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ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1145 www.ijaegt.com
Coupler Point Path Synthesis of Crank Rocker Mechanism
with Three Precision Positions and Unit Time Ratio
Dr. Khaled M. Khader
Department of Production Engineering & Mechanical Design
Faculty of Engineering, University of Menoufia
Shebin El-kom, Menoufia, Egypt
khkhm62@hotmail.com
Abstract – Crank rocker mechanisms have a wide engineering
applications, hence, its design attracted most researchers
attention. Path synthesis of mechanism's coupler point with a
definite precision positions associated with an optimal
transmission angle and unit time ratio of mechanism is the most
important and complicated part of mechanism design. There is a
lack of a computerized mechanism design, this problem
motivates to design a fast software to help mechanical designers.
Developed software called (SYNTH-COUPLER LAB) created
as a fast instantaneous tool for synthesizing the coupler point's
path of Chebyshev crank rocker mechanism for three precision
positions, satisficing optimum range of transmission angle and
unit time ratio of mechanism using Visual Basic programming
language. The software is helpful for mechanical designers and
researchers through providing an instantaneous calculations of
suitable mechanism links for generating a coupler point's path
which has the desired precision positions and satisfies an optimal
transmission angle and unit time ratio. Also, the software affords
an attractive clear animation of the synthesized mechanism
simultaneously with linkages ratios calculations.
Index Terms – Path Synthesis, Transmission Angle, Mechanism,
Design Techniques.
I. INTRODUCTION
An optimization problem presented in [1]; using Powell
technique in order to minimize an objective function of both
stroke and time ratio of the planar mechanism. Also,
mechanism synthesis dealing with the important design
parameters stroke, time ratio and transmission angle is
presented in [2] as a method for synthesizing three types of
planar mechanisms achieving the requests for stroke,
transmission angle and time ratio. An advanced motion
synthesis is presented in [3] using novel family of linkages.
A new synthesis approach for generating two coupler
precision positions of planar four bar mechanisms is
introduced in [4]. Synthesis dealing with the geometric
methods of planar linkages with three precision points is
presented in [5]. Also, synthesis of planar mechanism for
particular three coupler positions which can be accomplished
with transmission angles less than 18 % of its optimum value
of 900 is presented in [6]. While, synthesis of planar four bar
mechanism for certain four coupler positions presented in [7].
An advanced synthesis for pick and place jobs with
guiding positions of planar mechanism is presented in [8]. A
point to point path generation of an optimal synthesis is
presented in [9] for crank rocker mechanism. Synthesis of four
bar mechanism using the generalized methodology is presented
in [10]. A non-conventional approach method is presented in
[11] for path generation of planar mechanism using Harmony
search method to find the suitable mechanism dimensions
associated with an error minimization. A special adjustable
four bar linkages is presented in [12] to generate a desired
accurately continuous paths using a continuous controlled
adjustment for one independent parameter of four bar linkages.
Also, path generation using genetic algorithm of compliant
mechanisms is presented in [13].
Chebyshev and Evan mechanisms are planar mechanisms
have a certain coupler point (at its coupler link or at the
extension of this link). Aforementioned coupler point can be
moved through a requested path which can be used in the
industrial applications. Many researches deal with the path
synthesis of coupler point of Chebyshev and Evan mechanisms
as the synthesis of four-bar mechanisms for straight line
coupler curves in [14]. A function generators by Freudenstein-
Chebyshev used for synthesizing four bar mechanisms as in
[15]. Moreover, a motion generation of crank-crank linkages
using Chebyshev mechanism is introduced in [16]. Analysis
and design dealing with a walking robot of low cost and easy
operated leg using Chebyshev mechanism is presented in [17].
As well as, optimization problem dealing with the walking
machine design of eight-bar leg mechanism is presented in
[18].
Researches deal with optimal transmission angle in
addition to mechanism's stroke and time ratio considered as the
most important parameters through synthesizing planar crank
rocker mechanism. The transmission angles of a planar
mechanism is the smaller angle between the coupler link and
the output link. Furthermore, the reasonable fluctuation of the
transmission angle value from 900 is significant for providing a
good characteristics to mechanism motion as in [19]. Synthesis
a polynomial function generation of four bar mechanism
dealing with maximum and minimum transmission angles is
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1146 www.ijaegt.com
presented in [20]. The analytical synthesis of the motion
between two small separated positions with minimum and
maximum transmission angles of crank rocker mechanisms is
presented in [21]. In addition to, designing four bar
mechanism with function generators with an optimum angle, in
a mini-max sense, when their extreme values have variations
are equally around 900 as in [22]. As well as, nomograms for
synthesizing crank rocker mechanism with a definite desired
optimum range of transmission angle presented in [23] for
helping the mechanical designers.
On the other hand, other researches deal with Computer
Aided Design (CAD) in order to facilitate the designers jobs as
designing spherical mechanism using CAD software in [24],
while, [25] presented a computer aided position analysis and
modelling of crank and slotted lever mechanism. A multi-stage
gearboxes software for designing gearboxes is presented in
[26] providing the software's user with an easy interface menus
in order to quickly help the designers. As well as, software
called (SYNTH-MECH LAB) shown in [27] for synthesizing
crank rocker mechanism with the selected optimum range of
transmission angle for helping the mechanical designers.
Aforementioned software provides the designers with an
instantaneous calculations of suitable mechanism links ratios
for a definite synthesized transmission angle range.
This paper presents analytical modelling for synthesizing
the coupler point's path for including three desired precision
positions and satisfying the unit time ratio of crank rocker
mechanism.
First precision position of the coupler point occurs at the
first extreme position of rocker link. While, the second
precision position of the coupler point occurs at the second
extreme position of rocker link. Final third precision position
occurs when crank angle equals 00 which corresponds the
minimum transmission angle. Developed software called
(SYNTH-COUPLER LAB) is also created as a fast
instantaneous tool for synthesizing the coupler point's path of
Chebyshev crank rocker mechanism for three precision
positions satisfying optimal transmission angle and unit time
ratio using Visual Basic programming language.
II. COUPLER POINT'S PATH COORDINATES
Chebyshev planar crank rocker mechanism (ABCD) with
extended coupler link (BP) is indicated in Fig. 1 as follows;
Fig. 1 Coupler point (P) of Chebyshev crank rocker mechanism
where coupler point (rP) coordinates can be formulated as;
sinsin
coscos
2
2
PPy
PPx
rrr
rrr (1)
The coupler link angle (β) of crank rocker mechanism
which is indicated in Fig. 2 can be calculated as follows;
)2()(where,
)(where,
1124
1142
(2)
Fig. 2 Crank rocker mechanism
Also, angle (ϕ) of rocker link can be calculated as follows;
)2()(where,
)(where,
11311
11311
(3)
Where, (ɵ1) is the angle of the fixed link (r1), the length
(L) in addition to angles (τ1, τ 2, τ 3, τ 4 and transmission angle
μ) can be written as follows;
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1147 www.ijaegt.com
43
224
231
3
24
2231
4
4
23
2241
3
2
21
2221
2
1
22
2211
1
212
22
1
2cos
2cos
2cos
2cos
2cos
cos2
rr
Lrr
Lr
rLr
Lr
rLr
Lr
rLr
Lr
rLr
rrrrL
(4)
Where; fixed, crank, coupler and rocker links lengths are
r1, r2, r3 and r4, respectively. The rocker link (r4) which is
indicated in Fig. (2) has two extreme positions of its oscillating
motion through angle (00≤ψ≤3600). Also, minimum of
transmission angle (μ) occurs at (ψ=00). For assuring
mechanism continues motion without mobility problems
through passing at the three aforementioned precision
positions, the linkages lengths must be satisfying the following
constrains;
0
0
0
0
0
0
2143
3421
4321
3241
4231
1432
rrrr
rrrr
rrrr
rrrr
rrrr
rrrr
(5)
III. MODELLING FOR SYNTHESIZING THREE PRECISION
POSITIONS ON THE COUPLER POINT'S PATH
The desired coupler point's path of crank rocker
mechanism can be synthesised in order to pass through critical
precision positions and satisfy the unit time ratio. Generally,
designers are looking forward a desired path passes through
some critical precision positions of mechanism as the two
positions correspond to the two extreme positions of rocker
link, and the precision position corresponds the minimum
transmission angle. These three precision positions are
indicated in Fig. 3 as follows;
Fig. 3 Three precision positions of the crank rocker mechanism
The coordinates of the three precision positions and the
crank rocker mechanism's geometry is indicated in Fig. 4 as
follows;
Fig. 4 Coordinates of precision positions and geometry of mechanism
First precision position coordinate (x1, y1) of the coupler
point occurs at the first extreme position of rocker link at the
crank angle (ψ1). While, second precision position coordinate
(x2, y2) of the coupler point occurs at the second extreme
position of rocker link at the crank angle (ψ2+1800). Finally,
the third precision position coordinate (xm, ym) occurs when the
coupler angle (βm) corresponds to the minimum transmission
angle (μmin).
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1148 www.ijaegt.com
Satisfying the unit time ratio condition of crank rocker
mechanism which indicated in Fig. 4, leads to the following
equation;
21 (6)
Hence;
21 tantan (7)
Therefore;
11
22 y
x
xy (8)
Regarding Fig. 4, the equation dealing with the first
precision position (x1, y1) can be written as follows;
21
212 yxrrP (9)
Where (rP) is the total length of the extended coupler link
(BP). Also, equation dealing with the second position (x2, y2)
can be written as follows;
22
222 yxrrP (10)
Hence;
21
21
1
22 )( yx
x
xrrP (11)
Adding (9) to (11) leads to the following relation;
21
21
1
221
21 )(2 yx
x
xyxrP (12)
Therefore;
21
21
1
21 )2
( yxx
xxrP
(13)
Subtracting (11) from (9), the following equation can be
written as;
21
21
1
212 )
2( yx
x
xxr
(14)
Regarding the first extreme position of rocker link in Fig.
4, (cos ψ1) can be written as follows;
)(2
)(cos
231
24
223
21
21
21
11
rrr
rrrr
yx
x
(15)
Hence;
1231322
322
21
24 cos)(22 rrrrrrrrr (16)
Also, regarding the second extreme position of rocker link
in Fig. 4, (cos ψ2) can be written as follows;
)(2
)(cos
231
24
223
21
2rrr
rrrr
(17)
Hence;
2231322
322
21
24 cos)(22 rrrrrrrrr (18)
From (16) and (18) the following equation can be written
as;
113 cosrr (19)
Where (cos ψ1= cos ψ2) which satisfies the unit time ratio
condition of crank rocker mechanism. By substituting (19) in
(15) the following equation can be written as;
)(2
)(cos
231
24
223
21
1
31
rrr
rrrr
r
r
(20)
Hence; 2
322
21
24 rrrr (21)
The pervious relation guarantee the condition of optimal
transmission angle range has maximum and minimum values
which have variations equally around 900 as in [22], [23] as a
result of unit time ratio of mechanism.
By substituting (19) in (21) the following equation can be
written as;
122
122
21
24 cos rrrr (22)
Regarding the third precision position corresponds
minimum transmission angle in Fig. 4, (cos βm) can be written
as follows;
321
24
23
2212
)(2
)(cos
rrr
rrrr
r
rx
P
m
(23)
Hence;
mrrrrrrrrr cos)(22 213212
322
21
24 (24)
By substituting (19) in (24) the following equation can be
written as;
mm rrr
rrrrrr
coscos2coscos2
2cos
12112
1
21122
122
21
24 (25)
From (22) in (25) the following equation can be written as;
)1cos(cos2
)coscos(cos20
121
2111
2
m
m
rr
r
(26)
Hence;
211
11
cos)cos(cos
coscos1rr
m
m
(27)
By substituting the pervious equation in (19), the
following equation can be written as;
1211
13 cos
cos)cos(cos
coscos1
rr
m
m
(28)
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1149 www.ijaegt.com
By substituting (27) in (22), the following equation can be
written as;
12
21
122
114 cos
)coscos1(
cos)cos(cos1
m
mrr (29)
Now, the mechanism lengths (r1, r2, r3, r4) in addition to
the length of the extended coupler link (rP) can be calculated
using (27), (14), (28), (29) and (13) respectively only
depending on (x1, y1, x2 and xm ). Satisfying a unit time ratio of
mechanism guarantee the condition of optimal transmission
angle range have variations equally around 900 . The minimum and maximum values of transmission angle
(μmin and μmax ) are written as in [20], [28].
43
21
43
22
21
24
23
maxmin,2
cosrr
rr
rr
rrrr
(30)
IV. SYNTH-COUPLR LAB SOFTWARE
The developed software called (SYNTH-COUPLER
LAB) directly synthesizes the crank rocker mechanism lengths
(r1, r2, r3, r4) in addition to the length of the extended coupler
link (rP) only depending on (x1, y1, x2 and xm). The developed
software guarantee synthesizing the coupler point's path
including the three desired precision positions indicated in Fig.
5 as follows;
Fig. 5 Path of the coupler point
A. Software Welcome Menu:
The software welcome menu is indicated in Fig. 6. This
menu includes two buttons. The first button allows the
software's user for synthesizing the crank rocker mechanism
with a desired three precision positions. While, the second
button allows the software's user for selecting any lengths of
crank rocker mechanism linkages in addition to the extended
coupler length and showing mechanism positions parameters
in addition to showing its motion.
Fig. 6 Welcome menu of SYNTH-COUPLER LAB software
B. Software Menu of Mechanism's Synthesis:
The flow chart of the first part of SYNTH-COUPLER LAB
software of mechanism's synthesis shown in Fig. 7 as follows;
Fig. 7 Flow chart of first part of SYNTH-COUPLER LAB software
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1150 www.ijaegt.com
The software menu of mechanism's synthesis is indicated in
Fig. 8. This menu provides software's user with an attractive
interface for allowing him to directly select the desired vales of
three precision positions (x1, y1, x2 and xm) using an easy scroll
bars as indicated in Fig. 9.
Fig. 8 Software menu of mechanism's synthesis
Fig. 9 Scroll bars for selecting desired values
C. Menu of Mechanism's Positions Analysis and its Motion:
Fig. 10 shows the flow chart of second part of SYNTH-
COUPLER LAB software of mechanism's positions analysis
and its motion as follows;
Fig. 10 Flow chart of second part of SYNTH-COUPLER LAB software
The menu of the second part of the developed software is
indicated in Fig. 11 which provides software's user with an
attractive interface for allowing him to directly select the
desired crank rocker mechanism linkages lengths (r1, r2, r3, r4
and rP) and the crank angle (ψ) in addition to fixed link angle
(ɵ1).
Aforementioned menu in Fig. 11 includes button ("Press to
draw cycle") for showing attractive animation for a complete
turn of the crank. Another buttons are included for presenting
the rocker angle (ϕ), the coupler angle (β), the transmission
angle (μ), the rocker angular velocity ( ϕ' ), the coupler angular
velocity (β'), the rocker angular acceleration ( ϕ" ) and the
coupler angular acceleration (β'') through a complete turn of
the crank as shown in Fig. 12.
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1151 www.ijaegt.com
Fig. 11 Menu of mechanism's positions analysis and its motion
Fig. 12 Graphs of SYNTH-COUPLER LAB software
V. RESULTS
Using the developed software menus, the software's user
can select coordinates (x1=115.42 cm, y1=49.14 cm) of the first
precision position corresponds the first extreme position of the
rocker link. Also, software's user can select the horizontal
distance equals (38.12 cm) of the second coordinate (x2) from
(x1) corresponds the second extreme position of the rocker
link, this leads to (x2=77.3 cm and y2=32.91 cm). Software's
user can select the third coordinate (xm=107.5 cm) corresponds
the minimum transmission angle (μmin), this leads to (ym=58.62
cm). The corresponding synthesised values (r1, r2, r3, r4 and rP)
calculated using (27), (14), (28),(29) and (13) respectively
leading to calculated values (r1, r2, r3, r4 and rP) are; r1=58.5
cm, r2=20.72 cm, r3=53.85 cm, r4=30.89 cm and rP=104.72
cm, the values μmin=43.20 and μmax=136.80 means that (μmin +
μmax=1800) and the extreme values of transmission angle (μ)
have variations are equally around 900.
The transmission angle (μ) and the rocker angle (ϕ) can be
calculated also using (3), (4) and (30) for each crank angle (ψ).
The relation between (ψ) and both (μ & ϕ) is shown in Fig. 13
as;
Fig. 13 Relation between (ψ) and both (μ & ϕ)
Coupler point (rP) coordinates (rPx , rPy) can be calculated
also using (1), (2) and (4) for each crank angle (ψ). The
relation between (ψ) and (rPx , rPy) is shown in Fig. 14 which
shows the three desired precision positions are coinciding with
the path of the coupler point of the synthesized mechanism as
follows;
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I Vol-03, Issue-09, September 2015
1152 www.ijaegt.com
Fig. 14 Relation between (ψ) and both (rPx , rPy)
VI. CONCLUSION
The analysis and modelling are presented for synthesizing
the coupler point's path of crank rocker mechanism for
including three desired precision positions and satisfying the
unit time ratio of mechanism. As well as, developed software
(SYNTH-COUPLER LAB) created in this paper as a fast
instantaneous tool for synthesizing the coupler point's path of
Chebyshev crank rocker mechanism for three precision
positions satisfying optimal transmission angle and unit time
ratio using Visual Basic language. The created software is
helpful for mechanical designers, engineers and researchers
through providing an instantaneous calculations of suitable
mechanism links for generating three precision positions at the
coupler path and satisfies optimal transmission angle and unit
time ratio conditions in addition to affording clear animation
of the synthesized mechanism simultaneously with linkages
ratios calculations.
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I Vol-03, Issue-09, September 2015
1153 www.ijaegt.com
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