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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.
REFERENCES
References
-&kade, R.I., Mohan Rao, -4.V.: and Sandor, G.N., 1975, "Optimum Synthesis of Four-Bar and Offset Slider-Crank Planar and Spatial Mechanisms Csing the Penalty Function -4pproach With Inequality and Equality ConstraintsYl ASME Journal of Engineering for Industry, Vol. 97. pp. 785-790.
-Amarnath, C., and Gupta, B.K., 1978, "Xovel Cam-Linkage Mechanisms for 'vlulti- ple Dwell Generation", Cams and Cam Mechanisms, The Institution of Mechanical Engineers, London, pp. 123-127.
Angeles, J., 1982, Spatial Kinematic Chains. Analyszs, Synthesis, Optimization, Springer-hrlag, Xew York.
Angeles, J.. 1997, Fundamentals of Robotic Mechanical Systems. Springer-Verlag, Sen- York.
-Angeles. J. and Bernier, -4.. 1987, "-4 General Method of Four-Bar Linkage Mobility -4nalysis": ASME Journal of Mechanisms, Transmissions, and Automation in Design. \ Q I . 109. pp. 197-203.
Angeles, J. and Bernier, -4.. 1987-a, "The Global Least-Square Optimization of Funct ionGenerat ing Linkages" , ASME Journal of Mechanisms, Transmissions, and .4utomation i n Design, Vol. 109: pp. 204-209.
.Ingeles, J . and Lapez-Cajun. C.S.: 1991, Optimzza tion of Cam Mechanisms, Kluwer Academic Publishers, Dordrecht.
.\rakawa, A., Emura, T., Hiraki, M., and Wang? L.. 1993, "Control of a Quadruped Robot using Double Crank-Slider Mechanism" , Proceedings of the IElEE International Conference on Systems, A4an and Cybernetics, Vol. 4, pp. 157-162.
Berzak, X.: 1982, "Optimization of Cam-Follower Systems with Kinematic and Dy- ~ a m i c Constraintsy , Journal of Mechanical Deszgn, Vol. 104: pp.29-33.
Carnahan . B. et al., 1969, Applied Numerical Methods, John Wiley and sons Inc.: Sew York.
Chan, Y.W. and Kok, S.S., 1996, "Optimum Cam Design", International .Journal of Cornputer Applications i n Technologg, Vol. 9, No. 1, pp. 34-47.
REFERENCES
Cordes, S., K. Berns, M. Eberl, and W. Ilg, 1997, "On the Design of a Four-Legged Walking -Machinen , Interncztional Conference on Advanced Robotics, Montereyr July 7-9: pp. 65-70.
Fenton, R.G.. 1966-a, "Reducing Noise in Cams", Machine Design, Vol. 38, Xo. 8. pp. 187-190.
Fenton, R. G. , 1966-b, "Determining Minimum Cam Size" , Machine Design, Vol. 38, Xo. 2, pp. 155-158.
Fenton, R.G., 1975, "Optimum Design of Disc Cams" , Proceedings of the 4th World Congress on Theoq of Machines and Mechanisms, Newcast leupon-Tyne, September 3-13, Vol. 4, pp. 781-784.
Goldstine, H.H.: 1977, A History of Numertcal Analysis. From the 16th Through the 19th Century, Springer-Verlag, New York.
Gonzalez-Palacios, M.-A. and -Angeles, J., 1993, Cam Synthesis, Kluwer -1cademic P u blishers, Dordrecht.
Gonzalez-Palacios, M.-A. and -Angeles. J . , 1998. "The Design of a Xovel Mechanical Transmission for Speed Reduction" , ASME Design Engineering Teclmical Confer- ences: Atlanta, September, pp. 13-16.
Gradshteyn, I.S. and Ryzhik, I.M., 1965, Table of Integrals, Sertes, and Products, Fourt h Edition. Academic Press Inc., Xew York.
Hamniing, R.W., 1962, Numerical Methods for Scientists and Engineers. McGraw- Hill Book Co., Xew York.
Hartenberg, R.S. and Denavit, J., 1964, Kinematic Synthesis of Linkages, McGraw- Hill Book Co., New York.
Hirschhorn. J . , 1962, "Pressure ,Angle and Minimum Base Radius'' , Machine Design. \01. 34, pp. 191-192.
Hrones, J.,4., 1948, ;',An Analysis of the Dynamic Forces in a Cam-Driven System", Transactions of the ASME, Vol. 70, pp. 473482.
Huebotter, H..A., 1923, Mechanics of the Gasoline Engine, McGran--Hill Book Co.. Sen- York,
.Jcffrcy, -4.: 1995, Handbook of Mathematical Formulas and Integrals, Academic Prcss Inc.. California.
.Johnson. R.C.? 195.5: "Cam Design", Machine Design, Vol. 27, 30. 11, pp. 195-204.
.Johnson. R.C., 1956, "Cam Profiles", Machine Design, Vol. 28, 30. 25, pp. 129-132.
Liu, Z., 1993: Kinematic Optimization of Linkages. PhD Thesis, Department of Me- cllanical Engineering, McGill University, Montreal.
Loeff, L. and Soni, A H . , 1975, "Optimum Sizing of Planar Cams", Proceedings of the
REFERENCES
4 th World Congress o n Theory of Machines and Mechanisms, Newcastle upon-Tyne, September 8-12, Vol. 4, pp. 777-780.
Xlaxwell: R.L., 1960: Kinematics and Dynamics of Machinery: Prentice-Hall Inc., Englewood Cliffs.
l,lilligan? M. and Angeles, J., 1993, "The Synthesis of a Transmission for Rectify- ing and Reducing a Periodic Speed from a Turning Shaft", Proceedings of the Ninth World Congress on the Theory of Machines and Mechanisms, -4ugust 29-September '2: Milan, Italy, Vol. 1, pp. 615619.
llorgan. =.\.P. and Wampler, C.W. II, 1990, "Solving a Planar Four-Bar Design Prob- lem C'sing Continuation"! Transactions of the ASME, Vol. 112, pp. 544-550.
Reichardt. J., 2978, Robots. Facts, Fiction and Prediction, Penguin Books, Xew York.
Ri\.in, E.I., 1988, Mechanical Design of Robots, McGraur-Hill Book Co.. Xew York.
Rot hbart, H.A. , 1936, Cams: Design, Dynarnics, and Accuracg, John Wiley and sons Inc.. Xew York, pp. 313-316.
S e l b S.M .$ 1973, CRC Standard of Mathematics Tables, The Chemical Rubber Co.. Cleveland.
Shigley: J.E. and Gicker? J.J.. 1980, Theory of Machines and Mechanisrns, McGraw- Hill Book Co., Xew York.
S tadier, W.: 1995? Analytical Robotics and Mechatronics, McGraw-Hill Inc., Xew l'ork.
S toer, J. and Bulirsch, R., 1993, Introduction to Numerical Analysis, Second Edition, Springer-Verlag, Xew York.
Terauchi, Y. and El-Shakery, S._\., 1983, "-4 Cornputer--4ided Method for Optimum Design of Plate Cam Size Avoiding Undercutting and Separation Phenornena-1", Mechanism and Machine Theory, Vol. 18. pp. 157-163.
Ifkmpler, C.W., Morgan: =\.P., and Sommese, -4.J.. 1990: "NumericaI Continuation lfethods for Solving Polynomial Systems Arising in Kinematics", Journal of Mechan- icnl Design. Vol. 112, pp. 59-68.
\Vilson, C.E. and Sadler, J.P., 1993, Kinematics and Dynamics of Machinery, Harper Collins College Publishers? New York.
\ \ u . CH.J., 1998, A Modular Approach to the Sgnthesis of Quick-Return Mechanisrns. 11. Erig. Thesis, Depart ment of Mechanical Engineering, McGiIl Universits Mont reaI.
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