geometric design of planar four-bar mechanismsdigitool.library.mcgill.ca/thesisfile106588.pdf ·...

Geometric Design of Planar Four-Bar

Mechanisms

Mahya Shariatfar

Master of Engineering

Department of Mechanical Engineering

McGill University

Montreal, Quebec

December 2011

A thesis submitted to McGill University in partial fulfillment of the requirements

of the degree of Master of Engineering

c©Mahya Shariatfar, 2011, All rights reserved

Dedication

To my parents.

i

Abstract

Outstanding issues concerning systematic design procedures for planar four-bar mech-

anisms, specifically three aspects hitherto not addressed, are treated in this thesis. These

include

• Advance-retire crank angle ratio for crank-rockers,

• Angle ratio for double-rockers and,

• Lead-lag angle ratio for double-cranks.

This thesis aims to introduce a systematic method to determine parameters of planar

four-bar mechanisms which specify a unique mechanism design. The design procedures in-

troduced are based on the geometric properties of mechanisms. The symmetry of these

geometric formulations makes it easy to find a unique mechanism design for defined design

parameters. Moreover, the relations derived can be conveniently programmed and imple-

mented in “Computer Aided Design (CAD)” algorithms.

The equations, which are revealed for the first time in this thesis, can be used to deter-

mine a unique planar linkage design for specified design parameters. These design parameters

are such characteristics of the mechanisms which are of particular importance to the designer,

i.e., link lengths, assembly sequence and choice of fixed link. All design parameters and de-

sign formulations are dimensionless, giving this opportunity to the designer to select ratios

instead of size.

Crank-rocker, slider-crank, double-rocker, and double-crank mechanisms are studied in

this thesis. The design parameters are: The ratio of coupler length to crank throw radius

ii

and the time ratio for crank-rocker and slider-crank; oscillating angles of two rockers and

the ratio of two rocker link lengths for double-rocker; and specified minimum and maximum

leads and the angles at which the driver link has these extreme leads relative to the driven

link for crank-crank mechanisms.

Each chapter is devoted to the design of one of three types of planar four-bar mechanisms.

Crank-rocker and slider-crank mechanisms are studied in one chapter since their design

procedures are mainly the same. Double-rocker and double-crank mechanisms are the others.

For each type of mechanism, the design parameters are introduced, then the design process

and design equations are provided and clarified with proper figures and plots. Finally, the

design process is illustrated with numerical examples and validated by Grashof’s criterion.

Explanation of the step-by-step process to obtain each mechanism design is clearly illustrated

with appropriate diagrams.

iii

Resume

Les questions en suspens concernant les procedures systematiques de la conception du

mecanisme planaire de quatre-barres, et specifiquement trois aspects jusqu’ici non-abordes,

sont traitees dans cette these. Ceux-ci incluent la conception du

• Rapport des angles avance-retour pour des mecanismes a manivelle-balancier,

• Rapport des angles pour des mecanismes a double-balancier et,

• Rapport des angles avance-traıne pour des mecanismes a double-manivelle.

Une methode systematique est presentee afin de determiner les parametres des mecan-

ismes planaires de quatre-barres qui donnent lieu a de mecanisme unique. Les procedures

de conception presentees sont basees sur les proprietes geometriques des mecanismes. La

symetrie de ces formulations geometriques facilite la recherche d’un mecanisme unique pour

des parametres de conception donnes. De plus, les equations de conception, proposees dans

cette these, peuvent etre commodement programmees dans les algorithmes de “Conception

Assistee par Ordinateur (CAO)”.

Les equations, qui sont presentees pour la premiere fois dans cette these, peuvent etre

aussi utilisees afin de determiner un mecanisme planaire unique pour des parametres de

conception donnes. Ces parametres de conception sont les caracteristiques les plus impor-

tantes des mecanismes dans l’etape de la conception, c’est-a-dire, la longueur des maillons, la

sequence de montage et le choix du maillon fixe. Le fait que tous les parametres et formula-

tions de conception soient a dimensionless fournit au concepteur l’opportunite de selectionner

les rapports plutot que les dimensions absolues.

iv

Les mecanismes a manivelle-balancier, tiroir-manivelle, double-balancier, et double-

manivelle sont etudies dans cette these. Les parametres de conception sont: Le rapport

de la longueur de coupleur au rayon de manivelle et le rapport de temps pour les mecan-

ismes manivelle-balancier et le tiroir-manivelle; des angles d’oscillation de deux balanciers

et le rapport de deux longueurs des balanciers pour le mecanisme a double-balancier; et des

avances minimum et maximum spceifies et les angles extremes auxquels la barre entraınante

a la barre entraınee pour des mecanismes a manivelle-manivelle.

Chaque chapitre est consacre a la conception d’un type de mecanismes planaires de

quatre-barres. Les mecanismes a manivelle-balancier et tiroir-manivelle sont etudies dans

un chapitre commun puisque leurs procedures de conception sont essentiellement les memes.

Pour chaque type de mecanisme, les parametres de conception sont presentes, puis, le proces-

sus et les equations de conception sont fournis et clarifies par des images et des graphes ap-

propries. En conclusion, le processus de conception est illustre par des exemples numeriques

et valide par le critere de Grashof. L’explication etape par etape du processus de conception

afin d’obtenir chaque mecanisme est clairement illustree par des diagrammes appropries.

v

Acknowledgments

I would like to thank my supervisor, Professor Paul Zsombor-Murray, who supported

and guided my research at McGill. He tried to make me “think geometrically”. This thesis

could not have been finished without his continuous support.

I thank Prof. Jorge Angeles for his valuable critique and comments. I would also like

to address thanks to my lab-mates, CIM members, and all my good friends specially Payam

Rahimi Vahed and Vahid Raissi Dehkordi, who helped greatly in translating the abstract

into French. I appreciate the help of Mrs. Joyce Nault in guiding me in the procedure of

submitting my thesis.

Finally, I am thankful to my parents, who were always at my side during my study,

and I would like to express my deepest thanks to my love Amir Hossein, who is my greatest

source of encouragement.

vi

Table of Contents

Dedication ii

Abstract iv

Resume vi

Acknowledgments vii

List of Figures x

List of Tables xi

List of Symbols xii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Crank-Rocker and Slider-Crank 8

2.1 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Design Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Design Example Using Text Book Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

vii

2.2.2 Design Example Using Three Circle Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Double-Rocker 18

3.1 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 The Design Process of “Elbow Extended-Elbow Folded” Case . . . . . . . . . . . 20

3.1.2 The Design Process of “Elbow Extended-Elbow Extended” Case . . . . . . . . 28

3.1.3 The Design Process of “Elbow Folded-Elbow Folded” Case . . . . . . . . . . . . . . 33

3.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Design Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Double-Crank 50

4.1 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Parameters α1, α2, Lmin, and Lmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Design Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusions 56

References 58

viii

List of Figures

1.1 Crank-rocker four-bar mechanism design according to [22] . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 a) Pivot A remains on its circumscribing circle subtended on chord CC ′ b)

Designing a crank-rocker with a geometric approach based on intersection of

three circles: k1, k2, and k3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Crank-rocker four-bar mechanism design according to [22] . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Crank-rocker four-bar mechanism design verification for three circle method. . . 16

3.1 Three individual cases embedded within extreme excursions a)“elbow extended-

elbow folded”case b)“elbow extended-elbow extended”case c)“elbow folded-

elbow folded” case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Two circles and a common chord in “elbow extended-elbow folded” case with

rockers rotating in unison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Two circles and a common chord in “elbow extended-elbow folded” case with

rockers counter-rotating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Two circles and a common chord in “elbow extended-elbow extended” case

with rockers rotating in unison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Two circles and a common chord in “elbow extended-elbow extended” case

with rockers counter-rotating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Two circles and a common chord in “elbow folded-elbow folded” case with

rockers rotating in unison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

3.7 Two circles and a common chord in “elbow folded-elbow folded” case with

rockers counter-rotating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8 The steps to draw a double-rocker mechanism in its extreme excursions“elbow

extended-elbow folded” with rockers rotating in unison. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.9 Double-rocker mechanism within extreme excursions “elbow extended-elbow

folded” with α = 20◦, β = 30◦, and ρ1 = 0.4 a) under rotation in unison b)

under counter-rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.10 Double-rocker mechanism within extreme excursions “elbow extended-elbow

extended” with α = 20◦, β = 30◦, and ρ1 = 0.55 a) under rotation in unison

b) under counter-rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.11 Double-rocker mechanism within extreme excursions“elbow folded-elbow folded”

with α = 20◦, β = 30◦, and ρ1 = 0.7 a) under rotation in unison b) under

counter-rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.12 Common chord and oscillatory angles in“elbow folded-elbow folded”case with

rockers rotating in unison when q > p and q > s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.13 Double-rocker mechanism within extreme excursions“elbow folded-elbow folded”

with α = 20◦, β = 30◦, and ρ1 = 0.3 a) under rotation in unison b) under

counter-rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Double-crank mechanism in its parallel and antiparallel configurations . . . . . . . . . 51

4.2 Coupler link length of BC > AB fixed link length for α1 > π . . . . . . . . . . . . . . . . . . . 54

4.3 Parallel and antiparallel poses for Lmin = 30◦, Lmax = 80◦, and α1 = 120◦ . . . . . 54

x

List of Tables

1.1 Sorting planar four-bar linkages according to Grashof’s criterion [4], [5] . . . . . . . . 4

3.1 The parameters of double-rocker mechanism in terms of specified design pa-

rameters for three different oscillatory motion configurations . . . . . . . . . . . . . . . . . . . . 40

xi

List of Symbols

a1,2 Compressed parameter expression

k Compressed parameter expression

k1,2,3 Construction circles

l The fixed link length

p The length of the link adjacent to the ground in a planar four-bar

mechanism which is also the driven link

q The coupler link length

rM,N Construction circle radii

r1 The shortest link length in a planar four-bar mechanism

r2,3 The length of a link in a planar four-bar mechanism which is neither

the longest nor the shortest link

xii

r4 The longest link length in a planar four-bar mechanism

s The length of the link adjacent to the ground in a planar four-bar

mechanism which is also the driver link

t Compressed parameter expression

A(′) Left anchor revolute centre

B(′) Right anchor revolute centre

C(′) Right coupler revolute centre

D(′) Left coupler revolute centre

I1,2 Line parallel to the driven link constructed through the anchor

revolute centre of the driver link

L The angular lead of the driver link relative to the driven link

Lmax Maximum lead

Lmin Minimum lead

M Construction circle centre

N Construction circle centre

xiii

O Origin

Q The time ratio of a planar four-bar mechanism

α The swing angle of the driver link

β The swing angle of the driven link

δ The swing angle of the rocker link

ζ Compressed parameter expression

η Compressed parameter expression

λ Arbitrary choice of direction in construction

µ Compressed parameter expression

ρ1 Length ratio of driven link to driver link in planar a four-bar mechanism

ρ2 The length ratio of coupler link to driver link

ρ3 The length ratio of fixed link to driver link

φ Half of the angular difference between forward and return stroke crank

arcs in a crank-rocker mechanism

xiv

Chapter 1

Introduction

1.1 Background and Motivation

Reuleaux describes a mechanism as an “assemblage of resistant bodies, connected by

movable joints, to form a closed kinematic chain with one link fixed and having the purpose

of transforming motion.” A kinematic chain is a particular layout of links and joints when

the fixed link is unspecified [1]. A kinematic chain is called a mechanism when the fixed

link is chosen. This fixed link is considered as a reference frame to measure the motion of

any points on the linkage [1]. In order for a mechanism to execute a motion desired by its

designer, relative motions between links must be constrained. The required relative motions

between links are obtained by an appropriate choice of the number of links and the types of

joints that connect the links [1]. Considering the relative motions between links, mechanisms

may be divided into some subgroups like planar, spherical, cylindrical, Schonflies, and spatial

categories. In the case of planar mechanisms all points on every rigid body move so as to

remain on fixed, parallel planes. In planar mechanisms, the loci of all points on the links

can be described as planar curves that are parallel to a common plane [1]. The planar

four-bar is the most common mechanism type. Any given example may be categorized

as belonging to one of four sub-types, viz., crank-rocker, slider-crank, double-rocker, and

double-crank. Planar four-bar linkages are applied widely in machinery. A crank-rocker

linkage is a practical means of converting continuous rotary motion to oscillatory rotation. If

1

the oscillation moves points in a straight line, the mechanism is a slider-crank. This linkage

is called a quick-return mechanism if the time necessary for the working stroke is greater

than the time required for the return stroke. Quick-return mechanisms have been commonly

used to enhance machine productivity. There are many applications which require a working

stroke with high loading and a return stroke with small loading. Examples include cut-off

saws and shapers. The planar four-bar crank-rocker linkage has been widely utilized for such

a conversion between parallel shafts. Double-rocker or double-lever mechanisms are mainly

used for rocking motion like in Ackermann steering, rocking chairs, and pantographs. The

latter are copying devices [2] commonly used in drafting, sculpturing and minting. Double-

crank or drag-link mechanisms are mainly utilized as a coupling between two parallel non-

coaxial shafts to convert a uniform motion to a nonuniform motion of the driven shaft. This

nonuniform continuous output rotation is used as a flexible nonuniform input source for other

mechanism loops which generate nonuniform and discontinuous motions [3].

Therefore, because of the wide application and important role of four-bar mechanisms

in general and quick-return mechanisms in particular, the design, as opposed to analysis, of

four-bar mechanisms is still an open topic and is deemed to be worthy of further study. Since

the motions of all elements of a planar mechanism may be viewed in true size and shape

from a single direction and all motions can be represented graphically in a single view [1],

graphical methods are well suited to their solution. As a result, most of the work done to

aid in the design of planar four-bar mechanisms are based on graphical methods hence not

immediately implementable as “Computer Aided Design (CAD)” algorithms. For this reason

there remains a need to adapt these methods or develop new ones to this end.

1.2 Literature Review

Much research has been devoted to the investigation of four-bar mechanisms in general

and quick-return mechanisms in particular. One of the most important issues in this area

2

is the mobility problem. Grashof was the first who introduced a simple rule to investigate

the rotatability of links in four-bar linkages in 1883. Chang et al., introduced the principles

of Grashof’s criterion in [4] as follows. Consider a four-bar kinematic chain where r1 is

the shortest link length, r4 is the longest link length, and r2 and r3 are the link lengths

of the other two links. In summary, r4 > r3 ≥ r2 > r1. Grashof declared that in case

r4 + r1 ≤ r2 + r3, then at least one link can fully rotate with respect to the other three links

of the linkage, and if r4 + r1 > r2 + r3, then no link can make a full rotation. The necessity

and sufficiency of Grashof’s inequality for the existence of at least one fully-rotatable link in

a four-bar linkage was proved by Paul in [5]. If a linkage satisfies Grashof’s criterion, then it

is called a Grashof linkage; otherwise it is called a non-Grashof linkage. Moreover, a linkage

is identified as a Grashof neutral linkage or a change point mechanism if r4+r1 = r2+r3. In a

change point mechanism, it is not possible to uniquely determine all instantaneous centres of

the mechanism [6], and then the output behaviour of the mechanism will not be determined.

A Grashof linkage belongs to one of the following three different categories [4]: First,

the linkage is a crank-rocker mechanism when the shortest link is ground-adjacent. In this

case, the shortest link can make a full rotation and the other ground-adjacent link will be

a rocker making oscillating motion. Second, the linkage is a double-crank mechanism when

the shortest link is fixed. In this case, both ground-adjacent links can make full rotations.

And third, the linkage is a double-rocker mechanism when the shortest link is the coupler.

In this case, both ground-adjacent links will have oscillating motion, and the coupler can

make a full rotation.

If a four-bar linkage is a non-Grashof linkage, then its three moving links can have

oscillating motions and it is called a triple-rocker mechanism. Table 1.1 indicates a summary

of Grashof’s criterion.

Much research has been devoted to the investigation of four-bar mechanisms in general

and quick-return mechanisms in particular. To briefly mention a few, Freudenstein expressed

the classical problem of function generation and introduced a vector of dimensionless design

3

Table 1.1: Sorting planar four-bar linkages according to Grashof’s criterion [4], [5]

Case r1 + r4 Vers. r2 + r3 Shortest Link Class1 < Ground-Adjacent Crank-Rocker

2 < Frame Double-Crank3 < Coupler Double-Rocker4 = Any Change Point5 > Any Triple-Rocker

variables, called the Freudenstein parameters, to synthesize a four-bar linkage for a prescribed

relation between the input and output angles [7]. The Whitworth mechanism was employed

by Dwivedi [8] to implement a high velocity impacting press. Suareo and Gupta [9] designed

a spatial RSSR quick-return mechanism. Galerkin’s technique was used by Beale [10] to

study the stability and dynamics of a flexible link employed in a quick-return mechanism.

Fung [11], [12], [13], [14] and Lin [15], [16] used different control methods to examine the

response of a quick-return mechanism in the presence or absence of a flexible link. To

estimate the numerical solutions of a flexible quick-return mechanism, Ha [17] outlined a

finite difference technique using fixed and variable grids. The coupling influence of a geared

rotor on a quick-return mechanism experiencing three-dimensional vibration was examined

by Chang [18]. Chiang [19] studied double-rocker mechanisms as function generators and

considers the design of double-rocker mechanisms as a special case of creating a function

generator to match two determined velocity ratios. Khare and Dave [20] presented a method

for the synthesis of the planar four-bar double-rocker mechanism for determined extreme

positions with optimum transmission characteristics. Al-Dwairi investigated the relationships

among space occupation, extreme transmission angle, and the generated maximum delay in

double-crank mechanisms in [21], which constitutes the basis of the design method provided

for double-crank in this thesis.

Notwithstanding current activity in advanced dynamics of four-bar mechanisms, there

still remain some simple kinematic issues to be resolved. For example, in a recent text by

Uicker et al. (2011) [22], the method to design crank-rockers for a specified crank angle ratio

depends on a construction involving an arbitrary choice of direction. Thus, the length of

4

the rocker arm, hence the angle through which it rotates depends on this arbitrarily chosen

direction.

In the method outlined in [22], to determine the parameters of a crank-rocker mechanism

for specified rocker length p, rocker swing angle δ, and time ratio Q = (π + φ)/(π − φ), first

the rocker p is drawn to a suitable scale in its extreme positions indicating the desired swing

angle of δ. As shown in Fig. 1.1, a line of arbitrary direction, λ, is constructed through C.

Then, a line parallel to λ is drawn through C ′. Next, the angle φ, from line λ through point

C, is measured off. The intersection of this line with the line parallel to line λ indicates the

location of point A. The crank link length and the coupler length can be obtained from the

following equations: AC = q + s and AC ′ = q − s. Then the values of s and q will be: s =

(AC − AC ′)/2 and q = (AC + AC ′)/2.

q

s

p

φ

B

C C’

δ

A

D

D’

λ

Figure 1.1: Crank-rocker four-bar mechanism design according to [22]

The arbitrary direction of this line affects the final linkage design of the mechanism

5

obtained, and the link lengths of the linkage would differ for a different choice of direction

of this line. Furthermore, the underlying geometry, obscured by recipe-like instructions, is

thus unavailable for calculation in, say, a CAD algorithm.

1.3 Objectives

The objective of this research is to propose a systematic design method for each of the

four types of planar four-bar mechanisms. A unique mechanism is produced for any given

design specification. The method relies on the algebra of line and circle intersections and

reflects the procedure of well conceived procedures of constructive geometry. There are four

length parameters, six assembly modes and four frame selection criteria to be considered in

any design of a planar four-bar mechanism. Design equations, pertaining to link angular

displacement ratios in a motion cycle, will be derived. Although planar four-bar mechanisms

have been extensively studied, it is contended that the aspects to be dealt with herein are

essentially novel and original.

1.4 Thesis Outline

This thesis is organized as follows. Chapter 2 proposes a new and unified way to design

the quick-return cycle of either a crank-rocker or a slider-crank mechanism. Two design

equations are introduced which result in a unique mechanism configuration for two different

design parameter set selections. It is believed that Chapter 2 is indeed an improvement on

Uicker and Pennock’s design method. The new method is compared to that presented in [22]

in order to demonstrate the improvement that has been achieved.

In Chapter 3, a novel design method for double-rocker mechanisms is investigated. In a

planar four-bar double-rocker mechanism, the oscillation intervals of two links adjacent to the

6

fixed frame are identified in terms of usual “elbow folded” and “elbow extended” singularities

which occur when three revolute joints are in a line. These singular positions for a double-

rocker mechanism decompose into three cases depending on whether each rocker link is

in “elbow folded” or “elbow extended” singularity configuration at the limits of oscillating

intervals. Three cases “elbow extended-elbow folded”, “elbow extended-elbow extended”,

and “elbow folded-elbow folded” of the mechanism configuration are completely studied.

Moreover, each case is divided into two different sections when two rockers rotate in unison

and when two rockers counter-rotate. Each part contains a numerical example to illustrate

the design procedure.

Chapter 4 is about designing double-crank mechanisms. This chapter is built upon the

work of Al-Dwairi [21] so as to develop a novel design process to obtain a unique double-

crank mechanism for two specified angles at which the chosen maximum and minimum

angular leads of input angle to output angle occur. A numerical example is also provided to

illustrate the design process.

Finally, a summary of findings with respect to the project’s objectives and their im-

provements on previous design methods is provided in Chapter 5 as conclusions.

7

Chapter 2

Crank-Rocker and Slider-Crank1

A novel design method to determine parameters of a planar crank-rocker mechanism

is revealed. For a given crank angle ratio, which defines the ratio of rocker working stroke

to rocker return stroke, there is a one-parameter solution set. E.g., to design for a chosen

ratio of coupler length to crank-throw radius, one may use the implicit quartic function

that is revealed for the first time in this thesis to determine a unique planar crank-rocker

mechanism. This quartic function is described in terms of the ratio of coupler length to

crank-throw radius, the angular difference between forward and return stroke crank arcs,

and a polar angle which is measured from the centre of the unit circle that contains the end

points of the oscillating rocker tip arc and all possible crank centres. Due to the symmetry

of this geometric formulation, it is easy to find a unique mechanism for a defined crank angle

ratio and the ratio of coupler length to crank-throw radius.

2.1 Design Process

The time ratio, or crank angle ratio, of a mechanism is defined by Q = α/β where α is

the working stroke crank angle, and β is the return stroke angle. The time ratio angles are

defined by two singularities, where the crank and coupler are collinear. These singularities,

ADC and D′AC ′, define the oscillation limit of rocker indicated with C and C ′. Since

1This chapter is largely adapted directly from [26]

8

B

A

A

A

D

D

C’C

DA’D D

R

c

h

2

O

2D’

D’

D’

φφ

(a)

O

C’C

c

A

O’

CNCM11h

O

O’ C’C

D

D’

A

y O

k1

k3k2

φ/2

φ

φ

φ

φ

(b)

Figure 2.1: a) Pivot A remains on its circumscribing circle subtended on chord CC ′ b)Designing a crank-rocker with a geometric approach based on intersection of three circles:k1, k2, and k3

9

α + β = 2π, then α = π + φ and β = π − φ, and consequently Q = (π + φ)/(π − φ).

The design parameters are ρ2 = q/s, where q = coupler length of the mechanism and s

= crank-throw radius of the mechanism, and angle φ. As indicated in Figs. 2.1(a) and 2.1(b),

φ is the angle between ADC and D′AC ′. A constant angle φ, at vertex A of triangle ACC ′,

is maintained if A remains on the circumscribing circle of this triangle which is subtended

on chord CC ′. A must lie on this circle centred at O. When designing a slider-crank, CC ′ is

the gamut of the piston or wrist pin centre.

Consider the configurations elbow folded and elbow extended pattern of D′AC ′ and

ADC which respectively define lengths s and q in terms of the long AC = q + s and short

AC ′ = q − s rays which are subtended from chord CC ′. Consider s → 0; then, AC = AC ′

and the triangle ACC ′ becomes isosceles. Concerning Fig. 2.1(b), to find yO, the distance

OO′ from the centre of the chord CC ′ to the circumscribing circle centre O, one may invoke

the following chain of logic applied to another isosceles triangle AOC ′, whose equal sides of

length R are the segments AO and C ′O.

∠C ′OO′ + ∠AOC ′ = π, ∠OAC ′ = ∠OC ′A = φ/2 (2.1)

∠OAC ′ + ∠OC ′A + ∠AOC ′ = π, ∴ ∠C ′OO′ = φ

We may also normalize all distance parameters by setting R = 1 which helps find the unique

mechanism configuration, given designer chosen values of φ and ρ2; thus, yO = cos φ and

c = sin φ. Choosing the origin at O, all points A, a one-parameter set, can be represented

on the intersection of any pair of the three following circles k1, the circle centred at C ′ of

radius q − s, k2, the circle centred at C of radius q + s ,and k3, the circle centred at O of

radius R = 1, shown on the left of Fig. 2.1(b).

k1 : (x − c)2 + (y − yO)2 − (q − s)2 = 0 (2.2)

10

k2 : (x + c)2 + (y − yO)2 − (q + s)2 = 0 (2.3)

k3 : x2 + y2 − 1 = 0 (2.4)

Now, parameters x, y, and q will be eliminated from k1, k2, and k3. By proper substi-

tutions two implicit quartic equations will be obtained; the first in parameters φ, θ, and s2

and the second in parameters φ, θ, and ρ2, where θ is the polar angle measured from the

centre of the unit circle.

To eliminate x2 and y2 we form the subtractions k1 −k3 and k2 −k3. Next, q is removed

from these two differences. Then, the substitutions x = cos θ, y = sin θ, c = sin φ, and

yO = cos φ are made in the resultant term. After some algebraic simplification, a quadratic

in s2 is obtained.

s4 + 2(sin θ cos φ − 1)s2 + cos2 θ sin2 φ = 0 (2.5)

This first design equation expresses s2 as a quadratic in φ and θ.

One may prefer to select ρ2 instead and use the following relation. By forming the

difference k1 − k2 and substitution for s2 in terms of ρ2, obtained from the previous quartic

equation in s2, we obtain a design equation that expresses the parameter ρ2 as a function of

φ and θ.

cos2 θ sin2 φρ22 + 2(sin θ cos φ − 1)ρ2 + cos2 θ sin2 φ = 0 (2.6)

For a chosen value of rocker swing angle δ, the rocker link length can be obtained as

p =c

sin δ2

=sin φ

sin δ2

(2.7)

Eqs.(2.5) and (2.6) may also be used to design a slider-crank mechanism. In a slider-

crank mechanism the wrist pin that connects the coupler to the sliding block moves on a

straight line between C and C ′ of length 2c = 2 sin φ. The designer is thus able to obtain a

11

desired stroke η by multiplying link lengths s and q, obtained through Eqs.(2.5) and (2.6),

by ǫ = η/2 sin φ.

2.2 Design Examples

Here we establish the parameters of a planar four-bar crank-rocker mechanism with

given rocker length and rocker swing angle using the method suggested in [22]. Then we

solve a similar problem with the design method introduced in this thesis, which is to de-

termine the parameters of a planar four-bar crank-rocker mechanism with determined ratio

of coupler length to crank-throw radius and crank angle ratio. For brevity, the units of

length-parameters, e.g., inches, are not shown.

2.2.1 Design Example Using Text Book Method

It is desired to determine the parameters of a crank-rocker mechanism with rocker length

p = 4 and rocker swing angle δ = π/4. Also, a time ratio Q = 1.5 is selected [22].

To solve the problem, first one calculates φ = π/5 with the aid of equation Q = (π +

φ)/(π − φ). Then, by applying a suitable scaling, the rocker link with length p = 4 is

drawn in its extreme positions indicating the desired swing angle of δ = π/4 between these

two extreme positions. As shown in Fig. 2.2, a line λ on C is constructed in an arbitrary

direction. Then, a line parallel to λ is drawn through C ′. Next, another line on C is laid off

at an angle φ from λ whose intersection with the line on C ′ and parallel to λ locates A.

The crank link length and the coupler length can be obtained by measuring the lengths

of segments AC ′ = q − s and AC = q + s in Fig. 2.2.

q + s = 5.23, q − s = 3.93 ⇒ s = 0.65, q = 4.58

12

B

C C’

δB

C C’

δ

B

C C’

δ

A

q

s

φ

p

B

C C’

δ

A

φ

B

C C’

δ

AD

D’

λ λ

λ

Figure 2.2: Crank-rocker four-bar mechanism design according to [22]

The resulting mechanism is not unique because of the arbitrary direction of λ. This

produces a different link length set for every choice of this direction. Automating the process

would require choosing a fixed direction for no good reason or, similarly, using random

number generation for this purpose. The step by step design procedure is indicated in

Fig. 2.2.

2.2.2 Design Example Using Three Circle Method

Consider that it is desired to determine the parameters of a planar four-bar crank-rocker

mechanism with the ratio of coupler length to crank-throw radius ρ2 = q/s = 4 and crank

angle ratio Q = 1.5.

To solve the problem, first one calculates φ = π/5 with the aid of equation Q = (π +

φ)/(π − φ). Then, the differences k1 − k3 and k2 − k3 are formed incorporating ρ2 = q/s.

13

2cx + c2 − 2yOy + y2O − (ρ2 + 1)2s2 + 1 = 0

and

−2cx + c2 − 2yOy + y2O − (ρ2 − 1)2s2 + 1 = 0

Then, by inserting the numerical values we obtain

2 + 1.175570505x − 1.618033989y − 25s2 = 0

and

2 − 1.175570505x − 1.618033989y − 9s2 = 0

Eliminating x between these two equations and between the first and k3 yields a linear

and a quadric equation in y and s2.

4.70228202 − 3.804226068y − 39.96939716s2 = 0

and

2.618033988 − 6.472135956y − 100s2 + 4y2

+80.90169944s2y + 625s4 = 0

Solving these two equations simultaneously for s2 results in four real roots.

s2 = ±0.1420388931, ±0.02371638781

Only the positive ones are acceptable, thus

s = ±0.3768804759, ±0.154

Since s represents a length, the negative values obtained are discarded; moreover, between

14

two positive values obtained only one is acceptable because for s = 0.154, the segments AC ′,

C ′C, and CA cannot make a triangle in Fig. 2.1(b), and consequently a crank radius of 0.154

cannot achieve the rocker stroke implied by Q. Thus,

s = 0.3768804759 ⇒ q = 4s = 1.507521904

Substituting the value of s = 0.3768804759 into the linear equation in y and s2 results

in a unique value of y; by substituting the value obtained for y and s into either one of

equations k1 − k3 or k2 − k3, a unique value for x will be obtained.

y = −0.2562747041, x = 0.9666039945

The swing angle of the rocker may also be important to the designer. If the desired

swing angle of the rocker is δ = π/4, then according to Figs. 2.1(a) and 2.1(b) the proper

corresponding rocker length will be

p =c

sin δ2

=sin φ

sin δ2

= 1.53596

To validate the result by Grashof’s criterion, the fixed link length should also be calcu-

lated. The fixed link length is equal to the distance between points A and B. According to

Fig. 2.1(a), the coordinates of point B are

B : (0, cos φ − p cosδ

2) = (0,−0.6100250127)

Since l =√

(xA − xB)2 + (yA − yB)2, then

l = 1.029301979

According to Table 1.1, since the sum of the longest and the shortest link lengths, p+s =

1.912840476, is smaller than the sum of the other two link lengths, q + l = 2.536823883, and

15

C C’

A

B

s

p

q

D

D’

O

φ

C’

C’ C’

C

C C

O

B

A

B

A

φ

x

y y

x

Figure 2.3: Crank-rocker four-bar mechanism design verification for three circle method

the shortest link is ground-adjacent, so the obtained mechanism is a Grashof crank-rocker

with a crank link of length s.

Constructing this mechanism configuration is outlined in Fig. 2.3. First, the unit circle

is sketched to an appropriate scale, and the location of points C and C ′ are identified by

measuring angles −φ and φ from the y-axis of a coordinate system centred at point O, the

centre of the unit circle. Next, the points A and B are located; the coordinates of point A

are obtained above and the location of point B is the intersection of y-axis and two circles

centred at points C and C ′ of radius p. Finally, by connecting points A and B to points C and

C ′ and verifying the coupler link and crank link on the figure, the mechanism configuration

is obtained in its extreme postures.

16

Plotting all this on the unit radius circle shown in Fig. 2.3 verifies the design, and

obviously, the design is unique based on conscious choice of the designer.

17

Chapter 3

Double-Rocker

A novel design method to determine the parameters of a planar double-rocker mechanism

is proposed. For given oscillating angles of the links adjacent to the fixed link, there is a

one-parameter solution set. E.g., to design for a chosen ratio of the rocker link lengths, one

may use the quadratic equation that is revealed for the first time in this thesis to determine

a unique double-rocker mechanism configuration, given oscillating angles of two rockers, the

ratio of two rocker link lengths, and by considering the relative direction of rotation of rocker

links.

3.1 Design Process

In a planar four-bar double-rocker mechanism, the oscillation interval limits α and β

of two links adjacent to the fixed link, called rockers, are identified in terms of usual elbow

folded and elbow extended singularities which occur when three revolute joints are in a

line. These singular positions for a double-rocker mechanism decompose into three different

cases depending on whether each rocker link is in elbow folded or elbow extended singularity

configuration at the limits of the intervals of oscillation.

The first case happens when the mechanism oscillates between two singularity conditions

elbow folded and elbow extended. This case is indicated in Fig. 3.1(a).

The second case happens when the mechanism oscillates between two elbow extended

18

D1

C2

C1

D2

A B

s

p

q

l

(a)

D1

C2

C1D2

A B

s p

q

l

(b)

D1

C2

C1

D2

A B

s

p

q

l

(c)

Figure 3.1: Three individual cases embedded within extreme excursions a) “elbow extended-elbow folded” case b) “elbow extended-elbow extended” case c) “elbow folded-elbow folded”case

19

singularities. This case is indicated in Fig. 3.1(b).

The third case happens when the mechanism oscillates between two elbow folded singu-

larities. This case is indicated in Fig. 3.1(c).

Any of these three cases is divided into two different subsets depending on whether the

two rockers counter-rotate or turn in unison. In what follows, the geometric design algorithm

of these six cases will be introduced.

3.1.1 The Design Process of “Elbow Extended-Elbow Folded” Case

Here, we establish the parameters of a planar four-bar double-rocker mechanism oscil-

lating between singularity conditions “elbow extended” and “elbow folded” for determined

values of angles α, β, and the ratio of two rocker link lengths, p/s, first for the case when

the rockers rotate in unison and second for the case when the rockers counter-rotate.

A: Rotation in Unison

Consider the design of a double-rocker mechanism with determined values of α, β, and

the ratio of two rocker link lengths p/s.

Considering Fig. 3.2, angle α is subtended on two rays of length s and s − q while the

other angle β is subtended on rays of length p and p + q, where s and p are two rocker

link lengths and q is the coupler link length. Angle α sweeps to an elbow folded limit

while β sweeps to an elbow extended limit. The angles are subtended on a common chord

represented by the segment D1C2 in both cases. The length of segment D1C2 is indicated

by 2c. Applying trigonometric relationships to triangle AD1C2 and BD1C2, two following

equations are obtained.

(p + q)2 + p2 − 2(p + q)p cos β − (2c)2 = 0 (3.1)

20

D1 C2

A

B

M

N

O

A’

B’

D1 C2

A

B

p+q p

s

s - q

α

β

Figure 3.2: Two circles and a common chord in “elbow extended-elbow folded” case withrockers rotating in unison

(s − q)2 + s2 − 2(s − q)s cos α − (2c)2 = 0 (3.2)

Subtracting Eq.(3.2) from Eq.(3.1) and applying algebraic simplifications produce

(a1s + a2p)q − a1s2 + a2p

2 = 0 (3.3)

which is a linear equation in q, and where a1 = 1 − cos α and a2 = 1 − cos β.

Next, dividing Eq.(3.3) by s2 produces a normalized linear equation of ρ2 in terms of ρ1,

α and β as

(a1 + a2ρ1)ρ2 + a2ρ21 − a1 = 0 (3.4)

⇒ ρ2 =a1 − a2ρ

21

a1 + a2ρ1

21

where ρ2 = q/s, ρ1 = p/s, a1 = 1 − cos α and a2 = 1 − cos β.

Obviously, via Eq.(3.4), for certain values of ρ1, α, and β, a unique value for ρ2 will be

obtained. It is still important to note that any combination of the values of ρ1, α, and β do

not necessarily result in a proper coupler link length of a double-rocker mechanism, and in

order to have a feasible double-rocker mechanism, the resulting value for ρ2 should be a real

positive number.

According to Eq.(3.4), since ρ1 is considered to have a positive value, ρ2 is always a real

number unless the angles α and β are zero simultaneously, which for sure is not a desired

design case. For ρ1<√

(1 − cos α)/(1 − cos β), ρ2 always assumes a positive real value and

consequently the mechanism is feasible.

In the next step, to find the locations of the revolutes at A and B, consider a frame with

origin O at half chord D1C2 of length c. Positive x-axis is horizontal to the right, and y-axis

is vertical upward. Considering Fig. 3.2 again, the centre of the circle subtending angle α

on the circumference is centred at M(xM , yM) = (0, c cot α) and has radius rM = c csc α.

The centre of the circle subtending angle β on the circumference is centred at N(xN , yN) =

(0, c cot β) and has radius rN = c csc β. The equations of the two above mentioned circles

are respectively given by

x2A + (yA − c cot α)2 −

c2

sin2 α= 0 (3.5)

x2B + (yB − c cot β)2 −

c2

sin2 β= 0 (3.6)

The next step is to express the locations of points A and B in terms of p, q, and s. A

can be represented by the intersection of two circles k1, a circle centred at D1 of radius s,

and k2, a circle centred at C2 of radius s − q.

k1 : (xA + c)2 + y2A − s2 = 0 (3.7)

22

k2 : (xA − c)2 + y2A − (s − q)2 = 0 (3.8)

Subtracting Eq.(3.7) from Eq.(3.5) and Eq.(3.8) from Eq.(3.5) produce a pair of linear

equations in xA and yA. The sum of these two equations yields an equation of yA in terms

of s and q.

− 2yAc cot α − 2xAc − 2c2 + s2 = 0 (3.9)

− 2yAc cot α + 2xAc − 2c2 + (s − q)2 = 0 (3.10)

− 4yAc cot α − 4c2 + s2 + (s − q)2 = 0 (3.11)

Solving Eq.(3.11) yields yA. Substituting this value into Eq.(3.9) gives xA.

yA = −sin α(4c2 − 2s2 + 2sq − q2)

4c cos α, xA = −

q(−2s + q)

4c(3.12)

Similarly, B can be represented by the intersection of two circles k3, a circle centred at

C2 of radius p, and k4, a circle centred at D1 of radius p + q.

k3 : (xB − c)2 + y2B − p2 = 0 (3.13)

k4 : (xB + c)2 + y2B − (p + q)2 = 0 (3.14)

Subtracting Eq.(3.13) from Eq.(3.6) and Eq.(3.14) from Eq.(3.6) produce a pair of linear

equations in xB and yB. Their sum yields an equation of yB in terms of p and q.

− 2yBc cot β + 2xBc − 2c2 + p2 = 0 (3.15)

23

− 2yBc cot β − 2xBc − 2c2 + (p + q)2 = 0 (3.16)

− 4yBc cot α − 4c2 + p2 + (p + q)2 = 0 (3.17)

Solving Eq.(3.17) yields yB. Substituting this value into Eq.(3.15) gives xB.

yB = −sin β(4c2 − 2p2 − 2pq − q2)

4c cos β, xB =

q(2p + q)

4c(3.18)

Recall from Eq.(3.1) and Eq.(3.2), c can be represented with one of the following equa-

tions respectively.

c =1

2

√

(p + q)2 + p2 − 2(p + q)p cos β (3.19)

c =1

2

√

(s − q)2 + s2 − 2(s − q)s cos α (3.20)

By substituting either value of c of Eq.(3.19) and Eq.(3.20), here Eq.(3.20), in the equa-

tions obtained for xA, yA, xB, and yB, and then dividing them by s, normalized coordinates

of A = f(ρ2, α) and B = g(ρ1, ρ2, β) are obtained as

(xA, yA) = (ρ2(2 − ρ2)

2t,sin α(1 − ρ2)

t) (3.21)

(xB, yB) = (ρ2(2ρ1 + ρ2)

2t,tan β(ρ2

1 + ρ2ρ1 + (ρ2 − 1)(1 − cos α))

t) (3.22)

(xA, yA) and (xB, yB) are normalized coordinates of A and B respectively, xA = xA/s, yA =

yA/s, xB = xB/s, yB = yB/s, ρ1 = p/s, ρ2 = q/s, and t =√

ρ22 − 2ρ2(1 − cos α) + 2(1 − cos α).

24

B: Counter-Rotation

This case is very similar to the case when two rockers rotate in unison, except that the

centres of two subtending circles lie on opposite sides of segment D1C2. Compare Fig. 3.3

with Fig. 3.2.

D1 C2

A

B

M

N

O

B’

p+q p

s

s - q

α

β

A’

D1 C2

A

B

Figure 3.3: Two circles and a common chord in “elbow extended-elbow folded” case withrockers counter-rotating

Now, consider the design of a double-rocker mechanism with determined values of α, β,

and the ratio of rocker link length, p/s, and when rockers counter-rotate. Here, Eqs.(3.1),

(3.2), (3.3) and (3.4) can be used without modification. So,

(a1 + a2ρ1)ρ2 + a2ρ21 − a1 = 0 (3.23)

⇒ ρ2 =a1 − a2ρ

21

a1 + a2ρ1

25

where ρ1 = p/s, ρ2 = q/s, a1 = 1− cos α, a2 = 1− cos β, and ρ1<√

(1 − cos α)/(1 − cos β).1

In the next step, to find the locations of revolutes A and B, consider a frame with origin

O at half chord D1C2 of length c. Positive x-axis is horizontal to the right, and y-axis is

vertical upward. Considering Fig. 3.3, the centre of the circle subtending angle α on the

circumference is centred at M(xM , yM) = (0,−c cot α) and has radius rM = c csc α. The

centre of the circle subtending β on the circumference is centred at N(xN , yN) = (0, c cot β)

and has radius rN = c csc β. The equations of the above mentioned circles are respectively

given by

x2A + (yA + c cot α)2 −

c2

sin2 α= 0 (3.24)

x2B + (yB − c cot β)2 −

c2

sin2 β= 0 (3.25)




k1 : (xA + c)2 + y2A − s2 = 0 (3.26)

k2 : (xA − c)2 + y2A − (s − q)2 = 0 (3.27)

Subtracting Eq.(3.26) from Eq.(3.24) and Eq.(3.27) from Eq.(3.24) produce a pair of

linear equations in xA and yA. The sum of these two equations yields an equation of yA in

terms of s and q.

2yAc cot α − 2xAc − 2c2 + s2 = 0 (3.28)

1Refer to section 3.1.1.A

26

2yAc cot α + 2xAc − 2c2 + (s − q)2 = 0 (3.29)

4yAc cot α − 4c2 + s2 + (s − q)2 = 0 (3.30)


yA =sin α(4c2 − 2s2 + 2sq − q2)

4c cos α, xA = −

q(−2s + q)

4c(3.31)



k3 : (xB − c)2 + y2B − p2 = 0 (3.32)

k4 : (xB + c)2 + y2B − (p + q)2 = 0 (3.33)


linear equations in xB and yB. The sum of these two equations yields an equation of yB in

terms of p and q.

− 2yBc cot β + 2xBc − 2c2 + p2 = 0 (3.34)

− 2yBc cot β − 2xBc − 2c2 + (p + q)2 = 0 (3.35)

− 4yBc cot α − 4c2 + p2 + (p + q)2 = 0 (3.36)

Solving Eq.(3.36) yields yB. Substituting this value into Eq.(3.34) gives xB.

27

yB = −sin β(4c2 − 2p2 − 2pq − q2)

4c cos β, xB =

q(2p + q)

4c(3.37)

Recall from Eq.(3.1) and Eq.(3.2), c can be represented with one of the following equa-

tions respectively.

c =1

2

√

(p + q)2 + p2 − 2(p + q)p cos β (3.38)

c =1

2

√

(s − q)2 + s2 − 2(s − q)s cos α (3.39)

By substituting either value of c of Eq.(3.38) and Eq.(3.39), here Eq.(3.39), in the equa-

tions obtained for xA, yA, xB, and yB, and then dividing them by s, normalized coordinates


(xA, yA) = (ρ2(2 − ρ2)

2t,−

sin α(1 − ρ2)

t) (3.40)

(xB, yB) = (ρ2(2ρ1 + ρ2)

2t,tan β(ρ2

1 + ρ2ρ1 + (ρ2 − 1)(1 − cos α))

t) (3.41)



ρ22 − 2ρ2(1 − cos α) + 2(1 − cos α).

Obviously, the coordinates of B here are the same as when two rockers rotate in unison,

and the coordinates of A are the mirror image of the coordinates of A in previous section,

when rockers rotate in unison, with respect to y-axis.

3.1.2 The Design Process of “Elbow Extended-Elbow Extended” Case

Here, we establish the parameters of a planar four-bar double-rocker mechanism oscillat-

ing between the singularity conditions“elbow extended”and“elbow extended”for determined

28

values of α, β, and the ratio of two rocker link length p/s, first for the case when two rockers

rotate in unison and second for the case when two rockers counter-rotate.


D1 C2

A

B

M

N

O

A’

B’

D1 C2

B

p+q p

s s+q

α

β

A

Figure 3.4: Two circles and a common chord in “elbow extended-elbow extended” case withrockers rotating in unison

Considering Fig. 3.4, angle α is subtended on two rays of length s and s + q while the

other angle β is subtended on rays of length p and p + q, where s and p are two rocker link

lengths and q is the coupler link length. Angle α and β both sweep to an elbow extended

limit. The angles are subtended on a common chord represented by the segment D1C2 in both

cases. The length of segment D1C2 is indicated by 2c. Applying trigonometric relationships

in triangles AD1C2 and BD1C2, the two following equations are obtained.

(p + q)2 + p2 − 2(p + q)p cos β − (2c)2 = 0 (3.42)

29

(s + q)2 + s2 − 2(s + q)s cos α − (2c)2 = 0 (3.43)

Subtracting Eq.(3.43) from Eq.(3.42) and applying algebraic simplifications produce a

linear equation in q as

(−a1s + a2p)q − a1s2 + a2p

2 = 0 (3.44)

where a1 = 1 − cos α and a2 = 1 − cos β.

Next, dividing Eq.(3.44) by s2 produces a normalized linear equation of ρ2 in terms of

ρ1, α and β as

(−a1 + a2ρ1)ρ2 + a2ρ21 − a1 = 0 (3.45)

⇒ ρ2 = −a1 − a2ρ

21

a1 − a2ρ1


Obviously, by means of Eq.(3.45), for chosen values of ρ1, α, and β, a unique value for

ρ2 will be obtained. It is still important to note that any combination of the values of ρ1, α,

and β do not necessarily result in a proper coupler link length of a double-rocker mechanism,

and the resulting value for ρ2 must be a real positive number.

With the aid of simple algebraic operations, the value obtained for ρ2 is always feasible

but only if ρ1 ∈ (min{a1/a2,√

a1/a2}, max{a1/a2,√

a1/a2}), where a1 = 1 − cos α and

a2 = 1 − cos β.

In the next step, to find the locations of revolutes A and B, consider a frame with


is vertical upward. Considering Fig. 3.4, the centre of the circle subtending angle α on the

circumference is centred at M(xM , yM) = (0, c cot α) and has radius rM = c csc α. The centre

of the circle subtending angle β on the circumference is centred at N(xN , yN) = (0, c cot β)

30

and has radius rN = c csc β. The circle equations are given by

x2A + (yA − c cot α)2 −

c2

sin2 α= 0 (3.46)

x2B + (yB − c cot β)2 −

c2

sin2 β= 0 (3.47)

The next step is to express the points A and B in terms of p, q, and s. A can be

represented by the intersection of two circles k1, a circle centred at D1 of radius s, and k2, a

circle centred at C2 of radius s + q.

k1 : (xA + c)2 + y2A − s2 = 0 (3.48)

k2 : (xA − c)2 + y2A − (s + q)2 = 0 (3.49)



terms of s and q.

− 2yAc cot α − 2xAc − 2c2 + s2 = 0 (3.50)

− 2yAc cot α + 2xAc − 2c2 + (s + q)2 = 0 (3.51)

− 4yAc cot α − 4c2 + s2 + (s + q)2 = 0 (3.52)


yA = −sin α(4c2 − 2s2 − 2sq − q2)

4c cos α, xA =

q(2s + q)

4c(3.53)

31



k3 : (xB − c)2 + y2B − p2 = 0 (3.54)

k4 : (xB + c)2 + y2B − (p + q)2 = 0 (3.55)


linear equations in xB and yB. The sum of these two equations yields an equation of yB in

terms of p and q.

− 2yBc cot β + 2xBc − 2c2 + p2 = 0 (3.56)

− 2yBc cot β − 2xBc − 2c2 + (p + q)2 = 0 (3.57)

− 4yBc cot α − 4c2 + p2 + (p + q)2 = 0 (3.58)

Solving Eq.(3.58) yields yB, and then substituting this value into Eq.(3.56) gives xB.

yB = −sin β(4c2 − 2p2 − 2pq − q2)

4c cos β, xB = −

q(2p + q)

4c(3.59)

Recall from Eq.(3.42) and Eq.(3.43), c can be represented with one of the following

equations respectively.

c =1

2

√

(p + q)2 + p2 − 2(p + q)p cos β (3.60)

c =1

2

√

(s + q)2 + s2 − 2(s + q)s cos α (3.61)

32

By substituting either value of c of Eq.(3.60) and Eq.(3.61), here Eq.(3.61), in equations

obtained for xA, yA, xB, and yB, and then dividing by s, normalized coordinates of A =

f(ρ2, α) and B = g(ρ1, ρ2, β) are obtained as

(xA, yA) = (ρ2(2 + ρ2)

2t,sin α(1 + ρ2)

t) (3.62)

(xB, yB) = (−ρ2(2ρ1 + ρ2)

2t,tan β(ρ2

1 + ρ2ρ1 − (ρ2 + 1)(1 − cos α))

t) (3.63)



ρ22 − 2ρ2(1 − cos α) + 2(1 − cos α).

B: Counter-Rotation

Considering Fig. 3.5, for the case that the two rockers counter-rotate, the coordinates of

one of the revolutes at A or B —choose B— will remain the same as its coordinates in the

case “elbow extended-elbow extended with rockers rotating in unison”, and the coordinates

of the other revolute, i.e., A, will be the mirror image of its coordinates in the case “elbow

extended-elbow extended under rotation in unison” in respect to y-axis.2 So,

(xA, yA) = (ρ2(2 + ρ2)

2t,−

sin α(1 + ρ2)

t) (3.64)

(xB, yB) = (−ρ2(2ρ1 + ρ2)

2t,tan β(ρ2

1 + ρ2ρ1 − (ρ2 + 1)(1 − cos α))

t) (3.65)

3.1.3 The Design Process of “Elbow Folded-Elbow Folded” Case

Here, we establish the parameters of a planar four-bar double-rocker mechanism oscil-

lating in the singularity condition “elbow folded-elbow folded” for chosen values of α, β, and

2Refer to section 3.1.1.B

33

D1 C2

A

B

M

N

O

B’

p+q p

s

s+q

α

β

A’

D1 C2

A

B

Figure 3.5: Two circles and a common chord in “elbow extended-elbow extended” case withrockers counter-rotating

the ratio of two rocker link lengths p/s, first for the case when two rockers rotate in unison

and second for the case when two rockers counter-rotate.


Considering Fig. 3.6, angle α is subtended on two rays of length s and s − q while the

other angle β is subtended on rays of length p and p − q, where s and p are two rocker link

lengths and q is the coupler link length. Angles α and β both sweep to an elbow folded

limit. The angles α and β are subtended on a common chord represented by the segment

D1C2 in both cases. The length of segment D1C2 is indicated by 2c. Applying trigonometric

relationships in triangles AD1C2 and BD1C2, the two following equations are obtained.

(p − q)2 + p2 − 2(p − q)p cos β − (2c)2 = 0 (3.66)

34

D1 C2

A

B

M

N

O

A’

B’

D1 C2

A

B

p - q p

s

s - q

α

β

Figure 3.6: Two circles and a common chord in“elbow folded-elbow folded”case with rockersrotating in unison

(s − q)2 + s2 − 2(s − q)s cos α − (2c)2 = 0 (3.67)

Subtracting Eq.(3.67) from Eq.(3.66) and applying algebraic simplifications produce a

linear equation in q as

(a1s − a2p)q − a1s2 + a2p

2 = 0 (3.68)

where a1 = 1 − cos α and a2 = 1 − cos β.

Next, dividing Eq.(3.68) by s2 produces a normalized linear equation of ρ2 in terms of

ρ1, α and β as

(a1 − a2ρ1)ρ2 + a2ρ21 − a1 = 0 (3.69)

35

⇒ ρ2 =a1 − a2ρ

21

a1 − a2ρ1


Obviously, by means of Eq.(3.69), for chosen values of ρ1, α, and β, a unique value for

ρ2 will be obtained. It is still important to note that any combination of the values of ρ1, α,

and β do not necessarily result in a proper coupler link length of a double-rocker mechanism,

and in order to have a feasible double-rocker mechanism, the resulting value for ρ2 should

be a real positive number.

With simple algebraic operations, the value obtained for ρ2 is always feasible but only if

ρ1 /∈ (min{a1/a2,√

a1/a2}, max{a1/a2,√

a1/a2}), where a1 = 1 − cos α and a2 = 1 − cos β.

In the next step, to find the locations of revolutes A and B, consider a frame with


is vertical upward. Considering Fig. 3.6 again, the centre of the circle subtending angle α

on the circumference is centred at M(xM , yM) = (0, c cot α) and has radius rM = c csc α.

The centre of the circle subtending angle β on the circumference is centred at N(xN , yN) =

(0, c cot β) and has radius rN = c csc β. The equations of these two above mentioned circles

are respectively given by

x2A + (yA − c cot α)2 −

c2

sin2 α= 0 (3.70)

x2B + (yB − c cot β)2 −

c2

sin2 β= 0 (3.71)




k1 : (xA + c)2 + y2A − s2 = 0 (3.72)

36

k2 : (xA − c)2 + y2A − (s − q)2 = 0 (3.73)



terms of s and q.

− 2yAc cot α − 2xAc − 2c2 + s2 = 0 (3.74)

− 2yAc cot α + 2xAc − 2c2 + (s − q)2 = 0 (3.75)

− 4yAc cot α − 4c2 + s2 + (s − q)2 = 0 (3.76)


yA = −sin α(4c2 − 2s2 + 2sq − q2)

4c cos α, xA = −

q(−2s + q)

4c(3.77)


C2 of radius p, and k4, a circle centred at D1 of radius p − q.

k3 : (xB − c)2 + y2B − p2 = 0 (3.78)

k4 : (xB + c)2 + y2B − (p − q)2 = 0 (3.79)


linear equations in xB and yB. Their sum yields an equation of yB in terms of p and q.

− 2yBc cot β + 2xBc − 2c2 + p2 = 0 (3.80)

37

− 2yBc cot β − 2xBc − 2c2 + (p − q)2 = 0 (3.81)

− 4yBc cot α − 4c2 + p2 + (p − q)2 = 0 (3.82)

Solving Eq.(3.82) yields yB, and then substituting this value into Eq.(3.80) gives xB.

yB = −sin β(4c2 − 2p2 + 2pq − q2)

4c cos β, xB = −

q(2p − q)

4c(3.83)

Recall from Eq.(3.66) and Eq.(3.67), c can be represented with one of the following

equations respectively.

c =1

2

√

(p − q)2 + p2 − 2(p − q)p cos β (3.84)

c =1

2

√

(s − q)2 + s2 − 2(s − q)s cos α (3.85)

By substituting either value of c of Eq.(3.84) and Eq.(3.85), here Eq.(3.85), in the

equations obtained for xA, yA, xB, and yB, and then dividing by s, normalized coordinates


(xA, yA) = (ρ2(2 − ρ2)

2t,sin α(1 − ρ2)

t) (3.86)

(xB, yB) = (−ρ2(2ρ1 − ρ2)

2t,tan β(ρ2

1 − ρ2ρ1 + (ρ2 − 1)(1 − cos α))

t) (3.87)



ρ22 − 2ρ2(1 − cos α) + 2(1 − cos α).

38

B: Counter-Rotation

Considering Fig. 3.7, for the case that the rockers counter-rotate, the coordinates of one

of the revolutes A and B —take B— will remain the same as its coordinates in the case

“elbow folded-elbow folded with rockers rotating in unison”, and the coordinates of the other

revolute, i.e., A, will be the mirror image of its coordinates in the case “elbow folded-elbow

folded with rockers rotating in unison” with respect to y-axis as follows,3

D1 C2

A

B

N

O

A’

B’

D1 C2

A

B

p - q p

ss - q

α

β

M

Figure 3.7: Two circles and a common chord in“elbow folded-elbow folded”case with rockerscounter-rotating

(xA, yA) = (ρ2(2 − ρ2)

2t,−

sin α(1 − ρ2)

t) (3.88)

(xB, yB) = (−ρ2(2ρ1 − ρ2)

2t,tan β(ρ2

1 − ρ2ρ1 + (ρ2 − 1)(1 − cos α))

t) (3.89)

3Refer to section 3.1.1.B

39

3.1.4 Summary

All design equations derived for six different cases are summarized in Table 3.1.4. Note

that for two rockers rotating in unison k = 1, and for two rockers counter-rotating k = −1,

while a1 = 1 − cos α, a2 = 1 − cos β, and t =√

ρ22 − 2ρ2(1 − cos α) + 2(1 − cos α).

Table 3.1: The parameters of double-rocker mechanism in terms of specified design parame-ters for three different oscillatory motion configurationsIndividual Cases ρ2 (xA, yA) (xB , yB)

Extended-Foldeda1−a2ρ2

1

a1+a2ρ1

∗

(ρ2(2−ρ2)2t

, ksin α(1−ρ2)

t) (ρ2(2ρ1+ρ2)

2t,

tan β(ρ2

1+ρ2ρ1+(ρ2−1)(1−cos α))

t)

Extended-Extended −a1−a2ρ2

1

a1−a2ρ1

∗∗

(ρ2(2+ρ2)2t

, ksin α(1+ρ2)

t) (−ρ2(2ρ1+ρ2)

2t,

tan β(ρ2

1+ρ2ρ1−(ρ2+1)(1−cos α))

t)

Folded-Foldeda1−a2ρ2

1

a1−a2ρ1

∗∗∗

(ρ2(2−ρ2)2t

, ksin α(1−ρ2)

t) (−ρ2(2ρ1−ρ2)

2t,

tan β(ρ2

1−ρ2ρ1+(ρ2−1)(1−cos α))

t)

∗ ρ1<√

(1 − cos α)/(1 − cos β)∗∗ ρ1 ∈ (min{a1/a2,

√

a1/a2}, max{a1/a2,√

a1/a2})∗∗∗ ρ1 /∈ (min{a1/a2,

√

a1/a2}, max{a1/a2,√

a1/a2})

3.2 Design Examples

Here, the procedure of obtaining the parameters of a planar four-bar double-rocker which

result in a unique mechanism design will be illustrated with numerical examples. The ratio

of link length ρ2 = q/s and ρ3 = l/s, and the locations of revolute joints A and B will be

obtained for specified values of ρ1 = p/s, and oscillating angles of rocker links α and β for

six different cases.

40

B

D1 D1

B

D1

B

C2

C2

A

D1

B

C2

A

D1

B

C2

A

C1

D2

k1

k3

k2

β

β

Figure 3.8: The steps to draw a double-rocker mechanism in its extreme excursions “elbowextended-elbow folded” with rockers rotating in unison

First, consider the case when the mechanism is desired to oscillate between the sin-

gularity conditions “elbow extended” and “elbow folded” under rotation in unison. The

design parameter α = 20◦ and β = 30◦. Thus, a1 = 1 − cos α = 0.0603073792 and a2 =

1 − cos β = 0.1339745960. Since ρ1<√

(1 − cos α)/(1 − cos β) = 0.6709250384, then any

value ρ1 < 0.6709250384 can be used. Let ρ1 = 0.4 for this example. Therefore, with the

aid of Table 3.1.4,

k = 1

ρ2 = 0.3412852800

(xA, yA) = (0.6394591798, 0.5089822290)

(xB, yB) = (0.4399824397, 0.3349414824)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 0.2647284483

According to Table 1.1 and considering s = 1, since the sum of the longest and the

shortest link lengths, s + l = 1.2647284483, is greater than the sum of the other two link

lengths, p + q = 0.7412852800, so the obtained mechanism is a non-Grashof double-rocker

(or triple-rocker).

41

D1

A B

s

s-q

p

q

D2

C1

C2

q

(a)

s

s-q

p

D1

D2

C1

C2

A B

q

p

(b)

Figure 3.9: Double-rocker mechanism within extreme excursions “elbow extended-elbowfolded” with α = 20◦, β = 30◦, and ρ1 = 0.4 a) under rotation in unison b) under counter-rotation

In order to construct the mechanism configuration in its extreme excursions, first draw

BD1 of length p + q to a suitable scale. Then measure angle β from segment BD1 at point

B and draw the line BC2 of length p. The location of point A will be the intersection of

three circles; the first circle, k1, is centred at D1 of radius s, the second one, k2 is centred at

point C2 of radius s − q, and the third one, k3, is centred at point B of radius l. The steps

of drawing the mechanism in its extreme excursions is illustrated in Fig. 3.8. The drawing

steps for other cases are similar.

If it is desired that the rockers counter-rotate for α = 20◦, β = 30◦, and ρ1 = 0.4 , then

k = −1

42

ρ2 = 0.3412852800

(xA, yA) = (0.6394591798,−0.5089822290)

(xB, yB) = (0.4399824397, 0.3349414824)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 0.8671782980


shortest link lengths, s + q = 1.3412852800, is greater than the sum of the other two link

lengths, p + l = 1.267178298, so the mechanism obtained is a non-Grashof double-rocker (or

triple-rocker).

Now, consider it is desired that the mechanism be bounded between two elbow extended

singularity configurations and the rockers rotate in unison. The design parameters α = 20◦

and β = 30◦. Thus, a1 = 1−cos α = 0.0603073792 and a2 = 1−cos β = 0.1339745960. Since

ρ1 ∈ (min{a1/a2,√

a1/a2}, max{a1/a2,√

a1/a2}) , then it can assume any value between

0.4501404072 and 0.6709250384. Let ρ1 = 0.55 for this example. Therefore, with the aid of

Table 3.1.4,

k = 1

ρ2 = 1.478479964

(xA, yA) = (1.762664372, 0.5810744359)

(xB, yB) = (−1.306603693, 0.3823825710)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 3.075692591


shortest link lengths, l + p = 3.625692591, is greater than the sum of the other two link

lengths, s+ q = 2.478479964, so the mechanism obtained is a non-Grashof double-rocker (or

triple-rocker).

For the case “elbow extended-elbow extended”, under counter-rotation with α = 20◦, β

= 30◦, and ρ1 = 0.55, with the aid of Table 3.1.4,

43

s

D1

BA

C1

C2

D2

p

q

(a)

D1

D2

C1

C2

A B

s

p

q

(b)

Figure 3.10: Double-rocker mechanism within extreme excursions “elbow extended-elbowextended” with α = 20◦, β = 30◦, and ρ1 = 0.55 a) under rotation in unison b) undercounter-rotation

k = −1

ρ2 = 1.478479964

(xA, yA) = (1.762664372,−0.5810744359)

(xB, yB) = (−1.306603693, 0.3823825710)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 3.216932679


shortest link lengths, l + p = 3.766932679, is greater than the sum of the other two link

44

lengths, s+ q = 2.478479964, so the mechanism obtained is a non-Grashof double-rocker (or

triple-rocker).

Finally, for the case “elbow folded-elbow folded” when the rockers rotate in unison with

angles α = 20◦ and β = 30◦. Thus, a1 = 1 − cos α = 0.0603073792 and a2 = 1 − cos β =

0.1339745960 . Since ρ1 /∈ (min{a1/a2,√

a1/a2}, max{a1/a2,√

a1/a2}), it may not obtain

any values between 0.4501404072 and 0.6709250384. Let ρ1 = 0.7 for this example. There-

fore, with the aid of Table 3.1.4,

k = 1

ρ2 = 0.1595279655

(xA, yA) = (0.4122283856, 0.8071918940)

(xB, yB) = (−0.2778405617, 0.5311817093)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 0.7432205420


shortest link lengths, s + q = 1.1595279655, is smaller than the sum of the other two link

lengths, p+ l = 1.443220542, and the coupler is the shortest link, so the obtained mechanism

is a Grashof double-rocker.

For the case “elbow folded-elbow folded”, if the rockers counter-rotate with α = 20◦, β

= 30◦, and ρ1 = 0.7, according to Table 3.1.4,

k = −1

ρ2 = 0.1595279655

(xA, yA) = (0.4122283856,−0.8071918940)

(xB, yB) = (−0.2778405617, 0.5311817093)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 1.505801797


shortest link lengths, l + q = 1.6653297625, is smaller than the sum of the other two link

45

lengths, p + s = 1.7, and the coupler is the shortest link, so the obtained mechanism is a

Grashof double-rocker.

A B

s-q

q

p-qp

D1

D2

C1C2

s

(a)

sp-q

ps-q

A B

D1

D2

C1

C2 q

q

(b)

Figure 3.11: Double-rocker mechanism within extreme excursions “elbow folded-elbowfolded” with α = 20◦, β = 30◦, and ρ1 = 0.7 a) under rotation in unison b) under counter-rotation

In such cases where at least there is one elbow folded singularity limit, such as “elbow

extended-elbow folded” and “elbow folded-elbow folded” cases, for some chosen values of ρ1,

the value obtained of ρ2 may be greater than ρ1 and/or greater than unity; equivalently

q might be greater than p and/or s. The mechanism configuration “elbow folded-elbow

folded” is illustrated in Fig. 3.12 when q > s > p and rockers rotate in unison. The only

difference between these two cases, the first one with at least one elbow folded singularity and

46

q greater than p and/or s and the second one with at least one elbow folded singularity with

q smaller than p and s, is that in the first case, the rockers do not oscillate in angles facing

the common chord D1C2 but rotate through angles α = ∠D1AD2 and β = ∠C1BC2; this

difference does not affect the design formulas for ordinary “elbow folded-elbow folded” and

“elbow extended-elbow folded” cases. This will be explained for the case “elbow folded-elbow

folded” in the following and it would be similar for the case “elbow extended-elbow folded”.

Consider Eqs.(3.66) and (3.67); as indicated in Eqs.(3.90) and (3.91), if we change the terms

p − q to q − p, s − q to q − s, α to α′, and β to β′, the two obtained equations describe

the relations among the sides of the triangles in Fig. 3.12 but still are equal to Eqs.(3.66)

and (3.67). Note that the angles α = π − α′ and β = π − β′.

sq-sp

q-p

AB

D1

D2

C1

C2

αα’

β‘β

Figure 3.12: Common chord and oscillatory angles in “elbow folded-elbow folded” case withrockers rotating in unison when q > p and q > s

(q−p)2+p2−2(q−p)p cos(π−β)−(2c)2 = (p−q)2+p2−2(p−q)p cos(β)−(2c)2 = 0 (3.90)

47

(q−s)2 +s2−2(q−s)s cos(π−α)−(2c)2 = (s−q)2 +s2−2(s−q)s cos(α)−(2c)2 = 0 (3.91)

To obtain equations similar to Eqs.(3.70) to (3.89), it is sufficient to select the coordinate

system, in Fig. 3.12, centred at half chord D1C2 of length c while positive x-axis is horizontal

to the left and y-axis is vertical downward. Fig. 3.13 indicates an example of a double-rocker

mechanism in “elbow folded-elbow folded” case for α = 20◦, β = 30◦, and ρ1 = 0.3. With

the aid of Table 3.1.4 for the case of rotation in unison,

k = 1

ρ2 = 2.398690758

(xA, yA) = (−0.2023337253,−0.2024236414)

(xB, yB) = (0.9128272845,−0.1332071549)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 1.117307030


shortest link lengths, q + p = 2.698690758, is greater than the sum of the other two link

lengths, l + s = 2.117307030, so the obtained mechanism is a non-Grashof double-rocker (or

triple-rocker). And for the case when two rockers counter-rotate,

k = −1

ρ2 = 2.398690758

(xA, yA) = (−0.2023337253, 0.2024236414)

(xB, yB) = (0.9128272845,−0.1332071549)

ρ3 =√

(xA − xB)2 + (yA − yB)2 = l/s = 1.164573789


shortest link lengths, q + p = 2.698690758, is greater than the sum of the other two link

lengths, l + s = 2.164573789, so the obtained mechanism is a non-Grashof double-rocker (or

triple-rocker).

48

s q-s

p

q-p

A B

D1

D2

C1

C2

(a)

sq-p

A B

p

q-s

D1

D2

C1

C2

(b)

Figure 3.13: Double-rocker mechanism within extreme excursions “elbow folded-elbowfolded” with α = 20◦, β = 30◦, and ρ1 = 0.3 a) under rotation in unison b) under counter-rotation

49

Chapter 4

Double-Crank

A novel design method to determine parameters of a planar double-crank mechanism is

revealed. A unique double-crank mechanism parameter set is obtained for specified minimum

lead = Lmin, maximum lead = Lmax, and chosen angles α1 and α2 at which the driver link

has respectively its minimum lead and maximum lead relative to the driven link.

4.1 Design Process

Fig. 4.1 shows a double-crank mechanism in two configurations ABCD and ABC ′D′,

where the segments AB, BC(′), C(′)D(′), and D(′)A indicate the fixed link, the driven crank,

the coupler link, and the driver crank of the mechanism respectively. In both configurations

ABCD and ABC ′D′, the coupler link, is parallel to the fixed link. The angles α and β

indicate the counter-clockwise angles of the driving crank and the driven crank from the

fixed link respectively.

In a double-crank mechanism, the driver link rotates at a constant speed while it is

followed by the driven link with a variable speed. Specifically, the angular difference, that

shall be called L, between the driver and driven cranks varies continuously from minimum

to maximum value. These occur, respectively, when the drag link, or coupler, achieves a

parallel or antiparallel configuration with respect to the fixed link [21], [27], [28].

According to Grashof’s criterion, in a planar four-bar linkage, both ground-adjacent

50

A B

CD

M

D’C’N

α1

α2

µ

Lmin

Lmax

I1

I2

Figure 4.1: Double-crank mechanism in its parallel and antiparallel configurations

links are fully rotatable if and only if the sum of the longest link length and the shortest link

length is smaller than the sum of two other link lengths and the fixed link is the shortest [4].

Thus, the coupler link length CD must be greater than the fixed link length AB. Considering

Fig. 4.1, it is concluded that because CD > AB, then the angle α is always greater than the

angle β; therefore, the driver crank AD always leads the driven crank BC [21].

L = α − β (4.1)

In the parallel configuration of the mechanism, the lead has its minimum value Lmin =

α1 − β1, and in antiparallel configuration of the mechanism, the lead has its maximum value

Lmax = α2 − β2.

In order to find a unique mechanism configuration for chosen design parameters, it is

desired to find the ratios between the driver crank length and the other three link lengths of

the mechanism, i.e., the driven crank length to the driver crank length ρ1 = p/s, the coupler

link length to the driver crank length ρ2 = q/s, and the fixed link length to the driver crank

length ρ3 = l/s. In order to find these ratios, the geometric relations among links in parallel

and antiparallel configurations are utilized.

51

In the parallel configuration of the mechanism, M is the intersection point of coupler

link CD with I1 which is a line constructed parallel to BC through A. Consider a coordinate

frame centred at A with x-axis horizontal to the right and with y-axis vertical upward; since

DC ‖ AB, then the y coordinates of D and M are the same. yD = s sin α1, and yM =

p sin(α1 − Lmin). Therefore,

p =s sin α1

sin(α1 − Lmin)⇒ ρ1 =

p

s=

sin α1

sin(α1 − Lmin)(4.2)

Similarly, in antiparallel configuration of the mechanism, N is the intersection point

of the line parallel to AB passing through D′ with I2, the line constructed parallel to BC ′

through A; since C ′D′ ‖ AB, then the y coordinates of D′ and N are the same. yD′ =s sin α2,

and yN = p sin(α2 − Lmax). Therefore,

p =s sin α2

sin(α2 − Lmax)⇒ ρ1 =

p

s=

sin α2

sin(α2 − Lmax)(4.3)

Consequently, by equating Eqs.(4.2) and (4.3),

sin α1

sin α2

=sin(α1 − Lmin)

sin(α2 − Lmax)(4.4)

Applying the cosine law to triangle ADM , the value of µ, which is the length of segment

DM , can be obtained.

µ =√

p2 + s2 − 2ps cos Lmin ⇒µ

s=

√

ρ21 + 1 − 2ρ1 cos Lmin (4.5)

In the triangle AD′N , the length of segment ND′, ζ, can be expressed, again using the

cosine law, as follows.

ζ =√

p2 + s2 − 2ps cos Lmax ⇒ζ

s=

√

ρ21 + 1 − 2ρ1 cos Lmax (4.6)

Now, according to Fig. 4.1, q = ζ − l and q = µ + l. Therefore,

52

q =ζ + µ

2⇒ ρ2 =

q

s=

ζ + µ

2s(4.7)

and

l =ζ − µ

2⇒ ρ3 =

l

s=

ζ − µ

2s(4.8)

Thus, the values of the ratios ρ1, ρ2, and ρ3 are obtained which are sufficient to obtain a

unique mechanism configuration for specified minimum lead = Lmin, maximum lead = Lmax,

and chosen angles α1 and α2 at which the driver link has respectively its minimum lead and

maximum lead relative to the driven link.

4.2 Parameters α1, α2, Lmin, and Lmax

It is important to note some conditions and geometric relations among parameters α1,

α2, Lmin, and Lmax. This means that any combination of these four parameters may not

necessarily result in a double-crank mechanism. Eq.(4.4), indicates the dependency among

the values of the parameters α1, α2, Lmin, and Lmax, i.e., the designer is free to choose the

desired values for three parameters, and the fourth one will be obtained. Furthermore, as

indicated in Fig. 4.2 the angle α1 should always be smaller than π; otherwise for angles α1 >

π, the length of the fixed link will be greater than the length of coupler link which, according

to Grashof’s criterion, does not result in a double-crank mechanism. Moreover, because the

parallel and the antiparallel configurations of the mechanism should occur on opposite sides

of the fixed link and α1 < π, hence α2 > π.

Finally, according to Eq.(4.2), since ρ1 > 0 and α1 < π, then α1 − Lmin < π. Similarly,

according to Eq.(4.3), since ρ1 > 0 and α2 > π, then α2 − Lmax > π.

53

B C

DA

α1

Lmin

Figure 4.2: Coupler link length of BC > AB fixed link length for α1 > π

4.3 Design Example

In the following example, the parameters ρ1, ρ2, and ρ3 will be obtained for a double-

crank mechanism with Lmin = 30◦, Lmax = 80◦, and α1 = 120◦.

A B

C

C’

D

D’

s p

q

Figure 4.3: Parallel and antiparallel poses for Lmin = 30◦, Lmax = 80◦, and α1 = 120◦

With the aid of Eq.(4.2), ρ1 = cos π/6 is obtained. Then, by substituting the value

obtained for ρ1 in Eq.(4.5) and (4.6), the values of parameters µ/s and ζ/s are obtained as

0.5 and 1.203840743 respectively. Thus, according to Eqs.(4.7) and (4.8), ρ2 = 0.8519203715

54

and ρ3 = 0.3519203715. By choosing a suitable value for parameter s, the lengths of other

links will be obtained.


shortest link lengths, s + l = 1.3519203715, is smaller than the sum of the other two link

lengths, p+q = 1.71794577, and the fixed link is the shortest link, so the obtained mechanism

is a Grashof double-crank.

To construct the mechanism in its parallel and antiparallel configurations, first draw AB

of length l. Then, measure angle α1 from AB through point A of length s. Since Lmin = α1

- β1, then β1 =90◦. Measure angle β1 from AB through point B of length p. Similarly, for

antiparallel configuration, first measure angle α2 from AB through point A. The value of

angle α2 is obtained from Eq.(4.4) equal to 134.9◦. Since Lmax = α2 − β2, then β2 =54.9◦.

Measure angle β2 from AB through point B of length p. Fig. 4.3 shows the mechanism

configuration for the obtained design parameters.

55

Chapter 5

Conclusions

It is believed that we have developed novel geometric design methods for planar four-bar

mechanisms and substantially improved on the procedure for motion cycle design that could

be found in [22], especially by unifying the methods used to arrive at the desired mechanism

properties. A fairly careful search of some of the less well-known German and Austrian

literature failed to reveal anything on motion cycle design pertaining to double-rocker and

double-crank four-bar mechanisms.

The introduced design procedures are based on the geometric properties of mechanisms,

and the symmetry of these geometric formulations makes it easy to find a unique mechanism

configuration for defined design parameters. Another important achievement of this thesis is

to provide procedures that go beyond graphical constructions. The relations derived can be

conveniently programmed and implemented in “Computer Aided Design (CAD)”algorithms.

It is also important to note that any combination of design parameter values which are

chosen by the designer does not necessarily result in a feasible mechanism configuration. To

obtain a feasible mechanism configuration, the designer must chose a proper set of values

for design parameters. Automatic input pre-checks are easily incorporated to omit spurious

solutions due to improper values or improper combinations of design parameters which would

produce meaningless results.

Moreover, the notion of expressing all parameters as dimensionless ratios confines the

process to that of choosing proportion, not size, and is therefore consistent with a very im-

56

portant principle of engineering design. This principle is implicit in many fields of mechanical

engineering: Consider the fundamental design correlations wherever one encounters, in vari-

ous technical fields, the numbers of Sommerfeld, Reynolds, Froude, Grashof, Prandtl, Mach,

and Freudenstein parameters.

Each different type of planar four-bar mechanisms is studied separately in one chapter.

The design parameters are introduced, then the design process and design equations are

provided and clarified with proper figures and plots. Finally numerical examples are provided

to illustrate each procedure, and the results are validated by Grashof’s criterion. Explanation

of the step-by-step process to obtain each mechanism configuration is clearly illustrated with

appropriate diagrams.

In addition to the strictly angular design parameters that were addressed in this thesis,

there remains the issue of design optimization in the light of these. Design optimization

might include aspects such as transmission angle and mechanism compactness. The designer

might prefer to apply rigorous optimization in this regard rather than to make a somewhat

arbitrary choice based, at least partially, on experience and whim. The implementation of

mechanism optimization methodology, after primary angular performance specifications are

met, would seem to be a natural sequel to the line of investigation that has been taken in

this work.

57

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