VIEWS: 4 PAGES: 16 POSTED ON: 3/23/2011 Public Domain
Grand Valley State University The Padnos School of Engineering OSCILLATION OF A TORSIONAL SPRING EGR 345 Dynamic Systems Modeling and Control Andrew Edler October 12, 1999 Lab Partner Brian Malkowski Fall 1999 Table of Contents Table of Contents _______________________________________________ 2 List of Figures and Tables ________________________________________ 3 Executive Summary _____________________________________________ 4 1. Introduction __________________________________________________ 5 2. Theory ______________________________________________________ 6 3. Apparatus ____________________________________________________ 8 4. Procedure ____________________________________________________ 9 5. Results _____________________________________________________ 10 6. Analysis and Interpretation ____________________________________ 10 7. Conclusions and Recommendations_____________________________ 11 Appendices ___________________________________________________ 12 List of Figures and Tables Table 1 - Frequency .......................................................................................................... 10 Table 2 - Percent Discrepancy .......................................................................................... 10 Figure 1 - System Diagram ................................................................................................. 6 Figure 2 - LabVIEW Diagram ............................................................................................ 9 3 Executive Summary The 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 collected using 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. 4 1. Introduction This 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. 5 2. Torsional Springs A 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 Diagram 2 d T J J dt T K K ( 0 ) J 1G T L J G (1) 2 d T 1 J L dt The previous equations describe the equality of the torque generated by the rotating mass and the opposing torque of the torsional shaft. 6 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 form h e At : J G J 1G 2 2 d d T 1 J J 0 or L dt L dt J 1G J 1G h e At 0 JA 2 Ai L LJ J 1G JG J 1G h cos( t ) i sin( 1 t ) where rad s LJ LJ LJ and 1 J 1G (2) f Hz 2 LJ where: J 1 first moment of area of the shaft G shear modulus of the shaft J polar moment of inertia of the rotating mass L Length of the shaft T torque angular acceleration angular displacement Knowing the material properties of the shaft is very important as its shear modulus can vary 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 of the rotating mass is very important. 7 3. Apparatus The 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. Table 1 - List of Apparatus Item Manufacturer Model Serial Number Digital Trainer Cadet -- GVSU 12196 GVSU 217126 Digital Multimeter Fluke -- GVSU 23166 GVSU 23117 Computer Dell -- GVSU 125 GVSU 124 Stop Watch -- -- GVSU 6-94 Hot Glue Gun -- -- GVSU 8841 Protractor -- -- -- Calipers Mitutoyo -- GVSU 2428301 Scale Ohaus -- -- 1K Potentiometer Bourns 53C1 -- 10K Potentiometer Bourns 3540S -- Table Clamp -- -- -- Steel Block (Used as -- -- -- the rotating mass) 2”x4”x7” Aluminum Rod -- -- -- (Used as the torsional spring) 19.09” x .249” dia. Wood Block (Used -- -- -- as the rotating mass) 1”x3.5”x24” PVC Tubing (Used -- -- -- as the torsional shaft) O.D.=.85” I.D.=.59” Length=24” 8 Figure 1 illustrates the setup of the system. The potentiometer was glued to the bottom of the rotating mass with the hot glue gun. The input and ground of the potentiometer were connected to the +5V and ground of the digital trainer. The output of the potentiometer was connected to a data acquisition card on the back of the Dell computer. 4. Procedure Two different pieces of apparatus were set up. One apparatus consisted of a PVC pipe as the torsional spring and a wooden board as the rotating mass. The PVC pipe was centered on the board and C-clamps were attached on each end of the board for added mass. The second apparatus consisted of an aluminum rod as a torsional spring and a block of carbon steel as the rotating mass. As mentioned previously, the apparatus design was set up as shown Figure 1 – System Diagram from above. A potentiometer was mounted to the bottom of the rotating mass at its center of rotation using a hot glue gun. A constant voltage was then applied to the potentiometer. As the mass rotated the wiper of the potentiometer changed the resistance, which changed the voltage. A LabVIEW program for the data acquisition was designed to record the change in voltage over time. From this data we can estimate the natural frequency. The diagram for the LabVIEW program is shown below in Figure 2. Figure 2 – LabVIEW Diagram 9 All measurements were taken with great care and with the highest precision the available equipment would allow. It is advised to use the highest accuracy available, as this will lower the experimental error. 5. Results The raw data was taken from LabVIEW and analyzed to determine the frequency at which the system oscillated. Raw data taken from LabVIEW for the system with the aluminum shaft can be seen in Appendix B. Sample calculations for the first moment of area of the shafts, polar moments of inertia for the rotating masses, theoretical frequency, and the frequency measured by using the stopwatch can be found in Appendix A. Table 1 – Frequency shows the natural frequency of each system as measured with the stopwatch and LabVIEW. The table also shows the theoretical frequency for comparison. Table 2 – Percent Discrepancy shows the percent discrepancy of the values measured with the stopwatch and by LabVIEW with the theoretical values. Frequency (Hz) Stopwatch LabVIEW Calculated PVC 0.56 0.625 0.287 Aluminum 2.91 2.86 2.963 Table 1 - Frequency % Discrepancy Stopwatch LabVIEW PVC 95.12 117.77 Aluminum 1.79 3.48 Table 2 - Discrepancy 6. Analysis and Interpretation The results from Section 5 tend to agree well with the theory from Section 2 for system with the aluminum shaft. However, the results for the system with the PVC shaft show that there is an unacceptable level of error from theory as seen in Table 2. The sources of error for the system with the PVC shaft are most likely due to the calculation of the polar moment of inertia of the rotating mass. The rotating mass for this 10 system was a wood board with C-clamps clamped at the end for added weight. There was some difficulty in accurately calculating the polar moment of inertia for a board with C- clamps on each end. The parallel axis theorem was utilized to calculate the polar moment, but the difficulty lied in calculating the polar moment around the clamps’ axis of rotation. Once the polar moment around the clamps’ axis of rotation was determined, the parallel axis theorem was used to determine the polar moment of the clamps about the torsional spring’s axis of rotation. Other sources of error for both systems include the accuracy to which time was measured for the stopwatch calculations. Since the time was measured pressing the start/stop button on the watch after 10 cycles were completed, there has to be some degree of human error. Also, error in the LabVIEW data could be due to inaccuracy of the potentiometers used. If the wipers in the potentiometers are damaged there may not be good correlation between the position of the rotating mass and the output voltage of the potentiometer. 7. Conclusions and Recommendations The objective of the experiment was simply to study torsional vibration. By performing this experiment, one learns how different materials, masses, and geometry will effect the natural frequency of this type of system. In conclusion, the analytical model with the aluminum shaft agreed to theory fairly well while the analytical model with the PVC shaft did not. Percent discrepancies for both models can be seen in Table 2. Possible causes of error are discussed in Section 6. 11 Appendix A PVC apparatus: Calculation of the first moment of area of the torsional shaf t : D PVCout 0.85in H PVC 24 in D PVCin 0.59in 4 D PVCout D PVCin Ix 64 4 D PVCout D PVCin Iy 64 J PVC Ix Iy 4 4 J PVC 4.48610 in Calulation of the polar moment of inertia of the rotating mass including the mas ses (clamps) at the ends of the board: M board 1.64lb L board 24 in W board 3.5 in 1 M board L board 2 I xxboard I xxboard 78.72 lb in 2 12 1 M board W board 2 I yyboard I yyboard 1.674 lb in 2 12 J board I xxboard I yyboard J board 80.394lb in 2 Now , conc ider the masses (clamps) M mass 4.20lb D 11 in ( 3.5 in ) ( 2 in ) 2 2 J0 M mass 12 M mass D 2 J mass J0 (Parallel ax is theorem) J t ot al J board 2 J mass J t ot al 1.10810 lb in 3 2 12 G PVC 5 lbf 5.00410 2 in 1 J PVC G PVC f PVC (2 ) H PVC J t ot al 1 f PVC 0.287s Stop watch calculation: Cycles 4 Time 7.14s Cycles f Time 1 f 0.56 s Stee l apparatus: Calculation of the f irst moment of area torsional shaf t : D rod 0.249in H rod 19.09in 4 J rod D rod 32 4 4 J rod 3.77410 in Calculation of the polar moment of inertia on the mass: M st 15.45lb L st 7 in W st 4 in 1 M st L st 2 2 J stmass W st 12 J stmass 83.688lb in 2 6 lbf G rod 3.8 10 2 in 13 14 1 J rod G rod f st (2 ) H rod J stmass 1 f st 2.963s Stop watch calculation: Cycles 10 Time 3.44s Cycles f Time 1 f 2.907s 15 Appendix B Time (s) Voltage (V) 0.05 0.01 0.1 0.12 0.15 0.281 0.2 0.352 *1st Peak 0.25 0.317 0.3 0.195 0.35 0.088 0.4 0.071 0.45 0.134 0.5 0.254 0.55 0.293 *2nd Peak 0.6 0.271 0.65 0.161 0.7 0.088 0.75 0.09 0.8 0.183 0.85 0.298 Voltage vs. Time (Aluminum Shaft) 0.4 0.35 0.3 Voltage (V) 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 Tim e (s) 16