THE SYNTHESIS OF SPEED-RECTIFYING

MECHANISMS FOR MECHATRONIC

APPLICATIONS

Oscar R. Navarro-Martinez

Department of Mechanical Engineering

McGill University, Montreal

-4 Thesis submitted to the Faculty of Graduate Studies and Research

in partial fulfilment of the requircments for the degree of

Master of Engineering

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ABSTRACT

ABSTRACT

Slider-crank mechanisms are widely used in reciprocating machinew where transfor-

mation from rotational into translational motion (or vice versa) is required. Moreover,

the use of these mechanisms is quite common in robotic and mechatronic systems

when comples motions are to be produced with rotational actuators. However. the

velocity ratio of the slider-crank rnechanism is configuration-dependent and thus-

elaborate algorithrns are required to precisely control its performance.

In this thesis, a planar Cam mechanism with an oscillating follower is proposed

as a device that renders the velocity ratio of the slider-crank mechanism constant, an

operation that is termed here uelocity-ratio rectification: i t is espected that the recti-

fication will ease the feedback control of the slider-crank mechanism in mechatronic

applications. -4 methodology for the optimization of t his mechanism is developed.

First. the performance of the slider-crank mechanisrn is analyzed and optimum geo-

rnctric parameters are obtained. Then, an expression for the input-output relation

of the Cam mechanism a t hand is derived, and the corresponding displacement pro-

gram of the follower is produced. In addition, an approach for the optimization of

planar Cam mechanisms with an oscillating follower is introduced, to minimize the

O\-erall size, while maintaining an acceptable force-transmission performance based on

boiinds on the pressure angle. -4 Graphical User Interface (GUI) is devcloped to allow

for the above-mentioned optimization in an interactive m o d e the GUI is successfully

used in an esample of the design of a reducer-rectifier Cam mechanism that rectifies

t he velocity ratio of the actuator of a robotic quadruped.

Les mécanismes bielle-manivelle sont couramment utilisés dans les machines réciprocantes.

où il est nécessaire de transformer un mouvement de rotation en mouvement de trans-

lation (ou vice versa). L'utilisation de ces mécanismes est égalment fréquente en

robotique et en mécatronique quand des mouvements complexes doivent étre realisés

à l'aide d'actionneurs rotatifs. Cependant, le rapport de vitesse dans le mécanisme

bielle-manivelle est function de la configuration du mécanisme, ce qui exige des algo-

rithmes élaborés pour commander leur performance avec précision.

Dans cette thèse, l'auteur propose un mécanisme plan à cames avec bras oscillant

pour rendre constant ledit rapport de vitesse. une opération appelée rectification de

vitesse, avec le but de faciliter la commande asservie du mécanisme bielle-manivelle.

d'intérêt particular en systèmes mécatroniques. Une méthodologie pour l'optimisation

de ces mécanismes est developpée. Premièrement. la performance des mécanismes

bielle-manivelie est analysée, obtenant ainsi ses paramétres géométriques optimaux.

Ensuite. une expression pour la relation d'entrée-sortie du mécanisme à cames en

question est obtenue et le correspondant programme de déplacement du bras oscil-

lant est produit. En outre. une méthode pour optimiser les mécanismes plans à

cames avec bras oscillant est présentée, pour minimiser la taille globale de l'ensemble.

tout en maintenant un rapport de transmission de force acceptable. Lne interface

graphique mise au point dans le cadre de cette thèse, permet ladite optimisation de

facon interactive: cette interface a été utilisée avec succès lors de l'optimisation d'un

mécanisme de rectification du rapport de vitesse de l'actionneur d'un quadripède.

ACKNOWLEDGEMENTS

ACKNOWLEDGEMENTS

I would like to thank Professor Jorge .Angeles, my thesis supervisor, for his support,

encouragement and guidance in the work reported in this thesis. His profound knowl-

edge in the area, and his concern with preparing high-quality researchers, played a

definite role during the course of my research.

-4s well, 1 would like to thank al1 my colleagues and friends at the McGill Centre

for Intelligent Machines (CIM), who shared with me their tirne and knowledge. Par-

ticularly, 1 would like t o thank my good friend Chu-Jen Wu for insightful discussions

which clarified many concepts around my research area.

1,lany thanks to Dr. Raymond Spiteri for his guidance on numerical analysis

issues. Thanks are also due to CI&I for the state-of-the-art facilities provided and for

the pleasant research environment.

1 am very grateful to -4ndrés Xavarro-Garcia, my father, for his constant en-

couragement and support, which gave me confidence and determination in complet-

irig this thesis. 1 am also indebted to my former teacher and friend, Dr. Max -4.

Gonz5lcz-Palacios, and his family, as well as to Manuel Cruz-Hernandez and Diana

Hern5nciez--4lons0, for making my stay in Montreal a pleasant esperiencc.

TABLE OF CONTENTS

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RESUME iii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES vii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF T-4BLES is

CH-APTER 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 . hlotivation 3 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 - 3. General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

- 3.1. Planar Cam with Oscillating Follower . . . . . . . . . . . . . . . . . I

- 3.2. Ball-Screw Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4. .\.lecharronic Applications 9

CHAPTER 2- Optimization of the Slider-Crank Mechanism . . . . . . . . . 10

. . . . . . . . . . . . . . . . . . . . . . 1 . In-Line Slider-Crank Mechanism 11

. . . . . . . . . . . 1.1. Connecting-Rod-Length-to-Crank-Radius Ratio 12

. . . . . . . . . . . . . . . . . . . . . . '2. Offset Slider-Crank ?vIechanisms 18

. . . . . . . . . . . 2.1. Connecting-Rod-Lengt h-to-Crank-Radius Ratio 21

CH-APTER 3. Cam-Follower Displacement -Analysis . . . . . . . . . . . . . . 25

1. I 0 Displacement Relation of the Cam Mechanism . . . . . . . . . . . . 26

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Numerical Results 31

. . . . . . . . . . . . . . . . . . . . . . CH-4PTER 4 . Displacement Program 34

. . . . . . . . . . . . . . . . . . . . . . . . . . . . CH-APTER 5 . Cam Design 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Pressure Angle 45

'2 . Cam-Size Minimization Under Pressure -Angle Constraints . . . . . . . . 45

. . . . . . . . . . . . . . . . . . . . . . . . 3 . Implementation and Results 32

. . . . . . . . . . . . . . . . . . . . . . . CH-APTER 6 . Concluding Remarks 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Conclusions 59

. . . . . . . . . . . . . . . . . . '2 . Recommendations for Future Research 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 62

LIST OF FIGU'RES

LIST OF FIGURES

Different postures of the FIK quadruped . . . . . . . . . . . . . 2

One leg in reptile-like position . . . . . . . . - . . - . - . . . . 3

One ball-screw unit driven by an electric DC motor . . . . . . 3

Current design of the slider-crank mechanism and its actuator:

(a) transmission layout; (b) velocity ratio , . . . . . . . . - . . 4

Perspective view of the slider-crank mechanism and its actuator 4

Proposed system: (a) layout: (b) rectified velocity ratio . . . . - -3

Perspective view of the proposed system . . . . . . . . . . . . . - 3

Disk Cam with oscillating roller-follower . . . . . . . . . . . . . 8

Antifriction bal1 (a) and roller (b) screw designs (Ricin, 1988) . 8

Geometry of the in-line slider-crank mechanism . . . . . . . . . I l

Range of motion of the input link for different in-line slidcr-crank

mechanisms . . . . . . . . . . . . - . . . . . . . . . . . . . . . 1.5

Variation of the transmission angle for different in-line slider-crank

mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Transmission defect for different in-line slider-crank mechanisms 17

Geometry of the offset slider-crank mechanism . . . . . . . . . 18

Two possible configurations of the offset slider-crank mechanism 19

vii

LIST OF FIGURES

Variation of the transmission angle for different offset slider-crank

mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Transmission defect for different offset slider-crank mechanisms 23

Proposed transmission device . . . . . . . . . . . . . . . . . . . 25

Behaviour of the function g ( x ) = -x2 i- 2ax - b' . . . . . . . . 31

Solution for ,0(1a): (a) lSt case; (b) Znd case . . . . . . . . . . . 33

45-6-7 polynomial . - - - . . . . . . . . . . . . . . . . . . . . . 38

Displacement program of the follower (a) lSt case: (b) case 41

Rectified displacernent and velocity ratio: lSt case . . . . . . . 42

Rectified displacement and velocity ratio: znd case . . . . . . . 13

Cam mechanism with oscillating roller-follower . . . . . . . . . 44

GUI at the beginning of the session . . . . . . . . . . . . . . . 52

GUI with al1 its functions enabled . . . . . . . . . . . - . . . . 53

- - Flowchart of the design procedure . . . . . . . . . . . . . . . . aa

Follower curves for the design parameters of Table 3.1 . - . . . 56

- - Contours for the first rise phase: -30" 5 a 5 30" . . . . . . . . 9 1

Contours for the first rise phase for different bounds of the pressure - - angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -3 i

Pressure-angle distribution for the two solutions of the probleni 58

Pitch curve of the minimum-size carn for the reducer-rectifier

mcchanism . . . . . . . - . . . . . . . . . . . . . . . . . . . . . 58

LIST OF T,QBLES

LIST OF TABLES

Results for different values of K . . . . . . . . . . . . . . . . . 20

Comparison of the two types of slider-crank mechanisms . . . . 24

. . . . . . . . . . . . . . . . . . . . . . . . . Design parameters 33

Optimum parameters for the minimum-size cam . . . . . . . . 5-5

CHAPTER 1. INTRODUCTION

CHAPTER 1

Introduction

Due to the advances in robot technology, mechanical systems are required to com-

pl- with a wide variety of increasingly dernanding specifications, the integration of

concepts from various individual disciplines thus being required. The conjunction of

concepts from computer science, dynamics, control, electronics and mechanical de-

sign. which is often referred t o as mechatronics_ is needed for the optimum design

of these systems. The espected result is mechanical systems with a host of features

likc high velocities, high accelerations. high accuracy for pick-and-place operations,

as w l l as precise motion control in the trajectory planned.

In particular, transmission systems play an import,ant role in robotic mechani-

cal systems because of weight and space considerations, i-e., p o w r sources are of-

ten located at some distance from the point of actual force application, and hence:

powcr-transmission devices are unavoidable. However, in some cases, the use of these

devices introduces a nonlinear relation between the input and output variables in the

transmission, and consequentl~ data-processing equipment and comples control al-

gorithms are required for the proper operation of the whole system. This thesis thus

proposes a novel transmission aimed a t the rectification of the input-output (I/O)

nonlinear relations in robotic and mechatronic systems.

1. Motivation

The motivation behind this thesis Iies in the use of mechanka1 transmissions in the

articulated links of robotic mechanical systems of a complexity similar to four-legged

walking machines. Shown in Fig. 1.1 is such a machine, currently under development

at Forchungszentrum Informatik Karlsruhe (FIK) , at Karlsruhe University. Gerrnany

(Cordes et al.. 199'7).

n B Mamfr&like iayaut of the legs

Figure 1.1: Different postures of the FIK quadruped

The transmission used in the machine of Fig. 1.1 consists of a DC motor. a

ball-screw unit, and a slider-crank mechanism. Figure 1.2 shows one of the legs of

the machine, u-here the slider-crank mechanisms are apparent; in this case. thesc

mcchanisms are driven by bail-screw actuators: a ball-screw unit driven by a DC

motor. as showi in Fig. 1.3.

The use of slider-crank mechanisms introduces a configuration-dependent rela-

tionstiip betiveen the velocities of the slider and the crank, and hence the control

Figure 1.2: One leg in reptile-like position

- - --

Figure 1.3: One ball-screw unit driven by an electric DC motor

strate= for the overall system becomes complicated and may require comples data-

proccssing hardware. A plot of the input-output (I/O) velocity ratio s'($~): as well

as the actual design of the transmission device used in the legs of the FIK quadruped

arc sketched in Figs. 1.4 and 1.5. It would be desirable to have a constant velocity

ratio that should lead to a simple control scheme. thereby allowing for an enhanced

performance of the overall machine. We cal1 the task of producing a constant velocit~.

ratio from a configuration-dependent ratio velocity-ratio rectification.

\Vc propose in this thesis to produce the velocity-ratio rectification via an inter-

mediate mechanisrn, namely, a planur cam-follower mechanism. Figure 1.6 shows a

Figure 1.4: Current design of t h e slider-crank mechanism and its actuator: (a) transmission layout: (b) velocity ratio

v/' Figure 1.5: Perspective view of the slider-crank mechanism and its actuator

general scheme of the proposed system with the corresponding rectification in the

1/0 \-elocity ratio, while a perspective view of the system is shown in Fig. 1.7.

2. Literature Review

Sonlinearity between the input and output variables of the slider-crank mecha-

nism of Fig. 1.5 has been handled either by means of a table look-up method (Arakatva

s

(a) (b 1

Figure 1.6: Proposed system: (a) layout; (b) rectified velocitj- ratio

Figure 1.7: Perspective view of the proposed system

et aI.. 1993): or by complex control strategies (Cordes et al., 1997). The introduction

of a planar Cam mechanism to rectify this nonlinear behaviour seems to be a novel

idca, and is claimed to be a major contribution of this thesis.

The combination of Cam mechanisms with linkages has been used in the past to

improve their independent performance or to produce motions with suitable dynamic

behaviour. For example, Rothbart (1956) proposed a variable-speed Cam mechanism,

in which the input of the Cam is the output of a Whitworth quick-return mecha-

nism, as a means of reducing the Cam size while increasing force transmission, while

-1marnat h and Gupta ( 1978) designed a novel cam-linkage mechanism for mu1 tiple

dwell generation. However, literature related t o the design of Cam mechanisms as

reducer-rectifier devices is rather scarce, to the best of our knowledge. One can only

mention the contribution of Milligan and Angeles (1995), who reported on the design

of a cam mechanism to recti- the output of a universal joint and, a t the same time,

reduce the speed. -4s well, Gonzalez-Palacios and Angeles (1998) designed a novel

mechanical transmission based on Cam mechanisms to overcome gear-transmission

drawbacks such as backlash and friction.

The cam-design problem has been given due attention by many a researcher. The

study of the dynamics and kinematics of these mechanisms has been studied in the

past to some extent ( Hrones, 1948; Johnson, 1955, 1956; Rothbart: 1956: Fenton,

1966-a: Berzak, 1982; Gonzalez-Palacios and -hgeles, 1993). Moreover, since large

carn volume implies large inertial forces that may result in high contact forces and

stresses between the cam and the follower, it is a n objective in Cam design to include

niinimization of carn mass by reducing its size.

Cam-site minirnization, hence. has also been estensively studied. First? simple

aIge braic (Fenton, 1966- b) and graphical (Hirschhorn, 1962) solutions were used. but

since the advent of digital computers, methods of higher cornpiexit? were developed

for cam-size minimization under pressure angle and contact stress constraints by

Fenton (1945) and LoefF and Soni (1975). More r e c e n t l ~ Terauchi and El-Shakery

(1983) and Chan and Kok (1996) reported progress in this area. Of relevance to

tiic work presented on this thesis are the works of -Angeles and Lopez-Cajun (1991).

wlio presented an estensive study for the optimization of planar carn mechanisms. and

that of \Vu (1998), who deveioped a unified approach for planar cam-size minimization

undcr pressure angle-bounds.

1.3 GEXERAL BACKGROUND

General Background

Different types of mechanisrns are involved in the transmission system proposed

in this thesis, and thus, a brief description of these mechanisms is needed t o fully

understand their purpose and importance. In this section Ive will Focus only on the

planar cam-follower mechanism and in the ball-screw unit, leaving the description

and analysis of the slider-crank mechanism for Chapter 2.

3.1. Planar Cam with Oscillating Follower. -4 Cam is a mechanical ele-

ment that drives another element, known as the follower, through a specified motion.

-4 Cam mechanism usually consists of: a Cam, normally driven by a known input

motion: a follower, whose motion can be arbitrarily described by a periodic function:

and a frame, in which the Cam and folloiver are supported. A type of Cam mechanism

contains a fourth element, a roller, attached to the follower. -4s the Cam rotates, the

roller rolls on the Cam profile, this rolling action helping reduce wear and, therefore,

the roller-follower is often preferred over followers that have sliding contact.

Figure 2.8 shows the type of Cam mechanism used in this thesis: a disk cam

and a roller-follower element with oscillating motion. During the rotation of the

cam through one cycle of input aot ion, the follower undergoes a periodic motion.

the input-output motion being described by the displacement program, which is the

starting point in Cam design. In the plot of the displacement program. the abscissa

rcprcsents one cycle of the input displacement, while the ordinate represents the

corresponding follower displacement, which is called rise if? roughly speaking, the

follower moves away from the Cam centre; dwell if the follower is a t rest: and return

if the follower moves towards the Cam centre.

3.2. Ball-Screw Units. -4 ball-screw unit is a power-transmission devicc

niainly used as a means of transforming rotary into t ra~s la tory motion. A bal1 screw

is siniply a screw that runs on bal1 or roIIer bearings, as shown in Figs. 1.9a and 6:

the transmission consists of screw lit nut 2: set of balls 3' or rollers 3"? and returning

tubes 4 to carry the rolling elements from one end of the nut to the other.

Figure 1.8: Disk Cam with oscillating roller-follower

Figure 1.9: -Antifriction bal1 (a) and roller (b) screw designs (Rivin, 1988)

Bal1 screws are often used in mechanical transmissions where friction is unde-

sirable. The design of these devices allows for replacing sliding friction with rolling

friction. and hence' their efficiency is 90%, or even higher. Other advantages of the

baIl screws are

0 high transmission ratio;

meh ina t ion of backlash by preloading of the nut;

Ion- friction losses; - ionr starting torque ;

accurate positioning and repeatabilitv

high transmitted forces at relatively sniall sizes; and - predictable life expectancy.

4. Mechatronic Applications

Contra- to industrial robots, that are instalied on a fived base, and hence, use

conventional industrial facilities for their power supply and their controls: mecha-

tronic systems, such as waiking machines, rovers, and the like, cal1 for autonomous

operations. Mechatronic systems, thus, carry their own power-supply and control

subsystems. In these cases, then, weight and speed of response become crucial design

criteria. To ease the control of these systems, and consequently, to lower the demands

on the control hardware, we propose to rectify the configuration-dependent velocity

ratios in their transmissions, by rneans of simple and reliable mechanisms. It can be

argued that the addition of a transmission stage will offset the benefit of a constant

velocity ratio. Nevertheless. current research work a t the Robotic Mechanical Systems

Laboratory of the McGill Centre for Intelligent Machines (CIM) aims a t developing

mechanical transmissions that integrate, in one single unit, two functions, speed re-

duction and speed rectification. Therefore, the transmissions that ive are proposing

here will be an integral part of the actuator. In other words, we aim at systems that

wili do away with conventional gear trains for speed reduction, while replacing t hese

with more efficient, stiffer, and more reliable and lighter multifunction transmissions.

CHAPTER 2. OPTIiMIZ.4TION OF THE SLIDER-CR4NK MECEGhJISM

CHAPTER 2

Opt imization of the Slider-Crank

Mechanism

I t is apparent from Fig. 1.6 that force and motion are transmitted to the load by means

of a shder-crank rnechanism. The dimensioning and selection of this mechanism is

thus crucial in the design of the overall transmission.

The slider-crank mechanism transforrns rotational into translational motion (or

vice versa): ît is, therefore, widely used in reciprocating machinery such as piston

engines. cornpressors, pumps: saws, etc.

Figure 2.1 shows the geometry of the basic slider-crank mechanism, ivbere O

represents the front view of the asis of the crankshaft; l2 the length of the crank: 1,

the length of the connecting rod: and point C? the wrist pin that joins the connecting

rod n i th the slider. Moreover, the angle 4 represents an angular position of the crank

corresponding to the displacement s of the slider.

In Fig. 2.1 the trajecton; of the wrist pin C is a line passing through the pin joint

centre 0: for this reason this layout is cailed an in-line slider-crank mechanism. If

the motion of the slider were offset mith respect to 0: the linkage would be an oflset

dider-crank mechanism.

i s _ I

Figure 2.1: Geometry of the in-line slider-crank mechanism

The objective of this chapter is thus to compare the performance of the in-line

and the offset linkages, so as to determine which one is better in terms of the force

or torque transmitted.

1. In-Line Slider-Crank Mechanism

For Our purposes, we consider the case in which the motion of the slider is the

input of the system and the output is the motion of the crank. Note. however, that

the alternative case could be readily analyzed in the same rnanner.

First. ive derive the necessary conditions t o ensure the mobility of the Iinkage

under study, for which an expression for the output variable 4 in terms of the input

variable s is required. Referring to Fig. 2.1, by application of the Iau- of cosines. we

have

1; + s2 - l : COS Q =

212s

Therefore, the output link mil1 have full mobility if

or.

hloreover. the maximum value for s is attained when the slider lies farthest from

the centre of rotation of the crank, and its minimum value when it lies closest to the

centre of rotation1, namely, s,, = I I + l2 and s,i, = 1 - 1 2 : respectively. Taking t his

into consideration, the inequalities of eq. (2.3) yield the condition

which is,

full mobi

of course, a necessary and sufficient condition

litÿ. In general, the in-line slider-crank mechan

for the output link to have

ism has an 11 larger than 1 2 .

but a special case, called the isosceles slider-crank mechanism, results when I I = 12.

1.1. Connecting-Rod-Lengt h-to-Crank-Radius Ratio. An important is-

sue to be analyzed is the ratio of connecting-rod length to crank radius. since the

cffect of the change in this parameter directly affects the mechanism performance. It

is known (Huebotter, 1923) that for the slider-crank or piston-crank mechanism used

in gasoline engines, a large ratio of connecting-rod length to crank radius will reducc

friction, wear. and vibration. However, if a power-to-weight factor is important. a

ratio in the neighbourhood of 3.5 is recomrnended.

In this section, we will obtain an optimum ratio r = l1/Z2 for which the resulting

mechanism will have the broadest range of motion of the output link. while maintain-

ing a good force or torque transmission. To this end, we first introduce the concept

of transmzssion angle, which is the angle p shown in Fig. 2.2. This arigle is often

used as an indes of merit for four-bar Iinkages, i.e., the srnaller the del-iation of the

transmission angle from a value of p = &go0, the better the linkage, based on the

quality of its forcc transmission. -4 transmission angle ranging from 45" 5 p < 135'

'The limiting positions of a slider-crank mechanism are called dead-center positions bottom dead center position when the slider is nearest to the crankshaft centre; and top dead center position wlien the slider is farthest from t h a t point.

2.1 IN-LINE SLIDER-CR4-W MECH.WSM

is usually satisfactory. Note tha t the transmission angle is a function of the linkage

input variable, and can be thus considered a local performance index.

, in important index in linkage optimization, that measures the performance of the

linkage globally, as opposed to locally, is the transmission quality. The transmission

quali ty was defined by .Angeles a ~ l d Bernier (1 987-a) as a positive-definite quantitlv,

riamely,

in which the mobility range of the input link is assumed to be [qo. ~ I ] . The trans-

mission quality can then be maximized by minimizing its complement. i-e. the trans-

mission defect. namely,

where. of course. q + qf = 1.

In our case, an espression for the cosine of the transmission angle can be obtained

by mcans of the law of cosines, Fig. 2.1, as

1; + if - s2 I + r2 - O* COS p = - -

21J2 2r

with the dimensionless variables r and o defined as

7 = / ? 0 = s/I2

Therefore, the transmission defect can be written as

2.1 IN-LINE SLIDER-CROX MECHA-WSM

and

The transmission defect can be finally expressed as

with Ci: for i = 1,2,3, being constants defined as

In order to find the optimum ratio r,, it is necessary to define the mobiiity range

of the input link, 4at Le., we need to set a criterion to determine the values 00 and

0 . W7e first note that these two values have to be chosen inside the interval bounded

for the lirniting positions of the slider, namelÿ,

However? this represents a problem because we would like to consider a widc

\-ariety of mechanisms and the range of motion of the slider would be different for

each mechanism. To gain insiglit into this problem, we plot the stroke of the slider

for different mechanisms in Fig. 2.2.

2- 1 IN-LINE SLIDER-CRANK

Figure 2.2: Range of motion of the input link for different in-line slider-crank mech- anisms

One way to deal with this problem is to replace a. and of in eq.(2.12) for a

corrcsponding expression in terms of the angular displacement of the crank: and so.

as far as the mobility condition l 1 3 12 is observed. the range of motion of the crank

[do. of j can be freely chosen.

From Fig. 2.1, it is apparent that

so tha t

and: therefore,

2.1 IN-LINE SLIDER-CRANK MECHANISM

lloreover, since the transmission angle becomes O" or 180" at the dead-centre

points. Le.. when 4 = 180" and Q = 0". respectively. we will limit the range of motion

of the crank to lie between these two values. Othemise, the dynamic action of a

flywheel would be required to further the motion. Figure 2.3 shows the variation of

tlic transmission angle as the rnechanism goes from the top dead centre position to

the bottorn dead centre position, for different mechanisms.

Figure 2.3: Variation of the transmission angle for different in-line slider-crank mechanisms

In order to define the range of motion of the crank, we refer to Fig. 2.3, from

where we choose for & the lowest value of qi for which p = 133" and for 4r the

highest value for which p = 45". Therefore, the range of motion can be set to be

2'2.5" 5 Q < 130.5"; we thus have al1 the necessary information to compute the

transmission quality for different values of r.

Figure 2.4 shows a plot of the transmission defect for different mechanisms,

whence it is apparent that as r increases, the transmission defect decreases until it

rcachcs a minimum. Moreover, frorn this figure, we know that minirnizing the trans-

mission defect implies having a connecting-rod-to-crank-radius ratio greater than 10.

which is of little practical use. However, it is apparent that the value of the transmis-

sion defect does not have a significant variation from the case in which r x 3 to the

case in which r = 15, and hence a value of r between 2.5 and 3.5 may be adopted.

Figure 2.4: Transmission defect for different in-line slider-crank rnechanisms

2.2 OFFSET SLIDER-CR4-NK PVIECHANIS&lS

2. Offset Slider-Crank Mechanisrns

Figure

-s------I

2.5: Geometry of the offset slider-crank mechanism

.-\ procedure similar to the one for the in-line type ni11 be followed to analyze the

offset slider-crank mechanism . First: we derive an expression for the output variable

O in terms of the input variable S. From Fig. 2.5: it is apparent that

e y = arctan -: h/12 = J(e/12)2 + (s/12)2 = d m l K = e/i2

S

Son-. ive can rewrite eq.(2.16) as

1 + (h/12)2 - r2 cos q5 cos 7 - sin & sin 7 = = C ( r 7 K! O )

2 ( W 2 )

and thus, by writing

2.2 OFFSET SLIDER-CR4ii hIECHAhqSMS

27 1 - r2 s i n 4 = - cos4 = -

1 + 7 * ' 1 + 7 2 ' 7- = t an ( g )

we end up with a quadratic espression for r, namely,

sin 3' C - cos -{ 7' + 2 7 t = O

C + cos y C + cos y

Hence, the output. variable can be obtained as a function of the input variable

froni the real roots of this quadratic expression. If we let

sin .Ï C - cos -/ -4 = . B =

C + COSY C + COS *i

we can readily compute the roots of the polynomial in eq.(2.19) as

with the possible output values for given e and s defining the two conjugate configu-

rations of the linkage, as seen in Fig. 2.6.

Configuration # 1 Configuration #2

Figure 2.6: Ttvo possible configurations of the offset slider-crank mechanism

I t is apparent that the output link has full mobilitÿ if the linkage discriminant

-4' - B is non-negative. However. in order t o derive this mobility condition in terms of

the linkage parameters 1 1 , l2 and e, we will refer t o the mobility of the slider rather than

2.2 OFFSET S L I D E R - C R M MECHA-MSMS

the mobility of the crank. To this end, we write a n expression for the displacement

of the slider as

s = 12 COS 4 + \/1: - (e + l2 sin 412

and thus. only if

Z: - ( e + h sin ei)' 2 O (2.22)

the position of the slider will be feasible. Using the maximum value of sin @ = 1 in

the above inequality. ive obtain

which is the mobi

At this point

lity condit ion sought .

it is convenient t o analyze how the offset e affects in the range

of motion of the crank. By the condition in eq.(2.23). it is knomn that e 5 il - l2

or K 5 r - 1, mith K defined in eq.(2.17). Upon knowing the allowable values of

the offset, 1i.e can proceed by setting an arbitrary value for r and obtain the range

of motion of the crank in which 45" 5 p < 135", for different values of I< within

t h e specified bounds. Table 2.1 shows the results for r = 3 and for the two possiblc

configurations of the offset siider-crank mechanism (Fig. 2.6).

Table 2.1: Results for different values of h'

2.2 OFFSET SLDER-CR4NK MECHA-NSMS

It is apparent from Table 2.1 that the best results, Le., the broadest range of

motion of the crank, is obtained for the second configuration wvith the maximum

offset allowved. Henceforth, the value of K will be considered to be the maximum

possible, Le., K = r - 1. Moreover, it is aIso apparent that for the first configuration

of the linkage, the range of motion of the crank increases as the offset decreases and.

therefore, the maximum range of motion will be obtained when K = 0: Le., when the

rnechanisrn is of the in-lzne type.

2.1. Connecting-Rod-Length-t0-Crank-Radius Ratio. .As in the case

of the in-line slider-crank mechanism, the performance of the offset linkage depends

directly on the ratio r: therefore, it is important to determine which d u e of this

relation optimizes the performance of the linkage. To this end, we first derive an

espression for the cosine of the transmission angle so that the transmission defect for

this type of linkage can be obtained. From Fig. 2.5: applying the law of cosines yields

1: + 1: - e2 - s2 COS p =

21J2

l,Ioreover, letting e = Z 1 - l2 and dividing the numerator and denominator of

ccl.(2-24) by 1 2 , we obtain

Therefore,

2 l 2 l 4 cos p = 1 - -a + -a r 4r2

Substituting the above equation into eq. (2.8): the resulting expression for the

transmission defect is obtained by integration, namellv,

2.2 OFFSET SLIDER-CMAW ,MECH.WISMS

where a. and uf can be readily derived from eq.(2.21), Le..

Finally. to define the range of motion [@O, 4,], we proceed as in the case of the

in-line type of linkage. Figure 2.7 shows the variation of the transmission angle for

difkrent offset slider-crank mechanisms; referring to this figure, the range of motion

of the crank can be chosen as 154" < 4 5 309".

Figure 2.7: Variaticn of the transmission angle for different offset slider-cran k mcch- anisms

2.2 OFFSET SLIDER-CR4-NK XfECHALU?SMS

In Fig. 2.8 a plot of the transmission defect for different offset linkages is displayed;

we observe that, similar to the in-line case, as the ratio r increases, the transmission

defect decreases, with its minimum value being reached when r is the ma-ximum value

considered. ,\gain, this ratio must be chosen according to the power-to-weight (or

size) requirement.

I

Figure 2.8: Transmission defect for different offset slider-crank mechanisms

Finally. w e can compare the two types of slider-crank mechanism. i-e.. the in-liiie

and the offset mechanisms. for which ive first refer to Figs. 2.4 and 2.8. Although

i t is apparent tbat the in-line mechanism has a lower transmission defect than the

offset mechanism (and, consequently, a higher transmission quality) , it is important

to notice that the range of motion of the crank, [do, dl], for the latter is greater and

t hus the transmission quality is espected to be lower. However: if the range of motion

of tlie crank for the in-line arrangement is chosen S U C ~ that A&n-Line = AQoffset.

thcn a difference of at most 3% in favor of the offset mechanism cornes apparent-

2.2 OFFSET SLIDER-CR4h7C MECEA.WSMS

Table 2.2 shows the transmission quality for different values of r for the two types of

Table 2.2: Cornparison of the two types of slider-crank mechanisms

r 2.0 2 3

3.5 4

Due to this slight difference in the performance of the in-line and the offset

arrangements of the slider-crank mechanism, the use of one or the other is equivalent'

Q

with the in-Iine linkage being the most frequently used. We base our transmission

in-line 0.429 0.441 0.450 0.451 0.453

design. in the balance of the thesis, on the in-line layout.

offset 0.429 0.445 0.455 0.461 0.467

'For the offset slider-crank mechanism the value of the offset is considered to be the masimuni possible. Le., A- = r - 1 .

CHAPTER 3. CAM-FOLLOWER DISPLACEMENT AKALYSIS

CHAPTER 3

Cam-Follower Displacement Analysis

The configuration-dependent 1/0 relation between angular velocities in a slider-crank

mechanism can be rectified by rneans of a Cam mechanism, as depicted in Fig. 3.1.

'\,Iore precisely, what we actuaIIy want is to be able to have this relation as a rediiction

l/:\Ï. for a n integer N > 1: of the input angular velocity: such that

The objective of this chapter is to obtain the 1/0 displacernent relation of the

cam rncchanism, j3 = O(.)? required to accomplish this task.

Figure 3.1: Proposed transmission device

Motor

r

C a m Mech. -

s

3.1 IO DISPLACE-'MENT RELATION OF TEtE CAM hlECH-4AISM

1. IO Displacement Relation of the Cam Mechanism

To obtain the 1/0 displacement relation of the Cam mechanism, an expression for

the velocity ratio of this mechanism will be derived first. From Fig. 3.11 and applying

the chain rule, i t is apparent that

so that, by virtue of eq.(3.1),

The above equation describes the velocity ratio for the Cam mechanism in terms

of the velocity ratio of the slider-crank mechanism, d$'/dqi, an expression for which is

deri\-ed below. In Chapter 2, the displacement of the slider for the in-line configuration

\vas found to be

and thus.

R-licrc .+ and 6 represent the velocity of the slider and the angular velocity of the

crank. respectively. Furthermore, the term in parenthesis in the numerator of eq.(3.5)

is equal to s: which by the law of sines can also be espressed as

sin@ - 4) s =

sin d

3.1 IO DISPL.4CEiMENT REL.4TION OF THE C14M 'VIECHANISM

We aIso know, from the geometry of the mechanism, that

so that. after substitution of eqs(3.6) and (3.7) into eq.(3.5), Ive obtain

s 12sin(0 - 4) - - * - = SI (d) O cos e

Furthermore. we can find via the law of sines an expression for sin(0 - d) in

terms of 8. namely. sin(B - 9) = (s sin 8)/z2. and substitute it into eq.(3.8), thereby

obtaining

i Y = s tan 6 O

l2 sin qi tan 6 =

Z2c0s4 - S

Finallc if we use the law of cosines to find an espression for cos d in tcrms of

il. l y and S . namely. cos 4 = (s2 + 15 - l?)/(2sZ2). and we also use t h e trigonometric

idcntity sin2 9 + cos' 4 = 1: ive obtain

3.1 IO DISPLACEMENT REL-4TION OF THE C-LM -;-ILECHANISM

Moreover, if the mechanism is driven by a screw actuator, then there exists a

linear relation between the displacement of the slider and the rotation of the screw,

ivliere p stands for the pitch of the screw. Hence if we substitute eq.(3.13) into

cq.(3.11) and let A, = ljlp, for i = 1: 2: we obtain, after some algebraic manipulations.

,d d p p J-0" 2 a p - 6 2 - = - - - rn d 4 b - p2 = B ' ( 4

which is the expression sought, with

Finallx substitution of eq. (3.14) into eq43.3) leads to

do -8 J-(04 - -a@ + b2) - - - - d7.b A T b - p' (3.16)

which is an o r d i n a n ~ differential equation ( O D E ) in the angular displacement 3 of the

scren-. The solution of the foregoing ODE is necessary to obtain a relation betwecn

the angular displacement of the Cam and the corresponding angular displacement of

the follower and: therefore: the information required for the displacement prograrn of

tlic folIower.

In order to integrate eq.(3.16), me will first write it in the form

3.1 IO DISPLACEMENT RELa4TION OF THE C-4M MECH-UWSM

where z = P2 and dx = -dg. The integration (Gradshteyn and R y h i k , 1965:

Jeffrey, 1995) of both sides of eq.(3.18) leads to

where Cint is a constant of integration, as yet to be determined. Moreover: using the

t rigonometric identity

~ q ~ ( 3 . 1 9 ) can be reduced to

J-z2 +2ax - b2 (X + 6) = 8ii2,A; sin (2; - - 2ci.t) =4(W). O x f O (3.20)

x

and thus: the constant of integration Cint can be readily computed by using the initial

conditions x(+~) = x0 in eq.(UO), namely,

1 (xo + b) J-xz + Paxo - b2 Cint = - - - arcsin v 2 8ii?Xfxo

Finally, eq.(3.20) leads to a quartic equation in x, namelu:

Howver: although the roots of a quartic equation can be found esplicitly using

Ferrari's formula (Selby, 1973), it will likely be too complicated to be of practical use.

cspccially as cornpared to a numerical approach.

3.1 IO DISPLACEMENT RELATION OF THE C.&M MECHANSM

,At this point, we have two options to numerically obtain the integral of the ODE

in eq. (3.16)? namelx

(i) Direct numerical integration; and

(ii) a continuation method based on the numerical solution of the quartic polyno-

mial obtained from the formal integration of the ODE.

It may even be convenient to make use of both alternatives to ensure accuracy

in the rcsults. Xote, however: tha t whichever approach is followed: comples solutions

may mise, n-hich will be of no practical use. One way to cape with this problem is to

find the interval of ,O for which the expression inside the radical of eq.(3.16) remains

positive. To this end. if ive let rc = 0': then? the condition

miist hold for al1 x. \iCé can find the estreme values of x if we regard the above

inequality as a quadratic equation such that

and thus: the lower and upper bounds can be determined by finding the roots of

eq. (3.24): namelx

,\ Ioreover, if

t hen

3.2 ~ M E F U C A L RESULTS

which ensures that, in fact, g(x) attains positive values in the interval 4a2 (A1 - A2)' 5

x 5 4 x 2 ( X , + Le.: it is zero a t the lower bound; then. g(x) increases up to a

rna,xirnum value when x = a, after which it decreases again until it attains a value of

zero, when the upper bound is reached. Figure 3.2 shows this behaviour.

Figure 3.2: Behaviour of the function g(z) = -2 + 2 c c - b2

I t isl then, possible to conclude that the interval of real solutions for the ODE in

cq.(3.16) is defined by

2. Numerical Results

Two numerical methods are proposed to obtain the integral of the ODE a t hand,

narnely. i) the direct numerical integration of the ODE by means of a Runge-Kutta

method, and ii) a continuation method based on the numerical solution of the quartic

polynomial in eq. (3.22).

In order to t ry the proposed methods to obtain f l (+ ) , it is necessary to determine

the parameters involved in both procedures. To this end, the following recommenda-

tions should be considered:

r The length o f the links of the slider-crank mechanism: Il and 12. -4lthough the

seiection of the dimensions of the linkage are case-dependent. it is convenient.

as seen in Chapter 2: to keep the ratio r = 1 in the neighbourhood of 3.

r The pitch of the screw: p. It is common knowledge that the torque T required

to produce a force F parallel to the displacement of the nut can be cornputed

as

n-here is the efficiency of the bal1 screw. It is apparent that the smaller the

pitch. the less torque is required, which is always desirable. However. since

the angular displacement of the followver driving the screw is limited. the pitch

must be selected so that a reasonable stroke of the nut is permitted. Hence.

this parameter is also case-dependent. and a balance between the requirements

of torque and displacement of the nut must be sought.

The initial condition for the ODE. To determine the initial condition for 3.

the only consideration is that O(&) has to be within the bounds established

in cq.(3.28). However, since the behaviour of the solution is unknown. it is not

recommended to chose the initial condition close to the bounds.

A safc rvay to chose ,O0 is to add 7r/2 rad to the lower bound of the interval. or

subtract the same value from the upper bound. In this way: since the folloiver

cannot move more than i-i/2: because of the nature of the mechanism. results

rvithin the specified interval for f l can be expected.

3.2 NUh4EFUCAL RESULTS

The two proposed methods were tested on different problems. In this section, the

resul ts of the problem described in Table 3.1 are discussed. Note that two cases are

considered, the first with Po = 2 x ( X 1 - A*) + 7r/2, the second with Ba = 2 j i ( A l + A?) -

- - - --

Table 3.1 : Design paramet ers

Po lSt case 1 21.2 rad

S-C mech. stroke 1 100 mm

L I 1 150 mm 12 1 30 mm

The results obtained when applying the two methods were essentially the same,

for both are identical u p t o the 13th decimal place. Hence, the accuracy of the solution

can be presumed. Figure 3.3 shows these results for the interval - N 5 @ < 7;.

Figure 3.3: Solution for P(zD): (a) 1'' case; (b) znd case

B-S unit brand THK mode1 pitch

vel. info. 1 3000 mm

BLR3232D 32 mm

6 N

50 rpm 60

l case 1 37.7 rad

CHAPTER 4. DISPLACEMENT PROGRA-M

CHAPTER 4

Displacement Program

In Chapter 3, two numerical procedures were proposed to obtain the relation between

the angular displacements of the follotver and the Cam. as required to rectify the

configuration-dependent 1/0 velocity ratio in the slider-crank mechanism. In this

chapter, ive complete the analysis that will allow the synthesis of the displacement

program needed for the design of the corresponding Cam.

First. we recali that the displacement program describes the motion of the follower

diiring the rotation of the Cam through one cycle. Now. if we look a t Fig. 3.3. i t

becomes apparent that even though a relation P ( 3 ) for one input motion cycle can be

obtained a t this point, it is not convenient to take only this information to represent

t he motions of the Cam system. since abrupt changes in the position, velocity and

;iccelcration will be unavoidable. There are, however, many possible cunes , also

rcfcrred to as follower motions. which can be used to create a blending motion so that

t h e resriiting dynamic performance is as smooth as possible.

To construct the displacement program of the follower we will rely. as before.

on the desired conditions for the motion of the slider-crank mechanism. I t is then

rcqiiired to create a motion that will take the crank smoothly frorn = do, its home

position. to a value 9 = 4o + a t which the crank should smoothly reach the

constant velocity rate d, = $ / N . Then. after a displacement A# with a constant

~.eIocity: the follower must mach a displacement d2 = ol + A@: where it decelerates

CHAPTER 4. DISPLACE-MENT PROGR4M

and reverses its speed to reach its home position with zero speed and zero acceleration.

This blending motion will then consist of three sections with the following boundarq-

conditions:

Furthermore, to generate the first and last sections of the motion, a polynomial

approach will be followed, i.e.: the appropriate motion curves will be synthesized

\vit h polynomial functions. In order to at tain the desired smoothness, a polynomial

of degree seven must be used. i.e.,

wi t h derivat ives

Finally, the total displacement of the crank \vil1 be divided in the form:

CH-QPTER 4. DISPLACE-MENT PROGR4M

-.sr < $ 5 -QI sevent h-degree polynomial

-1ii 5 . 5 St2 d = L ~ N

@ z I " b I r seventh-degree polynomial

The coefficients of the polynomial can be readily computed, since we have, for

each case, a system of eight equations, eqs.(4.2) to ( 4 . 3 ~ ) ~ evaluated at the corre-

sponding boundary values from eqs.(4.5), and eight unknowns, namelx a , 6 , . - . . h.

Sote, hoivever, that the conditions for the first interval are quite similar to those for

the Iast interval and thus, once the polynomial is defined for either one of the inter-

vals, the curve for the remaining one can be obtained by a simple change of variables

and linear transformations.

In order to ease the computation of the polynomial coefficients, we will impose

the conditions

arid then. solve the resulting system of equations for the first interval of the motion.

The coefficients that satisfy the system are:

CHAPTER 4. DISPLACEI'MENT PROGR4-M

n-here

and

e = f = g = h = o (4.8)

\i-hich implies that the conditions are attained with a (1-5-6-7 polynomial, namely,

as displaycd in Fig. 4.1.

CHAPTER 4. DISPLACElMENT PROGRAM

Figure 4.1: 4-5-6-7 polynomial

Finally, we have the following espressions for the complete interval -;F 5 li, 5 z:

m - 7 ï < q < 7 b l

(4.1 la)

(4.11 b)

( 4 . 1 1 ~ )

where

CHAPTER 4. DISPLACE-MENT' PROGR4M

and ai : . . .: di defined as in eqs. (4.6).

4 -zbi 5 q

and

and the rest of the parameters defined as for the same interval, with the dif-

ference that now e2 replaces Ilrl.

Once we have obtained the expressions for 4(@) and its derivatives, we can pro-

ceed with the analysis to obtain the functions that Rrill generate the displacement

program of the follower, Le., the expressions P($) for the three defined intervals. To

this end we refer to eq.(3.2), which is displayed below in a more convenient way:

Sote that this expression has the same form of

(4.17)

the ODE obtained in Chapter 3:

with the difference that now the function &(+) is given as explained above. -Again:

to obtain the relation ,8($) corresponding to each motion, one of the two numerical

methods proposed in Chapter 3 must be applied.

The direct numerical integration process does not suffer an' major changes, the

only difference is that, depending on the interval, the ODE will have to be adjusted

so that d(d) takes the corresponding form.

The case of numerical continuation is a bit more challenging. The basic idea is the

sarne as in the direct integration, with A ( 5 ) descrihed as in eq.(3.10). and reproduced

t,clow

and o(tb) defined according to the interval. Moreover, the constant of integration

dcpcnds o n the initial value of q!~ a t each intcrval. i.e., for the first interval Cint will

bc computed using the value of do = -rr: but for the second and third intervals: the

final value of & in the previous section will be taken as the initial condition. In a

similar \va!+. the initial conditions for p must be taken.

The above considerations were taken and the displacement program for different

cases were successfully constructed, obtaining, as expected, the same results indepen-

tfently of the method iised. Figure 4.2 shows the displacement programs for the two

cases listed in Table 3.1. Note that the values v1 and Sr2 nfere chosen such that 90%

CHAPTER 4. DISPLACEMEXT PROGR4M

of the input motion interval can be used to generate a constant velocity ratio d/6: i.c.: z/II = 7)2 = 2.83 rad.

Figure 4.2: Displacement program o f the follower (a) 1'' case: (b) Yd case

CHAPTER 4. DISPLACEMENT PROGR431

In order to verify that the displacement program obtained corresponds to a Cam

that fulfils the initial requirements, the displacement of the crank can be computed

bj+ means of the expression

which is obtained from the geometry of the slider-crank linkage. T h e results corre-

sponding t o the displacement programs of Fig. 4.2, are shown in Figs. 4.3 and 4.4. It

is apparent from these figures that, in fact, the desired rectification was achieved and

t hus. t he practicality of the proposed transmission device is confirmed.

Velwty Ratio

0. 1 8 I 1 i 1

Figure 4.3: Rectified displacement and velocity ratio: lSt case

CHAPTER 4. DISPLACEMENT PROGRAM

Figure 4.4: Rectified displacement and velocity ratio: 2nd case

CE4PTER 5. CAM DESIGN

CHAPTER 5

Cam Design

The final phase in the design of the transmission device shown in Fig. 3.1 is to find

the geometric parameters of the smallest cam-follower mechanism that will perform

t lie required rectification, while maintaining a good force transmission. The layout

of the Cam mechanism to be synthesized is shown in Fig. 3.1: the objective of this

cliaptcr being the development of a method for carn-szze minimization under pressure

angle constraints.

Figure 5.1: Cam rnechanism with oscillating roller-follower

5.2 CA-M-SIZE l4IMMIZATION UNDER PRESSURE ANGLE CONSTR4INTS

1. Pressure Angle

The pressure angle, designated hereafter by a, is an index of merit in Cam design

that determines how good the force transmission of the mechanism is. This index,

shown in Fig. 5.1, is defined as the angle between the force exerted by the Cam on

the follower, acting in a direction normal t o the Cam profile a t the contact point,

and the velocity of the aforementioned point on the follower. Obviously, as the cam

rotates, the point of contact changes, and consequentl_v, so does the pressure angle,

whosc ideal value is zero, i.e,, the srnaller the absolute value of a, the better the force

transmission. Hence, in order to keep the transmitted force of the mechanism within

acceptable limits, the pressure angle is to be bounded as -ahf ,l I 5 ab*.

The maximum pressure angle, ahr7 occurs when the first derivative of û with

respect to + vanishes. Finding the values of @ a t which the pressure angle attains its

extrema, is then the first problem to solve in cam-size minimization under pressure

anglc constraints.

2. Cam-Size Minimization Under Pressure Angle Constraints

The approach presented in this section is a streamlined version of the rnethod

proposed by Wu (1998). Since t a n a grows monotonicalIy with cr in the interval

- 7 / 2 5 a 5 ~ / 2 , this method proposes the extremalization of this function to ease

the procedure, as esplained below. In general, for any type of cam mechanism. the

tangent of the pressure angle can be espressed in the form

and thus.

d tan a(@) = O

ddJ

5.2 CAM-SIZE 0,lINlMIZ-4TION UNDER PRESSURE ANGLE CONSTRNXTS

yields the extremality condition for the pressure angle. Now, by taking the derivative

of both sides of eq.(5.l) with respect to @! we obtain

ancl obviously, for the zeroing of the foregoing espression: the difference inside the

brackets must vanish, i.e.,

u-hich is verified at values of ~ where la/ attains its maximum value q r . - in espression

for the tangent of the pressure angle for Cam mechanisms with an oscillating follower

\vas derived by Angeles and L6pez-Cajun (1991) so that, for our case

with ,3(-u) and ,Of(z$) defined as in previous chapters. and u representing the ratio el1

of the follotver arm length e and the distance 1 between the axes of rotation of the

cam and of the follower. The estremality condition of eq.(5.4) now takes the form

lloreover. the angular displacement of the follower? ,LI(@)-.). can be espressed as

the sum of a constant 13~ representing the value of ,!3(@) a t the lotvest position of the

follower. and a positive-definite function a(@), namel_v,

n-i t h derivatives

5-2 CAAM-SIZE MINIMIZATION UNDER PRESSURE ANGLE CONSTR4IXTS

Pt(@) = ut(@) @y@) = off (dJ)

Substitution of eqs.(5.7) and (5.8) into eq.(5.6) yields

whcre G- stands for the value of th a t mhich a attains an extremum. Final15 if w;

and i ~ : ; are the values a t which the pressure angle attains a maximum and minimum:

respectively: the evaluation of the extremality conditions and of eq.(5.5) at those

values leads to a system of four nonlinear equations in the four unknowns u. , f i l . 12;.

and v;! namel';

uai + 0; sin(& + 01) - tan anru; cos(,Of + 01) = O ( 5 . LOa)

uo: + o: sin(& + O*) + tan anfa; cos(@f + O?) = O (5.lOb)

u[1 + O;] - COS(,^^ + oI) - sin(@[ + ol) tan ad[ = O (5 .10~)

u[1 + 41 - COS(& + Q) + sin(& +- 0,) tan a,,[ = O (5.10d)

wi t h a, = o(~: ) . for i = 1 ? 2. and a: and a; defined likewise. The foregoing system

of ecluations was solved by Wu (1998) using the Sewton-Raphson mcthod. who thus

ohtainecl a Cam with the smallest radius of the base circle 6: as shown in Fig. 5.1.

An espression for O can be readily obtain from the geometry of the oscillating cam

mcchanism, namel_v,

b2 = e2 + 12 - 2el cos fi[

5.2 CAM-SIZE b-Z.4TIO bTirJDER PRESSURE ANGLE CONSTRmTS

Note, however, that the foregoing approach does not guarantee the finding of

a solution, for the procedure depends on the initial guess usec! to start Newton-

Raphson iterations. kloreover, the problem may ncjt have a feasible solution: which

this approach would be incapable to tell. One way of coping with this problem is

developed below.

Firs t , we espand

COS(^[ + ai) = COS fli COS oi - sin Ji sin ai

sin(Pl + ai) = sin pl cos ai - cos ,Si sin ai

and t hen we int roduce the well-known trigonometric identities

1 - B2 2B COS gl = - sin = rhere B tan (:) (5.13)

1; B2: 1 + 8 2 :

so that . by substitution of eqs.(5.12) and (5.13) into eqs.(5.10), after some algebraic

nianipulations, n-e can obtain a new systern of equations in the form

whcrc O is the four dimensional zero vector: and m,: for i = 1: 2: 3: are four-dimensional

1-ectors defined as

-5.2 CAhI-SIZE ~ Z . 4 T I O N UNDER PRESSURE ANGLE CONSTR4IXTS

\vit11 c, = cos ai and si = sin ai, for i = 1: 2, and T = tan a,\[. Furthermore, for the

system of eqs.(5.14) to have a nontrivial solution, the four minors of order three of

matr i s M rnust vanish? narnely?

Al G

5.2 CAM-SIZE 5msJIMIZATION UNDER PRESSURE ANGLE COXSTR4Ih'TS

u-here mij stands for the ith component of vector mj. It is apparent from the foregoing

equations. that al1 {Ai}: are quadratic in u. However, if the third column of each

3 x 3 submatris is subtracted from the first, then we will end up with al1 four minors

linear in u, namely,

(5.17,)

(a. 1 Tb)

(5 .17~)

( a . 1 ~ )

t hc rcmaining vectors being defined according to the columns of {Ai}': in eqs.(5.16).

11% thcn have four equations in the two unknowns +; and @;; namely,

or after espansion and simplification

Each of the foregoing equations defines a contour in the T$~-& lan ne'. whose

intersections, then, provide al1 real solutions for the problem a t hand. The two

unknowns, zb; and 6, are then computed from the above four equations iising a

least-square approach, which will render a robust solution while filtering roundoff

errors. At this point it is important to mention that, based on esperience. and

L:.; arc likely to be found in the rise phase and the return phase of the displacement

prograrn. respectivelu. Moreowr: multiple rise and return phases are possible, and if

tliis is the case: al1 rise and return phases must be analyzed to obtain al1 the solutions

for thc problem a t hand.

Finally: the remaining two unknowns. u and ,LIl: can be readily computed by

nieans of a Ieast-square approach as well, from the system of four eqs.(5.10).

'Pleasc note that Gl and Q2 were defined in Chapter 4 as the vaiues of @ at which the biending with the 4-5-6-7 polynomials occurs. There should be no confusion here with the notation in this chaptcr.

5.3 IMPLEMENT-4TION .WD RESULTS

3. Implementation and Results

In order to verify the results frcm the method proposed in the previous sections,

a graphical user interface (GUI) was implemented under UNIX environment using

lIATL=i\B 5.1. -At the outset, the window is divided, as shown in Fig. 5.2? into two

areas: the control area and the plotting area. Moreover, the control area will change

as the design process advances, Le., new functions will become available as needed.

Figure 5.3 shows the GUI with al1 its functions enabIed.

Figure 5.2: GUI a t the beginning of the session

Each of the controls serves a specific function:

Data: This menu allows for the sclection of the user data. When selected, a

l i s t -box containing three options will be displayed: the aforementioned options are

1I.lTL.AB files that the user must program to generate the information required for

the follower curves.

Follower curves: Once an option has been selected from the Data menu, the

user will be able to see? in the plotting area, the corresponding follower curves by

cntcring this menu, Le., a 1W.t-box will appear from where the options Displacement,

Iklocitg. Accelerution, and Jerk, can be selected. If Displacement is selected, the

nicnus Rise phase and Return phase will be enabled: otherwise they are inactive.

5.3 IMPLEMENT-4TION A-XD RESULTS

Figure 5.3: GUI with al1 its functions enabled

Rise phase: When selected, the displacement program on the plotting area will

bc dividcd by vertical dashed lines, indicating the phases of rise. Then. the user will

specify through a list-box. which phase to analyze.

Return phase: This menu is similar to the previous one. The return phases will

be indicated on the displacement program; the user must select one.

Pressure Angle Bounds: This option. represented in the GUI as 5 a 5. takes

on the desired maximum and minimum values for the pressure angle. When these

\-alues are specified. the push-button OK will be enabled: otherwise it rernains inac-

tive.

OK: If the user data. rise and return phases, and pressure-angle bounds are prop-

crIy sclected. ttien the user must press this button to continue the design procedure:

otherwise. the aforementioned information can be changed a t anytirne. Once OK is

pressed. the program will plot the contours in the .rL1-2C.! plane; the user will then

have the option to zoom in o r out the plot to verify if solutions exist. If no possible

solutions are apparent, the user must repeat the operation for different selections of

the design data.

Optimize: If solutions exist for the given problem, the user may continue by

pressing this button. After pressing Optimize, a point must be selected with the left

mouse button near one of the solutions. This will provide the information to start

t h e Icast-square approach t o solve for 7.l~; and tb;. and then. 4 and TL. in this order.

IYhen the solution is reached, the distribution of the pressure angle will be displayed

in the plotting area. and the push-button Cam Profile will becorne visible.

Cam profile: This is a double-purpose push-but ton. When first displayedl the

option for plotting the profile of the Cam mil1 be available if pressed. a layout of the

cani. as well as the optimum parameters pl. u: b / i , and the values of ~; and d ~ ; . will

be displayed in the plotting area. Furthermore, once this done. the option for a 3 0

üiew of the Cam will become available by pressing the same button.

The flowchart for the design procedure is shown in Fig. 5.4.

S e s t . ive present the cam-design process for the problem stated in Table 3.1.

AIoreover. the displacernent program is generated such that 80% of the input motion

interval d l be rectified. Figure 5.5 shows the follower cumes for this problem. as

displayed by the GUI.

In this case. the displacernent program is defined by a rise-return-rise motion

of the follower. Figure 5.6 shows the contour plot for the problem a t hand when

arlalyzing the first rise phase, and for -30" 5 û. 5 30". Furthermore, a closer look to

the solution for the foregoing problcm. as well as an esample in which no solutions

would bc found. arc shown in Fig. 5.7.

The behaviour of the contours is similar for the second rise phase. i.e.. only one

solution {vas found when analyzing this phase. Hence, we will have two solutions

for t.;. namely. one in the first risc phase and one in the second. The resulting

pressure-angle distribution for these two cases are shown in Fig. 5.8.

Figure 5.4: Flowchart of the design procedure

Sote that even though this method anticipates the esistence of solutions, it cannot

predict if the estreme values of ck will be global or local, as it can be learned from

Fig. 5.8. This, however, does not represent a real disadvantage, because al1 possible

solutions can be verified by the designer within a few seconds. Finally: the layout

of the shape of the cam for which the pressure angle will be bounded as specified is

stion-n in Fig. 5.9, while Table 5.1 includes the corresponding optimum parameters.

I Tlie above procedure, then, can be repeated until

Table 5.1: O ~ t i m u m aram met ers for

a satisfactory design is obtained.

the minimum-size Cam

Displacement prograrn Veloci ty program

Acceleration program Jerk prograrn

Figure 5.5: Fdlower curves for the design parameters of Table 3.1

The foregoing examples pertained to pressure-angle bounds of the form -o.if 5

a < a.\,. However, nonsymmetrical bounds of the form -al 5 a 5 a,: with inde-

pendent lower and upper bounds al and cru? may also bc specified.

5.3 IMPLEMENT-4TION AND REStiLTS

Arca of possible solutions: one amarcnt ~olution

Ara of possible solutions: no appmnt solution

Figure 5.6: Contours for the first rise phase: -30" < - a. 5 30"

. . . . . . . ._ . c . _i . .I. - -

, .

1 st rise phase: -30 <a< 30

Figure 5.7: Contours for the first risc phase

1 st rise phase: - 1 O g a g 1 0

for different bounds of the pressure angle

Global Local r Maximum

f st rise phase: -30 cas 30 2nd rise phase: -30 sa< 30

Figure 5.8: Pressure-angle distribution for the two solutions of the problem

Figure 5.9: Pitch curve of the minimum-size Cam for the reducer-rectifier mechanism

6.1 CONCLUSIONS

CHAPTER 6

Concluding Remarks

1. Conclusions

-4 methodology for the design of a speed reducer-rectifier Cam mechanism, to sim-

plify the control of slider-crank mechanisms used in robotic and mechatronic systems.

ivas de\.eloped. The first question that emerged when formulating the problem of the

design of the Cam mechanism was whether an in-line or an offset slider-crank linkage

stiould be used to ensiire optimum force-and-torque transmission characteristics. BI-

comparing the transmission quality of the two types of arrangements it was apparent

tha t the use of one or the other is equivalent; because of its compactness. the in-linc

configuration was selected.

The kinematics of the cam-driven in-line sfider-crank mechanism \vas analyzed.

and an ordinary differential equation (ODE). describing the angular velocity ratio

of the required cam-and-follower system, \vas derived. The integration of this ODE

was necessary to obtain the input-output (I/O) displacement relation and. hcnce.

the information required for the design of the Cam. Whcn the involved ODE was

iritcgrated symbolically, the problern became one of finding the roots of a quartic

polynornial. Although the four roots of a quartic polynomial can be obtained es-

pliuitly using Ferrari's formula, this approach was not considered t o be suitable to

obtain the displacement of the follower for the whole cycle of rotation of the Cam,

duc to the cumbersome expressions of these roots. Hence, two numerical approachcs

6.2 RECOh/Ih,lE,niDATIONS FOR FUTURE RESEARCH

were proposed, nameiy, i) the direct numerical integration of the ODE by means of

a Runge-Kutta method, and ii) a continuation method based on the numerical so-

lution of the quartic equation. The aforementioned methods were implemented; the

sarne results were obtained in both cases, and thus: the accuracy of the solution was

verified. The displacement program of the follcwer, then, was created using a blend-

ing motion such that a smooth dynamic performance was produced. Furthermore,

the practicality of the reducer-rectifier Cam mechanism was confirmed by analyzing

the displacement of the crank link of the slider-crank mechanism in which the input

motion was produced by the output of the Cam mechanism.

For the design of the oscillating Cam mechanism, a cam-size minimization method

was developed. This method allows for the design of a Cam with the smallest possible

base-circle radius, and wi t hin safe pressure-angle bounds. Moreover, t his method

is advantageous because the existence of solutions can be visually verified before

in\-esting precious resources in the search for an inexistent solution. A Graphical Cser

Interface (GUI). mhich allows the user to perform the optimization interactively. was

iniplernented. This GUI was successfully used to complete the design of the optimum

cam to reduce and rectify the angular velocity ratio of a given slider-crank rnechanism

and its actuator.

2. Recomrnendations for Future Research

As estensions to the present work, further research is recommended in the fol-

lowing directions:

( i ) -4 detailed analysis of the dynamics of the whole transmission device is re-

quired to deterrnine the effects of masses and forces in the reducer-rectifier

Cam mechanism.

(ii) Determining the minimum and rna~imum allowable speed reduction should

also be given due attention.

(ii i) Further research is needed regarding the separataon phenornenon, Le., the s e p

aration of members of the Cam mechanism, if the Cam is to be driven at high

6.2 RECOMIME~\~DATIONS FOR FUTURE RESEARCH

speeds. Also, the issue of contact stress between the Cam and its follower needs

to be analyzed.

(iv) The range of motion of the crank link of the slider-crank mechanism can be

enlarged by means of an amplifier mechanism. However, the practicality and

complexity of the resulting system is worth further investigation.

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