Link-Balance Design Report
1 November 2005
Johnny Najem
Abdulla Al-Mahmoud
Alan Jennings
Summary of Problem:
We are asked to develop a prototype of a counterweight product. This prototype
should be able to balance a two liter pop bottle over a 90 degree output link motion.
The bottle should remain in any position it is placed within this range. Our model
should be designed from four aluminum bar links and a Victor Rat-trap spring. We
are supplied with a 1.5” pulley that will convert the output link motion to a vertical
motion. We are asked to first develop an equation that relates the angle of the input
arm to the output arm angle. Using this relationship along with a FORTRAN program
supplied by the University of Akron, we are asked to find the lengths’ ratios of the
four bars. The lengths of the bars would be acceptable if the output arm angle
obtained by the program is within 3 % of error from the output arm angle determined
by the equation found in the beginning. After finding the lengths of the bars, we are
asked to have two full scale drawings of the working counterbalance. Afterwards, we
are asked to construct the mechanism along with writing a report that summarizes the
problem statement, steps, and results of the project.
Summary of Results:
In to have system equilibrium, the net energy of the counterbalance system must
remain constant. This means the work of raising the bottle must equal the negative
work accomplished by the spring. Since the bottle does a constant amount of work
per angle and the spring does a decreasing amount of work per angle, the angles must
not be linear. The equation of work accomplished was solved for the pretension of the
spring and the relationship of spring angle (α) to pulley angle (β).
2
2
Lb  in
0.05
181.5 0  181.5 0  
2
2
0
0
k 90   0  90   0  
deg


W d
4.55 Lb  3 in
To determined the lengths’ ratios of the four bars and the input arm and output
arm angles, we used a Fortran program (BAL04.exe) that computes those as a result
of initial input angle and a ratio of input arm to base length. These two variables were
varied by the univariate method to find the lowest error.
The maximum error between the actual output arm angle and the desired was 1%. We
drew the model using Pro-Engineer and constructed it. We were able to balance the 2
liter pop bottle over an 85 degree output link motion.
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 


 

The obtained results are:
Angles
α0 = 91.50
Input = 900
Lengths
Base = 4.303”
Input = 8.477”
α0 is measured from unstretched spring angle.
Other angles are measured from level and
are when the bottle is in the lowest position.
Output = 42.530
All lengths measure from centers of pins.
Linkage = 9.781” Output = 6.000”
Strategy of Solution:
To achieve a workable solution two things were desired: geometry producing low
error and manufactured accurately. The desired geometry was calculated by the energy
method. Using the energy method accounted for all internal forces and greatly simplified
the calculations. The univariate method was used to optimize geometry to low error. The
geometry was then scaled to lengths to be easily manufactured at high accuracy.
Our system consisted of a spring and a counterbalance weight. The linkage weight
was not included in analysis due to the complexity of variables it would add. As the
weight would rise, the system would gain gravitational potential energy. As the spring
would wind, the system would gain elastic potential energy. Matching the two forms of
potential energy would mean that energy would simply be transferred from one form to
the other. This is assuming that no other forces are acting on the system. However
friction reduces kinetic energy of the system. This means that as the energies were
transferred from one form to another, friction would act to stop the motion is a single
spot. For a counterbalancing system, some friction is ideal. Excessive friction, however,
would require excessive work to move from one stop to another.
Constraints specify what spring we should use, the weight of the bottle and the
desired motion. The undetermined constant of the equation is the preload of the spring
which is solved evaluating at the bottles bottom position.
The equation given in results shows the idea relationship between angles. Actual
four bar linkages cannot produce that relationship precisely though. A FORTRAM
program (BAL04.exe) computes actual four bar motions based on input to base length, an
initial input angle and two sets of angle changes. The program would then use these
constraints to determine a linkage that would give those four points precisely. Most other
intermediate angles were found to contain error.
The univariate method optimizes multivariable systems. The procedure is to
successively vary each parameter to achieve optimum holding all other variables
constant. This method was chosen since no direct equation could be easily formed
between error and parameters. The parameters varied were input angle and input to base
length ratio. The intermediate angle changes were not varied and there effect is unclear.
Having acceptable results of geometry, lengths were scaled and verified. Linkages
were scaled larger to make small precisions errors account for low ratio errors.
Conversely linkages were scaled smaller to help increase accuracy in measuring and
reduce weight error. Using a 6” output arm keep all the arms reasonable sizes. Drafting
the linkages in different positions showed that the program did in fact produce the desired
motion. Force analysis was also done on the draft to verify correct equation development.
Two typical results are shown in Figure 1.
Figure 1: Draft of four bar linkage showing two positions. Black lines are links, Green lines are
forces from torques, Red lines are aixial loads in the conecting linkage and Blue lines are axial forces
in input and output links.
Error analysis:
Error is due to un-modeled energy and discrepancies between actual and desired.
The two primary sources of error are gravitational potential due to the links and the actual
angles not exact to the desired angles. Shown in Figure 1, when the input arm points left,
center of mass of the links is lower than when the input arm points up. This shows that
the bottle will tend to be pulled up.
There are two reasons that the actual angle is different than the desired angle. The
first reason is that no four bar linkage can exactly describe the desired motion. Therefore
some error is always present in designs. Optimization was done to reduce this and results
are shown in Figure 2.
The second is that construction introduced error by lack of precision
measurements. Accuracy was increased through using reasonable linkage lengths and
tight connections. Linkages lengths were such that the distance between holes could be
measured with a single measurement. This reduced compounding error of measurement.
Actual accuracy was about 1/32” for critical measurements.
Friction reduced the effect of the error by damping out movement. If no friction
were present in the system, the system would oscillate. When the spring force is
excessive, the static friction resists the excess spring force.
Error was shown to be a factor but a viable prototype was still generated. Motion
was controlled through 850 with very minimal friction. To achieve a greater range, excess
friction could be introduced into the system.
%error
0.2%
0.0%
0
10
20
30
40
50
60
-0.2%
-0.4%
-0.6%
-0.8%
-1.0%
-1.2%
Input arm rotation (deg)
Figure 2: Error of actual angle to desired angle
70
80
90