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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. 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