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International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 9, September 2017, pp. 442–453, Article ID: IJMET_08_09_048
Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=9
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
SIX-BAR LINKAGE MECHANISM THROUGH
THE CONTOUR METHOD
Rame Likaj, Xhevahir Bajrami*, Astrit Sallauka
Faculty of Mechanical Engineering
University of Prishtina, 1000 Prishtina, Kosovo * Corresponding author: xhevahirbajrami@gmail.com
ABSTRACT
In this paper is presented a six-bar linkage mechanism of the pump for oil
extrusion. In this mechanism are introduced even higher kinematic pairs. Dimensions
and other incoming links will be adopted as necessary. For the six-bar linkage
mechanism will be carried out the kinematic analysis and for all linkages will be
presented the displacement, velocity and acceleration. The analysis will be performed
by Math Cad software, while kinetostatic analysis will be carried out using Contour
Method, comparing results of two different software‘s Math CAD and Working Model.
The simulation parameters will be computed for all points of the contours of
mechanism.
Keywords: Kinematics, Dynamic, Simulation.
Cite this Article: Rame Likaj, Xhevahir Bajrami and Astrit Sallauka, six-bar linkage
mechanism through the contour method, International Journal of Mechanical
Engineering and Technology 8(9), 2017, pp. 442–453.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=9
1. INTRODUCTION
This paper has been realized with two application software’s: Math Cad and Working Model.
In this paper is presented a four-bar linkage ABCD and a simple crank mechanism DEF as
shown in the figure below. In this mechanism firstly is determined DFy and 5θ . Derivative
DFy represents the speed of the slider F. From the given picture o3601.0 2 ≤≤ θ should be
determined. The masses are adopted since the moments of inertia need to calculate. The
kinematic part of the paper will be completed by finding the velocities and accelerations of
each point A, B, C, D, E, and F for the centers from C1 to C6. In this way are determined the
angular accelerations and velocities of the linkages 2, 3, 4 and 5. Whereas for the kinetostatic
part will be determined the reaction forces of the points: XA, YA, XB,YB, XC,YC, XD,YD,
XE,YE, XF,YF, N=XF6 and Mtr which are acting on the leading link 2. In the picture are
shown 5 bodies: AB, BC right triangle, rod EF, and slider bar F.
Rame Likaj, Xhevahir Bajrami and Astrit Sallauka
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Figure 1 Six bar-linkage mechanism ABCDEF
Given data:
56.725 ,4.62.4 ,2.4:,9.6
7.1,2[deg],360...deg5.0
2243
222
=+===
===
α
ωθ
Da rrr
r
(1)
Determination of angleα of Dr with 1x : Linkage I
→→→→
=++ Da rrrr 432
(2)
From then designed outline at x and y we have the following initial conditions:
)sin()sin()sin()sin(
)cos()cos()cos()cos(
443322
443322
αθθθ
αθθθ
⋅=⋅+⋅+⋅
⋅=⋅+⋅+⋅
Da
Da
rrrr
rrrr
(3)
Figure 2 Four-bar linkage ABCD and crank mechanism DEF
As shown in the Figure 2, there are 14 unknown sizes, 12 equations for the first four
bodies, ),( yxdy for the slider 14 equations.
Six-Bar Linkage Mechanism through the Contour Method
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2. THE ANALYSIS OF THE POSITIONS, VELOCITIES AND
ACCELERATIONS OF SIX-BAR LINKAGE MECHANISM
In figure 3 is presented the velocity plan for the points A, B, C and D. Also are presented
angular velocities and angular accelerations for the points A, B and C. After finding the
velocities and accelerations of these points, the displacements, velocities and accelerations of
these points with angles 3θ and 4θ in function of 2θ are shown graphically. The outline is
shown in this form:
[deg]90[deg],180 43 == θθ (3)
The vector equation without the contour I is:
x
y
r4
rp
r2
r3
Figure 3 Velocity plan for the points A, B and C
=
+
+
0
)sin(
)cos(
0
)sin(
)cos(
0
)sin(
)cos(
0
)sin(
)cos(
44
44
33
33
22
22
α
α
θ
θ
θ
θ
θ
θ
p
p
r
r
r
r
r
r
r
r
(4)
22242
2
2
42
22322
2
3
2242
42232
3
43
432
)(,)(
)(,)(
[deg]48.59[deg],96.3[deg],18.104
),,()(
ωθθθ
εωθθθ
ε
ωθθθ
ωωθθθ
ω
βθθ
θθθ
⋅=⋅=
⋅=⋅=
=−==
=
d
d
d
d
d
d
d
d
FindZgj
(5)
After calculation of the values by Math Cad software, in the following are shown
graphically the values of positions, velocities, and accelerations respectively for 333 ,, εωθ and
444 ,, εωθ :
Rame Likaj, Xhevahir Bajrami and Astrit Sallauka
http://www.iaeme.com/IJMET/index.asp 445 editor@iaeme.com
Figure 4 Diagrams for positions, velocities and accelerations
The position of the point B is calculated in direction of x and y :
]/[03.10)()(
)cos(,)sin(
)sin(),cos(
22
22
222222
2222
scmvvv
rvrv
ryrx
ByBxB
ByBx
BB
=+=
⋅⋅=⋅⋅−=
⋅=⋅=
θθ
ωθωθ
θθ
(6)
Figure 5 The position diagrams for the position equations of the point B in direction of x and y
]/[)()(
)sin(,)cos(
222
22
2222
2222
scmaaa
rvra
ByBxB
ByBx
θθ
ωθωθ
+=
⋅⋅−=⋅⋅−=
(7)
Angular velocity and acceleration for the point B:
][deg45.0[deg],87.0
,
244
2242
2
2
4242
4
=−=
⋅=⋅=
εω
ωθθ
εωθθ
ωd
d
d
d
(8)
Six-Bar Linkage Mechanism through the Contour Method
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3. THE EQUATIONS FOR POSITION (DISPLACEMENT), VELOCITY
AND ACCELERATION OF THE POINT C
In the following is shown the displacement of point C, along with velocities and accelerations
which are presented graphically by the diagrams via Math Cad program:
)(
)))(cos()sin((
0)();(
))(cos()sin(,0
))(sin()cos(0
2
223322
22
23322
23322
θ
ωθθθ
θθ
θθθ
θθθ
cyc
cy
cxccyc
cycx
cc
aa
rra
avv
rrvv
rryx
=
⋅⋅−⋅−=
==
⋅+⋅==
⋅+⋅==
(9)
Figure 6 Velocities and acceleration of point C
3.1. The expression for the middle point of the bars AB, BC and CD
In the following are extracted the displacement for the middle position of the bars AB, BC
and CD and another displacement which belong to these bars. Are also presented the
velocities and accelerations of these bars and their diagrams separately:
22
22
22
22
22
222
2
)cos(2
;)sin(2
)sin(2
);cos(2
:
ωθωθ
θθ
⋅=⋅−
=
==
ra
ra
ry
rx
AB
ycxc
cc
(9)
2223
33
2223
32
233
3233
3
))(cos(2
;))(sin(2
))(sin(2
));(cos(2
:
ωθθωθθ
θθθθ
⋅=⋅−
=
==
ra
ra
ry
rx
BC
ycxc
cc
(10)
2224
44
2224
44
243
4244
4
))(cos(2
;))(sin(2
))(sin(2
));(cos(2
:
ωθθωθθ
θθθθ
⋅=⋅−
=
==
ra
ra
ry
rx
CD
ycxc
cc
(11)
Rame Likaj, Xhevahir Bajrami and Astrit Sallauka
http://www.iaeme.com/IJMET/index.asp 447 editor@iaeme.com
x
y R4a =42
50
R4a =40
R5 =92
Fy
E
Fig. 7 Crank mechanism
3.2. The equations for position (displacement), velocity and acceleration of the
point E
For the point, E of Crank Mechanism determined the displacement, and in the following
pictures are shown velocities and accelerations in the direction x and y:
22244
22244
2442244
244244
))(sin(;))(cos(
2))(cos(;))(sin(
))(sin());(cos(
ωθθωθθ
ωθθωθθ
θθθθ
⋅−=⋅−=
⋅=⋅−=
==
bEybEx
bEbE
bEbE
rara
ryrx
ryrx
(12)
Figure 8 Diagrams of the velocities and acceleration for the E
4. KINETOSTATIC ANALYSIS OF THE SIX-BAR LINKAGE
MECHANISM
For the kinetostatic part will be presented the kinetostatic analysis of six-bar linkage
mechanism by Math Cad software. Moments of inertia, masses of bodies are used in the
following:
2555
2244
2333
2222
65432
12
1);2.14.2(
12
1
2
1;
2
1
807.9;5;15;20;15;10
rmJmJ
rmJrmJ
gmmmmm
CC
CC
⋅⋅=−⋅=
⋅⋅=⋅⋅=
======
(13)
Six-Bar Linkage Mechanism through the Contour Method
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Linkage I:
The equilibrium conditions for the point A are equal to zero. Six linkages are used for the
kinetostatic analysis. For the first linkage are given the following equilibrium conditions
XA,YA and XB,YB for the middle points of the linkage AB, point C2 and for the body mass
m2.
)sin(2
)cos(2
)cos(2
)cos(2
)();(
22
22
22
22
222222
θθθθ
θθ
rX
rY
rX
rYM
amYYamXX
BBAAtr
ycBAxcBA
−+−=
⋅=−⋅=−
(14)
x
y
XA
YA
C2 Cxc2
Cyc2
XB
YB
Figure 9 Linkage I–Equilibrium conditions for the point AB
Linkage II
Also, for the linkage II are written the equilibrium conditions BC, which are XB,YB , XC,YC
center C3 body mass BC and moment of inertia for the point C3.
))(sin(2
))(2
))(cos(2
))(sin(2
)(
)();(
233
233
233
233
23
233233
θθθθθθθθθε
θθ
⋅⋅−⋅⋅−⋅⋅−⋅⋅=⋅
⋅=−⋅=−
rX
rY
rY
rXJ
amYYamXX
BBCCC
ycBCxcBC
(15)
x
y
XB
YB
C3 Cxc3
Cyc3
XC
YC
Figure 10 Linkage II – Equilibrium conditions for the part BC
Linkage III
)()()(
)(
)(
2244244
244
244
θθθε
θ
θ
mDzbaC
yCEDC
xCEDC
rXrYEHrYrYJ
amYYY
amXXX
⋅+⋅−−−+⋅−=⋅
⋅=−+
⋅=−+
(16)
Rame Likaj, Xhevahir Bajrami and Astrit Sallauka
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x
y
XD
YD
C4 Cxc4
Cyc4
XC
YC
XE
YE
Figure 11 Linkage III of the points C, D, E and center C4
))(sin(2
)())(sin(2
)()(
)(
)(
255
255
255
244
255
θθθθθε
θ
θ
⋅⋅+−⋅⋅+=⋅
⋅=−+
⋅=−
rYY
rXXJ
amYYY
amXX
FEFEC
yCEDC
xCFE
(17)
x
y
XF
YF
C5 Cxc5
Cyc5
XE
YE
Figure 12 Linkage IV of the part EF in direction of x and y
Linkage V
))(( 626 gmamY yF ⋅+⋅−= θ (18)
x
y
F
XF YF
m6
Figure 13 Linkage V of the point F in direction of x and y
All unknown parameters for the linkages in both directions x and y are calculated by Math
Cad. From point A to the point Find the Transmission moment Mtr:
trFFEEDDC
CBBAA
MYXYXYXY
XYXYXFindSolve
⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅= (:)2(θ
(18)
In the following diagrams are given the values of all unknown parameters for the positions
(0; 0.5; 1) by Math Cad software. This it was really a tough work since the working process
by Math Cad was very slow especially to follow the procedure which is given by kinetostatic
analysis.
Six-Bar Linkage Mechanism through the Contour Method
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Rame Likaj, Xhevahir Bajrami and Astrit Sallauka
http://www.iaeme.com/IJMET/index.asp 451 editor@iaeme.com
Figure 14 Diagrams of kinetostatic analysis of the given mechanism for the points A, B, C, D, E, F
and transmission moment Mtr
5. SIMULATION FOR SIX-BAR MECHANISM BY WORKING MODEL
SOFTWARE
In the second part of this paper is carried out the simulation for all points A, B, C, D, E, F and
Mtr (time dependent)of the six bar linkage mechanism by Working Model, which is shown in
the following.
Figure 15 Diagrams in Working Model for angular velocity and acceleration 3, 4, 5.
Six-Bar Linkage Mechanism through the Contour Method
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Figure 17 Diagrams in Working Model for equilibrium points A, B, C, D, E, F and transmission
moment (Mtr) in direction y
Figure18 Diagrams for 3θ, 4θ
and 5θ(time dependent), under command RUN
6. CONCLUSIONS AND RECOMANDATIONS
By application of the software’s Math Cad or Matlab we have reached to carry out the full
analysis of this paper. In this paper are made the calculations of all positions (displacement)
for the whole mechanism, and also are determined the plans for velocities and accelerations
for each point.
However, in this paper are shown the outline planes of the mechanism same as the
diagrams for each linkage through Math Cad and Working Model software, Six-bar linkage
mechanism diagrams which are derived by Working Model are almost similar to the diagrams
derived by Math Cad, same as the derived results, Through Working Model software, are
derived the results of reactions from the equilibrium conditions of six bar linkage mechanism,
angular velocities and accelerations for the points 3, 4 and 5, for the angles 3θ , 4θ and 5θ in
time domain through the command run, as the general conclusion; the results derived by both
software, same as for their diagrams, for all points of the six bar linkage mechanism are
within the reasonable boundaries.
REFERENCES
[1] H.J. Sommer: Vector loops facilitate simple kinematic analysis of planar mechanisms,
either closed-chain or open-chain, Mechanical & Nuclear Engineering Faculty, USA,
www.mne.psu.edu/sommer/me50/
Rame Likaj, Xhevahir Bajrami and Astrit Sallauka
http://www.iaeme.com/IJMET/index.asp 453 editor@iaeme.com
[2] J. M. McCarthy and G. S. Soh, 2010, Geometric Design of Linkages, Springer, New York.
[3] Bajrami, X., et al. (2016, June). Genetic and Fuzzy logic algorithms for robot path finding.
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