A Pulley Syste Apparatus for a Laboratory Experience in Dynaics Chris J. Kobus and Yin-Ping Chang Oakland University, Rochester, MI 48309-4478 Eail: cjkobus@oakland.edu Abstract This paper describes a particle kineatics easureent experiental apparatus for the junior level Engineering Mechanics Dynaics course. The Dynaics course is geared to introduce students to fundaental principles of Kineatics and Kinetics of particles and rigid bodies, including displaceent/velocity/acceleration kineatics relationships and kinetics analyses through Newtonian nd Law, work/energy equations, and ipulse/oentu principles approaches. This experient presents a unique apparatus, designed and built at Oakland University by the senior students, to introduce particle kineatics properties easureent techniques by easuring ultiple particles positions, velocities, and accelerations in a string/pulley syste. The experient takes 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 are also presented in this paper. Introduction The goal of this project was to design, build and test an apparatus that would relate the theory taught in ME 3, the junior level Engineering Mechanics course at Oakland University, about pulleys to practical applications. Most students know the theory of pulleys but lack the experiential coponent. The best way for students to understand 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 retention through experiential learning. The experient initially covers basic concepts of kineatics of particles, focusing on the relative otion of two particles. This project then introduces a pulley syste that has a fixed input velocity, travels through the syste and has a easured output velocity. A linear 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 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 was used to run the syste was attached to the pulley on the otor creating four different possible velocities. 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. Copyright 0, Aerican Society for Engineering Education
Figure : Motor Mounting Assebly Figure : Box and Motor Assebly 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 object s output velocity. Copyright 0, Aerican Society for Engineering Education
Figure 3: Setup the Experient Theory For soe systes the otion of one particle is related to another particle s 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. Position coordinates are always referenced fro a fixed 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 easured in the direction of otion and have a positive direction (Figure 4), which is the first setup for the experient, and 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 4b). 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 and used as a base line for setup (Figures 5a and b). Copyright 0, Aerican Society for Engineering Education 3
Datu S a S Motor Figure 4a: Theoretical Scheatic of Setup Figure 4b: Experiental Scheatic of Setup Copyright 0, Aerican Society for Engineering Education 4
Datu S a S c S S o A C Motor Figure 5a. Theoretical Scheatic of Setup Figure 5b: Experiental Scheatic of Setup Copyright 0, Aerican Society for Engineering Education 5
For setup, the total lengths of the cords should be related to the position coordinates by equations and. L ( S S ) + ( S S ) = c 0 c = S + S0 S c Eq. L L = S c + S a Eq. Since the velocity of block A is desired and the velocity of the otor is known, equations and can be cobined. S c = S + S 0 L L = ( S + S L ) + S 0 a Eq 3. Solving for the cobined lengths of cord gives equation 4. L + L = S + S0 + 4 S a Eq 4. With L, L and S 0 constant, the derivative of equation 4 with respect to tie while S and S a easure the lengths of the changing segents of the cord yields equation 5. L L + 0 = V + 4 V V a V a S = S 0 + S a + 4 = 4 Eq 5. The equation gives that the velocity of block A is one fourth of the velocity of the otor and in the opposite direction. Conclusion The experient provides very good and consistent results. The initial velocity is dependent on the otor s speed and also the diaeter of the pulley which the cord is being wound. The otor s RPM rating is always constant and therefore ost of the error fro initial velocity is attributed to the diaeter of the pulley. Saller pulleys resulted ore error, it creates a question for students to figure out why the different diaeters relate to the accuracy of the results. Although the diaeter is given, when the cord winds it overlaps on top of itself causing the diaeter to increase as the experient runs. 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 errors resulted fro friction and huan error. Copyright 0, Aerican Society for Engineering Education 6
Acknowledgeent The authors would like to acknowledge two Oakland University engineering students, Mr. Cesare Sclafani and Ryan Stapleton who have participated in this particular project as a capstone design. Bibliography Chris Kobus is the Outreach and Progra Coordinator and Associate Professor of Mechanical Engineering at Oakland University in Rochester, Michigan. He received his Ph.D. (998) fro Oakland University. Dr. Kobus was hired by Oakland University in 998 and is a very active experiental and theoretical researcher. He teaches in the areas of therodynaics, statics and dynaics, vibrations, basic and advanced heat transfer and fluid echanics, fluid and theral syste design, alternative energy, nuclear energy and nuerical ethods. Dr. Kobus is also the Director of Engineering and Energy Education for the Clean Energy Research Center (CERC) that he founded in 00. 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 00 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. Copyright 0, Aerican Society for Engineering Education 7