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OPTIMAL KINEMATIC DESIGN FOR A NOVEL CVT SYSTEM BASED IN METAHEURISTICS

RAMIRO-ERNESTO VELEZ-KOEPPEL

ESCUELA DE INGENIERÍA DE ANTIOQUIA MECHATRONIC ENGINEERING

ENVIGADO 2010


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OPTIMAL KINEMATIC DESIGN FOR A NOVEL CVT SYSTEM BASED IN METAHEURISTICS

RAMIRO-ERNESTO VELEZ-KOEPPEL

Degree’s project in partial fulfilment of the requirements for the degree of:

Mechatronic Engineer

PhD. Efren Mezura-Montes

PhD. Edgar-Alfredo Portilla-Flores

ESCUELA DE INGENIERÍA DE ANTIOQUIA

MECHATRONIC ENGINEERING ENVIGADO

2010


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TABLE OF CONTENTS

Contents

INTRODUCTION ............................................................................................................. 16

1 PRELIMINARY ......................................................................................................... 18

1.1 PROBLEM CONTEXT AND CHARACTERIZATION .......................................... 18

1.1.1 Problem definition ....................................................................................... 18

1.2 PROJECT’s OBJECTIVES ................................................................................ 19

1.2.1 General Objective ....................................................................................... 19

1.2.2 Specific Objectives ..................................................................................... 19

1.3 CONCEPTUAL BACKGROUND ........................................................................ 19

1.3.1 Engineering Optimization Problem (EOP) ................................................... 19

1.3.2 Continuously Variable Transmission (CVT) ................................................ 24

1.3.3 Metaheuristics applied to engineering design ............................................. 26

1.3.4 Kinematic Model ......................................................................................... 33

2 PROJECT’S METHODOLOGY ................................................................................. 35

2.1 BIBLIOGRAPHIC REVISION ............................................................................. 35

2.2 KINEMATIC MODELLING ................................................................................. 35

2.3 STATEMENT OF THE PROBLEM ..................................................................... 36

2.4 OPTIMIZATION BASED ON METAHEURISTICS .............................................. 36

2.4.1 PROPOSED APPROACHES ...................................................................... 37

3 CRANK-ROCKER-SLIDER CVT SYSTEM OPTIMIZATION ..................................... 48

3.1 KINEMATIC DESIGN OPTIMIZATION .............................................................. 48

3.2 KINEMATIC MODEL OF THE RRRR MECHANISM .......................................... 50


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3.2.1 Tightening Points ........................................................................................ 55

3.2.2 Operation Quality ........................................................................................ 56

3.3 MULTI-OBJECTIVE OPTIMIZATION PROBLEM (MOP) ................................... 58

3.3.1 MOP Definition ........................................................................................... 58

3.3.2 CVT Optimization Problem definition .......................................................... 59

3.4 OBTAINED RESULTS ....................................................................................... 64

3.4.1 Multi-Objective DE/rand/1/bin ..................................................................... 64

3.4.2 Multi-Objective DE/rand/1/exp .................................................................... 67

3.4.3 Multi-Objective DE/best/1/exp ..................................................................... 70

3.4.4 ME-ABC ..................................................................................................... 73

3.4.5 Comparison ................................................................................................ 76

4 ADD-INS ................................................................................................................... 78

4.1 ARTIFICIAL NEURAL NETWORK MODELLING ............................................... 78

4.1.1 Statistical validation .................................................................................... 79

4.2 RECONFIGURABLE HARDWARE .................................................................... 81

4.2.1 C-Sections .................................................................................................. 82

4.2.2 Threaded rods ............................................................................................ 84

4.2.3 Screws, nuts and washers .......................................................................... 84

4.2.4 Coupling devices ........................................................................................ 84

4.2.5 Ball-bearings .............................................................................................. 85

4.2.6 Shafts and sir-clips ..................................................................................... 85

4.2.7 Mounting device ......................................................................................... 86

4.2.8 Finite Elements Analysis (FEA) .................................................................. 86

5 CONCLUSIONS ....................................................................................................... 87


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6 FUTURE WORK ....................................................................................................... 89

7 BIBLIOGRAPHY ....................................................................................................... 90

APPENDIX 1 SOLUTIONS PROVIDED BY ME-ABC ...................................................... 96

APPENDIX 2 ANALYSIS PARAMETERS INFORMATION ............................................ 117

Load Case Multipliers ................................................................................................. 117

Multiphysics Information ............................................................................................. 117

Processor Information ................................................................................................ 117

PART INFORMATION ................................................................................................... 119

Element Properties used for: .................................................................................. 119

Material Information ................................................................................................... 119

AISI 1045 Steel, cold drawn, 19-32 mm (0.75-1.25 in) round - Brick ....................... 119

LOAD AND CONSTRAINT INFORMATION ................................................................... 121

Loads ......................................................................................................................... 121

Load Set 1: Unnamed .................................................. Error! Bookmark not defined.

Constraints ................................................................................................................. 121

Constraint Set 1: Unnamed ..................................................................................... 121

Meshing Results ......................................................................................................... 123

SUPERVIEW PRESENTATION IMAGES ...................................................................... 125

Stress ......................................................................................................................... 125

Strain ......................................................................................................................... 126

Displacement ............................................................................................................. 126

Deformed Shape ........................................................................................................ 127


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LIST OF TABLES

Table 1 Parameters for Multi-Objective DE/rand/1/bin ..................................................... 39

Table 2 Parameters for Multi-Objective DE/rand/1/exp .................................................... 41

Table 3 Parameters for Multi-Objective DE/best/1/exp ..................................................... 43

Table 4 Parameters for ME-ABC ..................................................................................... 47

Table 5 Solutions from Multi-Objective DE/rand/1/bin ...................................................... 66

Table 6 Solutions from Multi-Objective DE/rand/1/exp ..................................................... 69

Table 7 Solutions from Multi-Objective DE/best/1/exp ..................................................... 72

Table 8 Solutions from ME-ABC ...................................................................................... 76

Table 9 Results for Shapiro-Wilk Test .............................................................................. 80

Table 10 Mann-Whitney-Wilcoxon test for Optimal and Network Population .................... 80

Table 11 Mann-Whitney-Wilcoxon test for filtered populations ......................................... 81

Table 12 963 Feasible and non-dominated solutions for CVT kinematic design ............... 96


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TABLE OF FIGURES

Figure 1 different, local and absolute optimal present in a function .................................. 20

Figure 2 In dark gray, feasible region looks smaller than search space in light gray determined by boundaries a-b and c-d ............................................................................. 21

Figure 3 Elements of a Bi-Objective Optimization Problem .............................................. 24

Figure 4 Crowding Distance for solution “i” in a bi-objective problem ............................... 24

Figure 5 Different CVT mechanisms ................................................................................ 25

Figure 6 DE/rand/1/bin algorithm ..................................................................................... 29

Figure 7 ABC Algorithm ................................................................................................... 32

Figure 8 Multi-Objective DE/rand/1/bin algorithm ............................................................. 39

Figure 9 Multi-Objective DE/rand/1/exp algorithm ............................................................ 40

Figure 10 Multi-Objective DE/best/1/exp algorithm .......................................................... 42

Figure 11 ME-ABC algorithm ........................................................................................... 46

Figure 12 Chain and gear mechanism ............................................................................. 48

Figure 13 Double pawl mechanism .................................................................................. 49

Figure 14 Slider and pawl propelled by a rocker .............................................................. 49

Figure 15 Basic crank-rocker-slider CVT basic diagram................................................... 50

Figure 16 Four-bar Mechanism Diagram ......................................................................... 50

Figure 17 Closed Circuit Diagram .................................................................................... 52

Figure 18 Maximum output angle, a tightening point for the mechanism .......................... 55

Figure 19 Minimum output angle, other tightening point for the mechanism ..................... 55

Figure 20 Maximum Transmission Angle ......................................................................... 57

Figure 21 Minimum Transmission Angle .......................................................................... 57

Figure 22 Solutions provided by Multi-Objective DE/rand/1/bin variant ............................ 64


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Figure 23 Optimal �1 for Multi-Objective DE/rand/1/bin ................................................... 65

Figure 24 Balanced solution for Multi-Objective DE/rand/1/bin ......................................... 65

Figure 25 Optimal �� for Multi-Objective DE/rand/1/bin .................................................. 66

Figure 26 Solutions Provided by Multi-Objective DE/rand/1/exp variant ........................... 67

Figure 27 Optimal �� for Multi-Objective DE/rand/1/exp ................................................. 68

Figure 28 Balanced solution for Multi-Objective DE/rand/1/exp ........................................ 68

Figure 29 Optimal �� for Multi-Objective DE/rand/1/exp ................................................. 69

Figure 30 Solutions provided by Multi-Objective DE/best/1/exp variant ............................ 70

Figure 31 Optimal �� for Multi-Objective DE/best/1/exp .................................................. 71

Figure 32 Balanced solution for Multi-Objective DE/best/1/exp ........................................ 71

Figure 33 Optimal �� for Multi-Objective DE/best/1/exp .................................................. 72

Figure 34 Solutions provided by ME-ABC ........................................................................ 73

Figure 35 Optimal �� for ME-ABC ................................................................................... 74

Figure 36 Balanced solution for ME-ABC ......................................................................... 75

Figure 37 Optimal �� for ME-ABC ................................................................................... 75

Figure 38 Comparison of different Pareto's Fronts ........................................................... 76

Figure 39 Artificial Neural Network model for CVT ........................................................... 79

Figure 40 Important measures for C-Section ................................................................... 82

Figure 41 Machining process for horizontal C-Sections ................................................... 83

Figure 42 Machining process for vertical C-Sections ....................................................... 83

Figure 43 Main measures and types of Ball-Bearings ...................................................... 85


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TABLE OF EQUATIONS

Equation 1 ....................................................................................................................... 22

Equation 2 ....................................................................................................................... 22

Equation 3 ....................................................................................................................... 22

Equation 4 ....................................................................................................................... 22

Equation 5 ....................................................................................................................... 22

Equation 6 ....................................................................................................................... 22

Equation 7 ....................................................................................................................... 22

Equation 8 ....................................................................................................................... 22

Equation 9 ....................................................................................................................... 23

Equation 10 ..................................................................................................................... 23

Equation 11 ..................................................................................................................... 23

Equation 12 ..................................................................................................................... 23

Equation 13 ..................................................................................................................... 24

Equation 14 ..................................................................................................................... 31

Equation 15 ..................................................................................................................... 34

Equation 16 ..................................................................................................................... 38

Equation 17 ..................................................................................................................... 41

Equation 18 ..................................................................................................................... 43

Equation 19 ..................................................................................................................... 43

Equation 20 ..................................................................................................................... 44

Equation 21 ..................................................................................................................... 44

Equation 22 ..................................................................................................................... 44


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Equation 23 ..................................................................................................................... 47

Equation 24 ..................................................................................................................... 47

Equation 25 ..................................................................................................................... 48

Equation 26 ..................................................................................................................... 49

Equation 27 ..................................................................................................................... 50

Equation 28 ..................................................................................................................... 51

Equation 29 ..................................................................................................................... 51

Equation 30 ..................................................................................................................... 51

Equation 31 ..................................................................................................................... 51

Equation 32 ..................................................................................................................... 51

Equation 33 ..................................................................................................................... 51

Equation 34 ..................................................................................................................... 51

Equation 35 ..................................................................................................................... 52

Equation 36 ..................................................................................................................... 52

Equation 37 ..................................................................................................................... 52

Equation 38 ..................................................................................................................... 52

Equation 39 ..................................................................................................................... 53

Equation 40 ..................................................................................................................... 53

Equation 41 ..................................................................................................................... 53

Equation 42 ..................................................................................................................... 53

Equation 43 ..................................................................................................................... 53

Equation 44 ..................................................................................................................... 53

Equation 45 ..................................................................................................................... 53

Equation 46 ..................................................................................................................... 53


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Equation 47 ..................................................................................................................... 53

Equation 48 ..................................................................................................................... 53

Equation 49 ..................................................................................................................... 53

Equation 50 ..................................................................................................................... 54

Equation 51 ..................................................................................................................... 54

Equation 52 ..................................................................................................................... 54

Equation 53 ..................................................................................................................... 54

Equation 54 ..................................................................................................................... 54

Equation 55 ..................................................................................................................... 54

Equation 56 ..................................................................................................................... 54

Equation 57 ..................................................................................................................... 55

Equation 58 ..................................................................................................................... 56

Equation 59 ..................................................................................................................... 56

Equation 60 ..................................................................................................................... 56

Equation 61 ..................................................................................................................... 56

Equation 62 ..................................................................................................................... 56

Equation 63 ..................................................................................................................... 56

Equation 64 ..................................................................................................................... 57

Equation 65 ..................................................................................................................... 58

Equation 66 ..................................................................................................................... 58

Equation 67 ..................................................................................................................... 58

Equation 68 ..................................................................................................................... 58

Equation 69 ..................................................................................................................... 58

Equation 70 ..................................................................................................................... 58


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Equation 71 ..................................................................................................................... 58

Equation 72 ..................................................................................................................... 58

Equation 73 ..................................................................................................................... 59

Equation 74 ..................................................................................................................... 59

Equation 75 ..................................................................................................................... 59

Equation 76 ..................................................................................................................... 60

Equation 77 ..................................................................................................................... 60

Equation 78 ..................................................................................................................... 60

Equation 79 ..................................................................................................................... 60

Equation 80 ..................................................................................................................... 60

Equation 81 ..................................................................................................................... 60

Equation 82 ..................................................................................................................... 60

Equation 83 ..................................................................................................................... 60

Equation 84 ..................................................................................................................... 60

Equation 85 ..................................................................................................................... 60

Equation 86 ..................................................................................................................... 61

Equation 87 ..................................................................................................................... 61

Equation 88 ..................................................................................................................... 61

Equation 89 ..................................................................................................................... 61

Equation 90 ..................................................................................................................... 61

Equation 91 ..................................................................................................................... 61

Equation 92 ..................................................................................................................... 62

Equation 93 ..................................................................................................................... 62

Equation 94 ..................................................................................................................... 62


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Equation 95 ..................................................................................................................... 62

Equation 96 ..................................................................................................................... 62

Equation 97 ..................................................................................................................... 62

Equation 98 ..................................................................................................................... 62

Equation 99 ..................................................................................................................... 62

Equation 100 ................................................................................................................... 62

Equation 101 ................................................................................................................... 62

Equation 102 ................................................................................................................... 63

Equation 103 ................................................................................................................... 63

Equation 104 ................................................................................................................... 63

Equation 105 ................................................................................................................... 63

Equation 106 ................................................................................................................... 63

Equation 107 ................................................................................................................... 63

Equation 108 ................................................................................................................... 63

Equation 109 ................................................................................................................... 63

Equation 110 ................................................................................................................... 63


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RESUMEN

Los sistemas de transmisión de potencia, ampliamente usados en diferentes aplicaciones

desde sencillos procesos industriales hasta vehículos sumamente complejos, han sido

sujetos de mejora en tiempos recientes. Una decreciente oferta de recursos energéticos

en la última década ha impulsado el desarrollo de mecanismos cada vez más eficientes en

el uso de la energía, sin embargo, mecanismos con alta eficiencia suelen ser difíciles de

construir o con restricciones de funcionamiento tales que los hace poco aptos para

muchas tareas. En el presente trabajo se desarrolla un sistema de transmisión de potencia

bastante novedoso, cuyo mecanismo posee gran simplicidad de la mano con un buen

desempeño. Inicialmente, el modelo cinemático de una Transmisión Continuamente

Variable (TVC) con mecanismo manivela-balancín-corredera es desarrollado, buscando

poder construir un modelo paramétrico de diseño del sistema mencionado. Una vez el

modelo cinemático se construyó, un problema de Optimización Multiobjetivo fue

desarrollado, con la finalidad de optimizar dicho modelo en pro del desempeño así como

de la eficiencia. Posteriormente, un conjunto de 4 diferentes metaheurísticas fue

empleado para la optimización de dicho mecanismo. Cabe mencionar que el algoritmo

presentado bajo el nombre ME-ABC constituye un nuevo mecanismo de optimización

multiobjetivo basado en colonia de abejas, único en su tipo, diseñado específicamente

para el presente trabajo. Los resultados obtenidos son comparados entre sí, logrando

evaluar el desempeño de diferentes algoritmos así como de los mecanismos por ellos

optimizados. Una recreación del modelo cinemático a través de Redes de Neuronas

Artificiales trató de ser implementada, y los resultados de esta se exponen con detalle.

Para concluir, una propuesta de hardware reconfigurable para mecanismos de barras

similares es presentada, de gran utilidad para poder verificar la precisión de los modelos

con prototipos reales.

Palabras Clave: Metaheurísticas, Optimización Multiobjetivo, Modelado Cinemático,

Síntesis de Mecanismos, Evolución Diferencial, Colonia Artificial de Abejas, TVC, Diseño

Paramétrico.


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ABSTRACT

Power transmission systems, widely used in different applications from simple industrial

processes to very complex vehicles, have been subjects of improvement in recent years. In

last decade, a decreasing supply of energetic resources has driven the development of

more energy efficient mechanisms however, energy efficient mechanisms are hard to

build or with working constraints that make them unsuitable for several tasks. In the

present work a novel power transmission system is developed. This mechanism is simple

to build together with a good performance. Initially, the kinematic model of a

Continuously Variable Transmission (CVT) with a crank-rocker-slider mechanism is

developed, aiming to build a parametric design model for the system. Once kinematic

model is built, a multi-objective optimization problem (MOP) is stated, with the intention

to optimize kinematic design pro of design and efficiency. After problem’s statement, a set

of 4 different metaheuristics was used to optimize mentioned mechanism. Algorithm

presented as ME-ABC is a new optimization algorithm based on bees, specifically designed

for this degree project. Obtained results are compared among them; an assessment of the

performance of the algorithms and of the mechanisms optimized by them is realized. A

recreation of the kinematic model through Artificial Neural Networks is tried to implement

and the results are exposed in detail. Finally, an approach of reconfigurable hardware for

bar mechanisms is presented, very useful to verify model’s accuracy against real

prototypes.

Key-Words: Metaheuristics, Multi-objective Optimization, Kinematic Modelling,

Mechanism Synthesis, Differential Evolution, Artificial Bee Colony, CVT, Parametric Design.

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INTRODUCTION

Computer aided design software may be one of the most important approaches in the

engineering design framework at this time. Those tools allow obtaining a wide range of

different models, to develop manufacturing processes, to manage life-cycle, etc. However,

some modern systems can be complex to design. Some modern systems can be

considered as mechatronic design, where not only mechanical parameters are important.

This approach, known as concurrent design, requires that mechanical and control systems

be designed as an integrated unit. Due to its huge complexity, before an optimal

concurrent design problem be stated, another kind of problem can be stated, where the

objective is to find the best possible combination of parameters that describe the system.

These types of problems are so complex too, but allow identifying system’s functionality

and reliability, some important information to sketch how the system works and to

estimate other models complexity.

The best possible combination of parameter can be found through kinematic design,

aiming to fulfil the positions and velocities for the selected system. Once a set of optimal

parameter that describe the system are obtained, several potential and convincing

solutions can be proposed before a prototype be build or a more complex model be

stated. . Moreover, the design must propose a set of performance functions and

constraints to quantify system’s behaviour. This means the creation of potential solutions

in the absence of a well-defined algorithm or predict configure the solution (Norton,

1997). An alternative approach to solve the design problem is to propose it as an

optimization problem, where an optimal parametric design is carried out in order to solve

the original problem (Portilla-Flores, Mezura-Montes, Alvarez-Gallegos, Coello-Coello, &

Cruz-Villar, 2007).

In other way, implementation of new computer algorithms based on metaheuristics,

provides powerful tools for the solutions of the resulting design problems. These

algorithms do not ensure obtaining the global optimum. However, they produce a good

solution in a reasonable time, as well as not requiring specific knowledge about the

problem to be solved (Reeves, 1993). The main advantage on using metaheuristics is that

they are based on a set of points initial vector or set of design variables which are

randomly generated and avoids the sensitivity to the initial search of the solution

(typically found in classical optimization methods) (Portilla-Flores, Mezura-Montes,

Alvarez-Gallegos, Coello-Coello, & Cruz-Villar, 2007).

One of the contributions of the present work is to optimize the output motion and energy

transmission of a continuously variable transmission, by designing a four-bar mechanism.

In order to carry out this mechanical design, a couple of functions and a set of constraints

17

are proposed. A contribution of this work is to compare the performance of some nature-

inspired metaheuristic algorithms for solving engineering optimization problems,

specifically, the Differential Evolution (DE) and the Multi-Objective Elitist Artificial Bee

Colony (ME-ABC) algorithms. The obtained solution will be a set of optimal geometric

parameters. The optimal geometric parameters adjust the link lengths of the four-bar

mechanism. Once the optimal set of parameter values is selected by the design engineer,

a virtual prototype of the whole mechanism is used aiming to validate results.

The whole work related with the project is exposed in chapter 3. This chapter begins with

explanation of the system to optimize. Once the system is explained, the next subchapter

explains, step-by-step, the way to obtain the kinematic model of the CVT system. The third

item of the chapter consists in the statement of the Engineering Optimization Problem

(EOP) oriented to the CVT, together with the meaning of each one of the equations

present in the problem. After EOP is stated, the different metaheuristics used to optimize

the kinematic design are described. Have to be noticed that the Swarm-Intelligence based

algorithm could be considered as a contribution of this work, as it is one of the first

approaches for solving multi-objective EOPs based in the so-called Artificial Bee Colony.

Finally, a set of results is presented, with important information related with the solutions.

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1 PRELIMINARY

1.1 PROBLEM CONTEXT AND CHARACTERIZATION

Optimization is one of the most studied fields in engineering, due to its benefits in

economic and technical issues. At this time, a huge group of research centres around the

world has oriented their efforts to develop and to renew this field of study. Engineering

optimization is a complex task that requires the achievement of a set of independent tasks

for the problem statement, constraint-handling, etc. These contribute to a success result

in the optimization problem.

Engineering Optimization has been useful in unnumbered applications through years,

from the easiest tasks as to select the best dealer of a product, to the complex machines

ever built, such as Large Hadron Collider (LHC) (Mokhov & Strait, 1996), the biggest

particle accelerator in Switzerland-France territory. Previous reasons become in a solid

argument to improve the performance of a wide variety of systems.

In the last two decades, a sensible growth in metaheuristics field together with an

increased computer’s performance and development of novel and robust algorithms, have

motivated to stand out them as a powerful tool for solving optimization problems

oriented to real world applications, which outperforms their impact in engineering world.

The remarkable strength of metaheuristics in comparison with mathematical methods lies

in the capacity to solve any kind of problem, different from traditional techniques that

need the function to be continuous along time and differentiable at each point. These

features turn these techniques as a suitable method to solve complex problems, whit

sharp nonlinearity, high dimensionality, a huge set of constraint furthermore of

differentiability and continuity.

Since a complex problem such as a CVT has to be solved aiming its optimization, and

considering some problematic features such as mentioned in above lines, metaheuristics

becomes in a strong candidate to solve complex problems as will raises for the CVT.

Together with complex problems, optimization becomes hard and sometimes infeasible

through traditional methods, but when these problems are solved, the significance of the

solution has a relationship with its complexity.

1.1.1 Problem definition

Is it possible to obtain an optimal kinematic design of a CVT, able to optimize transmission

performance and energy consumption in a quick and reliable way?

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1.2 PROJECT’S OBJECTIVES

1.2.1 General Objective

To develop a technique based in metaheuristics, oriented to optimize the kinematic design of a CVT.

1.2.2 Specific Objectives

o To obtain a kinematic model able to determine the position of the main elements

of the CVT along time.

o To formulate the statement of the EOP, including objective function(s) and

constraints.

o To solve the EOP at least with three different metaheuristics aiming to compare

their results.

o To observe and to measure the performance of the optimal design aiming to

determine technique’s reliability.

1.3 CONCEPTUAL BACKGROUND

1.3.1 Engineering Optimization Problem (EOP)

Derived from all disciplines in engineering, Engineering Optimization aims to solve

problems related with planning, design, construction, operation and other common

aspects in Engineering (Rao, 2009).

Normally, solving an EOP raises the need of a statement of the problem. An EOP must

contain all the variables involved in the problem represented as one or objective functions

belonging to a search space. More problems that are complex need to include a set of

constraints, with different possible features as well. Constraints determine the Feasible

Region (Rao, 2009).

Such as a CVT, many EOPs aim to find solutions to different type of instances, where the

most disseminated are Static Optimization Problems, where the considerations to find

solutions belongs to a unique instant. A novel kind of problems called Dynamic

Optimization Problems requires for their solutions a set of considerations that evolve the

behaviour of the system along time. This possibility turns the researcher able to evaluate

the performance of the system through a number of working cycles. This kind of

problems, although has a higher complexity level than Static Optimization Problems,

provides a huge set of information related with the system, which is why sometimes

20

makes it easier to establish decision makers, objective function(s), etc (Lin, Peng, Grizzle,

& Kang, 2001).

Engineering design optimization problems are normally adopted in the specialized

literature to show the effectiveness of new constrained optimization algorithms. Those

are sometimes preferred instead benchmark problems due to their inner complexity, real

assessment availability between other powerful reasons (Cagnina, Esquivel, & Coello,

2009).

As its name indicates, in an EOP the aim is to optimize “something” expressed through

one or several objective function.

1.3.1.1 Local Optimum

In numerical or combinatorial optimization problems, a Local Optimum consists in a

solution that is optimal within a neighbourhood of solutions. An optimum can be a

maximum or minimum in accordance with the statement of the problem (Rao, 2009).

1.3.1.2 Absolute Optimum

In contrast with Local Optimum, an Absolute Optimum called Global Optimum too (either

maximum or minimum) can be considered as the optimal solution among all possible

solutions (Rao, 2009). When the EOP is solved through metaheuristics, the locality of an

optimum is dependent on the neighbourhood structure, defined by the metaheuristic that

is used for optimizing the solution (Corne, Dorigo, & Glover, 1999). Figure 1 shows a

hypothetic 2-D function with different optimal. The objective function determines if the

optimal is a maximum or a minimum.

Figure 1 different, local and absolute optimal present in a function

21

1.3.1.3 Search Space

Usually denoted by �, it represents an n-dimensional space that includes all possible

values for each one of the variables present in the problem. In cases where the EOP has

no constraints, the optimal solution can be in any point into � (Rao, 2009). When the EOP

is stated with constraints, only a region called Feasible Region is able to contain the

absolute feasible optimum (Rao, 2009).

When a multi-objective optimization problem (MOP) is explored, one of the most

important differences with respect to single-objective optimization is that besides a

decision space �, a objective space � have to be considered too. Each solution present in � has a correspondent point in �. This mapping task is realized between an n-dimensional

vector (where n is the number of variables) and other m-dimensional vector (where m is

the number of objective functions) (Deb K. , 2002). Both spaces can be considered to

improve known solutions. A MOP will be detailed later in the document.

1.3.1.4 Feasible Region

Determined by constraints, it is smaller than the whole search space. In this zone the

feasible solutions are located, i.e. solutions that accomplishes each one of the constraints

(Rao, 2009). Figure 2 shows how a search space and its feasible region in a 2-D space can

be represented.

Figure 2 In dark gray, feasible region looks smaller than search space in light gray determined by boundaries a-b and c-d

1.3.1.5 Objective Function

The “subject” to be optimized, an objective function is expressed in terms of the problem

variables. The optimization of this function determines the optimization in the result

(Messac & Ismail-Yahaya, 2001). A single-objective function is expressed as:

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� � �� Equation 1

Where

�� = ��, ��, … , �� Equation 2

�� ∈ �� Equation 3

On the other hand, a MOP is expressed as:

� Φ�� = ��, ��, … , �� Equation 4

where

�� = � �� Equation 5

�� = ��, ��, … , �� Equation 6

�� ∈ �� Equation 7

In this case, the selection criteria perform in a different way due to close relationship

among solutions. In a multi-objective problem, the optimum solution consists on a set of

(“trade-off”) solutions, rather than a single solution as in global optimization (Mezura-

Montes & Coello Coello, A Survey of Constraint-Handling Techniques Based on

Evolutionary Multiobjective Optimization, 2007).

1.3.1.6 Constraints

As said in lines above, constraints draw the feasible region for the Optimization Problem.

A simple classification for constraints in Optimization Problems separates them in two

groups:

• The first group consists in inequality constraints. These are written as follows:

�� = � �� ≤ 0 Equation 8

• The second group includes equality constraints which handling is explained below

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ℎ� = � �� = 0 Equation 9

In optimization, a huge difference can be observed between constrained and

unconstrained problems. In unconstrained optimization the objective function can be

complex in its essence, but any point located in the search space is considered feasible i.e.

it is a valid solution. In contrast, constrained optimization, where the feasible region can

consist in a quasi-non-considerable piece of the search space, not every solution can be

considered feasible, i.e., some solutions may violate one (or more) constraint(s).

Therefore, the search behaviour between these two optimization problems may be quite

different.(Klein & Young, 1999)

1.3.1.7 Multi-Objective Optimization

A MOP including m kinds of objective functions can be defined in a simple manner:

"� Φ�� = $�� , �� , … , �� %&'()*+ � �� ∈ Γ - Equation 10

Where, �� ∈ Γ is a feasible region in the solution space � Γ ⊆ ��, and �� / = 1,2, … , 1�are

the m objectives to be optimized.

When more than one objective function are considered in an EOP, a MOP is then solved.

In such problem, a set of non-dominated solutions called Pareto Optimal Set includes each

one of those so-called non-dominated solutions in the problem. Denoting the feasible

region as Γ, the Pareto Dominance between solutions 2� and &� ∈ Γ in a minimization

problem is defined as follows:

∀� ∈ ℳ: �� 2�� ≤ �� &�� ∧ ∃� ∈ ℳ: �� 2�� < �� &�� Equation 11

where ℳ = 91,2, … , 1:. If Equation 11 is satisfied, 2� dominates &�, denoted as � s�� ≼ � 2�� (Sato, Aguirre, & Tanaka, 2008).

The Pareto Optimal Solutions 2� is defined as:

=∗ ≔ 92� ∈ Γ|¬∃s� ∈ Γ ∶ � s�� ≼ � 2��: Equation 12

where P* is usually unknown for real-world problems. The goal is to obtain a sub-optimal

Pareto set including sub-optimal trade-off solutions (Mezura-Montes, Portilla-Flores,

Coello, Alvarez-Gallegos, & Cruz-Villar, 2008).

Finally, a Pareto Front (CD) is defined as:

Equation 13 shows all elements described above and a sketch with the optimal solutions

for a hypothetical problem.

Figure 3

1.3.1.8 Crowding Distance

As a technique to improve solutions found for

the search in the scarcely explored zones of a Pareto

crowding distance consists in the summation of the sides of a hyper

the nearest 2 solutions to the current solution, as can be seen in

Figure 4 Crowding Distance for solution

1.3.2 Continuously Variable Transmission (

24

CD = EΦ�� |�� ∈ C∗ F

all elements described above and a sketch with the optimal solutions

hypothetical problem.

3 Elements of a Bi-Objective Optimization Problem

Crowding Distance

As a technique to improve solutions found for MOPs, the crowding distance aims to orient

the search in the scarcely explored zones of a Pareto front (Kukkonen & Deb, 2006)

crowding distance consists in the summation of the sides of a hyper-

the nearest 2 solutions to the current solution, as can be seen in Figure

Crowding Distance for solution “i” in a bi-objective problem

Variable Transmission (CVT)

Equation 13

all elements described above and a sketch with the optimal solutions

Objective Optimization Problem

s, the crowding distance aims to orient

(Kukkonen & Deb, 2006). The

-cuboid traced from

Figure 4.

objective problem

25

A huge set of researchers have oriented their efforts to obtain low energy consumption

together with a high mechanical efficiency in rotational propulsion systems. Continuous

Variable Transmissions (CVTs) have been developed to attend this premise, by offering an

infinite set of gear ratios within a range. Automotive industry, due to its requirement of

fuel economy without decreasing system performance, has the biggest group of different

configurations for CVTs. The Van Doorne belt or V-belt CVT is the most studied mechanism

(Setlur, Wagner, Dawson, & Samuels., 2003) (Shafai, Simons, NeFF, & Geering, 1995). This

CVT is built with two variable radii pulleys and a chain or metal-rubber belt. Due to its

friction-drive operation principle, the speed and torque losses of rubber V-belt are a

disadvantage. The Toroidal Traction-drive CVT uses the high shear strength of viscous

fluids to transmit torque between an input torus and an output torus. However, the

special fluid characteristic used in this CVT makes the manufacturing process expensive. A

pinion-rack CVT is built in with conventional mechanical elements as a gear pinion, one

cam and two pairs of racks. The conventional CVT manufacture is advantageous over

other existing CVTs. However, in the pinion-rack CVT, it has been determined that the

teeth size of the gear pinion is an important factor in the performance of the system

(Silva, Schultz, & Dolejsi., 1994). The crank-rocker-slider CVT is a novel mechanism (Gomez

& Ceballos, 2006) for power transmission based in a set of bar mechanisms. This kind of

CVT is less explored than other, due to its novelty. Figure 5 shows different configurations

for CVTs exposed above.

Figure 5 Different CVT mechanisms

1.3.2.1 Crank-Rocker-Slider CVT

This configuration of CVT called Crank-Rocker-Slider consists in a set of bar mechanisms

which acts simultaneously, amplifying or reducing the input speed. The bar mechanisms

transmit the movement to a chain system, the same that through a foremast is able to

maintain a rotational movement in the same way whole time (Gomez & Ceballos, 2006).

Due to its novelty, this mechanism is selected to be improved in kinematic issues through

nature-inspired metaheuristic algorithms.

26

1.3.3 Metaheuristics applied to engineering design

Derived from the Greek word “Εὑρίσκω” which means “to find” (Yang, 2008),

metaheuristics consists in computational methods oriented to solve optimization

problems by an iterative manner by improving a candidate solution.

Although metaheuristics have their origin around 50’s (Robbins & Monro, 1951), a

considerable growth in the number of techniques have happened in last two decades.

Independently of their novelty, have been well accepted by a huge group of scientist for

optimization, due to their capacity to solve problems where special conditions leave aside

traditional mathematical method such as linear programming, Lagrange methods, etc.

Through years, an increasing number of different metaheuristics have been developed, all

of them motivated by different reasons and oriented to the same goal: optimization

(Corne, Dorigo, & Glover, 1999). Aiming to perform an assessment of the accuracy to

optimize, many functions with known optimal points have been optimized using

metaheuristics through years, obtaining them the same results known as optimal points

(Ozturk, 2007). Once accuracy of different metaheuristics were tested in benchmark

functions, some researchers have proposed some approaches to solve real problems in a

wide range of applications: the first patent for an object with a non-human made design

was for an antenna, becoming in a simple design able to cancels the noise at specified

frequencies (Altshuler & S.Linden, 1998).

Such as above example, water reservoir problems (Haddad, Afshar, & Mariño, 2006), TSP

(Wong, Low, & Chong, 2009), routing problems (Teodorovic & Dell'Orco, 2005), optimal

paths (Nariman-Zadeh, Felezi, Jamali, & Ganji, Pareto optimal synthesis of four-bar for

path generation, 2009) and an unnumbered set of real problems have been solved with

metaheuristics as an Engineering Optimization Method.

Besides the advantages mentioned before, metaheuristics make no assumptions about

the problem being solved, reason that make them capable to solve problems with large

scale search spaces, high dimensionality, integer, NP-hard and NP-complete problems, etc

(Pedersen, 2010). Along years, some metaheuristics have been based in nature behaviours

and theories such as human evolution, annealing processes and animal behaviour among

others. They have been oriented to different goals like local or global optimization,

numerical optimization, combinatorial optimization, etc (Lin, Peng, Grizzle, & Kang, 2001).

1.3.3.1 Operators

Within metaheuristics, operators act as the heuristics in the problem solution. These are

used to improve solutions through different mechanism determined by the metaheuristic

(Talbi, 2009). Operators actually con determine the type of search reached, the treatment

for different point in the solutions, such as constraint-handling, etc (Blum & Roli, 2005).

27

1.3.3.2 Selection Criterion

Independently from the metaheuristic used, a selection criterion has to be applied to

select among candidate solution and its improved version. Selection Criteria becomes

efficient in accordance to the features of the problem, reason that raises the range of

selection criteria to be used by the same metaheuristic, depending on the problem

features (Nakrani & Tovey, 2004) (Deb, Pratap, Agarwal, & Meyarivan, 2002).

In accordance to problem’s features, some examples of selection criteria are: Pareto

Dominance (Srinivas & Deb, 1994), Goal Attainment Method (Liu & Chen, 1991), Deb’s

Rules (Deb K. , Multi-Objective Optimization Using Evolutionary Algorithms, 2002),

Crowding Distance (Saku & Kalyanmoy, 2006).

1.3.3.3 Constraint-Handling

Aiming to orient the algorithms in easiest way to feasible region within search space,

some constraint-handling technique had been developed. Constraint-handling technique

has an inseparable relationship with the deferent kind of constraints present in the

problem (Michalewicz, 1998).

A constraint-handling technique can be as simple as death penalty where infeasible

solutions are rejected (Michalewicz, 1998). However, there are more robust techniques

which includes dynamic handling of a tolerance for equality constraints (Zielinski,

Vudathu, & Laur, 2008)

1.3.3.4 Local Search

Those mechanisms oriented to improve a solution found through an exhaustive revision of

its neighbourhood can be grouped into local search (Kalogeraki, Gunopulos, &

ZeinalipourYazti, 2006). Such mechanisms often use information directly related with the

function to optimize such as its gradient, Llagrangian matrixes, etc. Local search is really

useful in specific regions of the search space previously explored by a global search

algorithm (Kalogeraki, Gunopulos, & ZeinalipourYazti, 2006).

1.3.3.5 Differential Evolution (DE)

DE is an (EA) where the decision variables are represented by real numbers, which is

convenient when it is applied to parametric design problems as the problem tackled in this

research. Moreover, its computational implementation is straightforward (Price, 1999).

DE is a population-based and directed search method. Like other EAs, it starts with an

initial population of vectors (solutions to the problem), which is randomly generated when

28

no preliminary knowledge about the solution space is available (Zaharie, 2007). After

initialization, the same process is repeated through all generation. This process consists in

the application of a mutation operator first, aiming to generate a trial population. With

trial and target populations, an offspring population can be generated through a crossover

operator. After mutation and crossover operators, a selection criterion has to be applied

to keep the best individual from one generation to another (Shahryar Rahnamayan, 2008).

DE is an evolutionary direct-search algorithm designed to solve optimization problems. DE

shares some features with other evolution-based algorithms; however DE does not use

probability functions to self-adapt its parameters as an Evolution Strategy (Schwefel,

1995) and it does not use binary encoding as Genetic Algorithms (Goldberg, 1989). The

search-engine of DE performs mutation based on the distribution of the solutions in the

current population. In this way, search directions and possible step-sizes depend on the

location of the individuals selected to calculate the mutation values (Portilla-Flores,

Mezura-Montes, Alvarez-Gallegos, Coello-Coello, & Cruz-Villar, 2007).

Four parameters must be defined in DE:

• Population size or NP.

• Number of generations, sometimes called MCN or MAX_GEN.

• Factor D ∈ G0,1H. This factor scales the value of the differences computed from

randomly selected individuals from the population.

• I� ∈ G0,1H parameter, which controls the influence of the parents on their

corresponding offspring.

Once DE has demonstrated its robustness, several DE variants have been proposed (Price

K. V., 1999) and a nomenclature codifying method was developed for the algorithm. The

name has 4 parts separated by a “/”. The original and most popular version is called

“DE/rand/1/bin" which pseudocode for a minimization problem is shown in . “DE" means

Differential Evolution, the word “rand" indicates that the individuals selected to compute

the mutation values are chosen randomly from the current population, “1" is the number

of pairs of solutions chosen to calculate the differences for the mutation operator and

finally “bin" means that a binomial crossover operator is used.

In Figure 6, “D” represents the dimensionality of the problem; both random numbers have

a Uniform distribution and “NP”, ”MAX_GEN”, “F” and “CR” are the parameters that

belongs to the algorithm mentioned few lines above.

29

Figure 6 DE/rand/1/bin algorithm

Some DE variants are:

• DE/rand/1/exp, where an exponential crossover operator is used.

• DE/best/1/exp, where besides an exponential crossover operator, the individual to

compute the mutation values is selected as the one with the best value so far in

the current population.

These and other variants are subsequent work of the original version, aiming to improve

its performance (MezuraMontes, Coello-Coello, & Velazquez-Reyes, 2006) In the case of

binomial crossover a component of the offspring is taken with probability CR from the

mutant vector “y” and with probability 1-CR from the target vector, “x”. The condition

“rand(0,1)<CR or j=k” within the “if” statement in binomial crossover ensures the fact that

at least one component is taken from the mutant vector as to prevent to generate an

exact copy of the target vector. This type of crossover is very similar with the so-called

uniform crossover used in evolutionary algorithms (Zaharie, 2007).

For exponential crossover, once the target and mutant vector have been selected from

current population, two integer and random numbers have to be generated. The first

number “b” is in the interval [1,n] where “n” is the length of solution vector, and indicates

30

the initial point for the replacement of mutant vector in target vector. The second number

is “l” in the interval [b,n] and represents the number of elements to substitute in the

mutant vector from target vector. The probability for crossover descents exponentially,

while in binomial crossover, this probability descents linearly (Zaharie, 2007).

The original DE algorithm and its variants were originally proposed to solve global

optimization problems, then, for solving constrained optimization problems, a constraint-

handling mechanism has to be added, such as described in 1.3.3.3.

1.3.3.6 Swarm Intelligence

In nature, swarms have fascinating behaviours, based in simple rules followed by each

member of the swarm, which allow accomplishing complex tasks by the whole group. SI is

defined by Bonabeau, Dorigo and Theraulaz as “any attempt to design algorithms or

distributed problem--solving devices inspired by the collective behaviour of social insect

colonies and other animal societies” (Bonabeau, Dorigo, & Theraulaz, 1999). At this point,

SI has some important characteristics, such as description of collective behaviour of

decentralized, self-organized and emergent systems (Schut, 2007).

In accordance to the above lines, SI is an AI technique for search/optimization, based on

the idea of "swarming" (as observed in bird flocking, fish schooling, bee foraging, etc). This

term was initially coined by (Beni & Wang, 1990) in the area of cellular robotics (i.e.,

robotics based on the theory of cellular automates) and has since then been worked out

by Marco Dorigo (Bonabeau, Dorigo, & Theraulaz, 1999), inventor of ant colony

optimization (ACO) -a metaheuristic for combinatorial optimization problems-.

In SI, the intelligence of swarming as observed in nature is used for solving practical

coordination problems. It is extremely successful in the area of intelligent robotics, and

really fruitfully in abstract problems, such as Travelling Salesman Problem (TSP) (Schut,

2007).

1.3.3.6.1 As part of Nature-Inspired methods, SI has a well structured and defined set of

steps oriented to get a feasible solution to optimization problems. At the

beginning randomly generated swarm is considered. Every member of the initial

population is a possible solution for the proposed problem, and has to be

evaluated in order to verify its accuracy and feasibility. After the evaluation,

those better solutions are selected and new solutions are generated around

prior selected members of the swarm. New possible solutions (members of the

swarm) are evaluated again and some of them are selected for the next

iteration, until max iteration's number has been achieved.Artificial Bee Colony (ABC)

31

One of the most popular algorithms based on bees for solving constrained optimization

problems is the Artificial Bee Colony (ABC) initially presented in (Karaboga, 2005) and

further tested in (Karaboga & Akay, 2009) and (Karaboga & Basturk, On the performance

of artificial bee colony (abc) algorithm, 2008). In (TSai, Pan, Liao, & Chu, 2009) an

improved version with a novel operator based on the Newtonian Law of Universal

Gravitation was proposed to improve the exploration ability of the ABC. The ABC

algorithm is based in the interaction of three types of bees: employed, onlooker and scout

bees. The number of employed bees usually is the same to the number of onlooker bees.

In contrast to other Evolutionary Algorithms (EAs) and swarm intelligence algorithms, the

solutions in the ABC are represented by the food sources. The bees act as operators over

the food sources trying to find the best one among them. In ABC, the number of employed

and onlooker bees is the same, and this number is equal to the number of food sources.

An employed bee is assigned to one of the sources. Upon reaching the source, the bee will

calculate a new solution i.e., fly to another nearby food source from it and retain the best

solution (in a greedy selection). The onlooker bees are allocated to a food source based on

their profitability. These onlooker bees use the same operator of the employed bee to

generate a new food source. When a source does not improve after a certain number of

iterations, it is abandoned and replaced by those found by a scout bee, which involves

calculating a new solution at random. Three parameters must be defined by the user: the

number of solutions (food sources) “SN”, the total number of cycles (iterations) of the

algorithm “MCN” and the number of cycles that a non-improved solution will be kept

before being replaced by a new solution generated by the scout bee mechanism “limit”.

The selection process takes place when the employed bees share the information of their

food sources in the hive by waggle dances, emulated as a fitness proportional selection.

Two replacement processes are required in ABC:(1) in the greedy selection between the

current food source and the new one generated either by an employed or an onlooker

bee and (2) when an employed bee abandon a food source which could not be improved

within “limit” cycles and a scout bee generates a new one by using a random process. The

original operator used by both, employed and onlooker bees, to generate a new solution 2�J is as follows:

2�,KJ = ��,KJ + M�,K ∙ $��,KJ − �P,KJ % Equation 14

where ��J represents the solution in which the bee is located at that moment, �PJ is a

randomly chosen food source (and different from ��J ), / = 91, ⋯ , RS:,j = 91, ⋯ , T: and M�,K is a random real number within [-1, 1] generated at random for every / = 91, ⋯ , RS:

and every ) = 91, ⋯ , T:.

However, for the ABC version proposed in (Karaboga & Basturk, Artificial bee colony(abc)

optimization algorithm for solving constrained optimization problems, 2007) for

32

constrained real-parameter optimization a modified version was employed and a new

parameter 0 ≤ MR ≤ 1 was considered.

The detailed pseudocode of the original ABC algorithm is presented in Figure 7.

Figure 7 ABC Algorithm

Based on preliminary observations to the original ABC modified to solve constrained real-

parameter optimization problems, it was noticed that the algorithm converges faster to a

promising region of the search space, causing, sometimes, premature convergence.

Therefore, in order to improve its performance in constrained numerical search spaces,

five modifications were proposed. The expected effect of all changes is to slow-down

convergence by slowing-down the replacement process of the food sources generated by

the onlooker bees, modifying the operator of the employed bee and giving the scout bee

more chances to generate good solutions in the neighborhood of the best solution so far.

33

Finally, a dynamic mechanism was added to the tolerance for equality constraints with the

intention to facilitate the algorithm to approach the tiny feasible region in this type of

problems and a local search mechanism was considered to improve those best solutions

found. Attending previous comments, E-ABC algorithm, presented in (Mezura-Montes &

Velez-Koeppel, Elitist Artificial Bee Colony for Constrained Real-Parameter Optimization,

2010) was developed.

1.3.4 Kinematic Model

A Kinematic Model is able to determine the position of all elements into a mechanism

based in the position of an element where the power input acts in the crank-rocker slider

CVT (Willis, 2006). To transform the behaviour of a mechanism into equations becomes in

a powerful method that allows:

• Evaluate critical points for the mechanism

• Perform an analysis for mechanical efficiency

• Suggest modifications aiming to improve mechanism

• Evaluate dimensional considerations

Some works deals about the kinematic optimization of mechanical systems. In Mundo

(D.Mundo & Yan, 2007) a method for the kinematic optimization of transmission

mechanism is proposed. There, a motion control of a ball-screw transmission mechanism

is developed. The kinematic characteristics of the ball-screw mechanism are analyzed by

means of non-dimensional motion equations in order to formulate an optimization

problem. A genetic algorithm (GA) is implemented in order to optimize the objective

function and a penalty method is used to fulfill the design rules. Mermetas V. (Mermetas,

2004) presents an optimal kinematic design of planar manipulator with four-bar

mechanism. There, optimum link measurements of the manipulator that maximize the

local mobility index depending on the input link location are founded. Also, design charts

for the optimum manipulator design are obtained. In (Nariman-Zadeh, Felezi, Jamali, &

Ganji, Pareto optimal synthesis of four-bar for path generation, 2009) a hybrid multi-

objective genetic algorithm (GA) is used for Pareto optimum synthesis of four-bar linkages

considering the minimization of two objective functions simultaneously. The obtained

Pareto fronts demonstrate that trade-offs between these two objectives can be

recognized so that a designer can optimally compromise for the selection of a desired

four-bar linkage.

34

1.3.4.1 Freudenstein’s equations

Ferdinand Freudenstein is widely acknowledged to be the father of modern kinematics of

mechanisms and machines. The Equation 15 is known as the Freudenstein Equation and is

readily applicable to kinematics analysis of four-bar mechanisms. From known links

lengths and the input angle U, the output angle V can be found. Using the well-known

tangent half-angle trigonometric formulas for sine and cosine of angle V, it is possible to

show that there are two possible V's for a given angle U (Ghosal, 2010).

W� ∙ +�& U� − W� ∙ +�& V� + WX = cos U − V� Equation 15

35

2 PROJECT’S METHODOLOGY

Each one of the subsequent subchapters is used to describe the methodology in its

corresponding stage of the project.

2.1 BIBLIOGRAPHIC REVISION

An exhaustive search of different applications for metaheuristics and CVTs as well as

development of mentioned items has to be realized. In the case of metaheuristics, a

considerable amount of information related with their applications, efficiency, accuracy,

applications will be collected. A wide set of information related with mechanical

modelling, multi-objective optimization, constraint-handling and other fundamental topics

has to be read, aiming to obtaining a foundation for the development of the project.

Together with bibliographic revision, continuous meetings with experts in these topics

have to be scheduled aiming to keep focused in the right development of the model as

well as the techniques to optimize it.

2.2 KINEMATIC MODELLING

Due to the importance of the development of a kinematic model which can reach the

proposed objectives, this stage was achieved through a detailed construction of models

for different mechanisms. Although complete system may be novel and complex, this can

be separated in some simplest and more known mechanisms. Based on previous

bibliographic revision, each one of the required models had to be built and verified.

All models were recreated with a programming language, with the goal that they could be

used and modified in an efficient manner each time that something was needed from

them.

During this process, several variables had to be identified, aiming to describe the system in

terms only of significant variables. This step requires a wide knowledge of the mechanism,

then, the accompaniment of PhD. Edgar-Alfredo Portilla-Flores is actually determinant for

a well achievement of this stage and therefore subsequent stages.

36

2.3 STATEMENT OF THE PROBLEM

Once kinematic model was completed, the EOP has to be stated. This problem belongs to

multi-objective optimization branch, making it be solved through different techniques that

used for solving Mono-Objective Optimization problems.

The development of objective functions present in the problem aims to optimize the

performance of the system together with its efficiency. Those functions have to work as

the main indicators of the crank-rocker-slider CVT.

A set of constraints has to be designed to state working conditions for the crank-rocker-

slider CVT. With constraints some of the aims are:

• Achieve symmetric displacement in the mechanism.

• Avoid possible failures in its operation.

• Determine the dimensions of the system.

• Ensure a minimum efficiency level.

2.4 OPTIMIZATION BASED ON METAHEURISTICS

A revision of different techniques for constraint-handling, selection of solutions and other

related topics has to be performed with the intention to achieve a good assembly of

techniques (specially developed for multi-objective optimization) for solving the problem.

The support of PhD. Efren Mezura-Montes becomes crucial because some special

characteristics inherent to the problem may turn it harder to solve.

The problem tackled in this work is solved by using different methods; however a

reference point has to be taken. Ten independent runs were executed by each one of the

algorithms proposed. As suggested in the specialized literature, the final Pareto’s Front

belonging to each technique consists in a filtered front extracted the front obtained in

each independent run computed.

Looking for a fair comparison of the performance of each algorithm used in the

optimization process, each one of the used metaheuristic algorithms executes an equal

number of evaluations of the objective functions. The total number of executed

evaluations was 1,000,000 with the aim to give each algorithm enough time to excel. The

selection criteria are the same for each algorithm.

The features of the computer platform used are a 2.8 GHz Core 2 Duo processor, 3 GB of

RAM, under Windows ® XP with Matlab®7.9.0 R2009b.

37

2.4.1 PROPOSED APPROACHES

As it was pointed out before, metaheuristic acts as a black-box to solving problems, which

does not require any special feature related with the problem to be solved. This behaviour

can be translated as the metaheuristic does not need linearity in both constraints and the

objective functions. In the same way, the objective functions do not have to be continuous

and differentiable along the search space. These and other special advantages turn a

metaheuristic as a strong candidate to solve the design problem proposed.

In order to solve the MOP previously proposed, a set of 4 different metaheuristics have

been adapted, based on separated principles among them, looking for obtaining a robust

orientation in the solution of the problem.

2.4.1.1 Selection Criterion

All the algorithms use a common selection criterion. This is structured by two stages:

In the first stage, the algorithm aims to explore the search space by generating vectors

from the current generation to improve a set of non-dominated vectors. In the first level,

the selection criteria can be expressed as:

• Between two feasible solutions, the non-dominated solution is chosen.

• If one solution is feasible and the other one is infeasible, the feasible one is

chosen.

• Between two infeasible solutions, the one with the lower summation of constraint

violation is chosen.

Pareto dominance was the criterion utilized to select between two feasible solutions. A

vector of objectives M�� ; � = GM�, M�, ⋯ , M�H� dominates in accordance with Pareto, a

vector M′�� ; � = GM′�, M′�, ⋯ , M′�H�, if and only if M�� ; � is partially less than M′�� ; �,

that is expressed as M�� ; � ≼ M′�� ; �. At the end of each generation, the feasible and

non-dominated solutions are extracted out to an external file. This file contains the Pareto

optimal set, and its members constitute the so-called Pareto front.

In the second stage, the aim is to keep the algorithm focusing in previously found non-

dominated solutions, where the solutions chosen will be those non-dominated vectors in

scarcely explored regions of the Pareto front, based in crowding distance. To ensure that,

those regions with less solutions found will be more visited aiming to find new solutions

able to add non-dominated members in empty regions. Non-dominated solutions are

saved in an external file. This file is updated each generation with the set of feasible and

non-dominated solutions found.

38

Both stages depend of “NS” factor, a relation between the current evaluation or

generation and the total number of evaluations or generations.

2.4.1.2 Differential Evolution (DE)

2.4.1.2.1 Multi-Objective DE/rand/1/bin

As one of the more explored metaheuristics, DE has showed a remarkable performance in

solving optimization problems. Based in a randomly generated initial population of size

SN, the algorithm generates a new solution based on each current solution by selecting

three different members of the population (W� ≠ W� ≠ WX), by means of a mutation

operator. The new solution for each current solution (called target vector) is known as a

trial solution, from where the trial population is generated. A remarkable modification of

the original algorithm was the introduction of parameter NS (which value is in the interval

[0, 1]) that states if the algorithm jumps from the first to the second stage of the selection

criteria. While the number of the current generation or evaluation divided by the total

number of generations or evaluations respectively is less than NS, the first stage of the

selection criterion have to be used, else, the second selection criterion is stated. The

mutation operator can be written as:

2�,KJ = � _,KJ + D ∙ `� a,KJ − � b,KJ c Equation 16

Where F is a random number in the interval [0, 1] described in 1.3.3.5, different for each

individual in the population, i.e., is not a parameter tuned by user, it is modified

programmatically for each member of the population.

Once mutation is completed, the crossover operator is applied (binomial crossover is used

in this case) and it works as follows: between two mutant individuals, with the same

probability, each one of the variables of �� is swapped into a new individual called trial

vector (also called offspring).

The crossover operator aims to construct an offspring by mixing components of the target

vector and that generated by the mutation operator (Zaharie, 2007). In this case, the

parameter “CR” mentioned in 1.3.3.5 has a value of 0, i.e. the whole set of members of

the current population will be crossed. The aim of this total crossover is to achieve a

population with a huge variability and to generate non-dominated solutions to be

extracted in the external file.

When mutation and crossover are completed, selection criterion takes place, as said in

subchapter 2.4.1.1, is composed by two stages. The complete pseudo-code for MO

DE/rand/1/bin is shown in Figure 8.

39

This variant of DE/rand/1/bin uses the same operators along the whole process,

regardless of NS that affects only the selection criteria to develop the stages mentioned

lines above.

Figure 8 Multi-Objective DE/rand/1/bin algorithm

Parameters used for this algorithm are presented in Table 1. Just with 2 parameters, this

kind of algorithms can be easily tuned.

Table 1 Parameters for Multi-Objective DE/rand/1/bin

NS 0.8

40

SN 200

CR 0

2.4.1.2.2 Multi-Objective DE/rand/1/exp

In a similar way that DE/rand/1/bin, the DE/rand/1/exp uses the same mutation and

selection operators, but crossover operator has an important modification in the

crossover operator, mentioned in Error! Reference source not found. .

The complete pseudo-code for DE/rand/1/exp for MOP is presented in Figure 9

Figure 9 Multi-Objective DE/rand/1/exp algorithm

41

In Table 2 the parameters used to tune Multi-Objective DE/rand/1/exp are shown. Are the

same used in Multi-Objective DE/rand/1/bin.

Table 2 Parameters for Multi-Objective DE/rand/1/exp

NS 0.8

SN 200

CR 0

2.4.1.2.3 Multi-Objective DE/best/1/exp

So close to DE/rand/1/exp, in DE/best/1/exp the target vector for the mutation operator

is the best solution achieved so far, aiming to generate a more diverse set of solutions

near the best one so far. Finally, this best solution �d,KJ is always the same for all mutant

vectors into a generation i.e., it is replaced until the end of each cycle in the algorithm. �d,KJ is the solution with smaller constraint violation and bigger crowding distance in the

current population.

The mutation operator based in the best solution can be expressed as:

2�,KJ = �d,KJ + D ∙ `� a,KJ − � b,KJ c Equation 17

In this approach, Equation 17 acts only if the relation between the current generation and

the total number of generations is greater than “NS”, in other case, the mutation operator

is the same that described in the section related with DE/ran/1/bin.

The complete pseudo-code for DE/best/1/exp for MOP is exposed in Figure 10

42

Figure 10 Multi-Objective DE/best/1/exp algorithm

Parameters used for this algorithm are presented in Table 3. As can be seen, whole set of

DE variants uses the same tune parameters.

43

Table 3 Parameters for Multi-Objective DE/best/1/exp

NS 0.8

SN 200

CR 0

2.4.1.3 Swarm Intelligence (SI)

2.4.1.3.1 Multi-Objective Elitist Artificial Bee Colony (ME-ABC)

This algorithm, proposed originally in (Mezura-Montes & Velez-Koeppel, Elitist Artificial

Bee Colony for Constrained Real-Parameter Optimization, 2010), was modified to solve

MOP. To the best of the authors’ knowledge, this is one of the first attempts to solve a

multi-objective CVT design with a bee-based algorithm. This proposed approach uses the

parameter NS (described in previous sections) that states if the algorithm jumps from the

first to the second stage of the selection criterion.

ME-ABC algorithm uses three different kinds of bees to improve the food sources, where

the solutions are located. An initial population of size SN is generated randomly. From this

initial population a set of operators is used looking for to improve the solution. The first

operator is the Employed Bee, whose task is the exploitation of known food sources. The

current employed bee is modified by using:

2�,KJ = ��,KJ + 2 ∙ M� ∙ $�P,KJ − ��,KJ % Equation 18

where 2�J is the new food source generated and �PJ is a randomly chosen food source

taken from the current population. The M� value is within the range [0,1] and it is

multiplied by two in order to increase the step-size of the operator (useful in wider search

spaces). This M� value is kept fixed for every) ∈ G1,2, ⋯ , TH. This operator works in the

same way along all evaluations.

Onlooker Bees are oriented to the exploration of food sources. When the relation

between the current evaluation number and the total number of evaluations is less than

NS, the operator used by Onlooker Bees is very similar that described previously, with a

variant in M:

2�,KJ = ��,KJ + 2 ∙ M�,K ∙ $�P,KJ − ��,KJ % Equation 19

44

When the aforementioned relation between current evaluation number and total number

of evaluations is greater than NS, a new location for Onlooker Bees is found as follows:

2�,KJ = �d,KJ + 2 ∙ M�,K ∙ $�P,KJ − �d,KJ % Equation 20

Unlike the modified employed bee operator, the onlooker bee operator generates a

different M�,K value, within [0, 1], for each / ∈ G1,2, ⋯ , RSH and ) ∈ G1,2, ⋯ , TH aiming to

generate a more diverse set of solutions near the best one so far. Finally, this best solution �d,KJ is always the same for whole onlooker bees i.e., it is replaced until the end of the

current cycle in the algorithm.

If a solution is not improved in a number of cycles called limit, this solution that belongs to

the population is replaced, but not in random way, by using the Equation 21:

2�,KJ = ��,KJ + M� ∙ $�P,KJ − ��,KJ % + 1 − M�� ∙ $�d,KJ − ��,KJ % Equation 21

The aim of this modified operator is to increase the capabilities of the algorithm to sample

solutions within the range of search defined by the current population.

This modified scout mechanism, based on an original proposal utilized in PSO (Lu & Chen,

2008) is employed in Elitist-ABC. Instead of generating a random food source (as in the

original ABC), the scout bee will generate a new food source2�,KJ by using the food source

subject to be replaced ��,KJ as a base to generate a new search direction biased by the best

food source so far �d,KJ and a randomly chosen food source�P,KJ

, as indicated in Equation

21, where the M� value is generated and fixed per each food source / ∈ G1,2, ⋯ , RSH.

Additionally to nature-inspired operators, two local search processes have been adapted.

The first local search process works when 30\%, 40\%, 50\%, 60\%, 70\%, 80\%, 90\%,

95\% and 97\% of total evaluations (Fes) have been reached, and its purpose is to improve

the best solution achieved so far by generating a set of 1000 new food sources in its

neighbourhood. Each one of the new solutions 2�,KJ is generated as shown in Equation 22.

2�,KJ = ��,KJ + M� ∙ WeTfG0,1H ∙ g� − h�� ∙ i 0.10.5 ∙ lm Equation 22

This local search process generates a different M� value, within [-1, 1], which is fixed for

each ) ∈ G1,2, ⋯ , TH. T is a counter, beginning at 1, increases in 1 each time this local

search works.

45

The other local search process consists in a slightly modification of each decision variable

present in �d,KJ, ) ∈ G1,2, ⋯ , TH of the best solution so far, by using a small step-size (1xE-5

times the size of the variable interval g� − h��) until either there is no improvement or

100 cycles had been computed per each variable. This local search works when 45\%,

50\%, 55\%, 80\%, 82\%, 84\%, 86\%, 88\%, 90\%, 91\%, 92\%, 93\%, 94\%, 95\%, 96\%,

97\%, 98\%, and 99\% of FEs have been reached.

Once Employed and Onlooker Bees have been generated and local search completed, the

selection operator takes place.

In the next figure (Figure 11), the complete pseudo-code of ME-ABC used in the MOP

problem is shown.

46

Figure 11 ME-ABC algorithm

47

In order to satisfy equality constraints a dynamic tolerance was introduced. The tolerance

is calculated as:

n = o 1 /� = 1*piq r�∙stuqv m /� 1 < < R0.0001 /� ≥ R - Equation 23

where E(t) is the current number of evaluations performed by ME-ABC, Ef is the total

number of evaluations to be computed and S is the evaluation number when the user

wants n becomes to 0.0001. S has to be lower or equal than Ef. dec, which controls the

speed to reduce n is given by Equation 24, where 9.21034 is the exponent used to get the

tolerance value of 1xE-4 i.e., *px.��yXz = 1 ∙ {pz:

f*+ = 9.21034 ∙ {�R Equation 24

The set of parameter used to find previous Pareto’s Front with ME-ABC is presented in

next table:

Table 4 Parameters for ME-ABC

NS 0.9

SN 200

ε 1

S 10% of Fes

FEs ratio to perform local

search

30%, 40%, 45%, 50%, 55%,

60%, 70%, 80%, 82%, 84%,

86%, 88%, 90%, 91%, 92%,

93%, 94%, 95%, 96%, 97%,

98%, 99%

Cycle limit for local search 100

Step-size variation for each

variable in local search

1E-5(Ui-LI)

3 CRANK-ROCKER-SLIDER CVT SYSTEM OP

3.1 KINEMATIC DESIGN OPT

In essence, the crank

transformation: in the first stage

transformed into a linear movement in the slider. In the second stage, the linear

movement from the slider

a chain and pawl system.

Let’s analyze in a back forward way

moved by a gear. The gear is moved by a chain, as can be seen in

configuration, the angular velocity in the gear can be calculated as:

Where �Jt�^, the angular velocity in the gear, in consequence, in the shaft too.

tangential velocity of the chain, located in gear’s pitch diame

of the gear.

The chain receives its movement from a pawl system. Assembled to the slider, the double

pawl mechanism describes a linear displacement in 2 w

this special type of movement, each one of the pawls is located 180° from another; with

this configuration, the chain is moved in the same way independent from the moment in

the slider, i.e. the movement in the gear and sha

time. Figure 13 shows how double pawl mechanism works.

48

SLIDER CVT SYSTEM OPTIMIZATION

KINEMATIC DESIGN OPTIMIZATION

In essence, the crank-rocker-slider CVT develops along 2 stage

transformation: in the first stage, the rotational movement from the power input is


from the slider is converted in rotational movement at the

system.

in a back forward way the crank-rocker-slider CVT system.

gear. The gear is moved by a chain, as can be seen in Figure

configuration, the angular velocity in the gear can be calculated as:

�Jt�^ = �u��WJt�^

, the angular velocity in the gear, in consequence, in the shaft too.

tangential velocity of the chain, located in gear’s pitch diameter. Finally,

Figure 12 Chain and gear mechanism


pawl mechanism describes a linear displacement in 2 ways. Aiming to take advantage of



the slider, i.e. the movement in the gear and shaft is clockwise or counter

shows how double pawl mechanism works.

g 2 stages of movement

, the rotational movement from the power input is


output shaft trough

system. The output shaft is

Figure 12. With this

Equation 25

, the angular velocity in the gear, in consequence, in the shaft too. �u��, the

ter. Finally, WJt�^, the radius


ays. Aiming to take advantage of



ft is clockwise or counter-clockwise all the

49

Figure 13 Double pawl mechanism

Movement in the slider is caused by W�, a bar that connects the rocker and the slider

assembled to the pawl system. In Figure 14 a simple diagram shows how the rocker gives

movement to the slider.

Figure 14 Slider and pawl propelled by a rocker

The position of the slider can be calculated with cosine’s law as follows:

W� = 2 ∙ W� ∙ +�& Uz� ± �4 ∙ W�� ∙ cos� Uz� − 4 ∙ W�� − W��2 Equation 26

Where W� is contained into Wz , and can be controlled at each instant, is a parameter not a

variable. W� , a known size bar coupling W�and the slider with pawl, is a parameter too, and

finally, Uz, the angle among W� or Wz and the horizontal axis, the only one variable.

The slider’s speed, and in consequence, the chain tangential velocity can be calculated as

the change in the position for small time intervals, let’s say:

50

�� = W� � − W� − 1� Equation 27

From equations shown before, can be noticed that the only variable to calculate the

velocity is Uz, i.e. by optimizing Uz, the output of the system will be optimized, it implies

that a kinematic model that analyzes Uz is enough to optimize the kinematic design, and

other parameters are not determinant in the solution. Based in this, the kinematic model

for the four-bar mechanism (Revolute-Revolute-Revolute-Revolute or RRRR mechanism

integrated by W�, W�, WX and Wz ) shown in Figure 15 will be developed in next chapter.

Figure 15 Basic crank-rocker-slider CVT basic diagram

3.2 KINEMATIC MODEL OF THE RRRR MECHANISM

As shown in Figure 16, this RRRR or Four-bar Mechanism is composed by a reference bar

(r1), a crank bar (r2), a coupling bar (r3) and a rocker bar (r4). A set of four important

angles have to be considered, beginning with θ1, described by the angle between

horizontal axis and r1, θ2 is the angle between horizontal axis and r2, θ3 is the angle

between horizontal axis and r3 and θ4 is the angle between horizontal axis and r4. The

angle marked as μ is the transmission angle. This special configuration is called crank-rocker.

Figure 16 Four-bar Mechanism Diagram

51

In this particular RRRR mechanism, the power input is coupled in D, and the output can be

located in any point along r4, in other words, between B and C. For this special case, the

mechanism's output is located at B. Points A and B have mechanical constraints, reason

which their movement are a circumference and an arc respectively, giving origin these to

crank-rocker's name. Due to the displacement of A and B, their tangential speed is

calculated as shown in next two equations.

�� = �� ∙ W� Equation 28

�d = �d ∙ Wz Equation 29

where ωin and ωB are the angular velocities of A and B respectively.

As can be seen, the coupling bar r3 describes a planar movement, and obeying to rigid

body kinematics, the movement equation for B is as follows:

�d�� = �� + ��d�� × ��d�� Equation 30

Where

�d�� = −�d ∙ sin Uz� � +�d ∙ cos Uz� � Equation 31

�� = −�� ∙ sin U�� +�� ∙ cos U�� Equation 32

�� = WX ∙ cos UX�� + &/T UX�� Equation 33

��d�� = −��d�� Equation 34

with �, � and �� as unit vectors in the positive directions of x, y and z axis respectively.

By replacing the last four equations in the equation that denotes �d�� , a simultaneous

equations system is constructed as shown in the lines below:

52

�−�d ∙ &/T Uz� = ��d ∙ WX ∙ &/T UX� − �� ∙ &/T U�� d ∙ +�& Uz� = �� ∙ +�& U�� − ��d ∙ WX ∙ +�& UX� � Equation 35

From the system of equations presented in lines above, the tangential speed in B is given

by:

�d = �&/T U�� − +�& U�� ∙ eT UX�&/T Uz� − +�& Uz� ∙ eT UX�� ∙ �� Equation 36

By replacing the first two equations where �� and �d are defined in Equation 36, where �d

is defined, the angular velocity in the rocker (ωB), in terms of the angular velocity at the

input (ωB) and a set of angles (θ2, θ3 and θ4), is stated by the following equation:

�d = iW�Wzm ∙ �&/T U�� − +�& U�� ∙ eT UX�&/T Uz� − +�& Uz� ∙ eT UX�� ∙ �� Equation 37

Looking for θ3 and θ4, which measurements indicate both angular positions, a closed

circuit equation has to be stated. In Figure 17, from the vector representation, the next

two equations can be extracted:

Figure 17 Closed Circuit Diagram

�� = W�� + WX�� Equation 38

53

�� = W�� + Wz�� Equation 39

By making equal both previous equations, the closed circuit equation is finally stated as

follows:

W�� + WX�� = W�� + Wz�� Equation 40

Where each one of the terms in the equation can be represented in polar coordinates as:

W�� = W� ∙ *�∙�_ = $W� ∙ cos U��%� + $W� ∙ &/T U��%� Equation 41

W�� = W� ∙ *�∙�a = $W� ∙ cos U��%� + $W� ∙ &/T U��%� Equation 42

WX�� = WX ∙ *�∙�b = $WX ∙ cos UX�%� + $WX ∙ &/T U��%� Equation 43

Wz�� = Wz ∙ *�∙�� = $Wz ∙ cos Uz�%� + $Wz ∙ &/T Uz�%� Equation 44

For θ3 and θ4, their correspondent Freundenstein's equations from the closed circuit

equation presented before are:

� ∙ +�& UX� + � ∙ &/T UX� + I = 0 Equation 45

� ∙ +�& Uz� + { ∙ &/T Uz� + D = 0 Equation 46

Coefficients denoted by A, B, C, D, E and F can be found through closed circuit equation.

Their correspondent equations are:

� = 2 ∙ WX ∙ $W� ∙ +�& U�� − W� ∙ +�& U��% Equation 47

� = 2 ∙ WX ∙ $W� ∙ &/T U�� − W� ∙ &/T U��% Equation 48

I = W�� + W�� + + X� − Wz� − 2 ∙ W� ∙ W� ∙ +�& U� − U�� Equation 49

54

� = 2 ∙ Wz ∙ $W� ∙ +�& U�� − W� ∙ +�& U��% Equation 50

{ = 2 ∙ Wz ∙ $W� ∙ &/T U�� − W� ∙ &/T U��% Equation 51

D = W�� + W�� + Wz� − WX� − 2 ∙ W� ∙ W� ∙ +�& U� − U�� Equation 52

For a given angle ϒ, its trigonometric functions sin(ϒ) and cos(ϒ) can be expressed as:

&/T �� = 2 ∙ eT `�2c1 + eT� `�2c Equation 53

+�& �� = 1 − eT� `�2c1 + eT� `�2c Equation 54

By applying previous two equations to both initial Freudenstein’s equations, the angular

positions for θ3 and θ4 are given by:

UX = 2 ∙ eW+eT �−� + √�� + �� − I�I − � ¡ Equation 55

Uz = 2 ∙ eW+eT �−{ + √{� + �� − D�D − � ¡ Equation 56

Considering that the input power is provided by an electric motor, the angular velocity at

the input can be assumed as constant, reason why the angular position for θ2 is given by:

55

U� = �� ∙ Equation 57

where t is time in consistent units with ωin.

3.2.1 Tightening Points

Along a whole cycle of r2, the RRRR mechanism has two important stages, where r2 and r3

are collinear. In both, one degree of freedom (DoF) (Tana, Xib, & Wangc, 2004) is drop and

match up with the distant points in B's trajectory, which implies than the maximum and

minimum angles of the output θ4 max and θ4 min becomes present, as can be seen in Figure

18 and Figure 19 respectively.

Figure 18 Maximum output angle, a tightening point for the mechanism

Figure 19 Minimum output angle, other tightening point for the mechanism

A detailed observation of RRRR mechanism at both stages reveals that not only the

maximum and minimum angles of the output (θ4 max and θ4 min) are present. It also can be

noticed that the transmission angle (μ) has a behaviour in the same way that output angle

(θ4), i.e. when θ4 increases, μ increases too, and vice versa.

From the tightening points (Figure 18 and Figure 19), θ4 max and θ4 min can be calculated as

follows:

56

For θ1 < 0

Uz ��¢ = £ − �|U�| + eW++�& �W�� + Wz� − WX − W��2 ∙ W� ∙ Wz ¡� Equation 58

Uz �� = £ − �|U�| + eW++�& �W�� + Wz� − WX + W��2 ∙ W� ∙ Wz ¡� Equation 59

For θ1 = 0

Uz ��¢ = £ − eW++�& �W�� + Wz� − WX − W��2 ∙ W� ∙ Wz ¡ Equation 60

Uz �� = £ − eW++�& �W�� + Wz� − WX + W��2 ∙ W� ∙ Wz ¡ Equation 61

For θ1 > 0

Uz ��¢ = £ + �|U�| + eW+ + �& �W�� + Wz� − WX − W��2 ∙ W� ∙ Wz ¡� Equation 62

Uz �� = £ + �|U�| + eW++�& �W�� + Wz� − WX + W��2 ∙ W� ∙ Wz ¡� Equation 63

With the output located at B, the longer the trajectory of this point the higher the output

level, i.e. the maximization of θ4 implies the maximization of the output in the RRRR

mechanism.

3.2.2 Operation Quality

As it was said before, transmission angle (μ) becomes helpful to identify, within a set of

solutions for the RRRR mechanism, the most efficient movement transmission. This angle

is usually defined as the absolute value of the difference between the angles among the

coupling and rocker bar with the horizontal axis. Once the linkage completes a whole

57

cycle, the linkage describes an angle that changes along time. The transmission angle can

be calculates as:

¤ ��; � = |UX − Uz| Equation 64

Due to power to move the rocker comes from the coupling bar to the rocker bar, the angle

between both has to stay near to £ 2¥ rad as most time as possible, looking for to take

advantage of the energy provided by the power input. When the angle between rocker

and coupling bar is £ 2¥ rad, the rotational momentum of r4 respect to C becomes in a

maximum.

Values of μ above and below £ 2¥ rad can be accepted, as long as they stay close, because

a value of μ extremely smaller or bigger and any imperfection, such as additional friction,

can block the mechanism. To avoid these problems, μmax as well as μmin must be located as

near as possible to £ 2¥ rad. The maximum and minimum angles for μ are present when

the crank is collinear with the reference bar, as shown in Figure 20 and Figure 21.

Figure 20 Maximum Transmission Angle

Figure 21 Minimum Transmission Angle

58

Derived from the diagrams in Figure 20 and Figure 21, the angles μmax and μmin can be

calculated as:

¤��¢ = eW++�& �WX � + Wz� − W� − W��2 ∙ WX ∙ Wz ¡ Equation 65

¤�� = eW++�& �WX � + Wz� − W� + W��2 ∙ WX ∙ Wz ¡ Equation 66

3.3 MULTI-OBJECTIVE OPTIMIZATION PROBLEM (MOP)

3.3.1 MOP Definition

The design problem of the four-bars mechanism can be stated as follows:

� M�� ; � = GM�, M�, ⋯ , M�H� Equation 67

M�� = � ��; � Equation 68

�� = G��, ��, ⋯ , ��H� Equation 69

Subject to:

�P �� ≤ 0 Equation 70

�¦ ��; � ≤ 0 Equation 71

ℎ§ �� = 0 Equation 72

59

where φi are each one of the objective functions to optimize, �� ∈ �� is the vector of

solutions �� = G��, ��, ⋯ , ��H� and each xj, ) = 1, ⋯ , T is bounded by lower and upper

limits hK ≤ �K ≤ gK which define the search space S, �P �� with � = 1, ⋯ , ¨ is the set of

static inequality constraints, �¦ ��; � with © = 1, ⋯ , W is the set of dynamic inequality

constraints, ℎ§ �� with = 1, ⋯ , & is the set of static equality constraints.

3.3.2 CVT Optimization Problem definition

3.3.2.1 Objective Functions

As mentioned in sections related with tightening points and operation quality, the crank-

rocker-slider CVT design problem needs to satisfy two objectives. The first objective

consists in the maximization of the output, aiming to obtain the maximum speed from the

transmission system. This objective function is stated as follows:

M� �� = max ∆Uz�� Equation 73

Where ∆Uz can be calculated as:

∆Uz = Uz®¯° − Uz®±² Equation 74

The second objective is to take advantage of the energy provided by the power input,

keeping the transmission angle μ as close as possible to £ 2¥ rad.

For mentioned purpose, the next equation is stated:

M� �� = min i`¤��¢ − £2c� + `¤�� − £2c�m Equation 75

3.3.2.2 Constraint Selection

3.3.2.2.1 System Size

The maximum length that can achieve each one of the bars in the RRRR mechanism has to

be constrained, obeying to the space required by the system. The size of each bar is

limited to be between 0.05 m and 0.50 m and the angle between horizontal axis and the

reference bar is limited between £ 4¥ and −£ 4¥ rad.

Based in the lines written before, the constraints related with the size of the system are:

60

�� = 0.05 1 − W� ≤ 0 Equation 76

�� = W� − 0.5 1 ≤ 0 Equation 77

�X �� = 0.05 1 − W� ≤ 0 Equation 78

�z �� = W� − 0.5 1 ≤ 0 Equation 79

�� = 0.05 1 − WX ≤ 0 Equation 80

�³ �� = WX − 0.5 1 ≤ 0 Equation 81

�´ �� = 0.05 1 − Wz ≤ 0 Equation 82

�µ �� = Wz − 0.5 1 ≤ 0 Equation 83

�x �� = − £4 Wef − U� ≤ 0 Equation 84

+ �y �� = U� − £4 Wef ≤ 0 Equation 85

3.3.2.2.2 Grashof’s Law

In a four-bar linkage which describes a planar displacement, the summation of shortest

and largest bars has to be less or equal that the summation of other bars if a relative

rotational displacement between two elements is required, is the affirmation of Grashof's

61

law. Denoting as s and l the shortest and largest bars respectively, and as p and q the

other two bars in a RRRR mechanism, Grashof's law can be expressed as:

& + © ≤ + ¨ Equation 86

With input and output located at D and B respectively, the statement for Grashof's Law

can be written as follows:

�� = W� + WX − W� + Wz� ≤ 0 Equation 87

In addition, to ensure that proposed solutions obey Grashof's Law for a crank-rocker

mechanism two more constraints have to be accomplished:

�� = W� − WX ≤ 0 Equation 88

��X �� = Wz − WX ≤ 0 Equation 89

Once previous constraints have been satisfied, the result is a crank-rocker mechanism

with the input coupled in D and output in B.

3.3.2.2.3 Transmission Angle

Looking for a transmission angle able to avoid failures related with the locking of the

system, a dynamic inequality constraint that has to be satisfied the whole cycle is

determined. With a transmission angle greater or equal than £ 4¥ rad at each moment of

the cycle, this kind of failures are avoided. The dynamic inequality constraint is stated as:

�� , � = £4 Wef − ¤ ��; � ≤ 0 Equation 90

3.3.2.2.4 Rocker’s displacement symmetry

With an output which behaviour is symmetric, two important advantages have to be

noticed. The first one consists in an equal momentum required at each one of the

tightening points. The other one is represented in an equal displacement through both

axes in a planar movement. To ensure this equality constraint, the next equation is stated:

ℎ� �� = £ Wef − Uz®¯° − Uz®±² = 0 Equation 91

62

3.3.2.3 Statement of the Problem

Let be the design variables vector ��:

�� = GW�, W�, WX, Wz, U�H� = G��, ��, �X, �z, ��H� Equation 92

The MOP is stated as follows:

� M�� ; � = GM�, M�H� �� ∈ �� Equation 93

With

M� �� = max ∆Uz�� Equation 94

M� �� = min i`¤��¢ − £2c� + `¤�� − £2c�m Equation 95

Subject to:

�� = 0.05 1 − W� ≤ 0 Equation 96

�� = W� − 0.5 1 ≤ 0 Equation 97

�X �� = 0.05 1 − W� ≤ 0 Equation 98

�z �� = W� − 0.5 1 ≤ 0 Equation 99

�� = 0.05 1 − WX ≤ 0 Equation 100

�³ �� = WX − 0.5 1 ≤ 0 Equation 101

63

�´ �� = 0.05 1 − Wz ≤ 0 Equation 102

�µ �� = Wz − 0.5 1 ≤ 0 Equation 103

�x �� = − £4 Wef − U� ≤ 0 Equation 104

��y �� = U� − £4 Wef ≤ 0 Equation 105

�� = W� + WX − W� + Wz� ≤ 0 Equation 106

�� = W� − WX ≤ 0 Equation 107

��X �� = Wz − WX ≤ 0 Equation 108

�� , � = £4 Wef − ¤ ��; � ≤ 0 Equation 109

ℎ� �� = £ Wef − Uz®¯° − Uz®±² = 0 Equation 110

where φi are each one of the objective functions to optimize, �� ∈ �� is the vector of

solutions �� = G��, ��, �X, �z, ��H� and each xj, ) = 1, ⋯ ,5 is bounded by lower and upper

limits hK ≤ �K ≤ gK which define the search space S, �P �� with � = 1, ⋯ , 13 is the set of

static inequality constraints, �¦ ��; � with © = 1 is the dynamic inequality constraint and ℎ§ �� with = 1 is the static equality constraint.

64

3.4 OBTAINED RESULTS

3.4.1 Multi-Objective DE/rand/1/bin

The set of solutions shown in Figure 22 were found by using Multi-Objective

DE/rand/1/bin. A total amount of 325 solutions were obtained by this algorithm with 10

independent runs. With an average time per run of 221 seconds, this one was the faster

than other metaheuristics.

For all Pareto Fronts presented, the horizontal axis represents the first objective function ( M� �� = max ∆Uz��) and vertical axis the second one (M� �� = min i`¤��¢ − ¶�c� +`¤�� − ¶�c�c). Each point in the graph is a feasible and non-dominated solution for the

problem.

Figure 22 Solutions provided by Multi-Objective DE/rand/1/bin variant

At next, a part of the results obtained will be drawn. Aiming to expose the behaviour of

the proposed solutions, 3 graphs will be shown for each algorithm, the one with optimal M�, the one who optimizes M� and a solution in the midpoint of the Pareto’s Front. The

horizontal axis of each graph represents the time in milliseconds, and the vertical axis

represents the output for Uz in continuous line and ¤ in discontinuous line.In Figure 23 the

solution with optimal M� found through Multi-Objective DE/rand/1/bin is shown. This

solution can achieve a big output speed, but its efficiency is lower than other solutions

proposed by the same algorithm.

Figure 23

Solution shown in Figure

Figure 22. This solution has balance between

Figure 24 Balanced solution for

65

23 Optimal �� for Multi-Objective DE/rand/1/bin

Figure 24 belongs to the midpoint of the Pareto’s Front exposed in

This solution has balance between M� and M�.

Balanced solution for Multi-Objective DE/rand/1/bin

DE/rand/1/bin

belongs to the midpoint of the Pareto’s Front exposed in

DE/rand/1/bin

66

The last solution for Multi-Objective DE/rand/1/bin presented in Figure 25 has an efficient

performance; however its output speed is lower than other solutions provided.

Figure 25 Optimal �� for Multi-Objective DE/rand/1/bin

From Figure 23, Figure 24 and Figure 25 it can be observed an interesting behaviour

among both objective functions. Performance and efficiency are opposite in this problem,

i.e. an increasing performance implies a reduction in efficiency and vice versa. With

antagonist objective functions, the decision maker turns harder because the chosen

solution improves an objective function in detriment of another one.

In Table 5 a set of 10 solutions uniformly extracted from Pareto’s Front that belongs to

Multi-Objective DE/rand/1/bin are shown. These different configurations are non-

dominated among them, i.e. from a mathematic perspective, each solutions is as good as

others.

Table 5 Solutions from Multi-Objective DE/rand/1/bin

r1 r2 r3 r4 Θ1 Φ1 Φ2

0.499 0.05 0.4991 0.2503 -0.2482 0.1724 0.210166

0.4989 0.0682 0.4994 0.2536 -0.2458 0.31637 0.282654

0.4975 0.0796 0.4985 0.2504 -0.2374 0.445944 0.342679

0.4968 0.0894 0.4969 0.2503 -0.2381 0.566847 0.397513

0.498 0.097 0.4984 0.2504 -0.2333 0.671977 0.447469

0.4974 0.1038 0.4975 0.2507 -0.2324 0.772939 0.495104

67

0.4982 0.1095 0.499 0.2502 -0.2256 0.87044 0.541829

0.4963 0.115 0.4964 0.2501 -0.2269 0.966531 0.58678

0.4958 0.1251 0.4962 0.2516 -0.222 1.146404 0.673957

0.4994 0.1318 0.4994 0.2511 -0.2177 1.288526 0.739068

3.4.2 Multi-Objective DE/rand/1/exp

The set of solutions shown in Figure 26 was found with technique described before as

Multi-Objective DE/rand/1/exp. With an average time of 226 seconds per run, Multi-

Objective DE/rand/1/exp is so close to faster algorithm. 339 feasible solutions were found

by using Multi-Objective DE/rand/1/exp variant in ten independent runs.

Figure 26 Solutions Provided by Multi-Objective DE/rand/1/exp variant

In Figure 27 the solution with optimal M� found through Multi-Objective DE/rand/1/exp is

shown. This solution can achieve a big output speed, but its efficiency is lower than other

solutions proposed by the same algorithm, in a similar way that other metaheuristics.

68

Figure 27 Optimal �� for Multi-Objective DE/rand/1/exp

The solution shown in Figure 28 belongs to the midpoint of the Pareto’s Front exposed in

Figure 26. Its output is not the biggest, but its efficiency is not lowest.

Figure 28 Balanced solution for Multi-Objective DE/rand/1/exp

69


performance, however its output speed is lower than other solutions provided by Multi-

Objective DE/rand/1/exp.

Figure 29 Optimal �� for Multi-Objective DE/rand/1/exp


Multi-Objective DE/rand/1/exp are shown. These different configurations are non-

dominated among them.

Table 6 Solutions from Multi-Objective DE/rand/1/exp

r1 r2 r3 r4 Θ1 Φ1 Φ2

0.4994 0.0502 0.4995 0.2512 -0.2489 0.172603 0.210972

0.4924 0.0682 0.4925 0.2514 -0.2485 0.321998 0.285209

0.4982 0.0809 0.4982 0.251 -0.2417 0.457995 0.34579

0.4974 0.0899 0.4979 0.25 -0.2356 0.575194 0.401912

0.4971 0.1003 0.4972 0.2505 -0.2338 0.720127 0.470061

0.4972 0.1088 0.4972 0.2509 -0.2309 0.852615 0.532909

0.4957 0.1159 0.4957 0.2503 -0.2274 0.981209 0.594009

0.4986 0.1207 0.4988 0.2512 -0.2236 1.062921 0.632589

0.4981 0.1255 0.499 0.2503 -0.2168 1.167465 0.682526

70

0.4995 0.1317 0.4995 0.2506 -0.2171 1.291724 0.740151

3.4.3 Multi-Objective DE/best/1/exp

With Multi-Objective DE/best/1/exp, a set of solutions very similar to Multi-Objective

DE/rand/1/exp and DE/rand/1/bin was obtained, as can be observed in Figure 30. An

average time of 283 seconds per run, locates Multi-Objective DE/rand/1/exp in a mid-

point in comparison with other techniques used, such as DE variants and ME-ABC. 289

solutions were found after 10 independent runs.

Figure 30 Solutions provided by Multi-Objective DE/best/1/exp variant

In Figure 31 the solution with optimal M� found through Multi-Objective DE/best/1/exp is

shown. This solution is very similar to other solutions found through other DE variants.

71

Figure 31 Optimal �� for Multi-Objective DE/best/1/exp


Figure 30. Its behaviour develops between a medium speed output and a medium

efficiency.

Figure 32 Balanced solution for Multi-Objective DE/best/1/exp

72


performance; however its output speed is lower than other solutions provided by Multi-

Objective DE/best/1/exp.

Figure 33 Optimal �� for Multi-Objective DE/best/1/exp


Multi-Objective DE/best/1/exp are shown. These different configurations are non-

dominated among them.

Table 7 Solutions from Multi-Objective DE/best/1/exp

r1 r2 r3 r4 Θ1 Φ1 Φ2

0.4992 0.05 0.4993 0.2507 -0.2485 0.17187 0.21022

0.4951 0.0644 0.4951 0.2501 -0.2471 0.288991 0.266631

0.4993 0.0775 0.4995 0.2511 -0.2415 0.41889 0.327441

0.4988 0.0871 0.4989 0.2517 -0.2395 0.530842 0.380864

0.4988 0.0959 0.499 0.251 -0.235 0.652601 0.438018

0.4967 0.1027 0.4975 0.2511 -0.2305 0.754533 0.488597

0.499 0.1101 0.4994 0.251 -0.2277 0.87411 0.543141

0.496 0.1154 0.4966 0.2512 -0.2258 0.966211 0.588567

0.4996 0.1246 0.4999 0.2518 -0.2213 1.133329 0.666193

0.4982 0.1311 0.4988 0.251 -0.2157 1.276535 0.734353

73

3.4.4 ME-ABC

The set of solutions presented in Figure 34 were found with technique called ME-ABC.

With an average time of 682 seconds per run, E-ABC takes much time than other used

techniques such as DE variants. An amount of 963 feasible and non-dominated solutions

were found through ME-ABC variant at the end of 10 independent runs, becoming this in

the algorithm with the largest set of solutions.

Figure 34 Solutions provided by ME-ABC

In Figure 35 the solution with optimal M� found through ME-ABC is shown. This solution is

very similar to other solutions found through DE variants.

74

Figure 35 Optimal �� for ME-ABC

It becomes evident from Figure 35 that this solution outperforms the solution with biggest

output speed found at this time; however, a remarkable inefficiency in the system has to

be noticed, in comparison with previously presented solutions.


Figure 34. Its behaviour develops between a medium speed output and a medium

efficiency for this Pareto’s Front, but a detailed observation can reveal that a solution

located in the midpoint of the set provided by ME-ABC is so close to the optimal M�

provided by different DE variants.

75

Figure 36 Balanced solution for ME-ABC

The last solution for ME-ABC presented in Figure 37 has an efficient performance;

however its output speed is lower than other solutions provided by ME-ABC and DE

variants.

Figure 37 Optimal �� for ME-ABC

76


ME-ABC are shown. These different configurations are feasible and non-dominated among

them.

Table 8 Solutions from ME-ABC

r1 r2 r3 r4 Θ1 Φ1 Φ2

0.4991 0.05 0.4991 0.224 -0.2208 0.213099 0.204074

0.489 0.0501 0.489 0.1144 -0.1055 0.831846 0.438181

0.4888 0.05 0.4888 0.099 -0.0876 1.130803 0.58076

0.4997 0.05 0.4997 0.0909 -0.0761 1.365834 0.694508

0.4855 0.0505 0.4856 0.0874 -0.0722 1.527858 0.774823

0.4991 0.0503 0.4991 0.0838 -0.0673 1.666694 0.842401

0.4973 0.0503 0.498 0.0814 -0.0535 1.786777 0.901772

0.4806 0.05 0.4808 0.0787 -0.0601 1.905828 0.960843

0.4995 0.05 0.4996 0.0768 -0.0567 2.019307 1.016404

0.4993 0.0502 0.4993 0.0755 -0.0566 2.124529 1.068666

3.4.5 Comparison

As it can be seen in Figure 38, results achieved by using ME-ABC outperforms results

obtained based in DE variants. A remarkable difference in the span reached by ME-ABC

and DE variants can be observed. A more crowded set of solutions for M� and M�in

consequence is discovered by ME-ABC.

Figure 38 Comparison of different Pareto's Fronts

77

The whole set of proposed solutions from ME-ABC consists in feasible and non-dominated

solutions, acting as a reliable configuration for a RRRR mechanism and the CVT, where the

relationship between both, the output and the energy consumption is well known. All

these solutions are shown in Appendix 1.

With the set of solutions obtained, over 900 different configurations were obtained; i.e.

metaheuristic algorithms had act as reliable method to optimize the kinematic design

model of the CVT system.

78

4 ADD-INS

4.1 ARTIFICIAL NEURAL NETWORK MODELLING

An approach to obtain a model with a similar behaviour to the obtained with the

kinematic model was tried to be recreated. The aim of this model is to render in a really

simple way (through summations and products) a complex model such as the kinematic

model of the crank-rocker-slider CVT.

As the available data, the training data set was constituted by solutions present in the four

Pareto’s Fronts that belongs to each one of the metaheuristics used. Have to be noticed

that this training set may be is not representative, since it belongs to a very particular kind

of solutions, with particular features. To get good result in training stage a more varied set

of training had to be used.

To train Artificial Neural Networks, was used “Instrumento E” from the Engineer Jorge

Andres Fernandez Jimenez (Fernandez-Jimenez, 2009). This is practical and powerful

software developed for Neural Network Training among other tasks. The selected training

method for this case was Generalized Delta Rule (Otair & Salameh, 2005), a type of Back-

propagation Training.

Only mentioned training method was exhaustively used for this case, looking for receive a

good amount of information related with shown behaviour through different topologies.

This can be useful to perform an assessment of the complexity in completing training

process.

After trying many different topologies, a high sensitivity to iterative learning factor (·) was

advised. Depending of very slight variations in the iterative learning factor the training

process can be completed in a successfully or unsuccessfully manner. By using an adaptive

algorithm able to auto-adjust the value for the iterative learning factor in accordance with

their development along the training process the results can be improved. By using a fixed ·, this highly non-linear model can be hard to recreate. Anyway, an acceptable

approximation to the kinematic model of the crank-rocker-slider CVT system was found

(Figure 39), by considering all setbacks explained before. Algorithms based in populations

such as those previously used to optimize kinematic design model have a good

performance optimizing this kind of models (Kurban & Besdok, 2009). Due to its nature,

those algorithms usually do not have difficulties facing complicated functions as could be

observed in the previous chapter, then, by using metaheuristic algorithms a more

accurate can be found.

79

Figure 39 Artificial Neural Network model for CVT

4.1.1 Statistical validation

A statistic validation for the Artificial Neural Network model was performed, aiming to

establish some objective reference to compare both models. By using R®2.11.0 the whole

statistical analysis was made.

After any comparison can be made, an analysis of the populations has to be performed,

looking for to identify the distribution that they belong.

A Shapiro-Wilk test for normality was applied to both populations (Guner & Johnson,

2007), with the intention to know if they belong to normal distribution or not. In this case,

the null hypothesis ℎy is that each one of the populations describes a normal distribution.

The confidence level used to know if available populations have a normal distribution was

of the 99%. Although the confidence level may be hard to achieve, the accuracy required

by the model needs of this confidence level.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.20.40.60.81

Φ2

Φ1

Artificial Neural Networks Modelling

Original

Artificial

Neural

Networks

80

Table 9 Results for Shapiro-Wilk Test

Shapiro-Wilk Test Optimal Population Network Population

W 0.9610 0.9619

p-value 2.20E-16 2.20E-16

Null Hypothesis Rejected Rejected

In Table 9, results for normality test are shown. As can be seen, both Optimal and Neural

Network populations do not have a normal distribution. The small amounts of p-value and

W are enough to reject the null hypothesis ℎy: the populations belong to Normal

distributions.

With non-Normal populations to compare, a nonparametric statistic technique is used to

contrast both populations among them (Wasserman, 2005). The selected nonparametric

statistical technique was Mann-Whitney-Wilcoxon test. This technique is one of the best-

known non-parametric significance tests (Dewan & Rao, 2002). The aim is to perform

independent observations of involved populations and later compare the parameters

from both populations aiming to know if the populations’ distributions are similar.

The results for the Mann-Whitney-Wilcoxon test with a confidence level of 99% as was

explained lines above are presented in Error! Not a valid bookmark self-reference.. For

that mentioned test, null hypothesis ℎy is: the distributions of both groups are equal.

Table 10 Mann-Whitney-Wilcoxon test for Optimal and Network Population

Mann-Whitney-Wilcoxon Test

Both Populations

W 24.318

p-value 0.9528

Null Hypothesis Rejected

With those obtained results, it is clear that both populations have not the same

distribution in accordance with confidence level: however, with a lower level of

confidence these groups may be match, but the stated problem needs a really accurate

model. If this model would be used, the members of the population located in the upper

and lower side have to be unconsidered, as can be seen en Figure 39. With a set of 180

81

members of both populations extracted from the analysis, some flattering results for a

new Mann-Whitney-Wilcoxon test are obtained. Error! Not a valid bookmark self-reference. contains new results.

Table 11 Mann-Whitney-Wilcoxon test for filtered populations

Mann-Whitney-Wilcoxon Test

Both Populations

W 24285

p-value 0.9922

Null Hypothesis Accepted

With those new results, the obtained model can be accepted with a 99% of confidence.

Considering these results and the possibility to improve the Artificial Neural Network

model through other training methods, this can be a reliable technique to obtain

computational low-cost models for different systems. In this case, the neural network

model takes 30% less time than physical model constructed with equations.

4.2 RECONFIGURABLE HARDWARE

With the intention to evaluate real physical prototypes with different configurations, an

approach for a reconfigurable RRRR mechanism is proposed. A detailed observation of the

system leads to the fact that this reconfigurable hardware is not only able to optimize a

four-bar mechanism or a crank-rocker-slider CVT as shown previously; a four linkage

planar robot is obtained too (Soh & McCarthy, 2007), very useful in some mechatronic

topics also of the proposed for this degree project.

One of the premises for this approach is the simplicity in the system, together with a good

performance, an easy achievement of the elements that make up and a modular design

that allows modifying and expanding the mechanism as desired.

Most pieces of this mechanical approach are available in local markets, with the aim to

achieve those whenever be needed. Basic components for the reconfigurable hardware

proposal are listed below:

• C-Sections

• Threaded rods

• Screws, nuts and washers

82

• Coupling devices

• Ball-Bearings

• Shafts and sir-clips

In the next subchapters, each one of these items will be described.

4.2.1 C-Sections

Used to build a frame for the mechanism, 4 pieces of C-Sections are need. These pieces

are disposed in pairs, located a couple in a perpendicular disposition respect to another

couple.

This is a totally commercial steel section with structural features (build in AISI 1045 or AISI

A-36), with an easy achievement everywhere, provided that its dimensions are not too

large or too short. A galvanized C-Section is recommended, with the intention to enlarge

life-cycle of the system in some aspects, and may be most important is an improved

corrosion resistance. In accordance with measures exposed in Figure 40, the selected C-

Section has as values: � = 10011, � = 5011, C= 1511 and e= 211. This C-Section

has an average weight of 3,19 �� 1¥ , with average D and D¹of 35,15 �� 11^2¥ and

45,70 �� 11^2¥ respectively, enough to satisfy norm ASTM A 653 G50 (USS, 2009).

Figure 40 Important measures for C-Section

In most cases these sections are sold with a longitude of 6 meters per section, enough to

obtain all necessary pieces for the frame. As said lines above, these pieces are disposed in

pairs and machining operations are the same for each section of the couple, reason that

turns them as perfect elements to detail as pairs.

83

First, horizontal duo is described. These C-Sections have a simple machining process,

which aim is to make a slot along each bar. This slot has to be machined from midpoint of

side A along C-Section as shown in Figure 41. The slot can be performed with a 15 mm

milling tool. The material for the sections, although is not so hard, a type of coolant has to

be used during machining process.

Figure 41 Machining process for horizontal C-Sections

Machining for vertical C-Sections pair is so close to process described before. Besides a

slot along the C-Section, 2 more slots have to be made in a perpendicular disposition

respect of longitudinal axis. These slots have the mission of coupling with horizontal duo.

Figure 42 show in a detailed way machining needs.

Figure 42 Machining process for vertical C-Sections

Elements for coupling C-Sections are described in coupling devices subchapter.

84

4.2.2 Threaded rods

These rods are useful to give longitude to links for the RRRR mechanism. With an external

diameter of ¼ in, these rods have can be achieved with a galvanized process, a midterm

corrosion protection, inclusive manufactured in stainless steel. Last rods are expensive

than galvanized, but their corrosion resistance is improved.

Aiming to lock threaded rods from axial displacement, some nuts and washer are added.

Threaded rods are who able the system to change its dimensions. Although threaded rods

may be most simple element in the reconfigurable hardware, these give the essence of

modularity and flexibility.

4.2.3 Screws, nuts and washers

All these elements are used for making junctions and giving adjustment to the system. To

joint C-Sections ¼ inch diameter screws will be used with its respective nuts and washers.

With auto-adjustable washers better results can be obtained. Many different materials are

used to manufacture these elements, but as in previous subchapters, galvanized and

stainless steel pieces are recommended.

A socket-head screw can be useful to avoid external manipulation but if it is not a difficult,

any kind of screw is helpful in the same manner.

The same screws, nuts and washers referenced above in accordance with structural

features can be used to assembly the coupling devices

4.2.4 Coupling devices

Although devices described below have different missions, their design is so similar i.e. the

first element described can work as raw material for other devices.

A base element consists in a rectangular piece of steel with 2 threaded holes of ¼ inch

diameter. The longitude is not as important as the width that has to be greater of 19 mm.

These coupling devices have the aim to give back the resistance lost by long and

continuous slots in C-Sections. Have to be disposed in pairs and can be joint with screws

described above.

85

Last coupling devices are so similar among them and the element described above. The

main differences consist in a cavity that allows a ball-bearing and a couple of passing holes

along coupling device. These holes are for threaded rods and once rods are inserted, can

be adjusted to coupling devices with nuts and washer. The size of the cavity depends of

the selected ball-bearing.

4.2.5 Ball-bearings

The aim of these common elements is to reduce friction coefficient among coupling

devices and shafts to allow a free movement condition as close to ideal as possible.

Figure 43 Main measures and types of Ball-Bearings

Figure 43 shows main dimensions to consider in a deep groove ball-bearing. This is the

most common, simple and cheap ball-bearing in market that turns this as a perfect

candidate to stated needs. If it is possible, a sealed deep groove ball-bearing can be used,

aiming to avoid continuous lubrication. The sealed ball-bearing symbol is shown in Figure

43 too.

Selected dimensions for this case are: � = 19 11, f = 5 11 and � = 6 11. This ball-

bearing has an average dynamic and static load resistance of 2340 N and 950 N

respectively. Usually these elements are designed to work over 10,000 rpm, beyond from

system’s frequency.

4.2.6 Shafts and sir-clips

Used to connect two coupling devices giving them a condition of relative movement,

shafts are fundamental system’s parts. Those have to be made of steel. Although type of

steel is not fundamental, compatibility and corrosion resistance are important affairs to

consider.

86

External diameter for shafts is 5 mm, enough for coupling into ball-bearings. With the

intention to constraint axial displacement of shafts respect to ball-bearings, each shaft has

to have 4 external sir-clips.

4.2.7 Mounting device

This is a simple piece with shape of “U” which external measure is the same of the C-

Section and internal measure is the same of the width of coupling device. The aim of this

tool is locate coupling devices in an equidistant point from “B” sides of the C-Section.

4.2.8 Finite Elements Analysis (FEA)

Presented in Appendix 2

87

5 CONCLUSIONS

Kinematic models are helpful to perform an assessment of some complex systems, due to

the big amount of information that can provide about different aspects related with the

studied system. Instead of proceeding to develop complex models such as dynamical and

control system modelling, a kinematic model can be a little less complex to generate and it

might offer arguments to determine the viability of the system before spending time and

monetary resources in a more specialized research.

The improvement of different kind of systems can be reached by means of some EOPs;

however, it has to be noticed that these problems could be hard to optimize due to their

constraints and objective function(s), so a wide knowledge of the problem by the

researcher is needed, aiming to select the correct optimization technique.

Although metaheuristics are a powerful tool for solving a wide variety of optimization

problems, their use have to be totally justified, trying to avoid solving problems with this

kind of techniques when can be perfectly solved through traditional methods.

Tailor-made metaheuristics, such as ME-ABC that combines local search methods with

global optimization techniques and some special operators can outperform other

metaheuristics in some specific problems; however, their robustness has to be analyzed

before these algorithms are used to solve other problems, because a good performance in

a unique problem doesn’t imply a good performance in every problem, based on the No

Free Lunch theorems for optimization (Wolpert & W.G.Macready, 1995).

Computer aided design actually offers a wide range of tools for the improvement of

systems in different disciplines, and optimization through metaheuristics can be

incorporated to this group of tools, in the understanding that the kinematic design of a

system can be optimized by using these novel techniques.

Nature-inspired algorithm, besides other metaheuristics, cannot offer certainty about the

solution obtained is an absolute optimal. Nonetheless, those algorithms can provide a

good sub-optimal solution in a fast, reliable, and robust way.

A detailed observation of the behaviour of the stated problem, based in their continuous

solving can lead to a simplified model, such as happened with the crank-rocker-slider CVT,

where the complete system could be solved through a easiest model, the RRRR or four-

bar mechanism; however, this algorithms can be applied in other kinds of mechanisms

such as planar, spherical and space-moving mechanisms.

88

By using metaheuristics for solving MOPs, in a short range of time, a huge set of solutions

can be found as showed in the present work. Although a big set of solutions can reveal

much information from the problem, the decision maker still requires choosing a unique

solution and this task is not necessarily easy. In this way, a wide knowledge of the problem

is still needed, aiming to select the solution with the best performance in accordance to

established needs.

89

6 FUTURE WORK

Once the kinematic model optimization has shown satisfactory results, the dynamic model

of the crank-rocker-slider CVT system can be developed, with the aim of a substantial

improvement of the system.

Test the performance of the ME-ABC algorithm in other problems to assets its robustness

in multi-objective optimization.

Evaluate the susceptibility of the ME-ABC algorithm to its parameters, and to perform a

fine tuning of the mentioned algorithm aiming to improve its performance in a huge set of

problems.

90

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96

APPENDIX 1 SOLUTIONS PROVIDED BY ME-ABC

Table 12 963 Feasible and non-dominated solutions for CVT kinematic design r1 r2 r3 r4 Θ1 Φ1 Φ2

0.4991 0.05 0.4991 0.224 -0.2208 0.213099 0.204074

0.4991 0.05 0.4991 0.2199 -0.2165 0.220838 0.204161

0.499 0.0509 0.4991 0.2149 -0.2106 0.239486 0.209158

0.4999 0.0509 0.4999 0.2111 -0.2067 0.24785 0.209341

0.4999 0.0508 0.4999 0.2017 -0.1968 0.269828 0.212349

0.4997 0.0501 0.4997 0.1895 -0.1841 0.296542 0.21609

0.4993 0.0501 0.4993 0.1867 -0.1813 0.305413 0.218469

0.4985 0.05 0.4985 0.1839 -0.1787 0.313437 0.220576

0.4993 0.05 0.4995 0.1771 -0.17 0.337898 0.228193

0.4993 0.05 0.4995 0.1769 -0.1698 0.338659 0.228428

0.4878 0.05 0.4878 0.1757 -0.1738 0.343643 0.232204

0.4986 0.0502 0.4986 0.1741 -0.1682 0.352373 0.232759

0.4851 0.05 0.4851 0.1712 -0.1698 0.362008 0.238676

0.4968 0.05 0.4968 0.1675 -0.1618 0.377661 0.241194

0.4985 0.05 0.4986 0.1652 -0.1582 0.388294 0.244888

0.4996 0.0509 0.4998 0.1666 -0.1584 0.39611 0.249644

0.4821 0.05 0.4821 0.1625 -0.1613 0.40201 0.253026

0.4756 0.0503 0.4756 0.1632 -0.1642 0.403814 0.255825

0.4992 0.0508 0.4994 0.1623 -0.1539 0.415843 0.256714

0.4792 0.05 0.4792 0.1593 -0.1587 0.418557 0.259651

0.4739 0.05 0.4739 0.1582 -0.1593 0.424692 0.263071

0.4987 0.0509 0.4989 0.1581 -0.1495 0.440266 0.266239

0.4771 0.0501 0.4771 0.1556 -0.1552 0.440793 0.268587

0.4956 0.0507 0.4966 0.1568 -0.1437 0.444928 0.270808

0.4996 0.05 0.4996 0.1516 -0.1439 0.461746 0.272294

0.4774 0.05 0.4774 0.1517 -0.1508 0.462093 0.276519

0.4988 0.05 0.4988 0.1492 -0.1416 0.47701 0.278586

0.4709 0.05 0.4709 0.1491 -0.1499 0.4789 0.284404

0.4934 0.05 0.4934 0.1465 -0.1402 0.495323 0.286967

0.4982 0.0502 0.4992 0.1468 -0.1318 0.498225 0.290763

0.4741 0.0503 0.4741 0.1468 -0.1462 0.500311 0.292891

0.4956 0.0523 0.4958 0.1525 -0.1438 0.501313 0.293423

0.4999 0.05 0.4999 0.1439 -0.1355 0.513508 0.293501

0.4782 0.05 0.4782 0.144 -0.1419 0.51371 0.297109

0.4982 0.0503 0.4991 0.1444 -0.1297 0.517326 0.298527

0.4814 0.0501 0.4814 0.143 -0.1398 0.523105 0.300618

0.4965 0.05 0.4965 0.1416 -0.134 0.530875 0.301342

0.4984 0.0501 0.4994 0.1409 -0.125 0.539607 0.307903

0.4992 0.05 0.4992 0.1393 -0.1308 0.548912 0.308655

0.4999 0.05 0.4999 0.1391 -0.1304 0.550507 0.309243

0.4999 0.05 0.4999 0.1389 -0.1301 0.552138 0.309946

97

0.4992 0.05 0.4992 0.1382 -0.1296 0.557934 0.312545

0.4992 0.05 0.4992 0.137 -0.1283 0.568042 0.316927

0.4997 0.05 0.4997 0.1364 -0.1275 0.573183 0.319098

0.4982 0.0525 0.4984 0.1433 -0.1329 0.573848 0.323603

0.4939 0.05 0.4939 0.1349 -0.1273 0.586644 0.325758

0.4968 0.05 0.4968 0.1347 -0.1264 0.588325 0.326107

0.4985 0.0525 0.4987 0.1413 -0.1306 0.590686 0.330809

0.4749 0.05 0.4749 0.134 -0.1315 0.595637 0.33238

0.4987 0.05 0.4994 0.1335 -0.1189 0.60012 0.333027

0.4835 0.0501 0.4835 0.1337 -0.1287 0.600526 0.3334

0.4992 0.05 0.4992 0.1327 -0.1236 0.606706 0.333901

0.4994 0.05 0.4994 0.1325 -0.1233 0.608595 0.334713

0.4991 0.05 0.4991 0.1321 -0.123 0.612429 0.336452

0.498 0.05 0.4987 0.1317 -0.117 0.617253 0.340687

0.482 0.05 0.482 0.1313 -0.1265 0.6209 0.34243

0.4985 0.0501 0.4992 0.1315 -0.1166 0.621789 0.342758

0.4969 0.0524 0.4969 0.1373 -0.1282 0.624091 0.344932

0.4813 0.0501 0.4813 0.131 -0.1263 0.626509 0.345134

0.4821 0.05 0.4821 0.1306 -0.1256 0.627809 0.345471

0.4992 0.05 0.4992 0.1299 -0.1205 0.63413 0.34612

0.4975 0.0522 0.4977 0.1357 -0.1248 0.63454 0.349747

0.4777 0.05 0.4777 0.1296 -0.1257 0.638091 0.350623

0.4947 0.0501 0.4955 0.1298 -0.1148 0.639111 0.351235

0.4907 0.051 0.4909 0.132 -0.1229 0.64014 0.351623

0.4989 0.0503 0.4989 0.1292 -0.1197 0.649445 0.353383

0.4996 0.05 0.4996 0.1277 -0.118 0.657041 0.356379

0.4959 0.0503 0.4959 0.1284 -0.1196 0.658013 0.357593

0.483 0.0501 0.483 0.1276 -0.122 0.661586 0.360538

0.4949 0.0507 0.4954 0.129 -0.1161 0.663406 0.361949

0.4832 0.0501 0.4832 0.127 -0.1212 0.668099 0.363438

0.4829 0.05 0.4829 0.1267 -0.121 0.668581 0.363572

0.4998 0.05 0.4998 0.1262 -0.1163 0.673413 0.363771

0.498 0.0501 0.4987 0.1265 -0.111 0.674009 0.366227

0.482 0.05 0.482 0.126 -0.1204 0.676381 0.367197

0.4815 0.05 0.4815 0.1257 -0.1202 0.679772 0.368788

0.4992 0.05 0.4992 0.1249 -0.115 0.688153 0.370544

0.4985 0.05 0.4985 0.1245 -0.1148 0.692809 0.372744

0.4995 0.0523 0.4997 0.1302 -0.1181 0.694283 0.376344

0.499 0.05 0.499 0.1237 -0.1138 0.702194 0.37698

0.4973 0.0502 0.4979 0.1241 -0.1092 0.704373 0.379932

0.4995 0.0522 0.4997 0.1287 -0.1165 0.708418 0.382639

0.4852 0.05 0.4852 0.1227 -0.1159 0.714795 0.384255

0.4791 0.05 0.4791 0.1227 -0.1174 0.71506 0.385081

0.4985 0.05 0.4991 0.1225 -0.1072 0.717625 0.38565

0.4998 0.05 0.4998 0.122 -0.1117 0.722819 0.386364

0.4859 0.0502 0.4859 0.1219 -0.1147 0.730801 0.391719

0.4923 0.0522 0.4925 0.1266 -0.1158 0.733567 0.394919

0.4963 0.05 0.4963 0.1205 -0.1108 0.74198 0.395553

98

0.4865 0.0503 0.4865 0.1211 -0.1136 0.744088 0.39786

0.4999 0.05 0.4999 0.12 -0.1095 0.748353 0.398142

0.4974 0.05 0.498 0.12 -0.1045 0.749427 0.40039

0.4827 0.05 0.4827 0.1199 -0.1133 0.750369 0.400849

0.4703 0.05 0.4703 0.1197 -0.1161 0.753563 0.403718

0.4891 0.05 0.4891 0.1192 -0.111 0.759396 0.404337

0.4971 0.05 0.4971 0.1183 -0.1082 0.771293 0.409056

0.4869 0.05 0.4869 0.1182 -0.1104 0.773083 0.410899

0.494 0.0511 0.494 0.1201 -0.1104 0.782771 0.415753

0.4925 0.0503 0.4927 0.1181 -0.107 0.784526 0.41642

0.4915 0.05 0.4915 0.1169 -0.1078 0.791144 0.418833

0.4894 0.05 0.4894 0.1165 -0.1079 0.796981 0.421756

0.4975 0.051 0.4977 0.1186 -0.1061 0.800664 0.424098

0.4903 0.05 0.4903 0.1159 -0.107 0.805693 0.425733

0.4934 0.05 0.4934 0.1157 -0.1061 0.808518 0.426761

0.4975 0.0502 0.4975 0.1156 -0.105 0.816792 0.430439

0.4912 0.05 0.4912 0.115 -0.1058 0.819065 0.4319

0.4837 0.05 0.4837 0.1148 -0.1072 0.822403 0.434173

0.4915 0.05 0.4915 0.1146 -0.1052 0.825124 0.43471

0.4836 0.05 0.4838 0.1144 -0.1048 0.828881 0.437677

0.489 0.0501 0.489 0.1144 -0.1055 0.831846 0.438181

0.4842 0.05 0.4842 0.114 -0.1061 0.834665 0.439862

0.4901 0.05 0.4901 0.1134 -0.1042 0.843823 0.443611

0.4887 0.05 0.4887 0.1132 -0.1042 0.847051 0.445255

0.4883 0.05 0.4883 0.1131 -0.1042 0.848659 0.446047

0.4993 0.0502 0.4997 0.1135 -0.0983 0.849872 0.446721

0.494 0.05 0.494 0.1128 -0.1027 0.853235 0.447693

0.4942 0.0526 0.4942 0.1184 -0.1077 0.858348 0.452365

0.4948 0.0503 0.495 0.1129 -0.1005 0.863 0.452921

0.4985 0.0501 0.4989 0.1124 -0.0972 0.86399 0.453353

0.4906 0.05 0.4906 0.1121 -0.1026 0.86475 0.453416

0.4946 0.0519 0.4946 0.1163 -0.1056 0.866293 0.455431

0.489 0.05 0.489 0.1116 -0.1023 0.873089 0.457488

0.4945 0.0506 0.4947 0.1129 -0.1004 0.874131 0.458445

0.4978 0.05 0.4982 0.1114 -0.0963 0.876852 0.459396

0.4946 0.05 0.4946 0.1112 -0.1007 0.879574 0.460072

0.4855 0.05 0.4855 0.1109 -0.1023 0.885026 0.463428

0.4957 0.0503 0.4957 0.1114 -0.1006 0.887538 0.463993

0.4938 0.05 0.4938 0.1105 -0.1001 0.891541 0.4658

0.4857 0.05 0.4857 0.1102 -0.1014 0.897062 0.469097

0.4934 0.05 0.4934 0.11 -0.0996 0.900238 0.469953

0.4973 0.0504 0.4977 0.1109 -0.0956 0.900684 0.471024

0.4928 0.0505 0.4928 0.111 -0.1006 0.902193 0.471335

0.4659 0.0502 0.4659 0.1104 -0.1059 0.902211 0.473525

0.4893 0.05 0.4893 0.1096 -0.1 0.907443 0.473706

0.4964 0.05 0.4967 0.1095 -0.0953 0.909533 0.474752

0.4953 0.0518 0.4953 0.1134 -0.1022 0.910407 0.476075

0.4942 0.0503 0.4944 0.11 -0.0972 0.912316 0.476273

99

0.4933 0.05 0.4933 0.1092 -0.0987 0.91441 0.476683

0.4998 0.0513 0.4998 0.1119 -0.0998 0.917095 0.478462

0.494 0.05 0.494 0.1089 -0.0982 0.919785 0.47918

0.4988 0.0517 0.4988 0.1125 -0.1005 0.922056 0.481213

0.498 0.0511 0.4982 0.1111 -0.0973 0.923934 0.482103

0.4881 0.05 0.4883 0.1087 -0.0971 0.924062 0.482105

0.4971 0.05 0.4971 0.1083 -0.0969 0.930621 0.484091

0.4909 0.05 0.4909 0.1081 -0.0979 0.934562 0.486451

0.4954 0.0501 0.4956 0.1081 -0.0949 0.938851 0.48862

0.4936 0.05 0.4936 0.1077 -0.0969 0.94192 0.489741

0.4992 0.05 0.4992 0.1076 -0.0957 0.943585 0.490108

0.4919 0.05 0.4919 0.1075 -0.097 0.945755 0.491699

0.4955 0.05 0.4957 0.1073 -0.094 0.949822 0.49376

0.4924 0.0504 0.4924 0.1081 -0.0974 0.950785 0.494362

0.4967 0.05 0.4967 0.107 -0.0955 0.95509 0.495783

0.4978 0.0502 0.4982 0.1073 -0.0913 0.958424 0.498245

0.4927 0.05 0.4927 0.1067 -0.0959 0.961031 0.49892

0.4833 0.05 0.4833 0.1067 -0.0978 0.961413 0.499838

0.4959 0.0518 0.4959 0.1105 -0.0987 0.962381 0.500678

0.4959 0.0501 0.4961 0.1063 -0.0927 0.973351 0.505025

0.4995 0.05 0.4995 0.1059 -0.0937 0.976469 0.505795

0.4967 0.05 0.4967 0.1058 -0.0941 0.978567 0.507

0.4982 0.0508 0.4982 0.1075 -0.0954 0.978674 0.507521

0.4978 0.05 0.4982 0.1058 -0.0896 0.979394 0.508101

0.4604 0.05 0.4606 0.1058 -0.0994 0.980597 0.511339

0.494 0.0509 0.4942 0.1074 -0.0939 0.985448 0.511528

0.4992 0.0502 0.4995 0.1058 -0.0903 0.987661 0.51189

0.496 0.0504 0.4962 0.1062 -0.0923 0.988076 0.512268

0.486 0.05 0.486 0.1053 -0.0956 0.98904 0.512805

0.497 0.0502 0.497 0.1055 -0.0936 0.993161 0.514106

0.4967 0.05 0.4967 0.1049 -0.0931 0.996773 0.515716

0.4956 0.0503 0.4958 0.1054 -0.0915 1.000018 0.517936

0.4973 0.0511 0.4973 0.107 -0.0948 1.001359 0.518649

0.4978 0.05 0.4982 0.1047 -0.0883 1.001741 0.518784

0.4904 0.05 0.4904 0.1046 -0.0939 1.003206 0.519251

0.4802 0.0501 0.4802 0.1048 -0.0961 1.003856 0.520408

0.4968 0.0518 0.4968 0.108 -0.0957 1.011008 0.523804

0.4784 0.0501 0.4784 0.1044 -0.096 1.012262 0.524567

0.4884 0.0513 0.4884 0.1069 -0.0963 1.012336 0.524714

0.4965 0.05 0.4965 0.104 -0.0921 1.015524 0.524721

0.4911 0.05 0.4911 0.104 -0.0931 1.015734 0.525203

0.4978 0.05 0.4982 0.1038 -0.0872 1.020626 0.527827

0.4944 0.05 0.4944 0.1036 -0.092 1.024113 0.528989

0.4919 0.05 0.4919 0.1034 -0.0922 1.028507 0.531273

0.4975 0.05 0.4975 0.1032 -0.091 1.032615 0.53286

0.491 0.0517 0.491 0.1067 -0.0953 1.033723 0.535036

0.4924 0.05 0.4924 0.1029 -0.0916 1.039352 0.536446

0.4925 0.05 0.4925 0.1028 -0.0914 1.041543 0.537492

100

0.4968 0.05 0.497 0.1026 -0.0882 1.046252 0.539817

0.4978 0.0504 0.4978 0.1033 -0.0908 1.048559 0.540773

0.4939 0.05 0.4941 0.1024 -0.0885 1.050809 0.542204

0.4921 0.05 0.4921 0.1023 -0.0909 1.052641 0.542855

0.4978 0.05 0.4982 0.1023 -0.0854 1.053366 0.543539

0.4984 0.0513 0.4986 0.1049 -0.0899 1.054664 0.544616

0.4929 0.0511 0.4931 0.1043 -0.0903 1.059014 0.546956

0.4896 0.0504 0.4896 0.1026 -0.0915 1.06449 0.548995

0.4916 0.05 0.4916 0.1017 -0.0903 1.066207 0.549417

0.4958 0.0508 0.4959 0.1033 -0.0898 1.067197 0.550308

0.4963 0.05 0.4963 0.1015 -0.0892 1.070601 0.551222

0.497 0.0501 0.497 0.1017 -0.0893 1.070681 0.551279

0.4971 0.05 0.4973 0.1015 -0.0868 1.071047 0.551725

0.4843 0.05 0.4843 0.1014 -0.0913 1.073375 0.553368

0.4973 0.0504 0.4973 0.102 -0.0894 1.077858 0.554912

0.4916 0.05 0.4916 0.1011 -0.0896 1.080028 0.556074

0.492 0.05 0.492 0.101 -0.0894 1.082344 0.557163

0.4929 0.0505 0.4929 0.1019 -0.09 1.085049 0.558732

0.4915 0.05 0.4915 0.1008 -0.0893 1.087049 0.559464

0.4946 0.0509 0.4947 0.1026 -0.0892 1.087848 0.560392

0.4731 0.05 0.4731 0.1007 -0.0926 1.090167 0.562249

0.4929 0.0519 0.4929 0.1045 -0.0923 1.090695 0.562374

0.4954 0.0518 0.4954 0.1042 -0.0915 1.092803 0.563148

0.4928 0.05 0.4928 0.1004 -0.0886 1.096466 0.563921

0.4958 0.0527 0.4958 0.1058 -0.0928 1.097888 0.566161

0.4979 0.0504 0.4981 0.101 -0.0858 1.101733 0.566715

0.4996 0.0513 0.4996 0.1026 -0.0892 1.106295 0.569047

0.4914 0.05 0.4914 0.0999 -0.0882 1.108539 0.569839

0.4989 0.05 0.4991 0.0998 -0.0845 1.111175 0.570962

0.4907 0.0535 0.4907 0.1068 -0.0945 1.112074 0.573885

0.4966 0.05 0.4966 0.0995 -0.0868 1.118103 0.574132

0.4958 0.0502 0.4958 0.0997 -0.0871 1.123073 0.576707

0.4987 0.05 0.4989 0.0993 -0.0839 1.123459 0.576908

0.4935 0.0503 0.4937 0.0998 -0.0852 1.126137 0.57871

0.4888 0.05 0.4888 0.099 -0.0876 1.130803 0.58076

0.4906 0.0503 0.4906 0.0995 -0.0877 1.133188 0.581985

0.4829 0.05 0.4829 0.0989 -0.0886 1.133546 0.582468

0.4939 0.0508 0.4939 0.1004 -0.0879 1.135453 0.583179

0.4988 0.0512 0.499 0.1011 -0.0853 1.13815 0.584727

0.4975 0.05 0.4975 0.0986 -0.0856 1.140542 0.584933

0.4935 0.05 0.4935 0.0986 -0.0863 1.140694 0.585249

0.4942 0.0502 0.4944 0.099 -0.0842 1.141115 0.58584

0.4972 0.0503 0.4972 0.0991 -0.0861 1.142978 0.58631

0.4876 0.0512 0.4877 0.1009 -0.0882 1.143302 0.587784

0.4987 0.0502 0.4987 0.0987 -0.0854 1.148047 0.588615

0.4936 0.05 0.4936 0.0982 -0.0858 1.150906 0.590187

0.4994 0.05 0.4996 0.0982 -0.0825 1.151202 0.590286

0.4756 0.0504 0.4757 0.099 -0.0887 1.151702 0.59214

101

0.4989 0.0502 0.4989 0.0984 -0.085 1.155737 0.592328

0.4992 0.05 0.4994 0.098 -0.0822 1.156375 0.592802

0.4847 0.05 0.4847 0.098 -0.0872 1.15642 0.593408

0.4998 0.0513 0.4998 0.1004 -0.0866 1.160068 0.59502

0.4866 0.05 0.4866 0.0977 -0.0865 1.164155 0.597032

0.4998 0.0513 0.4998 0.1001 -0.0862 1.16772 0.598725

0.4946 0.05 0.4946 0.0975 -0.0848 1.169097 0.59894

0.4983 0.0525 0.4983 0.1024 -0.0885 1.169282 0.60029

0.4899 0.05 0.4899 0.0974 -0.0855 1.17192 0.60059

0.4796 0.05 0.4798 0.0974 -0.085 1.172883 0.602003

0.4888 0.0501 0.4888 0.0974 -0.0857 1.177164 0.603257

0.4831 0.05 0.4833 0.0972 -0.0841 1.178048 0.604276

0.4968 0.05 0.497 0.0969 -0.0813 1.185539 0.607074

0.4907 0.0501 0.4909 0.0971 -0.0826 1.18565 0.607545

0.4951 0.0515 0.4952 0.0998 -0.0854 1.186089 0.608165

0.4859 0.05 0.4859 0.0968 -0.0855 1.188126 0.608687

0.4899 0.05 0.4899 0.0967 -0.0847 1.190676 0.609682

0.4957 0.05 0.4959 0.0966 -0.0811 1.193711 0.6111

0.4978 0.05 0.4978 0.0965 -0.0831 1.19582 0.611722

0.4872 0.0514 0.4873 0.0992 -0.0861 1.197104 0.613933

0.4894 0.05 0.4894 0.0963 -0.0843 1.201626 0.615025

0.4892 0.05 0.4892 0.0963 -0.0843 1.201634 0.61504

0.4996 0.0501 0.4996 0.0964 -0.0826 1.203824 0.615565

0.4861 0.0503 0.4861 0.0968 -0.0853 1.204005 0.616551

0.4976 0.05 0.4976 0.0961 -0.0827 1.206838 0.617086

0.4903 0.05 0.4903 0.096 -0.0838 1.209894 0.618985

0.4989 0.05 0.4989 0.0959 -0.0822 1.212355 0.619695

0.4789 0.05 0.4789 0.0959 -0.0857 1.213141 0.621246

0.4998 0.0501 0.4998 0.0959 -0.082 1.217724 0.62231

0.4905 0.05 0.4905 0.0957 -0.0834 1.218279 0.623046

0.497 0.0505 0.497 0.0966 -0.083 1.219808 0.623698

0.4892 0.05 0.4892 0.0956 -0.0835 1.221148 0.624515

0.4947 0.0501 0.4949 0.0958 -0.0803 1.221281 0.624592

0.4877 0.0502 0.4877 0.0959 -0.084 1.223605 0.625909

0.4914 0.0509 0.4914 0.0972 -0.0845 1.224765 0.626653

0.4906 0.05 0.4906 0.0954 -0.083 1.226761 0.627161

0.496 0.05 0.4962 0.0953 -0.0795 1.229963 0.628681

0.4996 0.05 0.4996 0.0952 -0.0813 1.232131 0.629275

0.4993 0.0505 0.4993 0.0961 -0.0821 1.233803 0.630374

0.4945 0.05 0.4945 0.0951 -0.082 1.235192 0.631041

0.4876 0.05 0.4876 0.095 -0.083 1.238339 0.63296

0.4984 0.053 0.4984 0.1007 -0.0861 1.239065 0.634369

0.4893 0.05 0.4893 0.0948 -0.0825 1.244066 0.635647

0.489 0.05 0.489 0.0948 -0.0826 1.244078 0.63567

0.4889 0.05 0.4889 0.0948 -0.0826 1.244082 0.635678

0.4996 0.0503 0.4996 0.0952 -0.0811 1.248681 0.63749

0.4946 0.05 0.4948 0.0946 -0.0789 1.25027 0.638628

0.495 0.05 0.495 0.0945 -0.0812 1.252619 0.639494

102

0.476 0.05 0.476 0.0945 -0.0844 1.253378 0.640953

0.4969 0.05 0.4969 0.0943 -0.0806 1.258451 0.642232

0.4997 0.05 0.4999 0.0943 -0.0777 1.258917 0.64256

0.4985 0.05 0.4985 0.0941 -0.0801 1.264338 0.645015

0.4887 0.05 0.4887 0.0941 -0.0818 1.264709 0.645724

0.4772 0.05 0.4772 0.0941 -0.0837 1.265175 0.646612

0.4863 0.0514 0.4863 0.0967 -0.0844 1.266362 0.64744

0.4898 0.0512 0.49 0.0963 -0.081 1.267416 0.647884

0.4964 0.0512 0.4964 0.0962 -0.0822 1.269494 0.648273

0.497 0.0501 0.497 0.094 -0.0802 1.27306 0.649396

0.4954 0.05 0.4954 0.0937 -0.0802 1.27648 0.651094

0.4995 0.0502 0.4995 0.094 -0.0797 1.278662 0.652047

0.4886 0.05 0.4886 0.0936 -0.0812 1.279775 0.653064

0.4996 0.0509 0.4998 0.0951 -0.0781 1.285812 0.656123

0.492 0.05 0.492 0.0933 -0.0802 1.288817 0.657286

0.4998 0.0502 0.4998 0.0936 -0.0792 1.290845 0.657972

0.499 0.05 0.499 0.0932 -0.079 1.291637 0.658298

0.4953 0.0542 0.4955 0.1011 -0.084 1.291962 0.661132

0.4994 0.0506 0.4994 0.0941 -0.0796 1.298532 0.661945

0.4998 0.05 0.4998 0.0929 -0.0785 1.300922 0.662788

0.4766 0.05 0.4766 0.0928 -0.0822 1.304956 0.665999

0.4963 0.0509 0.4963 0.0944 -0.0803 1.306684 0.666234

0.4922 0.05 0.4923 0.0926 -0.0781 1.310925 0.668166

0.4974 0.0517 0.4974 0.0957 -0.0812 1.312544 0.669444

0.4882 0.05 0.4882 0.0925 -0.0799 1.313943 0.669738

0.4973 0.05 0.4973 0.0924 -0.0783 1.316775 0.670651

0.4994 0.0501 0.4994 0.0925 -0.078 1.319432 0.671893

0.4986 0.0507 0.4986 0.0936 -0.0791 1.319927 0.672473

0.4926 0.05 0.4926 0.0922 -0.0788 1.32334 0.674093

0.4985 0.05 0.4985 0.0921 -0.0778 1.326333 0.675259

0.4928 0.0501 0.4929 0.0923 -0.0775 1.326379 0.675722

0.4914 0.0504 0.4914 0.0928 -0.0795 1.327925 0.676595

0.482 0.05 0.482 0.092 -0.0803 1.330193 0.677995

0.4998 0.0504 0.4998 0.0926 -0.0779 1.334029 0.679149

0.4978 0.0516 0.4978 0.0948 -0.0801 1.3347 0.680173

0.4881 0.05 0.4881 0.0918 -0.0791 1.336439 0.680718

0.4974 0.0508 0.4974 0.0932 -0.0787 1.338591 0.681693

0.4986 0.05 0.4986 0.0916 -0.0771 1.342583 0.683193

0.4961 0.0501 0.4963 0.0917 -0.075 1.34606 0.68527

0.4915 0.0512 0.4915 0.0937 -0.08 1.346461 0.686039

0.4987 0.0501 0.4987 0.0916 -0.0771 1.348638 0.686195

0.4961 0.05 0.4961 0.0914 -0.0773 1.349264 0.686579

0.4978 0.0503 0.4978 0.0919 -0.0774 1.350904 0.687443

0.4985 0.05 0.4987 0.0913 -0.0742 1.353111 0.688546

0.4884 0.05 0.4884 0.0912 -0.0783 1.356197 0.69035

0.4978 0.0503 0.4978 0.0917 -0.0772 1.357518 0.690675

0.4975 0.0505 0.4975 0.092 -0.0775 1.359742 0.691873

0.4984 0.0501 0.4984 0.0912 -0.0766 1.361948 0.692716

103

0.4997 0.05 0.4997 0.0909 -0.0761 1.365834 0.694508

0.4989 0.0522 0.4989 0.0949 -0.0796 1.366664 0.696012

0.4915 0.0509 0.4915 0.0925 -0.0788 1.367677 0.696245

0.4945 0.0502 0.4945 0.0911 -0.077 1.371612 0.697677

0.491 0.05 0.491 0.0907 -0.0772 1.372929 0.698396

0.4871 0.05 0.4871 0.0907 -0.0779 1.373079 0.698664

0.4757 0.05 0.4757 0.0907 -0.0797 1.37354 0.699485

0.4826 0.051 0.4826 0.0925 -0.0802 1.374115 0.699906

0.4746 0.0508 0.4747 0.0921 -0.0798 1.375952 0.701236

0.4968 0.0511 0.4968 0.0925 -0.0778 1.379642 0.701921

0.4763 0.05 0.4763 0.0905 -0.0794 1.38034 0.702773

0.4989 0.0507 0.4991 0.0917 -0.0741 1.382618 0.703272

0.4816 0.05 0.4816 0.0904 -0.0784 1.383556 0.704064

0.4985 0.05 0.4985 0.0903 -0.0756 1.386356 0.704605

0.4932 0.0504 0.4932 0.091 -0.077 1.387468 0.705589

0.4957 0.05 0.4957 0.0902 -0.0759 1.389919 0.70648

0.4834 0.05 0.4836 0.0902 -0.0752 1.39105 0.707824

0.4998 0.0506 0.4998 0.0912 -0.0761 1.392815 0.707983

0.4943 0.0508 0.4943 0.0915 -0.0772 1.395168 0.709491

0.4788 0.0502 0.4788 0.0904 -0.0787 1.396226 0.7105

0.4994 0.0502 0.4994 0.0903 -0.0753 1.39891 0.710801

0.4937 0.05 0.4939 0.0899 -0.0732 1.40111 0.712231

0.4987 0.0508 0.4987 0.0913 -0.0762 1.401892 0.712569

0.4977 0.0502 0.4977 0.0902 -0.0755 1.402468 0.712621

0.4963 0.0504 0.4963 0.0905 -0.0759 1.404667 0.713855

0.4892 0.05 0.4894 0.0898 -0.0737 1.404804 0.714253

0.47 0.05 0.4701 0.0898 -0.0782 1.405267 0.715376

0.4983 0.0516 0.4983 0.0926 -0.0773 1.406908 0.715415

0.4781 0.0506 0.4781 0.0908 -0.079 1.407527 0.716259

0.495 0.0511 0.4951 0.0916 -0.0757 1.410731 0.717295

0.4982 0.0502 0.4982 0.0899 -0.075 1.413023 0.717766

0.493 0.0502 0.4931 0.0899 -0.0745 1.413543 0.718346

0.4994 0.0502 0.4994 0.0898 -0.0747 1.416532 0.71943

0.4941 0.05 0.4941 0.0894 -0.0752 1.418164 0.720381

0.4868 0.05 0.4868 0.0894 -0.0763 1.418441 0.720863

0.4878 0.0518 0.4879 0.0926 -0.0776 1.420061 0.72255

0.4843 0.05 0.4843 0.0893 -0.0766 1.422126 0.722788

0.4913 0.0513 0.4913 0.0916 -0.0774 1.423109 0.723541

0.4852 0.0512 0.4852 0.0914 -0.0782 1.424071 0.72427

0.4852 0.0501 0.4854 0.0894 -0.0738 1.425629 0.724676

0.4926 0.0504 0.4928 0.0899 -0.073 1.426711 0.724987

0.4938 0.0501 0.4945 0.0894 -0.0657 1.427053 0.725509

0.4864 0.0502 0.4865 0.0895 -0.075 1.4281 0.725784

0.4964 0.0504 0.4964 0.0898 -0.075 1.42949 0.726004

0.4969 0.0503 0.4969 0.0896 -0.0748 1.430222 0.726294

0.4933 0.0502 0.4934 0.0894 -0.0738 1.431442 0.727091

0.4978 0.0524 0.4978 0.0933 -0.0777 1.43238 0.728272

0.4991 0.0501 0.4991 0.0891 -0.074 1.435289 0.728589

104

0.4883 0.05 0.489 0.089 -0.066 1.435292 0.729748

0.4998 0.0504 0.4998 0.0895 -0.0742 1.44022 0.731107

0.499 0.0504 0.4992 0.0895 -0.0716 1.440915 0.731642

0.4996 0.0507 0.4996 0.09 -0.0746 1.441521 0.731887

0.4995 0.0506 0.4995 0.0897 -0.0743 1.445946 0.734015

0.4771 0.0502 0.4771 0.089 -0.0772 1.446335 0.735088

0.4893 0.05 0.4895 0.0886 -0.0722 1.448134 0.735442

0.4986 0.0515 0.4987 0.0912 -0.0743 1.45007 0.736551

0.4892 0.0505 0.4892 0.0894 -0.0756 1.450796 0.73682

0.4989 0.0512 0.4989 0.0906 -0.0751 1.452107 0.737327

0.4873 0.05 0.4873 0.0884 -0.075 1.454953 0.738717

0.499 0.0501 0.499 0.0885 -0.0733 1.457415 0.739441

0.4972 0.0514 0.4973 0.0908 -0.0741 1.458162 0.740533

0.4911 0.05 0.4913 0.0883 -0.0715 1.45923 0.74079

0.4998 0.0504 0.4998 0.0889 -0.0734 1.462327 0.741945

0.4861 0.0501 0.4863 0.0884 -0.0723 1.462334 0.742587

0.4981 0.0525 0.4983 0.0926 -0.0741 1.463994 0.743918

0.4989 0.0513 0.4989 0.0904 -0.0748 1.465899 0.74413

0.4786 0.05 0.4786 0.0881 -0.076 1.466549 0.744814

0.475 0.0502 0.475 0.0884 -0.0768 1.468748 0.746163

0.4847 0.05 0.4849 0.088 -0.0721 1.47079 0.746747

0.4899 0.0524 0.4899 0.0922 -0.0776 1.471669 0.747883

0.4979 0.0503 0.4979 0.0884 -0.0732 1.474534 0.747972

0.4965 0.0513 0.4966 0.0901 -0.0734 1.477429 0.749956

0.499 0.05 0.499 0.0877 -0.0723 1.480991 0.750964

0.4988 0.05 0.4988 0.0877 -0.0724 1.480998 0.750976

0.4853 0.0501 0.4855 0.0879 -0.0718 1.481307 0.751913

0.4901 0.0517 0.4902 0.0907 -0.0749 1.481631 0.752494

0.4783 0.0507 0.4783 0.0889 -0.0765 1.483108 0.753265

0.494 0.0503 0.4947 0.0882 -0.0638 1.484832 0.75381

0.4981 0.0502 0.4981 0.0879 -0.0726 1.486876 0.753976

0.4996 0.0504 0.4996 0.0882 -0.0726 1.488818 0.754949

0.4869 0.0501 0.4871 0.0877 -0.0713 1.488928 0.755571

0.4994 0.0524 0.4994 0.0917 -0.0755 1.489559 0.756186

0.4978 0.0501 0.4978 0.0876 -0.0723 1.491676 0.756302

0.4961 0.0505 0.4961 0.0883 -0.0732 1.49186 0.756636

0.499 0.0507 0.499 0.0886 -0.073 1.493729 0.757512

0.4959 0.05 0.4959 0.0873 -0.0723 1.496578 0.758746

0.4985 0.0503 0.4985 0.0878 -0.0723 1.497515 0.759222

0.4861 0.05 0.4863 0.0873 -0.071 1.497661 0.759842

0.495 0.0504 0.495 0.0879 -0.0729 1.500572 0.760916

0.4989 0.051 0.4991 0.0889 -0.0704 1.50313 0.762383

0.4973 0.0513 0.4974 0.0894 -0.0724 1.503826 0.762856

0.4887 0.0501 0.4889 0.0873 -0.0705 1.504421 0.763081

0.4825 0.05 0.4825 0.0871 -0.0741 1.50492 0.763434

0.4956 0.0508 0.4963 0.0885 -0.0636 1.507052 0.764813

0.4861 0.05 0.4861 0.087 -0.0734 1.508721 0.765135

0.4854 0.05 0.4854 0.087 -0.0735 1.508748 0.76518

105

0.4802 0.05 0.4802 0.087 -0.0743 1.508954 0.765518

0.4981 0.0519 0.4981 0.0903 -0.0743 1.509195 0.765653

0.4901 0.051 0.4902 0.0887 -0.0728 1.510847 0.766492

0.4873 0.05 0.488 0.087 -0.0634 1.511316 0.766933

0.496 0.05 0.4962 0.0869 -0.069 1.513019 0.766937

0.4989 0.0505 0.4989 0.0877 -0.072 1.515097 0.767921

0.4896 0.05 0.4896 0.0868 -0.0726 1.516521 0.768807

0.4973 0.0514 0.4974 0.0892 -0.0721 1.518276 0.769984

0.4954 0.0504 0.4961 0.0875 -0.0625 1.518844 0.770423

0.4865 0.05 0.4867 0.0867 -0.0701 1.521359 0.771438

0.4932 0.0518 0.4932 0.0898 -0.0745 1.521864 0.772044

0.4999 0.0504 0.4999 0.0873 -0.0714 1.523999 0.772209

0.4951 0.05 0.4951 0.0866 -0.0716 1.524317 0.7724

0.4764 0.0502 0.4771 0.087 -0.0649 1.525704 0.774561

0.4855 0.0505 0.4856 0.0874 -0.0722 1.527858 0.774823

0.4808 0.0507 0.4808 0.0877 -0.0746 1.529595 0.775925

0.4917 0.0501 0.4924 0.0867 -0.0622 1.530077 0.775957

0.4964 0.05 0.4964 0.0864 -0.0711 1.532337 0.776286

0.4987 0.0514 0.4987 0.0888 -0.0728 1.533517 0.777347

0.494 0.05 0.494 0.0863 -0.0713 1.536484 0.778423

0.4952 0.0515 0.4952 0.0889 -0.0733 1.536555 0.779033

0.4878 0.05 0.488 0.0863 -0.0694 1.537452 0.779273

0.4953 0.051 0.4953 0.0879 -0.0724 1.541839 0.781412

0.4991 0.0501 0.4991 0.0863 -0.0705 1.543369 0.781637

0.4987 0.0515 0.4987 0.0887 -0.0726 1.544349 0.782707

0.4891 0.0503 0.4891 0.0866 -0.0722 1.545626 0.783246

0.4993 0.0502 0.4995 0.0864 -0.0677 1.547076 0.783595

0.4878 0.05 0.4885 0.0861 -0.062 1.547605 0.784663

0.4967 0.0502 0.4974 0.0864 -0.061 1.54912 0.785107

0.4826 0.05 0.4826 0.086 -0.0727 1.549199 0.785158

0.4888 0.05 0.4895 0.086 -0.0617 1.551682 0.786615

0.4934 0.0508 0.4935 0.0873 -0.0707 1.552533 0.786709

0.4996 0.0529 0.4996 0.0909 -0.0742 1.553054 0.787525

0.476 0.05 0.476 0.0859 -0.0735 1.553593 0.787612

0.4991 0.05 0.4991 0.0858 -0.07 1.556859 0.788228

0.4979 0.05 0.4979 0.0858 -0.0702 1.556902 0.788297

0.487 0.0502 0.487 0.0861 -0.072 1.559169 0.789946

0.4877 0.05 0.4884 0.0858 -0.0616 1.560023 0.790746

0.4932 0.0507 0.4933 0.0869 -0.0703 1.561851 0.791251

0.4947 0.0506 0.4947 0.0867 -0.0713 1.562571 0.791456

0.4922 0.0504 0.4929 0.0864 -0.0615 1.563551 0.792442

0.4944 0.05 0.4944 0.0856 -0.0704 1.565375 0.792604

0.4996 0.0511 0.4998 0.0875 -0.0684 1.565626 0.793051

0.4991 0.0501 0.4998 0.0858 -0.0599 1.566739 0.793601

0.4765 0.0503 0.4765 0.0861 -0.0735 1.566749 0.794178

0.4965 0.05 0.4965 0.0855 -0.07 1.569499 0.794547

0.4915 0.0505 0.4922 0.0864 -0.0615 1.570737 0.796031

0.4951 0.05 0.4951 0.0854 -0.0701 1.573768 0.796702

106

0.4967 0.05 0.4969 0.0854 -0.0669 1.574454 0.797066

0.4903 0.0503 0.4903 0.0859 -0.0712 1.574576 0.797417

0.4973 0.0512 0.498 0.0875 -0.0616 1.574763 0.798035

0.4811 0.0508 0.4818 0.0868 -0.0634 1.576019 0.799198

0.4827 0.0507 0.4827 0.0865 -0.0728 1.578496 0.799836

0.4982 0.0509 0.4982 0.0868 -0.0707 1.579693 0.799848

0.4853 0.0509 0.4853 0.0868 -0.0726 1.580188 0.800637

0.4925 0.05 0.4932 0.0853 -0.0602 1.580867 0.800753

0.4916 0.0508 0.4918 0.0866 -0.0686 1.581894 0.801249

0.4926 0.0507 0.4926 0.0864 -0.0712 1.582304 0.801281

0.4982 0.0509 0.4982 0.0867 -0.0706 1.583879 0.801906

0.4987 0.05 0.4987 0.0851 -0.0692 1.586404 0.802775

0.4916 0.05 0.4918 0.0851 -0.0673 1.58742 0.803641

0.4993 0.0517 0.4995 0.088 -0.0687 1.587466 0.804024

0.4977 0.0501 0.4977 0.0852 -0.0694 1.58948 0.804366

0.4995 0.0502 0.4995 0.0853 -0.0692 1.592454 0.805798

0.498 0.0502 0.498 0.0853 -0.0694 1.592507 0.805882

0.4961 0.05 0.4961 0.0849 -0.0693 1.5951 0.807154

0.4873 0.05 0.4874 0.0849 -0.0691 1.595806 0.807892

0.4912 0.05 0.4914 0.0849 -0.0671 1.596043 0.807893

0.4868 0.0506 0.4868 0.0859 -0.0715 1.596471 0.808447

0.4962 0.05 0.4962 0.0848 -0.0692 1.599426 0.809279

0.4992 0.0517 0.4994 0.0877 -0.0683 1.599977 0.810171

0.4989 0.05 0.4989 0.0847 -0.0687 1.603678 0.811267

0.4977 0.05 0.4977 0.0847 -0.0688 1.60372 0.811334

0.496 0.05 0.496 0.0847 -0.0691 1.603781 0.81143

0.4942 0.05 0.4944 0.0847 -0.0664 1.604611 0.811983

0.4992 0.0532 0.4992 0.0901 -0.073 1.605683 0.813501

0.481 0.0502 0.4813 0.085 -0.0671 1.607299 0.813978

0.4775 0.05 0.4782 0.0847 -0.0615 1.607393 0.814383

0.4825 0.0515 0.4832 0.0872 -0.0631 1.60947 0.815791

0.4936 0.05 0.4936 0.0845 -0.0691 1.612622 0.815873

0.4981 0.0527 0.4983 0.0891 -0.0695 1.61266 0.816841

0.49 0.0519 0.4901 0.0877 -0.0709 1.614312 0.817638

0.4927 0.0506 0.4934 0.0855 -0.0599 1.61632 0.818351

0.4876 0.05 0.4885 0.0845 -0.0568 1.616607 0.81866

0.4819 0.0501 0.4822 0.0846 -0.0665 1.617331 0.818822

0.4804 0.05 0.4804 0.0844 -0.0709 1.617528 0.818825

0.4959 0.0502 0.4959 0.0847 -0.0689 1.61867 0.818837

0.4903 0.0501 0.491 0.0846 -0.0594 1.618673 0.81941

0.4979 0.0503 0.4979 0.0848 -0.0687 1.62166 0.82027

0.4965 0.05 0.4967 0.0843 -0.0655 1.622119 0.8205

0.4991 0.0501 0.4991 0.0844 -0.0682 1.624331 0.821463

0.4866 0.0502 0.4869 0.0846 -0.0657 1.624624 0.822247

0.4845 0.05 0.4845 0.0842 -0.0701 1.626237 0.822937

0.4973 0.0502 0.4973 0.0845 -0.0685 1.627468 0.823114

0.491 0.0501 0.4917 0.0844 -0.059 1.627502 0.823711

0.4989 0.05 0.4989 0.0841 -0.0679 1.630171 0.824309

107

0.4938 0.0506 0.4938 0.0851 -0.0694 1.630977 0.825129

0.4899 0.0519 0.49 0.0873 -0.0704 1.631411 0.826041

0.493 0.05 0.4937 0.0841 -0.0584 1.633263 0.826419

0.492 0.05 0.492 0.084 -0.0687 1.634903 0.826902

0.494 0.05 0.4942 0.084 -0.0655 1.635612 0.827232

0.4994 0.0524 0.4994 0.088 -0.0709 1.636872 0.828501

0.4923 0.05 0.493 0.084 -0.0583 1.637786 0.828665

0.4985 0.05 0.4985 0.0839 -0.0677 1.639172 0.828757

0.4953 0.05 0.4955 0.0839 -0.0652 1.64007 0.829374

0.4922 0.0504 0.4929 0.0846 -0.0589 1.641154 0.830471

0.4995 0.05 0.4995 0.0838 -0.0675 1.643661 0.83093

0.488 0.05 0.488 0.0838 -0.069 1.644079 0.831575

0.4952 0.05 0.4954 0.0838 -0.0651 1.6446 0.831606

0.4899 0.0501 0.4906 0.084 -0.0586 1.645503 0.832582

0.4931 0.05 0.4931 0.0837 -0.0682 1.648433 0.83352

0.4941 0.0504 0.4948 0.0844 -0.0583 1.650077 0.834773

0.4966 0.0501 0.4966 0.0838 -0.0678 1.65141 0.834892

0.4956 0.0503 0.4956 0.0841 -0.0681 1.653087 0.835829

0.4977 0.0502 0.4977 0.0839 -0.0677 1.654471 0.836395

0.4976 0.0503 0.4983 0.0841 -0.0575 1.655918 0.837469

0.4879 0.0501 0.4879 0.0837 -0.0689 1.656299 0.837632

0.4979 0.0502 0.4979 0.0838 -0.0675 1.659037 0.838637

0.4832 0.0503 0.4835 0.084 -0.0653 1.659344 0.839479

0.4909 0.0511 0.491 0.0853 -0.0683 1.660126 0.839799

0.4932 0.0503 0.4939 0.084 -0.0579 1.66067 0.839968

0.4973 0.05 0.4973 0.0834 -0.0672 1.662035 0.840063

0.4823 0.0508 0.483 0.0848 -0.0602 1.66292 0.841697

0.4882 0.0508 0.4883 0.0847 -0.0681 1.664626 0.842005

0.4991 0.0503 0.4991 0.0838 -0.0673 1.666694 0.842401

0.4922 0.05 0.4922 0.0833 -0.0678 1.666844 0.842622

0.4976 0.05 0.4978 0.0833 -0.0641 1.667446 0.842758

0.4975 0.0502 0.4975 0.0836 -0.0673 1.668259 0.843194

0.4897 0.0511 0.4898 0.0851 -0.0682 1.669221 0.84432

0.4971 0.0509 0.4978 0.0848 -0.058 1.669893 0.844563

0.488 0.0509 0.4887 0.0848 -0.0593 1.670289 0.845114

0.4966 0.0513 0.4969 0.0854 -0.0645 1.671338 0.845221

0.489 0.0502 0.489 0.0835 -0.0684 1.673204 0.845951

0.4998 0.0504 0.4998 0.0838 -0.0671 1.674394 0.846205

0.498 0.05 0.498 0.0831 -0.0668 1.67595 0.846893

0.4987 0.0502 0.4989 0.0834 -0.0639 1.678305 0.848135

0.4898 0.0511 0.4899 0.0849 -0.0679 1.678344 0.848804

0.4792 0.0501 0.4792 0.0832 -0.0695 1.679785 0.849542

0.4989 0.05 0.4991 0.083 -0.0635 1.681407 0.849584

0.4987 0.052 0.4987 0.0863 -0.0692 1.682231 0.850694

0.499 0.05 0.4997 0.083 -0.0559 1.683563 0.850884

0.4973 0.0505 0.498 0.0838 -0.0569 1.685205 0.851928

0.4944 0.0524 0.4945 0.0869 -0.0688 1.685815 0.85279

0.4987 0.0517 0.4988 0.0857 -0.0672 1.687174 0.853027

108

0.4967 0.0509 0.4974 0.0844 -0.0575 1.688327 0.853625

0.4982 0.05 0.4982 0.0828 -0.0664 1.69007 0.853845

0.4836 0.0517 0.4843 0.0857 -0.0606 1.690417 0.855468

0.495 0.0511 0.4953 0.0846 -0.0638 1.692814 0.855767

0.4965 0.051 0.4966 0.0844 -0.0664 1.693494 0.855964

0.4979 0.05 0.4979 0.0827 -0.0663 1.694833 0.856204

0.4971 0.05 0.4971 0.0827 -0.0664 1.694861 0.856246

0.4958 0.0501 0.4965 0.0829 -0.0561 1.696305 0.857293

0.4808 0.05 0.481 0.0827 -0.0656 1.696307 0.857599

0.4897 0.05 0.4904 0.0827 -0.0568 1.698174 0.858401

0.4999 0.05 0.4999 0.0826 -0.0659 1.699537 0.858452

0.4897 0.05 0.4897 0.0826 -0.0673 1.699902 0.859002

0.4881 0.05 0.4882 0.0826 -0.066 1.700372 0.8593

0.4965 0.0523 0.4965 0.0864 -0.0694 1.700467 0.859869

0.4947 0.0511 0.495 0.0844 -0.0636 1.702162 0.860376

0.4843 0.0504 0.4843 0.0832 -0.0685 1.703144 0.860954

0.4897 0.0503 0.4897 0.083 -0.0676 1.704568 0.86141

0.4992 0.0505 0.4992 0.0833 -0.0665 1.705722 0.861702

0.4992 0.0511 0.4994 0.0843 -0.0643 1.706264 0.862202

0.4895 0.0507 0.4895 0.0836 -0.068 1.70757 0.863045

0.4885 0.05 0.4892 0.0825 -0.0566 1.707817 0.86318

0.4978 0.0504 0.4985 0.0831 -0.0559 1.710435 0.864265

0.4872 0.05 0.4874 0.0824 -0.0643 1.710445 0.864308

0.4902 0.0502 0.4902 0.0827 -0.0672 1.710993 0.864521

0.4939 0.0503 0.4946 0.0829 -0.0562 1.712251 0.865263

0.4991 0.051 0.4993 0.084 -0.064 1.712621 0.8653

0.4913 0.05 0.4913 0.0823 -0.0667 1.714295 0.866036

0.4995 0.0511 0.4997 0.0841 -0.064 1.715711 0.866842

0.4897 0.0502 0.49 0.0826 -0.0625 1.717106 0.867588

0.499 0.0506 0.499 0.0832 -0.0663 1.718436 0.868013

0.4952 0.0502 0.4959 0.0826 -0.0557 1.718696 0.868347

0.4985 0.0502 0.4985 0.0825 -0.0658 1.720367 0.868842

0.4994 0.0501 0.4994 0.0823 -0.0655 1.722038 0.8696

0.4925 0.0509 0.4925 0.0836 -0.0675 1.723253 0.870731

0.4883 0.0502 0.489 0.0825 -0.0565 1.723849 0.871131

0.4791 0.05 0.4793 0.0821 -0.065 1.725362 0.871961

0.4962 0.0503 0.4969 0.0826 -0.0555 1.72668 0.872269

0.4901 0.0501 0.4901 0.0822 -0.0666 1.72726 0.872507

0.4936 0.0502 0.4943 0.0824 -0.0556 1.728503 0.873223

0.4841 0.0503 0.4841 0.0825 -0.0677 1.728934 0.873629

0.4986 0.0523 0.4988 0.0858 -0.0654 1.729016 0.873854

0.4938 0.0511 0.4941 0.0838 -0.0629 1.730711 0.874453

0.4932 0.0512 0.4939 0.084 -0.0571 1.730862 0.874745

0.4993 0.0503 0.4993 0.0824 -0.0655 1.733271 0.875213

0.4992 0.0503 0.4992 0.0824 -0.0655 1.733274 0.875218

0.4963 0.05 0.4965 0.0819 -0.0624 1.73453 0.87584

0.4927 0.0512 0.4934 0.0839 -0.057 1.735701 0.877139

0.4986 0.0509 0.4986 0.0833 -0.0663 1.737536 0.877549

109

0.4978 0.0509 0.4978 0.0833 -0.0664 1.737565 0.877592

0.4975 0.05 0.4977 0.0818 -0.0621 1.73944 0.878217

0.498 0.0504 0.498 0.0824 -0.0656 1.741407 0.879305

0.4947 0.0503 0.4947 0.0822 -0.0658 1.743297 0.880319

0.4924 0.0502 0.4931 0.0821 -0.0553 1.743334 0.880552

0.4991 0.05 0.4993 0.0817 -0.0618 1.744359 0.880586

0.4936 0.0511 0.4939 0.0835 -0.0625 1.745266 0.881623

0.4974 0.0502 0.4981 0.082 -0.0545 1.748096 0.882718

0.4914 0.0508 0.4914 0.0829 -0.0668 1.749357 0.883602

0.4987 0.0516 0.4987 0.0842 -0.0668 1.749641 0.883758

0.4887 0.0501 0.4894 0.0818 -0.0555 1.75026 0.884054

0.4996 0.0501 0.4998 0.0817 -0.0616 1.752541 0.884636

0.4974 0.0504 0.4981 0.0822 -0.0546 1.754472 0.885918

0.4886 0.05 0.4889 0.0815 -0.0613 1.755222 0.886318

0.4949 0.0515 0.4949 0.0839 -0.0671 1.756424 0.887205

0.487 0.0503 0.4877 0.082 -0.0558 1.756728 0.887363

0.4949 0.0503 0.4949 0.0819 -0.0654 1.758253 0.887689

0.4981 0.0507 0.4981 0.0825 -0.0655 1.760838 0.88899

0.4927 0.05 0.4927 0.0813 -0.0652 1.763876 0.890437

0.4982 0.0508 0.4982 0.0826 -0.0655 1.764001 0.890581

0.4882 0.05 0.4884 0.0813 -0.0627 1.764908 0.891093

0.4991 0.0503 0.4998 0.0818 -0.0539 1.766284 0.891634

0.4986 0.0523 0.4986 0.085 -0.0673 1.766418 0.892277

0.4899 0.05 0.4899 0.0812 -0.0654 1.769068 0.893097

0.4782 0.05 0.4782 0.0812 -0.067 1.769513 0.893745

0.4936 0.0502 0.4939 0.0815 -0.0605 1.771575 0.894264

0.4984 0.0507 0.4986 0.0823 -0.0621 1.771717 0.894331

0.4928 0.05 0.4935 0.0812 -0.0541 1.772162 0.89464

0.4978 0.0509 0.4978 0.0826 -0.0655 1.772193 0.89467

0.4787 0.05 0.4793 0.0812 -0.0576 1.77228 0.895159

0.4898 0.05 0.49 0.0811 -0.0622 1.775058 0.896038

0.4949 0.0501 0.4956 0.0813 -0.0539 1.775298 0.896144

0.4871 0.0502 0.4871 0.0814 -0.0659 1.775606 0.896491

0.489 0.0507 0.489 0.0822 -0.0663 1.776261 0.896921

0.4946 0.0506 0.4946 0.082 -0.0654 1.77794 0.897515

0.4907 0.05 0.4909 0.081 -0.0619 1.780165 0.898523

0.4938 0.0501 0.4945 0.0812 -0.0539 1.780476 0.898727

0.4962 0.0518 0.4962 0.0839 -0.0666 1.780532 0.899149

0.497 0.0504 0.4971 0.0816 -0.0631 1.782089 0.899396

0.4875 0.05 0.4882 0.081 -0.0545 1.782665 0.899988

0.4958 0.05 0.4958 0.0809 -0.0643 1.784272 0.900396

0.4827 0.05 0.4827 0.0809 -0.066 1.784753 0.901093

0.4973 0.0503 0.498 0.0814 -0.0535 1.786777 0.901772

0.4953 0.0502 0.496 0.0812 -0.0536 1.788798 0.902801

0.494 0.05 0.494 0.0808 -0.0644 1.789523 0.903048

0.4996 0.05 0.4998 0.0808 -0.0605 1.79019 0.903166

0.4895 0.0503 0.4902 0.0813 -0.0544 1.792271 0.904741

0.484 0.0501 0.484 0.0809 -0.0658 1.793128 0.905212

110

0.4961 0.0507 0.4968 0.0819 -0.0541 1.79452 0.905751

0.4996 0.0507 0.4996 0.0818 -0.0644 1.796317 0.906446

0.4989 0.0503 0.4996 0.0812 -0.053 1.797064 0.906779

0.4937 0.0508 0.4944 0.082 -0.0544 1.797822 0.90749

0.4918 0.0509 0.4918 0.0821 -0.0656 1.797838 0.907533

0.4988 0.0503 0.4988 0.0811 -0.0639 1.79905 0.90769

0.4903 0.0501 0.491 0.0808 -0.0537 1.801391 0.909131

0.4876 0.0501 0.4883 0.0808 -0.0541 1.801507 0.909281

0.497 0.0504 0.497 0.0812 -0.0642 1.802339 0.909407

0.4889 0.0501 0.4889 0.0807 -0.0648 1.803401 0.910107

0.4884 0.0502 0.4891 0.0809 -0.054 1.804717 0.910865

0.4885 0.05 0.4885 0.0805 -0.0647 1.80543 0.911089

0.4899 0.0501 0.4906 0.0807 -0.0536 1.806661 0.911736

0.4968 0.0504 0.4968 0.0811 -0.0641 1.807576 0.911999

0.494 0.0505 0.4947 0.0813 -0.0536 1.808942 0.912849

0.4955 0.0517 0.4958 0.0832 -0.0613 1.809006 0.913143

0.4964 0.05 0.4964 0.0804 -0.0635 1.810432 0.913291

0.4981 0.05 0.4982 0.0804 -0.0617 1.810807 0.913397

0.4996 0.0513 0.4998 0.0825 -0.0617 1.811153 0.91393

0.4969 0.0506 0.4976 0.0814 -0.0533 1.812045 0.91431

0.4975 0.0501 0.4975 0.0805 -0.0634 1.813648 0.914876

0.4945 0.0502 0.4952 0.0807 -0.0529 1.814997 0.915714

0.4958 0.05 0.4958 0.0803 -0.0635 1.815764 0.915944

0.4954 0.05 0.4954 0.0803 -0.0635 1.815778 0.915964

0.4852 0.0502 0.4852 0.0806 -0.0651 1.817347 0.917154

0.492 0.05 0.4927 0.0803 -0.0528 1.819214 0.917808

0.4815 0.0501 0.4815 0.0804 -0.0654 1.819551 0.918339

0.491 0.0502 0.4917 0.0806 -0.0532 1.820453 0.918515

0.4955 0.05 0.4957 0.0802 -0.0602 1.822001 0.918991

0.4985 0.0502 0.4985 0.0805 -0.0632 1.822184 0.919091

0.4931 0.0501 0.4938 0.0804 -0.0527 1.822434 0.919387

0.4998 0.0504 0.4998 0.0808 -0.0633 1.823309 0.919669

0.4909 0.0518 0.4916 0.0831 -0.0555 1.824582 0.921074

0.4878 0.0509 0.4881 0.0816 -0.0608 1.825381 0.92121

0.4976 0.0502 0.4976 0.0804 -0.0632 1.827561 0.921776

0.4938 0.0502 0.4938 0.0804 -0.0637 1.827696 0.921968

0.4983 0.0512 0.4983 0.082 -0.0644 1.82797 0.92228

0.4915 0.0511 0.4922 0.0819 -0.0543 1.828351 0.922674

0.4878 0.0509 0.4881 0.0815 -0.0607 1.830663 0.923812

0.4975 0.05 0.4977 0.08 -0.0597 1.832683 0.924197

0.4958 0.0502 0.4958 0.0803 -0.0633 1.832996 0.924519

0.4886 0.0513 0.4893 0.0821 -0.0548 1.834888 0.926055

0.4896 0.0511 0.4896 0.0817 -0.0652 1.835644 0.926333

0.4929 0.05 0.4929 0.0799 -0.0633 1.837366 0.926708

0.4929 0.0509 0.4936 0.0814 -0.0535 1.83774 0.927179

0.4999 0.0501 0.4999 0.08 -0.0625 1.840398 0.928012

0.489 0.0501 0.489 0.08 -0.0639 1.840785 0.928559

0.4984 0.0507 0.4991 0.081 -0.0524 1.841733 0.928899

111

0.4926 0.0501 0.4933 0.08 -0.0522 1.844024 0.930027

0.4837 0.05 0.4839 0.0798 -0.0612 1.844067 0.930271

0.4996 0.0524 0.4996 0.0836 -0.0653 1.844995 0.931033

0.4997 0.0501 0.4999 0.0799 -0.0592 1.846746 0.931099

0.4921 0.0502 0.4921 0.08 -0.0634 1.849396 0.932741

0.4917 0.0501 0.4924 0.0799 -0.0522 1.849519 0.93276

0.4865 0.05 0.4872 0.0797 -0.0526 1.851931 0.934086

0.4978 0.0503 0.4978 0.0801 -0.0627 1.852468 0.934105

0.4998 0.0502 0.4998 0.0799 -0.0623 1.854599 0.935063

0.4966 0.05 0.4973 0.0796 -0.0512 1.857009 0.93626

0.4996 0.0511 0.4998 0.0813 -0.0602 1.857573 0.936759

0.4932 0.0518 0.4932 0.0824 -0.0651 1.857886 0.937419

0.4975 0.0502 0.4975 0.0798 -0.0625 1.86018 0.937894

0.4815 0.0501 0.4822 0.0797 -0.0532 1.860958 0.938728

0.4814 0.0501 0.4821 0.0797 -0.0532 1.860963 0.938734

0.491 0.0507 0.4913 0.0806 -0.0591 1.861706 0.938935

0.4936 0.0501 0.4936 0.0796 -0.0628 1.862552 0.939159

0.498 0.0504 0.4987 0.0801 -0.0514 1.864599 0.940075

0.4997 0.05 0.4999 0.0794 -0.0586 1.865497 0.940324

0.4977 0.05 0.4979 0.0794 -0.0588 1.865569 0.940421

0.4997 0.0502 0.4997 0.0797 -0.0621 1.865632 0.940518

0.4939 0.0504 0.4939 0.08 -0.063 1.866873 0.941379

0.4989 0.05 0.4996 0.0794 -0.0506 1.868005 0.941603

0.4994 0.0502 0.4994 0.0796 -0.062 1.871196 0.943278

0.4993 0.0502 0.4993 0.0796 -0.062 1.8712 0.943283

0.4894 0.05 0.4896 0.0793 -0.0598 1.871454 0.94359

0.4975 0.05 0.4982 0.0793 -0.0506 1.873647 0.944425

0.4971 0.0502 0.4978 0.0796 -0.0509 1.87469 0.945011

0.4938 0.05 0.494 0.0792 -0.0591 1.876891 0.946132

0.4923 0.0501 0.493 0.0794 -0.0513 1.877177 0.946354

0.4936 0.05 0.4939 0.0792 -0.0575 1.877379 0.946359

0.4999 0.0504 0.4999 0.0798 -0.062 1.877762 0.946569

0.4908 0.0501 0.4908 0.0793 -0.0627 1.879383 0.947565

0.4877 0.05 0.4884 0.0792 -0.0517 1.879673 0.947698

0.471 0.0502 0.4717 0.0795 -0.0542 1.88145 0.949207

0.4869 0.0507 0.4871 0.0802 -0.0607 1.883447 0.949816

0.4953 0.05 0.496 0.0791 -0.0506 1.884991 0.950078

0.4864 0.0501 0.4864 0.0792 -0.0632 1.885175 0.950572

0.4928 0.0503 0.4928 0.0795 -0.0626 1.885922 0.950795

0.4947 0.05 0.4948 0.079 -0.0603 1.887673 0.951461

0.485 0.0506 0.4857 0.08 -0.0527 1.888394 0.952267

0.4997 0.0523 0.4997 0.0826 -0.0641 1.889686 0.953062

0.4958 0.0512 0.4965 0.0809 -0.0521 1.890823 0.953297

0.4905 0.0506 0.4908 0.0799 -0.0583 1.891696 0.953709

0.499 0.05 0.499 0.0789 -0.0613 1.892746 0.953882

0.496 0.05 0.4962 0.0789 -0.0584 1.893782 0.954403

0.4868 0.05 0.4875 0.0789 -0.0513 1.896715 0.956116

0.4982 0.0504 0.4989 0.0795 -0.0504 1.898137 0.956583

112

0.4994 0.0507 0.4994 0.0799 -0.0619 1.898863 0.957107

0.4986 0.0507 0.4986 0.0799 -0.062 1.898891 0.957145

0.494 0.05 0.4942 0.0788 -0.0585 1.899569 0.957322

0.4868 0.0503 0.4875 0.0793 -0.0517 1.900993 0.958315

0.4896 0.0511 0.4899 0.0805 -0.0588 1.902504 0.959227

0.4934 0.0504 0.4941 0.0794 -0.0509 1.904024 0.959629

0.4963 0.05 0.4963 0.0787 -0.0613 1.904287 0.95967

0.4947 0.0507 0.4947 0.0798 -0.0624 1.904689 0.960134

0.4806 0.05 0.4808 0.0787 -0.0601 1.905828 0.960843

0.4877 0.0509 0.4877 0.0801 -0.0635 1.905856 0.961002

0.489 0.0503 0.4897 0.0792 -0.0512 1.906607 0.961008

0.4999 0.0506 0.4999 0.0796 -0.0616 1.906894 0.961031

0.495 0.05 0.4957 0.0787 -0.05 1.907842 0.961342

0.4897 0.0503 0.4897 0.0791 -0.0625 1.908762 0.962182

0.499 0.05 0.499 0.0786 -0.0609 1.90996 0.962393

0.4948 0.05 0.4949 0.0786 -0.0598 1.910571 0.962771

0.4974 0.0509 0.4976 0.08 -0.0589 1.912107 0.963683

0.4962 0.0507 0.4969 0.0797 -0.0507 1.91383 0.964461

0.4988 0.0501 0.4995 0.0787 -0.0494 1.916812 0.965677

0.4982 0.053 0.4982 0.0832 -0.0645 1.917386 0.967014

0.4848 0.05 0.4855 0.0785 -0.051 1.919867 0.967578

0.4973 0.0508 0.4975 0.0797 -0.0586 1.920241 0.967672

0.4927 0.0505 0.4927 0.0792 -0.062 1.921048 0.968221

0.4866 0.0503 0.4869 0.0789 -0.0576 1.921877 0.968629

0.4947 0.0501 0.4954 0.0786 -0.0498 1.922794 0.968745

0.4997 0.0506 0.4997 0.0793 -0.0612 1.924115 0.969551

0.4936 0.05 0.4943 0.0784 -0.0497 1.925327 0.969997

0.487 0.0503 0.487 0.0788 -0.0624 1.926202 0.970889

0.4917 0.0504 0.4924 0.079 -0.0505 1.927123 0.971057

0.4951 0.05 0.4951 0.0783 -0.061 1.927564 0.971217

0.4848 0.05 0.4848 0.0783 -0.0623 1.927935 0.971723

0.4998 0.0501 0.4998 0.0784 -0.0604 1.930755 0.972684

0.4995 0.05 0.4995 0.0782 -0.0603 1.933294 0.973919

0.4979 0.05 0.4981 0.0782 -0.0572 1.934303 0.97436

0.4944 0.0501 0.4951 0.0784 -0.0495 1.93452 0.974531

0.4972 0.0504 0.4972 0.0788 -0.061 1.935009 0.974954

0.4825 0.05 0.4829 0.0782 -0.0558 1.935921 0.975569

0.4939 0.0501 0.4939 0.0783 -0.061 1.936846 0.975873

0.489 0.05 0.4897 0.0782 -0.0499 1.937284 0.976026

0.4971 0.0504 0.4978 0.0788 -0.0495 1.938579 0.976538

0.491 0.0508 0.4914 0.0794 -0.0557 1.938834 0.976983

0.4903 0.0505 0.4905 0.0789 -0.0586 1.939559 0.977337

0.4844 0.05 0.4844 0.0781 -0.0621 1.939741 0.977572

0.4956 0.051 0.4963 0.0797 -0.0505 1.941056 0.977977

0.4983 0.0515 0.4983 0.0804 -0.0621 1.942241 0.978827

0.4856 0.0521 0.4863 0.0814 -0.0533 1.942974 0.979574

0.4915 0.0506 0.4922 0.079 -0.0503 1.945492 0.980169

0.4839 0.05 0.4839 0.078 -0.062 1.9457 0.980533

113

0.4912 0.0503 0.4919 0.0785 -0.0498 1.947275 0.980968

0.4996 0.0506 0.4996 0.0789 -0.0607 1.947466 0.981103

0.4997 0.05 0.4997 0.0779 -0.0599 1.951108 0.982726

0.4988 0.05 0.4988 0.0779 -0.06 1.951139 0.982767

0.4968 0.0501 0.4975 0.0781 -0.0487 1.952211 0.983179

0.4945 0.05 0.4947 0.0779 -0.0572 1.952258 0.983332

0.4987 0.05 0.4994 0.0779 -0.0483 1.954744 0.984345

0.4971 0.0502 0.4978 0.0782 -0.0487 1.955568 0.984853

0.4939 0.0501 0.4946 0.078 -0.0489 1.958321 0.986274

0.4972 0.0503 0.4979 0.0783 -0.0488 1.95893 0.986536

0.491 0.0506 0.491 0.0787 -0.0615 1.95962 0.987375

0.4908 0.0506 0.4908 0.0787 -0.0615 1.959628 0.987385

0.4933 0.0504 0.494 0.0784 -0.0493 1.962456 0.988415

0.4899 0.05 0.4899 0.0777 -0.0608 1.96348 0.989138

0.4961 0.05 0.4963 0.0777 -0.0567 1.964237 0.989204

0.4817 0.05 0.4819 0.0777 -0.0585 1.964786 0.989913

0.4873 0.0509 0.4876 0.0791 -0.0573 1.965321 0.990229

0.4939 0.0501 0.4939 0.0778 -0.0604 1.96672 0.990649

0.4941 0.0517 0.4942 0.0803 -0.0607 1.966858 0.991121

0.4709 0.05 0.4716 0.0777 -0.0515 1.968031 0.99174

0.4972 0.05 0.4972 0.0776 -0.0598 1.969284 0.991792

0.4879 0.0501 0.4886 0.0778 -0.0493 1.970646 0.992523

0.4976 0.0503 0.4976 0.078 -0.06 1.973341 0.993874

0.4945 0.0503 0.4945 0.078 -0.0604 1.973449 0.994019

0.4999 0.05 0.4999 0.0775 -0.0593 1.975284 0.994683

0.498 0.0501 0.4987 0.0777 -0.0479 1.976301 0.995019

0.4961 0.05 0.4963 0.0775 -0.0564 1.976396 0.995213

0.4999 0.0515 0.4999 0.0798 -0.0611 1.97728 0.996104

0.4964 0.0514 0.4971 0.0797 -0.05 1.977762 0.996151

0.4985 0.0517 0.4987 0.0801 -0.0582 1.978964 0.996906

0.4894 0.0506 0.4897 0.0784 -0.0563 1.979188 0.996918

0.4888 0.0502 0.4895 0.0778 -0.0491 1.980078 0.997173

0.4986 0.0507 0.4986 0.0785 -0.0602 1.980696 0.997599

0.4999 0.05 0.4999 0.0774 -0.0592 1.981406 0.997713

0.4995 0.05 0.4995 0.0774 -0.0593 1.981419 0.997731

0.4884 0.0501 0.4891 0.0776 -0.049 1.982816 0.998505

0.4871 0.0509 0.4874 0.0788 -0.0569 1.983292 0.999105

0.4908 0.0514 0.4915 0.0796 -0.0505 1.983969 0.999377

0.4945 0.0515 0.4947 0.0797 -0.0583 1.984416 0.999662

0.4912 0.0501 0.4912 0.0775 -0.0603 1.985103 0.999824

0.4936 0.0511 0.4943 0.0791 -0.0497 1.985872 1.000144

0.4854 0.05 0.4856 0.0773 -0.0574 1.989081 1.001799

0.4912 0.0514 0.4919 0.0795 -0.0503 1.989941 1.002306

0.4902 0.05 0.4909 0.0773 -0.0484 1.991646 1.002776

0.4994 0.0507 0.4994 0.0783 -0.0599 1.992824 1.003577

0.4954 0.0516 0.4956 0.0797 -0.0581 1.99367 1.004235

0.4973 0.0502 0.4973 0.0775 -0.0595 1.994443 1.004296

0.4915 0.0503 0.4922 0.0777 -0.0485 1.99562 1.004782

114

0.4875 0.0503 0.4882 0.0777 -0.049 1.995793 1.004984

0.4995 0.0501 0.4995 0.0773 -0.059 1.997161 1.005551

0.4963 0.0504 0.497 0.0778 -0.048 1.998807 1.006245

0.499 0.0508 0.4997 0.0784 -0.0482 1.999896 1.006817

0.4986 0.0509 0.4988 0.0785 -0.0567 2.000556 1.00734

0.4972 0.0502 0.4972 0.0774 -0.0594 2.00064 1.007366

0.4991 0.0531 0.4993 0.0819 -0.0594 2.000893 1.008132

0.4902 0.0505 0.4909 0.0779 -0.0488 2.002454 1.008244

0.4897 0.0506 0.49 0.078 -0.0557 2.003616 1.008971

0.4912 0.0501 0.4912 0.0772 -0.0599 2.00367 1.009011

0.4985 0.0506 0.4992 0.078 -0.0478 2.005484 1.009531

0.4987 0.0501 0.4994 0.0772 -0.0471 2.00714 1.010206

0.4974 0.0507 0.4981 0.0781 -0.048 2.008908 1.011276

0.4997 0.0501 0.4997 0.0771 -0.0587 2.009628 1.011716

0.4972 0.0504 0.4979 0.0776 -0.0476 2.011176 1.012319

0.494 0.0505 0.4941 0.0777 -0.0582 2.011428 1.012802

0.4918 0.05 0.4918 0.0769 -0.0595 2.012766 1.013466

0.4973 0.0503 0.498 0.0774 -0.0473 2.014018 1.01369

0.4945 0.0511 0.4948 0.0786 -0.0555 2.014081 1.014142

0.4845 0.05 0.4848 0.0769 -0.0553 2.014596 1.014373

0.4956 0.0504 0.4963 0.0775 -0.0476 2.017494 1.015478

0.498 0.0509 0.498 0.0782 -0.0597 2.018077 1.016171

0.4995 0.05 0.4996 0.0768 -0.0567 2.019307 1.016404

0.4987 0.0502 0.4987 0.0771 -0.0588 2.019361 1.016591

0.4972 0.0503 0.4979 0.0773 -0.0472 2.0203 1.016791

0.4965 0.0511 0.4972 0.0785 -0.0484 2.022419 1.018072

0.4945 0.0504 0.4952 0.0774 -0.0476 2.023822 1.018628

0.4922 0.0507 0.4922 0.0778 -0.0601 2.024009 1.01922

0.4979 0.0509 0.4979 0.0781 -0.0596 2.024306 1.019257

0.4995 0.05 0.4995 0.0767 -0.0583 2.025157 1.019383

0.4995 0.05 0.4996 0.0767 -0.0566 2.025654 1.019544

0.4962 0.0502 0.4962 0.077 -0.0589 2.025768 1.019833

0.4915 0.0505 0.4922 0.0775 -0.048 2.027356 1.020482

0.4983 0.0501 0.4983 0.0768 -0.0585 2.028626 1.021161

0.4927 0.05 0.4934 0.0767 -0.0471 2.02922 1.021231

0.499 0.0508 0.4997 0.0779 -0.0474 2.030928 1.022117

0.4998 0.05 0.4998 0.0766 -0.0582 2.031525 1.022527

0.495 0.05 0.495 0.0766 -0.0587 2.031688 1.022741

0.4972 0.0503 0.4979 0.0771 -0.0469 2.032952 1.023031

0.4979 0.0511 0.4986 0.0783 -0.0479 2.03483 1.024152

0.4984 0.0501 0.4984 0.0767 -0.0584 2.035005 1.024317

0.4995 0.05 0.4995 0.0765 -0.0581 2.037945 1.025715

0.4996 0.05 0.4997 0.0765 -0.0563 2.038442 1.02587

0.4975 0.0503 0.4982 0.077 -0.0467 2.039315 1.026161

0.4844 0.05 0.4847 0.0765 -0.0547 2.040068 1.026959

0.4914 0.05 0.4921 0.0765 -0.0469 2.042095 1.027616

0.4999 0.05 0.4999 0.0764 -0.0579 2.044375 1.028889

0.4973 0.0503 0.498 0.0769 -0.0465 2.045731 1.029331

115

0.499 0.0501 0.499 0.0765 -0.058 2.047848 1.03066

0.4973 0.0506 0.498 0.0773 -0.0469 2.04958 1.031309

0.4964 0.0501 0.4971 0.0765 -0.0462 2.051796 1.032294

0.4956 0.0514 0.4956 0.0784 -0.0598 2.053782 1.034053

0.4914 0.0501 0.4914 0.0764 -0.0588 2.054592 1.034211

0.4864 0.051 0.4867 0.0778 -0.0554 2.054934 1.034531

0.4999 0.0507 0.4999 0.0773 -0.0585 2.055463 1.034569

0.4998 0.05 0.4999 0.0762 -0.0559 2.057873 1.035483

0.4894 0.051 0.4894 0.0777 -0.06 2.059614 1.037009

0.4988 0.0542 0.499 0.0826 -0.0594 2.059943 1.037627

0.4996 0.0507 0.4996 0.0772 -0.0584 2.061917 1.037773

0.4972 0.0511 0.4974 0.0778 -0.0557 2.063754 1.038679

0.4925 0.05 0.4925 0.0761 -0.0584 2.064161 1.03889

0.4707 0.0502 0.4707 0.0764 -0.0613 2.065339 1.040186

0.4954 0.0525 0.4955 0.0799 -0.0593 2.065757 1.040199

0.4991 0.0511 0.4991 0.0777 -0.0587 2.06908 1.041443

0.4907 0.052 0.491 0.0791 -0.0558 2.069307 1.041789

0.4993 0.05 0.4993 0.076 -0.0574 2.070508 1.041848

0.4936 0.0506 0.4936 0.0769 -0.0588 2.071691 1.042753

0.4991 0.0527 0.4992 0.0801 -0.0589 2.072203 1.043333

0.4971 0.0503 0.4971 0.0764 -0.0579 2.07436 1.043896

0.4902 0.0503 0.4905 0.0764 -0.0535 2.076212 1.044738

0.4991 0.05 0.4991 0.0759 -0.0573 2.077128 1.045133

0.4997 0.05 0.4998 0.0759 -0.0555 2.077621 1.045261

0.4976 0.05 0.4978 0.0759 -0.054 2.078223 1.045528

0.4995 0.0514 0.4995 0.078 -0.0588 2.079215 1.046532

0.4981 0.0503 0.4983 0.0763 -0.0542 2.081965 1.047446

0.4923 0.0513 0.4923 0.0778 -0.0595 2.082566 1.048368

0.4997 0.05 0.4998 0.0758 -0.0554 2.084271 1.048554

0.4937 0.05 0.4939 0.0758 -0.0543 2.085018 1.048993

0.4994 0.0501 0.4994 0.0759 -0.0572 2.087237 1.05016

0.488 0.0509 0.4881 0.0771 -0.0577 2.08923 1.051575

0.4942 0.05 0.4942 0.0757 -0.0576 2.090627 1.051951

0.4954 0.05 0.4956 0.0757 -0.0539 2.091643 1.052226

0.4991 0.0501 0.4991 0.0758 -0.0571 2.093936 1.053486

0.4971 0.0503 0.4971 0.0761 -0.0575 2.094244 1.053745

0.4988 0.0503 0.499 0.0761 -0.0538 2.095239 1.053995

0.4992 0.0504 0.4992 0.0762 -0.0573 2.097631 1.055393

0.4917 0.05 0.4919 0.0756 -0.0542 2.098502 1.055718

0.4932 0.0505 0.4932 0.0763 -0.0581 2.101292 1.057396

0.4999 0.05 0.4999 0.0755 -0.0567 2.103907 1.05838

0.4992 0.05 0.4994 0.0755 -0.0532 2.104989 1.058731

0.4994 0.0508 0.4995 0.0767 -0.0559 2.105247 1.059158

0.4996 0.0516 0.4996 0.0779 -0.0585 2.105508 1.059603

0.4998 0.0509 0.4999 0.0768 -0.0559 2.108665 1.060866

0.4959 0.0502 0.4959 0.0757 -0.0572 2.111008 1.062056

0.498 0.0502 0.4981 0.0757 -0.0552 2.111462 1.062115

0.4998 0.0507 0.4998 0.0764 -0.0573 2.114674 1.063899

116

0.499 0.0513 0.499 0.0773 -0.058 2.115181 1.06433

0.4997 0.05 0.4997 0.0753 -0.0564 2.117531 1.065136

0.4987 0.05 0.4987 0.0753 -0.0566 2.117564 1.065178

0.4957 0.05 0.4957 0.0753 -0.0569 2.117666 1.065307

0.4994 0.0501 0.4994 0.0754 -0.0565 2.121036 1.066906

0.4994 0.0503 0.4994 0.0757 -0.0567 2.12117 1.067025

0.4981 0.05 0.4981 0.0752 -0.0565 2.124448 1.068605

0.4999 0.0502 0.4999 0.0755 -0.0565 2.124509 1.06864

0.4997 0.0502 0.4997 0.0755 -0.0565 2.124516 1.068649

0.4993 0.0502 0.4993 0.0755 -0.0566 2.124529 1.068666

117

APPENDIX 2 ANALYSIS PARAMETERS INFORMATION

LOAD CASE MULTIPLIERS

Static Stress with Linear Material Models may have multiple load cases. This allows a

model to be analyzed with multiple loads while solving the equations a single time. The

following is a list of load case multipliers that were analyzed with this model.

Load Case

Pressure/Surface Forces

Acceleration/Gravity Displaced Boundary

Thermal Voltage

1 1 0 0 0 0

MULTIPHYSICS INFORMATION

Default Nodal Temperature 0 °C

Source of Nodal Temperature None

Time step from Heat Transfer Analysis Last

PROCESSOR INFORMATION

Type of Solver Automatic

Disable Calculation and Output of Strains No

Calculate Reaction Forces Yes

Invoke Banded Solver Yes

Avoid Bandwidth Minimization No

118

Stop After Stiffness Calculations No

Displacement Data in Output File No

Stress Data in Output File No

Equation Numbers Data in Output File No

Element Input Data in Output File No

Nodal Input Data in Output File No

Centrifugal Load Data in Output File No

119

PART INFORMATION

Part ID

Part Name Element

Type Material Name

1 Coupling device

21.par Brick

AISI 1045 Steel, cold drawn, 19-32 mm (0.75-

1.25 in) round

Element Properties used for:

• Coupling device 21.par

Element Type Brick

Compatibility Not Enforced

Integration Order 2nd Order

Stress Free Reference Temperature 0 °C

MATERIAL INFORMATION

AISI 1045 Steel, cold drawn, 19-32 mm (0.75-1.25 in) round - Brick

Material Model Standard

Material Source Algor Material Library

Material Source File C:\Program Files\ALGOR\MatLibs\algormat.mlb

Date Last Updated 2004/10/28-16:02:00

Material Description None

120

Mass Density 0.00000000785 N*s^2/mm/mm³

Modulus of Elasticity 205000 N/mm²

Poisson's Ratio 0.29

Shear Modulus of Elasticity 80000 N/mm²

Thermal Coefficient of Expansion 1.150000E-005 1/°C

121

LOAD AND CONSTRAINT INFORMATION

LOADS

Surface Forces

ID Description Part ID

Surface ID

Magnitude Vx Vy Vz

1 Unnamed 1 3 950 0 0 -1

CONSTRAINTS

Constraint Set 1: Unnamed

7.1.1.1 Surface Boundary Conditions

ID Description Part ID

Surface ID

Tx Ty Tz Rx Ry Rz

1 Unnamed 1 18 Yes Yes Yes Yes Yes Yes





Reaction Sums and Maxima for Load Case

122

Sum of applied forces

X-Force Y-Force Z-Force X-Moment Y-Moment Z-Moment

0.0000E+00 0.0000E+00 -9.5000E+02 0.0000E+00 0.0000E+00 0.0000E+00

Sum of reactions


-4.1706E-13 -3.8222E-13 1.1645E-12 0.0000E+00 0.0000E+00 0.0000E+00

Sum of residuals


-4.1706E-13 -3.8222E-13 -9.5000E+02 0.0000E+00 0.0000E+00 0.0000E+00

Sum of unfixed direction residuals


6.1394E-13 -2.6783E-13 1.7833E-12 0.0000E+00 0.0000E+00 0.0000E+00

Largest applied forces and moments

Node Node Node Node Node Node


0 0 1379 0 0 0

0.0000E+00 0.0000E+00 -3.9983E+00 0.0000E+00 0.0000E+00 0.0000E+00

Largest nodal reactions



7086 37 1168 0 0 0

7.8538E-01 1.3990E+00 -5.0153E+00 0.0000E+00 0.0000E+00 0.0000E+00

Largest nodal residuals



7086 37 1168 0 0 0

7.8538E-01 1.3990E+00 -5.0153E+00 0.0000E+00 0.0000E+00 0.0000E+00

Largest unfixed direction residuals



19791 24644 24664 0 0 0

2.6236E-13 2.9465E-13 5.4312E-13 0.0000E+00 0.0000E+00 0.0000E+00

123

MESHING RESULTS

Surface Mesh Statistics

Mesh operation Solid mesh

Final mesh size 0.87 mm

Elements created 10076

Solid Mesh Statistics

Mesh type Mix of bricks, wedges, pyramids and tetrahedra

Watertight Yes

Mesh has microholes No

Total nodes 28553

Volume 13300.1 mm^3

Total elements 32123

Tetrahedra Pyramids Wedges Bricks

Elements 9580 5211 1953 15379

Volume % 4.31 5.05 4.69 85.96

124

Tetrahedra Pyramids Wedges Bricks

Max. length ratio 346.5 79.6 4.9 4.6

Avg. length ratio 5.5 3.3 1.8 1.3

Avg. aspect ratio 1.3 1.3 1.1 1

Unconstrained aspect ratio 4.8 3.8 1.5 1.5

125

SUPERVIEW PRESENTATION IMAGES

STRESS

126

STRAIN

DISPLACEMENT

127

DEFORMED SHAPE

optimal kinematic design cvt - laniaemezura/util/files/thesis-ramiro.pdf · optimal kinematic...

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