→ − + ⋅− =

+ ⋅ −

4

4

3

4 23

4 3

3 2

(1 )

(1 ) (1 )

(1 )

i

ii

i

T i e

U e

V i e

α

αα

α

δ δ

δ δ

δ δ

= ⋅ + ⋅ −

= ⋅ − − ⋅ −

= − ⋅ − ⋅ −

e

PPP-

P

2 3

1Z

iλ δ δ

λ

→ − ⋅ +− =

⋅ +

4

4

3 4

2 4

3 2

(1 )

( 1)

i

i

T e

U e iV i

α

α

δ δ

δ δδ δ

= ⋅ − −

= ⋅ − − ⋅= ⋅ −

PP-P

P

3

2 3

(1 )iZi e α

δ λ δλ

→ − + ⋅− =

+ ⋅ −

3

3 3

3

4 2

3 4

2 3

( )

(1 )

( 1)

i

i i

i

T i e

U i e e

V e i

α

α α

α

δ δ

δ δ

δ δ

= ⋅ − ⋅

= ⋅ ⋅ − ⋅ −

= ⋅ − − ⋅

PPPP

2 3

1Z

iλ δ δ

λ

→ − ⋅ +− =

⋅ +

3 4

2 4

2 3

( )T iU iV i

δ δδ δ

δ δ

= ⋅ −= − ⋅ += + ⋅

34

CHAPTER 3

CADSYN MECHANISM DESIGN SOFTWARE

3.1 GENERAL

In this thesis the main task is to implement the synthesis theory on a user friendly

software. The CADSYN mechanism design software is written to provide the

mechanism designer, a platform that is kinematically similar to real working

environment of the mechanism with least user effort.

The user must decide on the type of the mechanism to be used and the positions of

the moving body. The software will provide the user all the possible mechanisms that

will satisfy these requirements. It is then the user who decides on the mechanism.

The software guides or warns the user of his/her choice.

The solutions and their all properties are stored under predefined type arrays. The

user has access to change or limit solutions. The calculation is done for only once so

even though it may take increasing amount of time with increased sample number, it

is done for only once.

As mentioned above the main task of this thesis is to implement the synthesis theory

to a computer. The design theory used is the same as shown in [11] Polat’s study

because there is not a vital theoretical change in dyadic approach. In addition, this

approach is the easiest way to adapt computerized design methodology which makes

it preferred. The study of Sezen [13] on the other hand is very illustrating on how to

implement synthesis software. In this study both of the studies are made use of

occasionally.

35

3.2 VISUAL BASIC PROGRAMMING UNDER AUTOCAD®

AutoCAD® is well known, easy to use and reliable CAD software. After versions of

2000, it present a project load option to the users so the user can write his/her own

functions and load it under a user defined button. This makes AutoCAD® very

powerful among other drawing softwares like CADKEY®. Some other software on

the market has this feature also.

The program written is specificly for AutoCAD®. Implementation of CADSYN can

be made to serve other drafting programs. This will not require a drastic change in

the algorithm of CADSYN if the program has Visual Basic® support. The synthesis

algorithm and visual interface will remain the same while the functions specific to

AutoCAD® has to be manipulated accordingly.

Visual Basic is easy to learn and very simple programming language. Furthermore it

is supported by AutoCAD® and there are certain functions specific to AutoCAD®, so

that the user has direct access to use the visual tools of AutoCAD®. There are several

other advantages of AutoCAD such as it has access to use programming functions of

Office Applications. This is important because mechanism design task can also be

continued for further analysis in Microsoft Excel® which is preferred by many

designers. This is done to make the program more versatile and useful. All the

properties of design parameters can be exported to Excel® by the use of these

functions.

36

3.3 THE CODE STRUCTURE OF CADSYN

The algorithm is written in different modules for Motion Generation, Path

Generation and Function Generation for four-bar synthesis. For slider crank

mechanism, similarly a different module is used for evaluation but some different

forms are utilized. Therefore according to the type of the problem different user

interfaces and user forms are utilized automatically. The code exceeds 5000 lines

which is less than Quadlink[13] (12000 lines). This is due to additional algorithm

used in Quadlink for visual interface construction.

There are several modules are used in the program. All the functions are written in

Macros and called from the corresponding button or action. Only simple visual

actions are written under buttons or tools because it is time saving and easy to

understand. Different procedures and functions are brought together in different

modules for easy understanding the nature of the software and for easy tracking of

errors. The modules and their contents are listed below;

- Basic: Covers the basic Excel functions; four bar, angle etc. (By E. Söylemez)

- Function_Generation: Covers the tools, visual actions, calculations for function

generation

- Motion_Generation: Covers the tools, visual actions, calculations for motion

generation

- Path_Generation: Covers the tools, visual actions, calculations for path

generation

- Global_Definitions: All the global definitions and types used are given here

- Input_MotionGen_Parameters: All the input values and actions are written and

input variables are assigned here.

- Math_Functions: All the mathematical functions are stored here.

- Mechanism_Functions: Mechanism functions to determine properties of a

mechanism, like Grashof type, Order etc. are here

- Visual_Instructions: Important visual actions requiring mathematical operations

like animation of the mechanism are written here.

37

There are several important global parameters used during synthesis some of them

are;

- Multiple_Position_Type: Stores the type of the position inputs, i.e. PPP, PP-P

- Multiple_Position_Number: Stores the total number of input positions, i.e. =3

- BPA: Burmester Point array storing the location of center and circle points

with beta values for four position inputs

- CPC1, CPC2: Circle & center point curves; *1 for first dyad, *2 for second

dyad for three position inputs

- Dyad1: The first link selected active and visible on screen

- Dyad2: The second link selected to construct mechanism seen on screen

In CADSYN, there are several forms that appear on the screen in order so this makes

the user less confused because he/she will be allowed to do a limited number of

actions.

In any of the forms if a warning like “CAD functions are active now!” is observed,

than that means the user has access to use any of the AutoCAD® function just like

he/she makes a drawing. This utility can be activated by pressing the button with

“ ” symbol.

3.4 MECHANISM DESIGN WITH CADSYN

There are two main toolboxes designed for the purpose of designing slider-crank and

four-bar mechanisms separately. The inputs and the structure is very much the same

for both. The user has to follow the steps mentioned below in order:

- If user has a technical drawing to be used during synthesis he/she should

open this document before start-up to work on.

38

- If user does not have an introductory technical drawing, no problem!

He/she may construct himself/herself after start up, after the program allows user

access.

- When the user decides on which mechanism to design and clicks on either

four bar or slider-crank button than the menu shown in Fig.3.1 is seen on the screen.

Figure 3.1 Form to input selection method

- After defining the synthesis method, the position input window appears

and the user is allowed to specify three or four positions by clicking on PP button for

infinitesimal and P-P button for finite positions. Also there is settings button to allow

the user to place some restrictions.

Figure 3.2 Form to input positions

- If the button with “P-P” caption is clicked, the user has to input the

position on screen either with mouse or by keyboard just like drawing a line in

39

AutoCAD®. The positions are indicated by lines of arbitrary length because, each

position must be defined by coordinates of a point and the angular orientation of a

line. The coordinates of the first point of the line drawn and the angle made by the

line w.r.to the horizontal axis of the drawing is selected as the parameter values that

define this position.

- If the button with “P-P” caption is clicked, than the form below appears

which lets the user to input vectors in x and y directions. Values should be entered to

the boxes in the form. If a non zero value is written into the boxes, these numbers

will be assigned to the corresponding infinitesimal position vector component.

Figure 3.3 Form to input infinitesimal position vector components

- If the settings button is pressed the figure below appears and asks the user

to enter any limitation to the solutions set. User has to click on the checkboxes first

to activate textboxes.

Figure 3.4 Form impose a limitation

40

- User can see and manipulate input positions by double clicking on the list

box shown in Fig. 3.3, then, the form in Fig. 3.5 will appear. The user can change the

position properties with the aid of this property of CADSYN.

Figure 3.5 Form to change position properties

- After entering the inputs, for three positions, a window shown in Fig.3.6

will appear twice for first and second dyads to enter input parameters. User can make

three different selections to get a single solution that are; inputting the exact of crank

rotations or inputting the exact location of center (Fixed pivot) point or circle point

(Moving pivot). Also a number of solutions can be obtained if a parameter is selected

to be varied over a range.

Figure 3.6 Form to input required data for three positions

41

- In four position synthesis, the form shown in Figure 3.6 will not be

appeared, rather a dialog box will appear asking the user that if he/she is satisfied

with the solution or not. If not and if the user has flexibility in changing the

positions, then he/she can use that position like a variable and resolve the problem

(2nd button from left in Fig.3.7 is for this purpose). User has immediate access to see

how the solution changes. In addition, user can add the new result to the solution set

(1st button from left in Fig3.7 is to for this purpose). If the solution curve is bounded

by a limiting window, user can easily define a window (1st button from left in Fig.3.7

is to for this purpose). Outside the window, all the points will be erased.

Figure 3.7 Additional options for four positions

- After having the solution data on screen, we need to make selection

among this data. Selection of mechanism is carried out in two parts.

1/2- 1st- 2nd Dyad selection buttons, 3/4-Back-Forward buttons during selection of dyads, 5- Step between dyad selection, 6-Save link, 7-Set active link, 8-Open link repository, 9-Filter for selection of 2nd Dyad, 10-Construct mechanism, 11-Open mechanism repository, 12-Exit, 13- Analyze mechanism, 14-Animate mechanism, 15-Settings, 16-Excel export, 17- Activate CAD functions, 18-Type of active(drawn) mechanism

Figure 3.8 Selection form to construct mechanisms

42

1. Select 1st Dyad (Driven Dyad):

- Be sure that the 1st Dyad button (1) is pressed.

- During selecting 1st Dyad two construction lines in green colour are drawn

indicating “Filemon Region”. If there exists a region (There may not be), then

a text is written to direct the user to select either circle or center points in that

region. Selection is done by backward and forward buttons (3 &4).

- After selecting the 1st dyad, user must make this selection “Active Link” by

pressing 7 (Fig.3.9).

- The active links are stored in “Link Repository” that may be needed in

future.

2. Select 2nd Dyad (Driving Dyad):

- After selecting 1st dyad, button for 2nd Dyad (2) must be pressed twice to

make 2nd dyad selection active.

- By pressing filter button (9), user can eliminate undesired solutions

- By pressing construct mechanism button (10) the mechanism will be

constructed and added to the mechanism repository.

- To see the stored mechanisms and active mechanism which is written in

capital letters, button must be pressed (11).

3. After selecting 1st and 2nd dyads and constructing mechanism; analysis (13),

animation (14) and Excel export functions become enabled.

- Link repository has some features to show properties of the selected link.

After double clicking the list box in link repository window, the properties of links

become apparent. The link repository is shown in Fig 3.9.

43

Figure 3.9 Link repository used to store link data

- Mechanism repository works similarly. The properties of mechanisms can

be reached from double clicking on the mechanisms listed in the list box in

mechanism repository form. The property window appears as given in Fig. 3.10.

User is allowed make decisions like drawing the mechanism and to drawing coupler

curve or not (If the mechanism has been drawn: active).

Figure 3.10 Mechanism data stored

- The filter selection allows the user to eliminate data for 2nd Dyad. Order

elimination of data points will be done according to Grashof type selection.

Therefore it is automatically set with the Grashof condition. There is also option to

eliminate branching solutions and the solutions with transmission angle less than a

specified value. Also a window can be defined to limit center or circle or both points.

The solution will be for 2nd Dyad only so when user goes back to 1st Dyad selection,

the whole solution (Burmester curve) will become visible again. This will be

achieved by drawing the solution to two different layers and making one visible the

other invisible. The Fig. 3.11 shows the filter tool which is very simple to use, if all

check boxes are set true that means the solution may be in all Grashof types.

44

Figure3.11 Filter form used differentiate solutions with order and branch problem

- After constructing a mechanism, excel export function becomes enabled.

The form in Fig.3.12 shows the parameters that can be transferred.

Figure 3.12 Problems that can be transferred to Excel for further analysis

- Similarly, an analysis form is designed shown in Fig.3.13, to give

immediate insight about transmission angle, angular velocity ratios, coupler and

displacements of the mechanism designed and selected active. It also makes a simple

acceleration analysis to provide the user initial information about the design.

45

Figure 3.13 Analysis form for simple kinematic entities

- The animation can be initiated by pressing the button 14 shown in Fig.3.8,

next the form in Fig.3.14 appears. It has simple features and functions.

- 1st button (1) is to move the driving link of the mechanism backward. The

step can be increased or decreased by slider in Fig.3.14.

- 2nd button (2) is to play the animation of the mechanism. One important

remark for playing option is that the animation will play for full cycle for

crank rocker and double crank mechanisms.

- 3rd button (3) is to move the driving link of the mechanism forward. The

step can be increased or decreased by 4th and 2nd buttons respectively.

- 4th button (4) is used to add “Block” drawing parts any of the moving

links.

- 5th button (5) is for exiting to selection form in Fig.3.14.

46

- Slider bar (6) is to increase or decrease the motion speed by changing

animation sample number.

1/3-Rotate crank one step backward/forward, 2-Play, 4-Attach drawings to links, 5-Exit, 6- Slider bar

for setting animation sample no., 7- Input link start angle; angle at initial position, 8-Input link end

angle; angle at last position, 9-Overall rotation of input link and its direction

Figure 3.14 Animation form used for mechanism analysis

- Designer usually would like to see the motion of actual links which may

have some other parts or assemblies attached on them. In order realize this “Add

Block” property is developed in CADSYN. To see their motion, a combining

operation is necessary. Here, after pressing the “Add Block” button, the form in

Fig.3.15 appears in which existing blocks in the drawing are listed to provide user

the readily available block drawings. If there is not, user can immediately define or

draw the block to be attached to any of the links of the mechanism designed. Then

the user can draw and attached the blocks at first position of the mechanism.

47

Figure 3.15 Block addition form

To define blocks;

- First draw or define block objects that are to be attached to the links of

the mechanism through activating AutoCAD functions with button;

→Draw → Block → Make

- By pressing the “Back” button, the blocks exisiting in the drawing will be

listed in the list box with the caption “Block” in Fig. 3.15.

- As mentioned above user can draw anything he/she wishes by pressing

the button with mouse icon. After pressing this button all CAD commands will

become active and the form below appears. In addition, some of the forms in

CADSYN in that it is saying “CAD functions are active at the moment!” allow this

utility also (Fig. 3.16).

Figure 3.16 Form used warn the user that the CAD functions are active

48

3.5 SYNTHESIS TYPES

The synthesis procedure and forms are not different for all cases. The interface is the

same for all cases. In only function generation case, one of the dyads will be drawn

already, the user will select only one dyad.

49

CHAPTER 4

TEST CASES

The program is tested for several test cases including the cases in Polat’s and Sezen’s

studies to make a comparison among the approaches and check the results. A few

more examples are added to apply additional features of the program.

4.1 TEST CASE 1

A pick and place mechanism is needed to transfer packages from one conveyor to the

other. This is a very common need in assembly lines. The parts are carried by means

of gripping clamps that are attached to the coupler of the four-bar.

4.1.1 The Design Constraints

As this is a repeating action, a continuous input is needed. Therefore a fully rotatable

crank of the four-bar mechanism is required. The angular orientations of the coupler

plane is important to prevent slippage of the packages and to gurantee a proper pick

and place action. This implies the synthesis method to be used is synthesis for motion

generation. One other requirement is that when the picking action takes place, in

order not to cause a damage to the packages, the coupler link is required to have zero

velocity at that instant. This means that a first order infinitesimally separated position

must exist at that position.

50

Figure 4.1 The design domain of test case 1

4.1.2 The Synthesis Procedure

According to the design specifications, a PP-P-P case motion generation is selected

as the synthesis case. The first position is taken at the picking position and by

examining the design space, the input parameters are selected as follows:

First Prescribed Position:

Cartesian coordinates and angle with X-axis: X1 = 88 Y1 = 4 θ1 = 300

Second Prescribed Position: Infinitesimally separated from the first position.

=⇒= =

→

22 0 δddδ αα

δ 0 + 0 i

Third Prescribed Position: Finitely separated from the second position.

Cartesian coordinates and angle with X-axis: X3 = 72 Y3 = -17 θ3 = 00

Or the dyad vector and rotation: 33 δ→

3= ⇒δ =δ 69-15 i α3= 150

Fourth Prescribed Position: Finitely separated from the third position.

Cartesian coordinates and angle with X-axis: X4= -260 Y4= -260 θ4= -150

51

Or the dyad vector and rotation: 44 4δ δ δ→

= ⇒ = -260 – 260i α4= -150

According to these input parameters, the selected mechanism among the synthesized

mechanisms is shown in Figure (4.2). The mechanism, coupler curve and the

prescribed positions are directly taken from the AutoCAD screen. The working plane

is copied as picture to AutoCAD screen and the positions are taken from the picture.

Figure 4.2 The synthesized mechanism for test case 1

The pivot coordinates relative to the first prescribed position are:

A0 =19.1 + 3.2 i (mm.)

A1 = 27.9 + 3.3 i (mm.)

B0 = 28.5 + 49.9 i (mm.)

B1 = 3.9 + 68.9 i (mm.)

52

The resulting mechanism, when scaled, is of crank-rocker type and has the link

lengths: a1 = 668.0 mm., a2 = 123.5 mm., a3 = 980.0 mm. and a4 = 434.0 mm.

Figure 4.3 The synthesized non-scaled mechanism properties for test case 1

Figure 4.3 shows the mechanism dimensions and other properties of the design.

4.2 TEST CASE 2

For the same problem of test case 1, at the point of placing (releasing) the packages

on the conveyor, the angular velocity of the coupler link may wanted to be

eliminated in order to maintain a smooth release of the packages. The design space is

the same as Figure (4.1).


For the same reasons as test case 1, a fully rotatable crank of the four-bar mechanism

is required and the angular orientations of the coupler plane is important. This

53

implies that the synthesis method has to be synthesis for motion generation. One

other requirement is that when the picking action takes place, in order not to cause

damage to the packages, the coupler link is required to have zero velocity at that

instant. This means a first order infinitesimally separated position must exist at that

position. And an additional requirement is to have zero angular velocity of the

coupler link at the point of release.


According to the design specifications, a PP-PP case motion generation is selected as

the synthesis case. The first position is taken at the picking position and by





iδddδ 0022 0 +=⇒= =

→

ααδ


Cartesian coordinates and angle with X-axis: X3 = 72 Y3 = -17 θ3 = 00

Or dyad vector and rotation: 33 3δ δ δ→

= ⇒ = -20 – 30 i α3 = -300

Fourth Prescribed Position: Infinitesimally separated from the third position.

=⇒= =

→

442

δαδδ ααd

d -1 + 0 i


mechanisms is shown in Figure (4.4). The pivot coordinates relative to the first

prescribed position are:

54

A0 = 63 – 19.8 i (mm.)

A1 = 71.8 – 11.4 i (mm.)

B0 = 46.2 + 16.5 i (mm.)

B1 = 79.1+ 6.9 i (mm.)

Figure 4.4 The synthesized mechanism for Test Case 2

The resulting mechanism is of crank-rocker type and when sacled the original link

lengths can be found as: a1 = 561.2 mm., a2 =170.4 mm., a3 = 275.8mm. and a4 =

480.2 mm.

4.3 TEST CASE 3

In the figee floating crane, the path of the load is required to be a close

approximation to a straight line. This is because the load carrying energy increases if

55

the load is elevated and lowered vertically hence an unnecessary potential energy

gain and loss appears. The design domain is shown in Figure (4.5).

Figure 4.5 The desired path of a body of Case 3


The motion is not repeated, hence there are no constraints on the Grashof type of the

mechanism. Due to the fact that the load is usually carried by means of a rope, the

angular orientation of the coupler plane is not important. However, the displacement

of the load may be correlated with the input link rotation. This implies that the

synthesis method has to be synthesis for path generation.


According to the design specifications, a P-P-P-P case path generation is selected as

the synthesis case. The first position is taken at the picking position and by


56


Cartesian coordinates: X1 = 102.7 Y1 = 85.5

Second Prescribed Position: Finitely separated from the first position.

Cartesian coordinates of coupler point X2 = 83.7 Y2 = 85.5

Or, dyad vector and rotation of the crank: -18.9 + 0 i β=⇒=→

222 δδ δ 2 = 22.460


Cartesian coordinates and angle with X-axis: X3 = 64.0 Y3 = 85.5

Or, dyad vector and rotation of the crank: -38.7 + 0 i β=⇒=→

333 δδ δ 3 = 54.220


Cartesian coordinates and angle with X-axis: X4= 43.3 Y4= 85.5

Or, dyad vector and rotation of the crank: -59.7 + 0i β=⇒=→

444 δδ δ 4 = 76.680


mechanisms is shown in Figure (4.6).


57


A0 = -8.2 – 43.8 i (mm)

A1 = 111.3 + 47.1 i (mm)

B0 = 28.3 + 14.6 i (mm)

B1 = 88.6 + 37.0 i (mm)

The resulting mechanism is of double-rocker type and has the link lengths: a1 = 67.6

mm, a2 = 150.1 mm, a3 = 24.8 mm. and a4 = 66.0 mm. It is not the same mechanism

as in Polat’s thesis but it is similar. The link ratios can be scaled to give the exact link

lengths. The analysis of the mechanism has been accomplished by using the

analysis option of CADSYN and the plots of the kinematic entities are given directly

as a screenshot of the analysis window of CADSYN in Figure (4.7).

Figure 4.7 The analysis window for the mechanism of test case 3

It can be observed from “Coupler Pt. Displacements vs. Crank Angle” plot that the

“yc” curve traces nearly a horizontal path.

58

4.4 TEST CASE 4

In meat cutting machines, the rocking knife which is circular in shape is used to cut

meat by moving it back and forth. The relative motion of this knife with respect to

the fixed frame (cutting table) is rolling without slippage where the motion is a force

closed motion. The geometry of the workspace is given in Figure (4.8). The motion

of the knife can be approximated by a properly designed four-bar mechanism.


There are no constraints on the Grashof type of the four-bar as the motion will be a

back and forth motion. The rotation of the rocking knife, hence the coupler plane is

important which implies that the synthesis method has to be synthesis for motion

generation. The aim is to approximate the straight line motion of the contacting point

of the rocking knife to the cutting table.

Figure 4.8 The working plane of the mechanism of test case 4

59


According to the design specifications, a PPPP motion generation is selected as the

synthesis case. The first position is taken at the current contacting position of the

rocker knife to the cutting table. When the cutting knife rotates α radians, according

to pure rolling conditions, the distance traveled on the cutting table will be s = α.r.

Hence, for finite rotations, the dyad vector will be:

rαδ −=→

where r = 500 mm.

Taking the successive derivatives, one can assign the prescribed positions as:


Cartesian Coordinates and angle with X axis: X1= 0 Y1= 0 θ1= 900


rdd

−=⇒= =

→

22 0 δαδδ α = - 500

Third Prescribed Position: Infinitesimally separated from the second position.

032

2

3 0 =⇒= =

→

δα

δδ αdd


043

3

4 0 =⇒= =

→

δα

δδ αdd



60

Figure 4.9 The designed of meat cutting mechanism

The coupler point is actually located at the inflection circle of the desired motion.

Therefore the straight line generated is a third order approximation.


A0 = 54.2 –176.8 i (cm.)

A1 = -27.1 + 13.4 i (cm.)

B0 = -52.2 – 178.9 i (cm.)

B1 = 26.1 + 14.4 i (cm.)


cm., a2 = 206.9 cm., a3 = 53.2 cm. and a4 = 208.6 cm.

61

4.5 TEST CASE 5

In this test case it is aimed to replace the epicyclic train of Figure (4.10) by a four-bar

mechanism. In order to accomplish this task, the functional relationship between the

arm and the planet gear of the gear train must be obtained in a limited range of

motion of the four-bar mechanism.


Since the desired functional relationship is only true for an interval, there are no

constraints on the Grashof type of the four-bar mechanism. To be more precise, the

functional relationship is desired to be approximated to the third order.

Figure 4.10 Epicyclic gear train of test case 5


According to the design specifications, a PPPP function generation is selected as the

synthesis case. According to pure rolling conditions of the gear pair, the linear

functional relationship between the input and output links are taken as:

62

βψr

rR +=

where β and ψ represent the rotation of the arm and planet gears respectively. Taking

the successive derivatives, one can assign the prescribed positions as:


Angle with X axis: θ1= 900


rrRδ

ddδ 22

+=⇒=

βψ = 3 + 0i

Third Prescribed Position: Infinitesimally separated from the second position.

002

2

=⇒= = 33 δddδ ββ

ψ + 0i


003

3

=⇒= = 44 δddδ ββ

ψ + 0i



This is a special case for case PPPP function generation where the center and cirlce

point curves degenerate into a line (the y-axis actually) and a circle. The Burmester

curves, coupler curve and the synthesized mechanism are directly taken from the

screen of CADSYN. Although it may not be clearly seen, the y axis contains the

degenerated center and circle point curves.


A0 = 87.18 –240.40 i

63

A1 = -261.54 –77.79 i

B0 = 0 – 13.0 i

B1 = 0 – 96.0 i


As this is a function generator mechanism, not the link lengths but the link length

ratios are important. Hence if the mechanism of Figure (4.11) is scaled with respect

to the output link length which is taken as 1 unit, the scaled mechanism link length

ratios are: a1 =2.03229, a2 = 4.63578, a3 = 3.24638 and a4 = 1.

4.6 TEST CASE 6

Mechanisms are very commonly used in the furniture industry. An interesting design

may be a shelf design for a cabinet shown in Figure (4.12). The shelf , when not used

as a writing table, rests in the cabinet and serves as a shelf. A four-bar mechanism

can be used to guide the shelf between these positions.

64



The pivots must be located in the cabinet and in the shelf position, no portion of the

links must be outside the cabinet boundaries. The mechanism need not to be a

specified Grashof type. The input will be through the shelf itself, hence transmission

angle optimization is not critical. As the orientation of the shelf is important, the

synthesis case has to be motion generation.


According to the design specifications, a P-P-P motion generation is selected as the

synthesis case. The origin of the coordinate system is taken at the down left corner of

the cabinet. The prescribed positions are selected as follows:



Second Prescribed Position: Finitely separated from the first position

Cartesian coordinates and angle with X-axis: X2 = 70 Y2 = 55 θ2 = -50


2= ⇒ =δ δ 70 + 55 i α2= -50

65




3= ⇒δ =δ 61+ 118 i α3= 00

This is a special case for P-P-P motion generation where the center and circle point

curves degenerate into lines. According to these input parameters, the selected

mechanism among the synthesized mechanisms is shown in Figure (4.13).



A0 = 24.0 –55.0 i (cm.)

A1 = 6.0 + 3.3 i (cm.)

B0 = 52.0 + 83.0 i (cm.)

66

B1 = 30.5 – 33.7 i (cm.)


cm., a2 = 54.8 cm., a3 = 39.0 cm. and a4 = 53.8 cm.

4.7 TEST CASE 7

It is desired to approximate the function y=2x2 + 4x + 1 in the interval 2 x 3. A

properly designed four-bar mechanism can accomplish this task.


This case is clearly a function generation case where the two crank rotations must be

correlated. There is no constraint on the Grashof type of the mechanism. Four

prescribed positions will be used.


According to the design specifications, a P-P-P-P function generation is selected as

the synthesis case. The interval for the input and output link rotations are ∆β =58°

∆ψ = 100°. In the selection of the prescribed positions, Chebyshev spacing formula

is used. The formulation is as follows:

⎥⎦⎤

⎢⎣⎡ −

−∆=−=∆ )2

)12(cos(121

0 njxxxx jj

π

jj xxx ∆+= 0

where:

j = 1,2,3,4

n = 4 ∆x = 3.0 - 2.0 =1.0

67

Hence: x0 = 2.00, x1 = 2.04, x2 = 2.30, x3 = 2.69, x4 = 2.96, x5 = 3.00. The

corresponding values of y are: y0 = 17, y1 = 17.4832, y2 = 20.78 , y3 = 26.2322, y4 =

30.3632, y5 = 31.

Converting the functional relation between x and y to input and output rotations:

)( 1xxx jj −

∆∆

=ββ , )( 1yy

y jj −∆∆

=ψψ

Hence the prescribed position data can be calculated as:



Second Prescribed Position: Finitely separated from the first position

Rotation of the input link: β2 = 15.08

Rotation of the output link: ψ2 = 23.549

Third Prescribed Position: Finitely separated from the second position


Rotation of the output link: ψ3 = 62.4930

Fourth Prescribed Position: Finitely separated from the third position


Rotation of the output link: ψ4 = 920




A0 = 0 + 0 i

68

A1 = 0 + 100 i

B0 = 49.93 – 52.74 i

B1 = 89.98 + 98.71 i


The resulting mechanism is of double-rocker type and has the link lengths: a1 =

72.63, a2 = 100.00, a3 = 90.00 and a4 = 156.65. In function generation the ratio of the

link lengths are the design parameters. Therefore the resulting mechanism can as

well be specified as: a1 = 0.4637, a2 = 0.6384, a3 = 0.5745 and a4 = 1 (units).

4.8 TEST CASE 8

It is desired to design a four-bar linkage to guide the bucket of a skid loader as it is

raised from its resting position to a specified height. The design domain is given in

Figure (4.15). The motion of the bucket will be controlled by another mechanism;

thus it is only the path of the end point of the linkage that carries the bucket that is of

interest. This path is desired to be an approximation to a stright line for two reasons.

The first and the most important reason is that, if this linkage is pivoted about a fixed

axis, the circular path its end point takes may result in the falling over of the skid

loader when the bucket contains heavy load and is raised high enough so that the

69

moment it creates can overcome the weight of the machine. The second reason is

that, when the loader is operated near a vertical obstacle, the bucket can only be

raised above if the path it takes is a straight line.



There are no constraints on the Grashof type of the mechanism. However there are

lots of design considerations including how the hydraulic drive for the linkage would

be placed, the placement of linkages in order not to impede the operator’s viewing of

his workspace etc. These are not directly related to the kinematics of the problem but

are things to consider during the selection of the proper mechanism among the

kinematic solutions to the problem.


According to the design specifications, a P-P-P-P path generation is selected as the

synthesis case. After various trials, the following input parameters are selected as the

prescribed positions data which give the most suitable Burmester curves for the

design.

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Cartesian coordinates: X1 = 0 Y1 = 0

Second Prescribed Position: Finitely separated from the first position.

Cartesian coordinates of coupler point: X2 = 0 Y2 = 63

Or, dyad vector and rotation of the crank: 0 + 63 i β=⇒=→

222 δδ δ 2 = 13.150


Cartesian coordinates and angle with X-axis: X3 = 0 Y3 = 188


333 δδ δ 3 = 43.130


Cartesian coordinates and angle with X-axis: X4= 0 Y4= 335


444 δδ δ 4 = 70.160

In fact, these prescribed position data has been reached after numerous trials in order

to have suitable Burmester curves. Among the synthesized mechanisms, the best

solution is selected not only for its kinematic properties but also for other criteria

given in the preceding paragraphs. The selected mechanism is shown in Figure (4.16)

with the design domain as the background image. The four positions are given in

different colors and the mechanism and coupler curve are directly taken from the

screen of CADSYN.


A0 = 121.6 +165.1 i (cm.)

A1 = 221.5 + 96.4 i (cm.)

B0 = 135.8 + 85.2 i (cm.)

B1 = 249.2 + 48.8 i (cm.)

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The resulting mechanism is of double-rocker type and has the link lengths: a1 = 81.1,

a2 = 121.2, a3 = 55.1 and a4 = 119.1 (all dimensions are in centimeters).


4.9 TEST CASE 9

The design of four-bar linkages for dead center positions can also be accomplished

by using the multiply separated position synthesis technique. The trick is that the

angular velocity of the output link changes sign at the dead center positions of a four-

bar. Hence at the dead center positions, the angular velocity of the output link must

be zero.

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At least four prescribed positions are needed for this task. These four positions

consist of two finitely separated positions and two infinitesimally separated positions

each of which belongs to one finitely separated position and is zero.


According to the design specifications, a PP-PP function generation is selected as the

synthesis case. The prescribed position data given as inputs are:




022 0 =⇒= = δddδ ββ

ψ + 0i


Rotation of the input link: β3 = 1900

Rotation of the output link: ψ3 = 530


044 2=⇒= = δ

ddδ βββ

ψ + 0i

The Burmester curves for the selected inputs and the selected mechanism can be seen

in Figure (4.17) as a screenshot from CADSYN. As it can be seen, this is a special

case where the center and circle point curves degenerate into lines and circles.


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A0 = -41.95 -55.65 i

A1 = 31.94 – 8.12 i

B0 = 200.0 – 100.0 i

B1 = 200.0 + 100.0 i

Figure 4.17 Synthesized mechanism and Burmester curves for test case 9

The resulting mechanism is of crank-rocker type and has the link lengths: a1 =

245.98, a2 = 87.86, a3 = 199.83 and a4 = 200 (all dimensions are in units). As this is a

function generation case, the link length ratios are important. Hence scaling the

mechanism by taking the output link length as 1 unit: a1 = 1.2299, a2 = 0.4393, a3 =

0.9992 and a4 = 1 units.

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CHAPTER 5

DISCUSSION AND CONCLUSION

5.1 SUMMARY AND DISCUSSION

In the present study, a visual, interactive computer program for the synthesis and

analysis of planar four-bar linkages is constructed and named CADSYN. This

current version of the program is capable of synthesizing four-bar and slider crank

mechanisms for three and four finitely and/or infinitesimally separated positions (or

multiply separated positions) for the common cases of motion generation (or rigid

body guidance), path generation with prescribed timing and function generation. The

program can also be used to analyze the synthesized or an independently given

mechanism for its basic kinematic entities (ie. translational and rotational

displacements, velocities and accelerations of the links). The designed mechanisms

can be visualized throughout their range of motion via the animation option of the

program. Most of the data for the curves, plots etc. can be exported as Excel® sheet

for further analysis.

The code is constructed as Visual Basic macros under AutoCAD®. Therefore the

AutoCAD® functions and its tools can be used which saved great amount of time for

constructing a new visual interface. Besides, synthesis task in AutoCAD® domain is

that through Visual Basic functions, programmer has easy access to drawing

commands or other properties of entities and it is very easy to understand and apply

these functions in Visual Basic language.

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The dyadic approach is used as the main synthesis technique and the graphical

techniques are used for the analysis of the mechanism. A complex design domain is

generated within the program. In solving the governing complex vector equations,

the calculations have been accomplished using analytical solution procedures. One of

the main features of this program is that it enables the user to control almost every

variable in the design. For instance, in three multiply separated position cases, it is

possible to generate the center and circle point circles and curves by fixing either one

of the scalar variables (ie. bl's). The sample point numbers used in plotting all of the

curves (i.e. the Burmester curves of the synthesis task and the coupler curves) can be

modified by the user. All of the data for these curves are stored in dynamic arrays,

which are created and destroyed when needed, during runtime.

The program has been tested for nine test cases involving real life examples of

multiply separated motion, path and function generation problems. Some of these

cases were selected from the special cases, which do not have straightforward

solutions as general synthesis problems. The program faced no problems in these test

cases. After every test case, the synthesized mechanism is analyzed by using the

analysis options of CADSYN. In most of the test cases, the synthesized mechanism

and the design domain are drawn together to give a better visualization of the

problem and the solution.

After the completion of the program, an intense debugging stage has also been

performed and the detected bugs have been removed. However, as every commercial

visual program of today’s programming world, it is expected for CADSYN to

possess some bugs. These undetected bugs are hoped to be detected by the users as

the program finds use in practical design problems, and will be corrected by current

programmer or future programmers.

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5.2 FUTURE RECOMMENDATIONS

The modular structure of the present code enables it to be improved quite easily with

the addition of new classes, modules and variables. Algorithms for multi-loop

linkage synthesis for different types of six-link mechanisms can be implemented by

using the triadic approach and/or the dyadic theory based on the relative motion

concept. Naturally, the visualization and analysis of these mechanisms will be harder

to implement and will require addition of other major classes. However the

mathematical and geometrical solution libraries of CADSYN can still be effectively

used.

In addition, the synthesis algorithms that is very straightforward for cams can be

applied as a project in AutoCAD® by making use of the graphical utilities of the

program. This will eliminate the drawing of cams after design as the output is in

drawing format.

The software does not list the possible outputs instead the designer fills that list.

CADSYN is designed to reach solutions with the user effort. Therefore design of an

interface simply listing mechanism solutions is not recommended to be done in

AutoCAD®. Because then there will not be use of any AutoCAD® functions. It can

be better designed in Visual Basic® or Delphi® which run much faster than the Visual

Basic® in AutoCAD®.

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REFERENCES

[1] Beloiu, A. S., Gupta, K. C., “A Unified Approach for the Investigation of Branch and Circuit Defects”, Mechanism and Machine Theory, Vol. 32, pp. 539-557, 1997

[2] Beyer, “Kinematic Synthesis of Mechanisms”, Chopmall and Hall Ltd., London, 1963 [3] Erdman, A.G., Sandor, G.N., “Mechanism Design, Analysis and Synthesis Volume 1”, Prentice-Hall, USA, 1984 [4] Erdman, A.G., Sandor, G.N., “Mechanism Design, Analysis and Synthesis Volume 2”, Prentice-Hall, USA, 1984 [5] Filemon, E., “Useful Ranges of Center Point Curves for Design of Crank and Rocker Linkages”, Mechanism and Machine Theory, Vol. 7, pp. 47-53, 1972 [6] Freudenstein, F., “An Analytical Approach to the Design of Four-Link Mechanisms”, Transactions of the ASME, Vol. 76, 1954, pp. 483-92 [7] Holte, J. E., “Two Precision Position Synthesis of Planar Mechanisms with Approximate Position and Velocity Constraints”, Ph.D. Thesis, University of Minnesota, December 1996

[8] Kaufman, R., “Mechanism Design by Computer”, Machine Design, Vol. 24, pp.94-100, 1978 [9] Mlinar, J.R., “An Examination of the Features of the Burmester Filed and the Linear Solution Geometry of Dyads and Triads”, Ph.D. Thesis, University of Minnesota, August 1997

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[10] Müller, H. R., “Kinematik Dersleri (Turkish Translation)”, Ankara Üniversitesi Fen Fakültesi Um 96 - Mat27, Ankara, 1963 [11] Polat, M., “Computer Aided Synthesis of Planar Mechanisms”, M.Sc. Thesis, METU, March 1985 [12] Prentis, J.M., “The Pole Triangle, Burmester Theory and Order and Branching Problems – I/II”, Mechanism and Machine Theory, Vol.26, pp. 19-39, 1991 [13] Sezen, S., “Development Of An Interactive Visual Planar Mechanism Synthesis And Analysis Program By Using Delphi Object Oriented Programming Environment”, M.Sc. Thesis, METU, June 2001

[14] Söylemez, E., “Mechanisms”, Middle East Technical University Publication No:64, Ankara, 1999 [15] Tesar, D., “The Generalized Concept of Three Multiply Separated Positions in Coplanar Motion”, Journal of Mechanisms, Vol.2, pp. 462-474, 1967 [16] Tesar, D., “The Generalized Concept of Four Multiply Separated Positions in Coplanar Motion”, Journal of Engineering for Industry, pp. 135-142, February 1969

[17] Ulushan, U., “Computer Aided Kinematic Analysis Of Multi-Loop Planar Mechanisms”, M.Sc. Thesis, METU, December 2001 [18] Waldron, K. J., “Graphical Solution of The Branch and Order Problems of Linkage Synthesis for Multiply Separated Positions", Journal of Engineering for Industry, pp. 591-597, 1977 [19] Waldron, K. J., “Location of Burmester Synthesis with Fully Rotatable Cranks”, Mechanism and Machine Theory, Vol.13, pp. 125-137, 1978 [20], WATT, Mechanism Design Suite, by Heron Technologies, http://www.heron.com

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