 Open Access
 Total Downloads : 504
 Authors : B. Ravi Shankar, B. Krishnamurthy
 Paper ID : IJERTV2IS90150
 Volume & Issue : Volume 02, Issue 09 (September 2013)
 Published (First Online): 06092013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Optimization of Shaking Force by Discretization of Links Mass of A Planar Four Bar Mechanism
Optimization of Shaking Force by Discretization of Links Mass of A Planar Four Bar Mechanism
B. Ravi Shankar*, Department Of Mathematics, Andhra Loyola Institute of Engineering And Technology, Vijayawada520008. INDIA
B. Krishnamurthy, Sasi Institute of Technology and Engineering, Tadepalligudem, West Godavari District. INDIA
Abstract
The reduction of the shaking force by redistributing the each link mass of 4bar mechanism in such a way that the sum of distributed masses are equal to the total mass of each link. The shaking force is minimized by two schemes one is by varying the coordinates of discretized mass concentrated points and other is redistributing the discretized mass magnititude keeping the coordinates constraint by using an optimization technique is Nelders simplex method.

Introduction
The minimizing of shaking force was done by discretizing the each movable link of a 4bar mechanism in some parts. However, instead of adding the counter weight for minimizing of shaking force, we can modify the shape of links by optimizing the mass concentrated coordinates of discretized mass of linkages. For optimizing coordinates an optimization technique Nelder Simplex Method is used. The main contribution of the present paper is the proof for planar mechanisms. Counterweight balancing can be reformulated as a convex optimization problem. However, instead of assuming a particular counterweight shape. The counterweight balancing problem is formulated as shaking force optimization problem by discretization.
The Nelders simplex search method has been known one of the top ten algorithms of the century [1, 2]. The first simplex algorithm has been introduced by
[3] as local search method by introducing a gradient activity on a function of problem to reveal the potential solution route [4]. The Nelder simplex method is simple to understand and fast to converge an optimization problem. However, the Nelder algorithm is sensitive to initial value [5]. For example in function optimization problem, different initialization produces different solution. In order to avoid this circumstance, there are two possible ways to initialize these values. First, a very careful initialization selection and second using random generated initialization.Balancing of shaking force in high speed mechanisms/machines reduces the forces transmitted to the frame, which minimizes the noise and wear and improves the performance of a mechanism. The balancing of shaking force has been studied by various researchers [620], and others. A considerable amount of research on balancing of shaking force and shaking moment in planar mechanisms has been carried out in the past [620]. In contrast to rapid progress in balancing theory and techniques for planar mechanisms, the understanding of shaking force and shaking
moment balancing of spatial mechanisms is very limited. Kaufman and Sandor [12] presented a complete force balancing of spatial mechanisms like (revolute spherical sphericalrevolute) RSSR and (revolutesphericalsphericalprismatic) RSSP. Their approaches are based on the generalization of the planar balancing theory developed by Berkof and Lowen [10], a technique of linearly independent vectors. Using the real vectors and the concept of retaining the stationary centre of total mass, Bagci has obtained the design equations for force balancing of various mechanisms

Optimization

Nelder simplex method:
The basic idea in the simplex method is to compare the values of the objective function at n+1 vertices of a general simplex and move this simplex gradually towards the optimum point during the iterative process .the moment of the simplex is achieved by using three operations know as reflection, contraction and expansion.

Reflection
If Xh is the vertex corresponding to the highest value of the objective function among vertices of a simplex, we can expect the point Xr obtained by reflecting the point Xh in the opposite face to have the simple value .if this is the case we can construct a new simplex by rejecting Xh from simplex and including new point Xr:Replection point is given by Xr = 1 + Xo Xh
Where Xh is the vertex corresponding to the maximum function value
f Xh = maxi=1to n+1f(Xi)
Xo is the centroid of all the points Xi except i=h and is given by,
Xo = 1 n+1 Xi
n i=1 i/=h
And >0 is the reflection coefficient defined as,
= distance between XrandXo distance between Xh and Xo
Thus Xr lie on the line joining Xh and Xo on the far side of Xo
If f( ) lies between f( ) and f( )where is the vertex corresponding to the minimum function value,
f Xl = mini=1to n+1f(Xi)
Xh is replaced by Xr and a new simplex is started

Expansion
If a reflection process gives a point for which f(Xr) < f(Xl), if the reflection produces a new minimum ,one can generally expect to dcrease the function value further by moving along the direction Xo and Xr .hence we expand Xr to Xe by the relation
Xe = Xr + (1 )Xo
Where is called the expansion coefficient define as,
= distance between Xeand Xo > 1 distance between Xr and Xo
if f(Xe) < f(Xl) ,we replace the point Xh by Xe and restrat the process of reflection .on the other hand ,if f(Xr)>f(Xl) ,it means that expansion process is not successful and hence we replace the point Xh by Xr, and start the reflection process again.

Contraction


If the reflection process gives a point Xr for which f(Xr) > f(Xi) for all I excepting i=h
,and f(Xr) < f(Xh ) , then we replace the point Xh by Xr .Thus the new Xh will be Xr. In this case, we contract the simplex method as follows:
Xc = Xh + (1 ) Xo
Where is called the contraction coefficient (0<=<=1), and is defined as, = distance between Xc and Xo
distance between Xh and Xo
if f(Xr )>f(Xh ) we still use the xc with out changing the previous point Xh .if the contraction process produce a point for which f(Xc) < min[f(Xh ), f(Xr)] ,replace the point Xh in 1, X2Xn+1 by and proceed with the reflection process again .on the other hand ,if f(Xc) >= min[f(Xh ),f(Xr)],the contraction process will be a failure and this case ,we replace all Xi by (Xi + Xl)/2 , and restart the reflection process.

Problem formulation

Formulation of the Problem for Planar Mechanism
= 2 + 2
= 1 1 2 2 3 3
= 1 1 2 2 3 3
Links angles 3
= 2 1 + 2 +2 +2
+
= 1
1
1
=
= 2 + 2 +2 1
= 1
2
= 3
2
= 4
2
2
1
2
2
= 1 1 3
1 3
Where
2 = 1 3 1
3
3 1
3 1
= 1
3
1 = 1 2 + 1
3 = 1 2 + 3
Link angular accelerations
= 2
+
3
3
2 1 1
2
1 3 3 1
1 3 1
= 3 + cos
2 + cos 2 cos 2
3 1 1 3
1 3 1 3
3 1 1 3 3
Accelerations of center of mass
1
1
x 1 = 2 p1 cos 1 q1 sin 1 1 p1 sin 1 + q1 cos 1
1
1
1 = 2 1 1 1 1 1 1 1 + 1 1
2 = 1 1 1 1 2 1 2 2 2 + 2 2 2 2 2
1 2
2 2
2 = 1 1 1 1 2 1 + 2 2 2 2 2 2 2 2 +
1 2
2 2
3
3
3 = 3 3 3 + 3 3 2 3 3 3 3
3
3
3 = 3 3 3 3 3 2 3 3 + 3 3

Formulation of the shaking force reducing Problem for Planar Mechanisms after discretization
When the links mass is discretized the expression for the shaking force is given by
F = F2 + F2
x y
Fx = n 1 m1i x 1i n 1 m2i x 2i n 1 m3i x 3i
i= i= i=
Fy = n 1 m1i y 1i n 1 m2iy2i n 1 m3iy3i
i= i= i=
n n n
m1i = m1 m2i = m2 m3i = m3
i=1
i=1
i=1
n n n
X 1i = X 1 X 1i = X 2 X 1i = X 3
i=1
i=1
i=1
n n n
Y 1i = Y 1 Y 1i = Y 2 Y1i = Y 3
i=1
i=1
i=1

Optimum variables
Each moving link of the mechanism, i.e. links i = (1,2,3) is discretized .The general choice of mass parameters for a planar mechanism are its mass mi , its center of gravity (COG) position (p,q) with respect to the local coordinate system of link i to mass concentrating point .
=
=
Mass constrain 1 = 1

Objective function
Instead of minimizing a weighted combination of the three balancing effect indices, the balancing tradeoff is controlled based on the following approach.
Minimize F Subjected to
The advantage of this approach is that the shaking force is minimized while the
designer directly controls, through the designerspecified upper bounds the maximum allowed increase > 1), or the minimum wanted reduction (< 1) of the shaking force


The origin and use of the force will now be explained by means of an example, using a mechanism with the following dimensions
Figure 1 slandered 4bar configuration
Link lengths is given as a1=50.8 mm
a2=101.6 mm
a3=152.4 mm a4=152.4 mm
Angular acceleration is given as 1=100 rad/sec

Results For SchemeI
In schemeI the magnititude of the discrete mass is fixed and varying the mass concentrated points i.e. coordinates of the discrete mass. Finding the set of coordinates which gives optimal forces
(degree)
Force (N)
0
108.4387
71.74577
108.1865
107.9968
109.8092
144.0359
108.8155
179.9947
106.7311
215.9478
100.2193
251.9926
105.5625
287.9686
62.0558
323.9503
133.3234
359.9894
108.4418
Table 1 Shaking Forces For Scheme I
p11
p12
p13
q11
q12
q13
0.19558
11.59256
1.15824
0.03048
0.58928
0.10668
0.57658
1.15824
2.31394
0.30988
0.39116
0.03048
0.96266
1.9304
3.08356
0.1651
2.81178
0.10922
1.34874
2.70002
4.24434
0.69342
1.46558
0.30988
1.73736
3.47218
5.01396
0.10668
0.52832
0.1651
2.1209
4.24434
6.16966
0.14224
0.35814
0.17526
2.50444
5.01396
6.94182
0.03048
0.04826
0.1651
2.89052
5.78104
8.09752
0.17526
0.3556
0.69342
3.27914
6.55574
8.87222
0.1651
1.00584
0.03048
3.47218
6.94182
10.02792
0.10922
0.44958
0.14224
Table 2 New CoOrdinates of Mass Concentration for optimum force

Results for Scheme II

In schemeII mass concentrated points i.e. coordinates of the discrete mass is fixed and varying the magnititude of the discrete mass. Finding the set of magnititude of mass which gives optimal forces.
input angle (degree) 
Shaking force F (N) 
0 
86.6824 
71.74577 
91.7267 
107.9968 
90.2019 
144.0359 
86.7403 
179.9947 
79.8967 
215.9478 
71.8965 
251.9926 
63.5197 
287.9686 
57.0764 
323.9503 
113.1934 
359.9894 
90.9896 
Table 3Shaking Forces for SchemeII
Masses for link1 (Kgs) 
Masses for link 2 (Kgs) 
Masses for link 3 (Kgs) 
1.21e4 
6.17e5 
1.08e5 
2.34e4 
1.80e4 
3.29e5 
1.87e4 
1.80e4 
1.41e5 
1.24e5 
9.57e4 
1.74e5 
4.41e–5 
2.628e4 
3.87e4 
7.44e5 
2.21e4 
8.21e5 
2.16e4 
2.88e4 
8.21e5 
4.36e5 
5.27e4 
2.39e5 
1.97e4 
3.88e5 
5.49e5 
6.51e4 
1.08e4 
2.85e5 
Table 4 New Set of Magnititude of Masses for optimal force
Conclusion
In this work a procedure to minimize the shaking force in a four bar mechanism is presented. The shaking force is minimized by the redistribution of the mass in all the three links except the first link that constitute the mechanism the redistribution is carried out by using two schemes. One scheme involves variation of the locations of the distributed masses that constitute each link. The other schemes carries out the redistribution by varying the magnitude of the discretized masses, in keeping their locations fixed. For determining the redistributed magnitude or the locations of discretized masses for minimum shaking forces, a non linear optimal method, Nelder Mead simplex is adopted.
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