Title: Mechanical ComponentsTrial 1Title: Mechanical ComponentsAuthor: Josh Hoekstra, Joel OostdykDate: 01/15/19Purpose:To study torsional oscillation using LabVIEW and computer data collection.Theory:Suppose a large symmetric rotating mass has a rotational inertia J, and a twisting rod has a torsional spring coefficient K. Recall the basic torsional relationships. We can calculate the torsional spring coefficient using the basic mechanics of materials Finally, consider the rotating mass on the end of a torsional rod.Equipment:- CDT (#217126)- DMM (#23116) Fluke 8010A- Wire Harness (16 pin)- 6 Craftsmen 6 inch clamps- 2 wooden blocks- 1 plastic rod (PVC)- Calipers (#2428301)- 2 x Potentiometer, Bourns 3540S-1-103 10K -5%- LabVIEW- 15 lb. steel block- 1 Aluminum rod- Protractor (no serial number)Procedure:Group 1 Procedure:1. A solid steel 15lb block was center drilled and tapped. A 0.249” diameter aluminum rod was threaded so that it would fit with the steel block. Clamps were used to hold the aluminum rod motionless while the torsional spring was oscillated.2. LabVIEW was set up to read the changing voltages caused by the change in the potentiometer’s resistance.3. Data was collected from the oscillating torsional spring using LabVIEW and was stored to Excel.Group 2 Procedure:To calculate the equation for the natural frequency for a rotating mass with a torsional spring the differential equation for the sum of the torques had to be solved.JJ1GL0Since the torsional spring will oscillate, the solution will have to reflect this.Choosing tcos A t( )Therefore A sin A t( )A2cos A t( )J A2J1GLcos A t( )0rewriting J A2J1GL0AJ1GJ LThe solution becomes:y C1cos A t( )The natural frequency is found byf12A12J1GJ L1. A PVC pipe was used as the torsional spring. One end of the pipe was fixed to a table. The other end of the pipe was fixed to a wood block of dimensions 1.75" X 3.5" X 24.0". C-clamps were used as equally spaced weights on either ends of the wood block to supply a moment. A potentiometer (pot) was fixed to the bottom of the boardcentered on the PVC pipe’s radius, and not allowed to oscillate with the PVC pipe.2. LabVIEW was set up to read the changing voltages caused by the change in the potentiometer’s resistance.3. Data was collected from the oscillating torsional spring using LabVIEW and was stored to Excel.Results:Group 1:Trial 1Trial 2Voltage vs. Time00.050.10.150.20.250.30.350.40 1 2 3 4Time (sec)Voltage (V)Voltage vs. Time-0.35-0.3-0.25-0.2-0.15-0.1-0.0500 0.5 1 1.5 2 2.5Time (sec)Voltage (V)Trail 3Group 2:Trial 1Voltage vs. Time-0.8-0.75-0.7-0.65-0.6-0.55-0.50 1 2 3 4 5Time (sec)Voltage (V)Trial 11.421.4251.431.4351.441.4451.451.4551.461.4650 500 1000 1500 2000 2500 3000Data PointsVoltage (V)Trial 2Trial 3Trial 21.411.421.431.441.451.461.471.480 500 1000 1500 2000 2500 3000Data PointVoltage (V)Trial 31.4151.421.4251.431.4351.441.4451.451.4551.461.4650 500 1000 1500 2000 2500 3000Data PointVoltage (V)Trial 4Big Deflection1.391.41.411.421.431.441.451.461.471.480 500 1000 1500 2000 2500 3000Data PointsVoltage (V)Calculations (Group 1):deg014294981114152resistance00.0520.1040.1730.2550.4070.529m slope deg resistance( )b intercept deg resistance( )f m degb0 50 100 150 2000.200.20.40.6resistancefdegm 3.466 103This shows that our potentiometer has a linear resistanceTo calculate the natural frequency and period, we first had to calculate the moment of inertia for the aluminum rod:J Ix Iy Ixr44Iyr.2492inJ1r42J1 1.571 1010m4Inertia of the metal block:a 7 inb 2 inc 4 inmb15.45 lbIxmba2b212Iymba2c212Izmbb2c212JwIx Iy IzJw0.052 kg m2J JwJ 0.052 kg m2Next, shear modulus was found:G 30 109PaThe Length was found to be:L 18.09 inL 0.459 mFinally the frequency can be calculated:f12J1 GJ Lf 2.235 s1T1fT 0.447 sCalculations (Group 2):First, it was necessary to determine the relationship between the deflection angles and the resistance values of the potentiometer.deg060120180240300resistance1.024.864.689.522.394.189m slope deg resistance( )b intercept deg resistance( )f m degb0 100 200 30000.511.5resistancefdegm 2.739 103This shows that our potentiometer has a linear resistanceTo calculate the natural frequency and period, we first had to calculate the moment of inertia for the PVC pipe:J Ix Iy Ixr44IyJr42ro.85 ri.59J1 JouterJinnerro42ri42J1 .03935 in4Inertia of the wood block:a 24 b 1.75 c 3.5 mb1.64Ixmba2b212Iymba2c212JwIx Iy( )Jw159.533lb in2Inertia of the weights:Implement the parallel axis theorem:Ix' Ix Ad2a 1.998 b 1.75 c 3.5 d 11m1 4.2m2 4.5A c bA c bIxm1 b2a212m1 d2Ix2m2 b2a212m2 d2Iym1 a2c212Iy2m2 a2c212Jc1Ix Iy( )Jc2Ix2 Iy2( )Jc1516.354Jc2553.236J JwJc1Jc2J 1.229 103J 1.659 103lbin2Next, shear modulus had to be calculated.E 2 G1( )E 3.45 109.38GE2 2( )G 1.25 109Nm2Unit conversion to SI:J1 .03935 in4.0254min4J1 1.638 108m4Sample Calculation of the Period and Frequency from the Graphs (Group 2)Given that the time step for the LabVIEW program was between 5.6 and 5.7 ms, the period could be calculated in the following manner:The period was calculated for all trials for the groups, and are tabulated below with their percent error:Group1 Frequency Period Percent Discrepancyt1 2.91 0.34 23.10t2 4.29 0.23 47.85t3 3.81 0.26 41.28Group2t1 1.47 0.68 6.09t2 1.66 0.60 5.63t3 1.68 0.60 6.72t4 1.47 0.68 6.09J 1189.31 lbin2.0254min21 kg2.205 lbJ .347979 m2kgL 24 in.0254minL .6096 mf12J1 GJ Lf 1.5636 s1T1fT 0.64 sTime for 8 cycles:6.524 sec1.092 sec80.679 sPeriod .679 sFrequency1PeriodFrequency 1.473 s1Percent Discrepency:pdFrequency 1.536 s11.536 s1100pd 4.118Discussion:By comparing the results from either group, it was apparent that the first group had larger percent errors than the second group. Possible reasons for this may include thelarger time step that the first group’s LabVIEW program was set up with. This in combination with faster oscillations of the block may account for a larger percent error. Also, although the torsional spring was presumed to be held securely by the vice, the graphs show evidence of multiple sine waves. The
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