Table of ContentsList of Figures and TablesExecutive Summary1. Introduction3. Apparatus4. Procedure5. ResultsTable 2 - Discrepancy6. Analysis and Interpretation7. Conclusions and RecommendationsAppendix AGrand Valley State UniversityThe Padnos School of EngineeringOSCILLATION OF A TORSIONAL SPRINGEGR 345 Dynamic Systems Modeling and ControlAndrew EdlerOctober 12, 1999Lab PartnerBrian MalkowskiFall 1999Table of ContentsTable of Contents_________________________________________________2List of Figures and Tables_________________________________________3Executive Summary_______________________________________________41. Introduction___________________________________________________52. Theory________________________________________________________63. Apparatus_____________________________________________________84. Procedure_____________________________________________________95. Results______________________________________________________106. Analysis and Interpretation______________________________________107. Conclusions and Recommendations______________________________11Appendices____________________________________________________12List of Figures and TablesTable 1 - Frequency............................................................................................................10Table 2 - Percent Discrepancy............................................................................................10Figure 1 - System Diagram..................................................................................................6Figure 2 - LabVIEW Diagram..............................................................................................93Executive SummaryThe objective of the experiment was to study torsional oscillation using LabVIEW and data collection. The main goal was to study the effects of applying a radial displacement to torsional systems consisting of varying materials. In particular, the focus of the experiment was to gather information on the frequency of the systems’ responses once they had been displaced.The systems consisted of a torsional spring (shaft) which was fixed at one end and suspended in a vertical position. A mass was fixed to the other end of the shaft. By applying a radial displacement to the mass on the end of the shaft and releasing it, the system began to oscillate. The spring constant was determined by measuring the amount of torque required to displace the system a given number of radians. Then, a potentiometer was calibrated and used to measure the position of the system. The potentiometer was connected to a data acquisition card in a PC and the data was collectedusing a software package called LabVIEW. The position was plotted and by knowing the time interval between data points on the plot, the frequency was determined. As a source of comparison, the frequency was also determined by measuring the amount of time for the system to complete ten cycles once it had been displaced. Two systems were used: one used an aluminum shaft and the other used a PVC shaft.The natural frequency of a torsional system was calculated and then compared to the measured results. As expected, the use of the computer proved to be more accurate than measuring the time for ten cycles when compared to the theoretical natural frequency.In conclusion, the results attained for the system using an aluminum shaft agree with theory better than those obtained with the PVC shaft do. An improvement for this experiment would be to use the same mass on the end of each shaft in order to eliminate any discrepancies over the mass’s inertia.41. IntroductionThis experiment was performed to gain a more complete knowledge of, and better understanding of the relationship between material stiffness, inertia and radial deflection. The study of these variables and how they interact with one another made it possible to understand how all three effect the natural frequency of oscillating systems.The calculations of the systems’ natural frequencies were done so under the assumption that the systems were undamped. Since all unforced systems will eventually come to rest, they have some internal damping. However, the internal damping for the systems used in this experiment are very small and can be ignored. In addition, the calculations of the polar moments of inertia of the masses on the end of the torsional shafts were calculated under the assumption that the density of each material was uniform throughout.52. Torsional SpringsA large symmetric rotating mass has a rotational inertia J, and a twisting rod has a torsional spring coefficient K. Refer to Figure 1. The basic torsional relationships are as follows:Figure 1 – System Diagram21dtdJLGJT( 0 )The previous equations describe the equality of the torque generated by the rotating mass and the opposing torque of the torsional shaft.6LGJTKKTdtdJJT102)(The natural frequency of an oscillating system can be found by solving the differential equation of motion (Equation (1)). This is done in the following derivation of a second order homogeneous differential equation where the assumed solution is of the formAthe:or whereandHzLJGJf121( 2 )where:1Jfirst moment of area of the shaftGshear modulus of the shaftJpolar moment of inertia of the rotating massLLength of the shaftTtorqueangular accelerationangular displacementKnowing the material properties of the shaft is very important as its shear modulus canvary greatly with different materials. Also, accurate measurement of the of the shaft dimensions and the dimensions of the rotating mass are important in order to lessen the error in calculating the first moment of area of the shaft and the polar moment of inertia ofthe rotating mass is very important.3. ApparatusThe items shown in Table 1 are the various apparati used in the experiment. More than one item was used where two different serial numbers are listed in the same box.721dtdJLGJT021dtdJLGJAtheLGJJA120 LJGJiA1)sin()cos(11tLJGJitLJGJhsradLJGJ1Table 1 - List of ApparatusItemManufacturer Model Serial NumberDigital Trainer Cadet -- GVSU 12196GVSU 217126Digital Multimeter Fluke -- GVSU 23166GVSU 23117Computer Dell -- GVSU 125GVSU 124Stop Watch -- -- GVSU 6-94Hot Glue Gun --
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