coupler point path synthesis of crank rocker mechanism...

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

[email protected]

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

mailto:[email protected]

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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;

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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).

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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)

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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

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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.

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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;

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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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