OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
curve i i
( Px , Py ) that has to be traced by the mechanism and the corresponding generated points i i
( Px , Py ) by the
d
d
designed mechanism. The desired points are a set of target points along the coupler curve indicated by the
designer and should be met by the coupler point of the mechanism and the generated curve is the curve that is
actually obtained by the coupler point of the designed mechanism. For minimizing the error between desired and
generated curves, the objective function is given by
1
Minimize F ( x)
= ⎡
N
∑ ⎢⎣
subject to h
j
( x)
≤ 0, x ≤ x ≤ x ,
where N = number of precision po int s
l
N
i=
1
i i 2 i i
( Px − Px ) + ( Py − Py )
j
d
u
g
d
2
g
⎤
⎥⎦
The inequality constraints ( h j
( x)
) consist of conditions of Grashof’s rule that specifies crank rocker situation.
x l and x u are respectively the lower and upper bounds of the design variables. They consist of link lengths L i
,
i i
( Px , Py ) and the structural angle, the angle that the line AP AB
makes with the line of centre of the coupler
d
d
to obtain the co-ordinates of the generated positions of the coupler point displacement analysis
as seen in Fig. 1
is essential.
The successful application of the developed methodology for synthesis of mechanisms is shown through some
case studies. All the data presented here are in a consistent set of units, i.e., all linear dimensions are in the unit
of length and the angular dimensions are in degrees unless otherwise stated. For each case study an intermediate
result and the final result are given. Finally, a summary of the results of all the cases indicating the values of the
design variables obtained through DE is presented. The optimization algorithm is applied using MATLAB
version 7.1. All the linkages shown in the following sections are obtained by using the design variables. Here
the technique of geometric centroid of precision points (GCPP) [12] has been used to define the initial bounds of
the design variables. The GCPP is obtained by evaluating the mean coordinates X and Y
cg
cg (mass center
position) of the desired precision points in X and Y directions respectively.
2.4. Results from Computer Programming (DE)
Input parameters
Number of population: 250
Maximum number of iteration: 1000
Minimum value to reach: ( 1.95) ∗1.
e − 0. 08
lmax Link length ratio: = 8
lmin
0
0
Limits of transmission angle: µ min = 15 and µ max = 165
Desired points
x : 26 23 20 17 14
d
y : 16 16 16 16 16
d
Incremental crank angles corresponding to the first precision point:
0 22 44 66 88
Considering following design variables as L 1 = x1
, L2
= x2
, L3
= x3
L4
= x4
, L5
= x5
, β = x6,
θ2
= x7
, θ1
= x8,
L0
= x9
, L6
= x10
We have optimized.
Limits of design variables ( x 1 to x 10 ) obtained from the GCPP technique:
X min : 0 0 0 0 0 0 0 0 -41.0574 -45.0574
X max : 61.0574 61.0574 61.0574 61.0574 61.0574 6.2832 6.2832 6.2832 81.0574 77.0574
Shape of coupler is in Fig. 2 approximately straight line instead of triangular. After optimization the structural
angle is 0.0305.
g
g
(5)
Table 1 shows the optimized values of variables where
θ =inputvalue
and we assume, θ = , L = L 0 .
2
1
0
0 6
=
248