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UNFORMATTED ATTACHMENT PREVIEW
4600:460 Concepts of Design Fall Semester, 2019 Project Statement Linkage-Balance Design Project Early version (before 2014) by Jon S. Gerhardt, Professor Modified by Shao Wang, Associate Professor, in Fall Semesters of 2014, 2015, 2016, 2017, 2018 and 2019 Department of Mechanical Engineering, University of Akron The Light Lift Latch Company is considering a new concept for the design of a counterweight product. You are asked to develop a prototype demonstrator that is required to balance a weight of 7 lbf over an angular range of -45° (below the horizontal plane) to +45° (above it) of the output link rotation, i.e., within this range, the weight will remain in any position where it is placed. It is desirable, but not required, for the upper boundary of balancing to reach an angle slightly greater than +45° because it would then give a larger design margin. The design will use a single Victor Rat-Trap spring and an aluminum four-bar linkage. The output link may have an additional structure with a hole (or pin) on which the weight is hung. The hole (or pin) is located at a radial distance of 1 1/4 inches from the center of rotation of the output link. As a further constraint, the input link will also go through a 90° span during the balancing phase that corresponds to the required angular range of the output link. Preliminary measurements indicate that the rat-trap spring is made of ten and three quarters turns of 0.070 inch diameter wire with a mean coil diameter of about 0.35 inches. Spring rates in the neighborhood of 0.05 to 0.055 pound-inch per degree have been measured, but should be verified with a formula. In the initial stage of the project, a relationship between the spring potential energy and the weight position should be developed. Based on this relationship, you are to derive an expression for the motion of the output link relative to the input link of the four-bar mechanism. We will assume that the friction in the mechanism will cover a 3% relative error in angular position, i.e., deviation in the angular position of the output link from the derived input-output relationship. (When this error is extremely small, it might even be possible to achieve a higher upper boundary of balancing than +45°.) The executable FORTRAN program BAL04.exe will be available in the selected department computer labs (specified in a Guidelines document for this project) and can also be loaded on your own computer to assist you in determining the linkage proportions based on your expression. The next phase of this project is to produce the drawings so as to make a working counterbalance. This will require multi-view drawings (part drawings and an assembly drawing) with supporting documentation. The final phase is to construct the mechanism. Materials to be provided include an eight by sixteen by ¾ inch wood base, one inch angle, ¾ and 1 inch by 1/8 inch thick aluminum stock and ¼ inch aluminum rod. The design must include the linkage construction, the attachment of the spring to the frame and input link, the attachment of the additional structure to the output arm and other pin joints. Some means of spring adjustment should be considered for last-minute tuning of the design. The linkage should not be assembled too tight and should be able to move freely when felt during an attempt to rotate it by hand, thus avoiding any fake testing result just due to a partially stuck linkage. A report as well as a working model (prototype) will be required. The report should contain who was responsible for each section of the work, a write-up of your input-output relationship and how you arrived at it, the drawings of the mechanism and the drawings needed to make the model. More requirements are given in the Guidelines document for this project. In addition to scientific and engineering accomplishment, the quality of technical writing will also be evaluated. 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INTRODUCTION —————————————————————————————————– 1 II. THEORETICAL DERIVATIONS ———————————————————————————- 4 III. SAMPLE CALCULATIONS ————————————————————————————– 13 IV. SEARCH METHOD(S) IN OPTIMIZATION —————————————————————— 17 V. COMPUTER ANALYSIS —————————————————————————————— 17 VI. SELECTION PROCESS AND CRITERIA ——————————————————————— 1 VII. EMBODIMENT DESIGN ——————————————————————————————- 4 VIII. MANUFACTURING ———————————————————————————————— 13 IX. DISCUSSION ——————————————————————————————————– 17 X. PART DRAWINGS ————————————————————————————————- 17 XI. ASSEMBLY DRAWING ——————————————————————————————- 13 XII. CONCLUSIONS —————————————————————————————————- 17 INTRODUCTION THEORETICAL DERIVATIONS To begin our calculations we began with an initial starting equation. This equations is a potential energy balance between a torsion spring and weight. This equations shows that the change potential energy of the spring and the change potential energy of the weight should cancel each other out no matter where they are in location. We then proceeded to expand this equation using the initial position of the linkage. Once we expanded the equation out we were able to see that all of the quantities were know but two. The initial unsprung angle of the spring and the initial angle of the linkage. We decided to solve for the initial unsprung angle of the spring. Through some algebra we were able to determine that angle. This equation is the equation for the initial unsprung angle. With the know values input the equation becomes the following. We then used the original energy balance equation but used an arbitrary point along the linkages rotation. This resulted in the following equations. When solved for beta prime you get the following. With the know values inputted into the above equation a resulting equation is obtained. All of these derived equations were then inputted into the excel file that was used to determine what design we used for the final prototype. SAMPLE CALCULATIONS SEARCH METHOD(S) IN OPTIMIZATION For our optimization method we decided to go with a univariate method. We began our process by jumping through a list of different options until we found a starting point that gave us practical answers from the FORTRAN program. This first point was an LR of .5, and initial angle of 135 ° and an initial alpha rotation of 45°. We then began running options and producing error data. With this data we were able to plot a curve and determine a minimum operating point for the error. Graph 1 shows this curve. Graph 1. Optimized plot of LR = .5 We then decided to move to a LR of .7. We used this ratio for our final prototype design. When optimizing the LR of .7 we followed the same strategy of LR of .5 and plotted the curve of the relative mean error. Graph 2 shows this plot. Graph 2. Optimized plot of LR = .7 COMPUTER ANALYSIS Fortran and Optimization The process for optimizing the four bar linkage system can be done through an iterative process to find the optimum base length, input link length and first and second input angle change. This process can be done by making an initial guess of each of these parameters and varying them to see if it raises or lowers the error sum of squares when input into the table in Figure ???. The inputs for this table come from a computational program, Fortran. Fortran allows 7 different inputs as shown in Figure ???. Each input is a different parameter of the system that can be varied to change the resulting output of the system. After an initial guess for each of these parameters is made, the inputs are ran through Fortran. The resulting output from Fortran gives length values for each of the bars in the system as well as a list of alpha and beta angles for the motion of the system. The alpha angle is the initial input angle of the system at the spring clamped end. The beta angle is the angle at which the output link (4) will be at for each alpha angle. Then using the equation derived previously, it is possible to find a value for β’ theoretical. Adding Ѱ, which was previously calculated by subtracting 𝝅/4 from the initial β actual value of the iteration, the sum becomes β theoretical. Now that both β actual and β theoretical have been found an error can be calculated. Error is calculated by taking the difference of β actual and β theoretical and dividing that by β theoretical. To get percent error this value must be multiplied by 100. The maximum percent error for an iteration indicates the maximum error of the system at any given angle of movement of the system. The lower the maximum error is, there is a better likelihood that the system will work correctly. The sum of squares calculation, which is the difference of β theoretical and β actual squared, helps with the iteration process because the closer it is to zero it means the system is optimized. SELECTION PROCESS AND CRITERIA Our selection process was based off our optimization process and what we thought would lead to the best linkage. We made the decision that we would use the LR of .7 and a delta beta 1 of 37°. We chose 37° because we found that it gave us a low relative mean error of .8%. It was later determined to not be the lowest but was only off by .2%. The lowest value was at 37.5°. We did not use 37.5 because we were unable to fully optimize because lack of time and the need to start the manufacturing process. This in some ways relates to real world applications were an engineer may not be able to produce the most optimized product because of project circumstances but still produces a product that is well within design standards. EMBODIMENT DESIGN Methods of Connections Pin & Clip To secure the links together so they can still pivot, we could use a pin and clip connection. This method will be very easy and quick to assemble and disassemble. The only downside could be that it may not be strong enough. Figure A Figure B Bolt & Nut Another method to connect to the links would be to use a bolt and nut method. This would create a very secure method of connecting the parts. A downside to this method would be the time to secure all the nuts and bolts. Figure C Figure D Screw To connect the rails to the baseplate, a good method would be to use a screw. This is a very easy and effective way. Since the baseplate will be wood, the screws will be very secure. Figure E Figure F Rivet Another method to connect the rails to the base plate would be to use a rivet. This would be a very clean and aesthetically pleasing method. The installation would not be as quickly as other methods. Figure G Figure H MANUFACTURING The manufacturing process began with using our data from the calculations to create CAD drawings for all of our linkages and other components. We used these drawings to create the linkages. Three of our main linkages were created with the inch by 1/8 inch bar stock. Our base link was created with the angle in order to secure the linkage to the base board. We also used the ¾ inch by 1/8 inch bar stock to create the triangle component that held the weight on the linkage. The linkages were marked up using Dykem steel layout fluid and then the shapes and holes that were to be cut/drilled were scratched into the Dykem using safety scribers and compasses. With the shapes of the linkages laid out we then began cutting and shaping all of our aluminum bar stock with the provided band saws. As we went we made sure to buff off all of the sharp edges with the sanders. Once the bars were shaped and even we then drilled the holes to connect all the linkages together. These holes were drilled to a size of ¼ inch to accommodate ¼ bolts or pins made from the ¼ rod stock. The base board was also altered to fit our design. One end was notched to allow clearance for the weight linkage to rotate the full range. Also there were holes drilled to anchor the linkage down and countersunk in the bottom so the nuts were flush. The base was also painted to give the linkage better aesthetics. Finally all pieces were combined together and the linkage was tested and adjusted using the adjuster mechanism that was installed in the base angle link. DISCUSSION PART DRAWINGS ASSEMBLY DRAWING CONCLUSIONS REFERENCES APPENDICES Glossary MULTI-VIEW DRAWINGS