A Unique String/Pulley System Apparatus Capstone Design Project for a Hands-On Senior-Level Laboratory Design Experience. Y. P. Chang 1.

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1 A Unique String/Pulley Syste Apparatus Capstone Design Project for a Hands-On Senior-Level Laboratory Design Experience Y. P. Chang Departent of Mechanical Engineering Oakland University Rochester, MI ABSTRACT This paper describes a particle kineatics easureent experiental apparatus for the junior level Engineering Mechanics Dynaics course. The Engineering Mechanics - Dynaics course is geared to introducing students to fundaental principles of Kineatics and Kinetics of particles and rigid bodies, including displaceent/velocity/acceleration kineatics relationships and kinetics analyses through Newtonian 2 nd Law, work/energy equations, and ipulse/oentu principles approaches. It has been the Mechanical Engineering Departent s philosophy that theory learned in the classroo be augented by experiential knowledge gained by laboratory experience. In this light, hands-on laboratory experients have been developed that are integrated with the course aterial. This paper presents a unique experiental apparatus, designed and built at Oakland University by the senior students, which is a precursor to the Capstone Design Project at Oakland University, to introduce students to particle kineatics properties easureent techniques to easure particle s positions, velocities, and accelerations in a string/pulley syste. The Capstone Design Project is geared to taking students through the entire taxonoy of the design process; fro knowledge, coprehension and application, to synthesis, analysis, and finally evaluation. The experient covers basic concepts of kineatics of particles, specially focusing on the relative otion of ultiple particles. Results of the students experiences will be presented in this paper. 2. PROJECT MOTIVATION The goal of this project was to design, build and test an apparatus that would relate the theory taught in ME 32, the senior level Engineering Mechanics Dynaics course at Oakland University, about pulleys to practical applications. Most students know the theory of pulleys and their basic design but few have seen the actual applications of the. The best way for students to understand how the theory of what they are learning is to see it being used in real world situations. This laboratory experient will let students see how sall changes in the setup of a pulley syste can affect the output, soething that can not be taught very well just fro a book. The knowledge of the workings of a pulley syste will help to cultivate the students engineering sense which can be an engineering student s greatest tool.

2 3. DESIGN OF EXPERIMENT The laboratory was developed in order to get a better understanding of the aterial being taught in the class. The experient covers basic concepts of kineatics of particles, focusing on the relative otion of two particles. This project is to create a pulley syste that has a fixed input velocity, travels thru the syste and has a easured output velocity. A otor controls the input velocity which is attached to a cord that goes through the designated syste that is attached to weight which is the output velocity. The goal was to create a highly versatile experient that could be changed easily and quickly. This was accoplished by having a varying input velocity and pulleys that could be placed anywhere in the plane of travel. For the varying velocity a pulley with four different diaeters was attached to a otor that has a given RPM rating. The cord that is used to run the syste is attached to the pulley on the otor creating four different velocities that can be used. The syste allows the pulleys to be arranged in any position desired, this was accoplished by ounting a slotted bar to the top and botto of the apparatus, which allows the pulleys to slide freely into any position. The design of the box was done in such a way that the experient would not be crowded. With a large height experients could be run longer because there is ore roo for the weight to travel. Also having a long width gives the ability to space out the pulleys so that the experient does not look cluttered and can be ore easily understood by the students. The cost of this experient was a defining factor in the design of the experient. The overall cost should be low enough that the experient is worth the university investing into it. One way cost was cut was by doing ost of the expensive labor portions by students for their Capstone Design Projects or by having it done free of charge through contacts in the industry. One thing that was at the center of the project design was that the experient apparatus be highly durable. Having an experient that had a liited nuber of trails was not acceptable. During the selection of aterials durability played a larger role in selection. High tension load cable, heavy duty anchor pulleys, an internal gearbox for the otor and solid pine for the box were all used because of their durability. With all of this in consideration any sketches and proposals were discussed until a final design was agreed upon. Coputer aided design software was used to create and odify our design. AutoDesk Inventor 8 software package was used for all of the design purposes. Creating 3-diesional solid odels of the apparatus allowed for coplete layout views that were fully diensioned. This ade the actual creating of all the coponents and assebly purposes uch easier. Outside fro the echanical side of the experient, designing on the electrical side was also encountered. Since the lab is run fro an electric otor, in order to create a user friendly setup which would be easy to use, a tier (Figure 8) was incorporated. This also helps prevent daaging of the setup; if the otor stays on for too long it could break rando coponents. Three On/Off switches were installed so that if the syste was in any danger of being daaged it could be easily shutoff. The whole design concept is to

3 take as any precautions as possible anyone could run the lab without the worry of having difficulties. 4. MANUFACTURING The construction of the lab took roughly a onth to coplete. The wooden box had to be glued together and then sanded for a sooth finish. The box was then stained with a dark color so the light colored coponents would vividly stick out. The box was coated with satin three ties while sanding in between each. The otor ounting assebly went together soothly after all the achining was copleted. A coat of paint was used to finish the coponents, giving the a professional look. The otor shaft only had a length of ½, the shaft had to be extended. A longer shaft was purchased and was attached to the otor using a role pin. Since the shaft was so long it needed to be supported because the load on it could the otor s perforance or even cause daage. The solution was using two bearing for support, ounting one on each side of the pulley to ake sure the shaft would stay fully constrained. The proble encountered was that the otor did not have a reverse so the cord was easily wound up, but getting it unwound was very difficult. The otor has a built in gear box to accoplish such a low RPM rating, therefore the shaft was very difficult to turn by hand. A hand crank was added which worked exceptionally for easy unwinding. Once the assebly was coplete the testing began. The testing process reinforced that the theoretical designed odel worked. Adding the crank to the shaft was the only proble that was found during the testing phase. The echanis worked exactly like it was designed. Figure : Motor Mounting Assebly

4 Figure 2: Box and Motor Assebly Figure 3: Pulley Specifications

5 5 VAC Gearhead 8.75 RPM CW Rotation High Torque 75 lb-in Merkle Korff SF Series Shaded pole otor..2 Aps, quick connect ale terinals for connection. Shaft end ounting with 4 tapped (0/32) holes in gear box. Shaft size /2" dia x 9/6" long with 3/6" dia hole through shaft for pinning. Rear shaft 3/6" dia x 3/4" long for fan attachent (not supplied). O/A size 5-/4" H x 3" W x 4" Deep. New. Figure 4: Motor Specifications Relay, Tie Delay. Tie Delay Relay, Coil Voltage 20 VAC, Contact For DPDT, Contact Current Rating Resistive 0 Aps, Maxiu Tie Range 999 Minutes, Miniu Tie Range 0.05 Second, Pins, Multi Tie Range and Multi Function Socket, Relay, Pins. Relay Socket, Nuber of Pins, Mounting DIN/Screw, Length 2.20 Inches, Depth 0.97 Inch, Width 2.33 Inches, Current Rating 0 Aps, Electrical Ratings. Figure 5. Tier Control Specifications and Required Accessory

6 Figure 6: Setup the Experient 5. OBJECTIVES This laboratory was designed so students could get a hands on introduction to kineatics of particles. Seeing concepts carried out in person helps to reinforce all the theoretical aterial learned fro the class. Using the experiental apparatus, two different setups each with four different velocities were conducted recording 0 trials for each, for a total of 80 trials. The objects change in distance was the only easureent fro each trial that had to be recorded. The tie for each trial was given, so the velocity was easily found. Theoretical data was then copared with the easured data and percent error was calculated. The lab shows how different pulley systes have a relationship with an objects output velocity. 6. THEORY For soe systes the otion of one particle is related to another particles corresponding otion. This type of otion is referred to as an Absolute Dependent Motion Analysis of Two Particles. This is evident in systes where particles are attached by a cord that will not stretch which is wrapped about a nuber of pulleys. The related otion of one particle to another can be solved using position coordinates.

7 Datu S a S Motor Figure 7: Theoretical Scheatic of Setup Figure 8: Experiental Scheatic of Setup

8 Position coordinates are always referenced fro a fixed point called a datu. The datu is used to cut the cord in a way that splits the pulleys directly in half so that only the linear portion reains. The position coordinates should be easure in the direction of otion and have a positive direction (Figure 7), which is the first setup for the experient, is used to deonstrate the theory of the position coordinates. Since the setup would look clustered if done exactly like the theoretical odel the experiental scheatic was altered slightly so that the experient is ore open. The datu theoretical cuts the cord so that only the vertical coponent of it reains, which allows for soe horizontal changes (Figure 8). This was one of the reasons why the box was designed with a larger width. The velocity ratio of one in the first setup gives that the weight and the block are oving with equal but opposite velocities, hence the velocity of the otor can be found fro setup one and used as a base line for setup 2 (Figures 9 and 0). For setup 2, the total lengths of the cords should be related to the position coordinates by equations and 2. L ( S S ) + ( S S ) = c 0 c = S + S0 2 S c Eq. L L2 = S c + 2 S a Eq 2. Since the velocity of block A is desired and the velocity of the otor is known, equations and 2 can be cobined. S S S L c = + L = ( S + S 2 L ) + 2 S a Eq 3. Solving for the cobined lengths of cord gives equation 4. L + 2 L2 = S + S0 + 4 S a Eq 4. Taking the tie derivative of equation 4 realizing that L, L 2 and S 0 reain constant while S and S a easure the lengths of the changing segents of the cord gives equation 5. L L + 2 t t 0 = V + 4 V 2 V a V a S = t S + t 0 S + 4 t a = 4 Eq 5. The equation gives that the velocity of block A is four ties the velocity of the otor and in the opposite direction.

9 Datu S a S c S S o A C Motor Figure 9. Theoretical Scheatic of Setup 2 Figure 0: Experiental Scheatic of Setup 2

10 7. ANALYSIS The experient gave off very good results, but soe error still occurred. The initial velocity is dependent on the otors speed and also the diaeter of the pulley which the cord is being wound. The otors RPM rating is always constant so ost of the error fro initial velocity is attributed to the diaeter of the pulley. Although the diaeter is given, when the cord winds it overlaps on top of itself causing the diaeter to increase as the experient runs. This can be noticed by looking at the difference in percent error of pulleys # through 4. Since pulley # is the sallest it has to wind up the ost aount of ties, which results in a large change in diaeter. Pulley #4 has very little overlap, so the given diaeter stays alost constant throughout, giving very accurate results. Other error was obtained fro friction and huan error. Although friction is an issue, it is neglected due to its sall influence over the syste. Huan error always plays a role in experients results. Huan error could consist of inaccurate easureents of change is distance and setup probles. For the experient to provide accurate results a good setup is necessary. The pulleys ust run parallel with one another; this is accoplished by lining the up by eye. Therefore the ore consideration taken in the setup and easureent of the experient the ore accurate the results will be. 8. ETHICS One of the ain thees of the project was that it ust be safe. It would be unethical to design and fabricate an experient, pass it on to the students knowing that it could cause the har. Every piece of the apparatus was designed with safety in ind. One of the first feature introduce to help ake the experient safer was the tier control box. Soon after construction was coplete it becae clear that the lab could danger itself if left on for too long, hence the tier control box. It allows for the experient to be run without fear of having to shut it off as long as the tier is set correctly. The cord used also went through soe odification. Originally a plain /6 th inch galvanized wire rope was used, but the cord could easily cut into students hands and splinter at points where it had been cut. To over coe this hazard a nylon coated cord was used. Also the cord itself was chosen for the fact that it has a 270 pound weight liit which far out cedes the weights actually used. The hand crank was added so that the students did not have to unwind the otor pulley by hand which easily tires the hand. Throughout the entire project design the students safety was put first. 9. CONCLUSION The laboratory perfored just as expected. The use of the step pulley for variable velocities, showed how the saller the pulley size the greater the percent error will be. Pulleys and 2 were the saller two and showed uch ore error than the larger two, pulleys 3 and 4. Pulleys 3 and 4 showed alost no error, this was due to a saller opportunity for overlapping of the cord. Since pulleys and 2 give off a good aount error, it creates a question for students to figure out why the different diaeters relate to the accuracy of the results. Overall the experient was great success, but for optial results pulleys 3 and 4 should be used.

11 0. ACKNOWLEDGEMENT: The author would like to acknowledge two students, Mr. Cesare Sclafani and Ryan Stapleton who have participated in this particular design project in winter 2005 seester at Oakland University. Their enthusias, creative thinking, and inquiring questions during their attepts to synthesize better designs, continually fuels the enthusias as teachers to discover and develop new ideas and ethods to enhance our effectiveness as engineering educators.. REFERENCES Hibbeler, R.C. (2004). Engineering Mechanics Dynaics, 0 th Education, Inc. edition, Pearson Beer, P.F., and Johnston, E.R. (2004). Vector Mechanics for Engineers Statics and Dynaics, 7 th edition, McGraw-Hill. ME 32 Fall 2004 Hoework on 9/7/2004. ME 32 Fall 2004 Quiz on 9/9/ PROFESSOR YIN-PING CHANG s VITA Yin-ping (Daniel) Chang received his B.S. and M.S. degrees fro Departent of Mechanical Engineering, National Sun-Yat-San University, Taiwan. He worked for Mitsubishi Motor Corporation (MMC), priarily focused on engine/transission design; Electric Vehicle/Hybrid Electric Vehicle (EV/HEV) developent; and Noise, Vibration, and Harshness (NVH) studies. He was also a new-engine developent project anager working with GM, Delphi, Sieens, and Lotus. Dr. Chang later studied transportation, specifically in FEM, coputational solid echanics, and vehicle/tire dynaics fields. Later working in the Vehicle Siulation Research Center, Pennsylvania Transportation Institute, the Pennsylvania State University since fall 999, Dr. Chang was doing research focused on both physical vehicle crash tests and virtual siulations. He was awarded a Graduate Teaching Fellowship and becae an instructor of the undergraduate courses Machine Dynaics, Finite Eleent Analysis, in Departent of Mechanical Engineering at Penn State University. He received his Ph.D. degree in 2002 and continues his research as an assistant professor at Oakland University, Rochester, Michigan. His current research interests include vehicle/tire dynaics, FEA coputational solid echanics, bioechanics, achine dynaics, achine design, and classical echanis synthesis and analysis

12 ME 32 : Dynaics and Vibrations Laboratory Experient # Assignent : Lab Handout Particles Kineatics : Absolute Dependent Motion Analysis of Two Particles Objective Figure. Scheatic of Setup The purpose of this experient is to introduce the principles of absolute dependent otion analysis of two particles. This principle is used to on probles of otion where one particle is dependent on another particle, i.e. a weight attached to a pulley by an

13 inelastic cord attached to a otor. As the otor pulls the cord the pulley syste can change the velocity of the weight. The setup of the syste deterines the resulting velocity of the weight. Givens Motor Pulleys Diaeters:. in 2..5 in in 4. 3 in Procedure Setup. With the experiental apparatus provide in the lab, set up the experient shown in Figure. 2. Attach cord to specified otor pulley and crank so that the cord bends around pulley. 3. Line up top right anchored pulley to specified otor pulley arked on the top T bar. 4. Turn on otor by oving switch push in the line. Pushing in the circle will turn off otor. 5. Check to ake sure tier s second dial is on one shot setting. 6. Input desired tie, aking sure tier s first dial is set accordingly. For the 9.99S setting an input of 0 0 will give a tie of second. For the 99.9S setting an input of 0 0 will give a tie of 0 seconds. NOTE: the first or second should be the only settings used. DO NOT USE the 999S, 99.9M, or the 999M settings. 7. Measure initial distance fro floor to weight. 8. Turn on tier control box by flipping switch fro Off to Tier On. 9. Start the otor by flipping Pull Tier On switch once. NOTE: if at any tie the cord begins to becoe over wound (i.e. a weight or a pulley starts to becoe pulled into an anchored pulley) flipping Tier On switch to the Off position will stop the otor. 0. Measure final distance aking sure to easure to the sae spot that was easured for the initial distance.. Wind weight back down using hand crank. 2. Repeat steps 7 through nine ties for a total of ten experiental data points.

14 Setup 2. With the experiental apparatus provide in the lab, set up the experient shown in Figure Coplete steps 2 through 2 for setup. Assignent Specifications Figure 2. Scheatic of Setup 2. Derive equations for the length of cord using position coordinates for setup Fro Setup calculate velocity of the weight and the otor pulley velocity for each otor pulley (neglect friction of the pulleys). Tabulate data for all trials and averages. 3. With the velocities of the otor pulleys and the given diaeters calculate the rp of the otor. 4. Using the velocities of the otor pulleys solve for the velocity of the weight in setup 2. Tabulate data for all trials and averages. 5. If the otor had 8.75 rp, calculate percent error fro the trial averages for both setups. 6. Explain what reasons could cause errors during the experient.

15 ME 32 : Dynaics and Vibrations Laboratory Experient # Assignent: Exaple Lab Report Particles Kineatics : Absolute Dependent Motion Analysis of Two Particles Subitted to: Prof. Yin-ping Chang Prepared by: Cesare Sclafani Ryan Stapleton

16 Introduction The purpose of this experient was to introduce kineatics of particles. The velocity of the weight was dependent on the velocity of the otor pulley. For each of the two setups a total of ten trials were conducted on each otor pulley. By using the experiental apparatus the velocity of the weights was related to the velocity of the otor using a velocity ratio. Theory The total lengths of the cords should be related to the position coordinates by equations and 2. L ( S S ) + ( S S ) = c 0 c = S + S0 2 S c Eq. L L2 = S c + 2 S a Eq 2. Since the velocity of block A is desired and the velocity of the otor is known equations and 2 can be cobined. S S S L c = + L = ( S + S 2 L ) + 2 S a Eq 3. Solving for the cobined lengths of cord gives equation 4. L + 2 L2 = S + S0 + 4 S a Eq 4. Taking the tie derivative of equation 4 realizing that L, L 2 and S 0 reain constant while S and S a easure the lengths of the changing segents of the cord gives equation 5. L L + 2 t t 0 = V + 4 V 2 V a V a S = t S + t 0 S + 4 t a = 4 Eq 5. The equation gives that the velocity of block A is four ties the velocity of the otor and in the opposite direction.

17 Results and Discussion Using the data for otor pulley four in setup one the experiental rps of the otor was found to be 8.8 revolutions per inute. The velocity for the weight at each pulley for each setup was calculated ten ties. The averages were also found and used to find the percent error of the apparatus. These data points can be seen in Tables through 4. For the first otor pulley in setup one had the highest percent error of any pulley in setup one (Table 2). This is due to the fact that as the otor turns the pulley the cord is wound around the pulley which increases its diaeter every tie it copletes a revolution. This increase in diaeter changes the velocity of the otor pulley because the velocity of the otor pulley is the angular velocity of the otor ultiplied by the diaeter. The change in velocity cause the weight s velocity to increase as the tie wears on. For otor pulley two in setup one the percent error is still high but is decreased because of the fact that it has a larger circuference and the cord is not wrapped around as any ties. Motor pulleys three and four, in setup one, have drastically saller percent errors because of even larger circuferences that will wrap the cord to the side not on top (Table 3). During setup every otor pulley has an increase in percent error due to longer running ties. Motor pulley one gives the highest percent error for the entire experient since it has a running tie of 28 seconds. This results in the cord wrapping itself around the pulley nuerous ties. The percent error increases for pulley two due to the sae wrapping effect as in pulley one (Table 4). Pulleys three and four also see an increase in percent error but not to the sae effect as the other pulleys (Table 5). A brief coparison of the experiental and theoretical values is listed in Table. Table. A Brief Coparison of the Experiental and Theoretical Values Setup Pulley Experiental Theoretical Error.25 in/s in/s 9.9% in/s in/s 6.24% in/s in/s 0.27% in/s in/s 0.27% Setup 2 Pulley ExperientalTheoretical Error in/s in/s 24.95% in/s in/s 0.4% in/s in/s 2.9% in/s in/s 2.90% Conclusion Overall this experient deonstrates the concepts of particle kineatics with clarity that is hard to find in the classroo. To be able to see how changing the syste changes the output gives a better understanding of the aterial. Using the experiental apparatus it was possible to calculate velocities of four different otor pulleys and the velocity of the dependent weight for eight different systes. It has shown that changing the diaeter of the otor pulley will change the velocity fro a constant.

18 Table 2. Setup Pulley Setup Pulley V easured= in/s V theoretical= in/s Trial Start Stop Displaceent Tie Velocity 4 in in in 0 sec.2250 in/s 2 4 in in in 0 sec.2875 in/s 3 4 in in in 0 sec.2875 in/s 4 4 in in in 0 sec.2875 in/s 5 4 in in in 0 sec.2250 in/s 6 4 in in in 0 sec.2250 in/s 7 4 in in in 0 sec.2875 in/s 8 4 in in in 0 sec.2250 in/s 9 4 in in in 0 sec.2250 in/s 0 4 in in in 0 sec.2250 in/s Average in 0 sec.2500 in/s Percent Error 9.9% Setup Pulley Theoretical vs Experiental Velocity.3.2. Velocity (in/s) Theoretical Experiental Trial Figure. Setup Pulley Experiental vs. Theoretical Velocity

19 Table 3. Setup Pulley 3 Setup Pulley 3 V easured = in/s V theoretical = in/s Trial Start Stop Displaceent Tie Velocity 4 in in in 8 sec in/s 2 4 in in in 8 sec in/s 3 4 in in in 8 sec in/s 4 4 in in in 8 sec in/s 5 4 in in in 8 sec in/s 6 4 in in in 8 sec in/s 7 4 in in in 8 sec in/s 8 4 in in in 8 sec in/s 9 4 in in in 8 sec in/s 0 4 in in in 8 sec in/s Average in 8 sec in/s Percent Error 0.27% Setup Pulley 3 Experiental vs Theoretical Velocity (in/s) Theoretical Experiental Trial Figure 2. Setup Pulley 3 Experiental vs. Theoretical Velocity

20 Table 4. Setup 2 Pulley 2 Setup 2 Pulley 2 V easured = in/s V theoretical = in/s Trial Start Stop Displaceent Tie Velocity 6 in in in 24 sec in/s 2 6 in in in 24 sec in/s 3 6 in in in 24 sec in/s 4 6 in in in 24 sec in/s 5 6 in in in 24 sec in/s 6 6 in in in 24 sec in/s 7 6 in in in 24 sec in/s 8 6 in in in 24 sec in/s 9 6 in in in 24 sec in/s 0 6 in in in 24 sec in/s Average in 24 sec in/s Percent Error 0.4% Setup 2 Pulley 2 Theoretical vs Experiental Velocity Velocity (in/s) Theoretical Experiental Trial Figure 3. Setup 2 Pulley 2 Experiental vs. Theoretical Velocity

21 Table 5. Setup 2 Pulley 4 Setup 2 Pulley 4 V easured = in/s V theoretical = in/s Trial Start Stop Displaceent Tie Velocity 6 in in.375 in 5 sec in/s 2 6 in in.375 in 5 sec in/s 3 6 in in.375 in 5 sec in/s 4 6 in in.375 in 5 sec in/s 5 6 in in.375 in 5 sec in/s 6 6 in in.375 in 5 sec in/s 7 6 in in.375 in 5 sec in/s 8 6 in in.375 in 5 sec in/s 9 6 in in.375 in 5 sec in/s 0 6 in in.375 in 5 sec in/s Average.375 in 5 sec in/s Percent Error 2.90% Setup 2 Pulley 3 Experiental vs Theoretical Velocity (in/s) Theoretical Experieental Trial Figure 4. Setup 2 Pulley 4 Experiental vs. Theoretical Velocity