Friday, April 5, 2019

Stainless Steel Rods: Young’s Modulus and Shear Modulus

Stainless Steel Rods y asidehfuls Modulus and Shear ModulusValentin HaemmerliExperimentally determine Youngs modulus, shear modulus and Poissons ratio of stainless steel terminals exploitation magnetostrictive sonorousnessAbstractYoungs modulus (E) and the shear modulus () of melt off stainless steel retinal magnetic poles, as well as Poissons ratio (), were experimentally ground by determining the longitudinal and torsional reverberant frequencies for different known durations of rods victimization magnetostrictive resonance. Youngs modulus was found to be 140 GPa 17 and shear modulus 59.2 GPa 5.7. Poissons ratio was found for the rods of varying length and three of these were within justly range at 0.230.07 for the 0.417m rod, 0.130.04 for the 0.411m rod and 0.110.03 for the 0.251m rod.IntroductionThis experiment aimed to determine Youngs modulus (E) and the shear modulus () of thin stainless steel rods, as well as Poissons ratio (), by finding the longitudinal and tors ional resonant frequencies for different known lengths of rods using magnetostrictive resonance. A drive curl connected to a indi guttert amplifier was utilize to vary the driving frequence and excite the steel rods. The vibrations of the steel rods delinquent to the changing magnetized field were measured using a binaural pick-me-up connected to an range.TheoryMagnetostriction is the effect detect when magnetic materials in an external magnetic field increase in length very slightly, due to the alignment of the microscopic domains. By rapidly reversing a magnetic field virtually a ferrous rod, such as the steel rods examined in this experiment, it is thinkable to induce vibration by the motion of the domains 1.Youngs modulus and the shear modulus of a material determine the frequency at which it resonates in different panaches. Solids can experience three main modes of vibration longitudinal, torsional, and flexural 2. The modes examined in this experiment ar longitu dinal and torsional. Longitudinal vibrations be stretching and contracting of the beam along its own axis 3, p. 182 of the material when a driving frequency is applied, while torsional is a twisting motion of the material. Youngs modulus determines longitudinal resonance and shear modulus determines torsional resonance. The raw(a) frequencies for longitudinal and torsional vibration of a steel rod ar given by, (1), (2)where , L is the length of the rod, and C are the wave velocities (3)and (4)respectively, where E and are Youngs and shear moduli and is density.These equations are used to relate f to 1/L and consequently find the elastic moduli.Poissons ratio, , is the ratio of assortment in dimensions laterally and longitudinally of a material placed under a uniform longitudinal tensile (compressive) load and is normally around 0.3 3, p. 4. Davis and Opat give this as, (5)where is given by 2 . (6)Method The system was adapted from that used by Davis and Opat in pliable vib rations of rods and Poissons ratio 2. Six stainless steel rods of varying lengths between 0.102 and 0.417 m were individually clamped at their centres by three pointed screws to reduce contact and thus damping. The rods were then positioned to pass through a drive coil, also close to their centre, and finally the stereo pickup arm stylus was positioned at the top of the rod, off centre on the flat end, as shown in send off 1. witness 1 Clamp stand with rod clamped in the centre, coil clamped slightly supra, and the stereo cartridge positioned above the rod to pick up vibrations. Foam used under clamp stand to onrush to reduce back ground vibrations.This positioning allowed for the detection of and distinction between longitudinal and torsional resonances. The both outputs of the stereo cartridge each respond to different component of motion of the stylus at 45 to the horizontal. mannikin 2 is a diagram of the stereo cartridge stylus and placement on the end of the rod from Davi s and Opat 2 which shows how it was possible to differentiate between longitudinal and torsional modes. Whenever resonance occurred and the two channels were in phase it was longitudinal as both directions of motion moved up and down at the same time. When resonance occurred out of phase it was torsional as the rotation meant the two directions of motion were outputting opposite signals.Figure 2 From Davis and Opats Elastic vibrations of rods and Poissons ratio 2. Stereo stylus design (a) and placement on the rod (b)The drive coil was connected to a power amplifier and the output frequency was varied. The two outputs of the stereo cartridge were connected to the two channels of an oscilloscope. In this way, it was possible to vary the frequency until the amplitude shown on the oscilloscope was a level high hat and record the frequency. This was repeated for rods of different length. Also recorded were the mass and diameter of each rod analysed in mark of magnitude to find the de nsity since each steel rod had slightly different composition.ResultsFigure 3 shows the resonant frequency plot against the interactionals of the lengths of steel rods. Also plotted is a line of best crack by least squares method with intercept 0 as a result of equation (1), if 1/L =0, f=0. The error bars on the frequency are the standard errors found by regression. Error in the equipment for frequency was 2Hz and insignificant compared to the large ergodic error. Error bars in the reciprocal length comes from the percentage error of the measurements due to an equipment error of 0.003m. As can be seen, the line of best fit is outside of the error boxes created by these errors and this suggests that the data is not very reliable and that there are not full points for the line of best fit to be very accurate.Figure 3 resonating frequencies ( kilohertz) of longitudinal vibrations for n=1 (fundamental) plotted against the reciprocals of the lengths of the rods (m-1).The gradient of the fitted line in Figure 3 is 2.095 kHz 0.129. Using equations (1) and (3) with n=1, this gives E=140 GPa 17 using steel =7970 kg m-3 3, p. 435, or using the average of the densities of steel recorded ( =8020 kg m-3 700) E=141 GPa 20.Similar to Figure 3, Figure 4 shows the fundamental resonant frequencies for torsional vibrations of the same rods.Figure 4 Resonant frequencies (kHz) of torsional vibrations for n=1 (fundamental) plotted against the reciprocals of the lengths of the rods (m-1).The gradient of the fitted line in Figure 3 is 1.363 kHz 0.066. Using equations (2) and (4) with n=1, this gives = 59.2 GPa 5.7 using steel =7970 kg m-3 3, p. 435, or using the average of the densities of steel recorded ( =8020 kg m-3 701) =59.6 GPa 7.8.Poissons Ratio () is found from the longitudinal and torsional resonant frequencies of the same rod and the same mode (n=1) using equations (5) and (6). This measurement varies for each rod, again evidence of a large random error in the resonan t frequencies. mesa 1 shows the different values of . The errors for Poissons ratio are calculated based on the random error in each of the resonant frequencies.Table 1 Poissons Ratio for different lengths of rods for mode n=1 from equations (5) and (6)DiscussionYoungs modulus and shear modulus are in the same order of magnitude as literature values, with experimentally determined E=140GPa compared to a literature value of around 180 GPa for stainless steel 4 or 194 according to Blevins 3. Experimentally determined shear modulus was found to be =59.2GPa compared to 77.2Gpa 5. The result for the shear modulus is more accurate, and this is confirmed by the smaller random error. The errors due to the equipment for these measurements are very small, since the frequency could be varied to within 0.1 Hz and differences in amplitudes on the oscilloscope could be observed within 2Hz. However, with only 5 points, and no modes higher than n=1 to confirm the resonant frequencies, as well as a limited few lengths, there are not enough points of data to go for a truly accurate result.For Poissons Ratio, there is a large renewal between the values for each rod, which is in part linked to the large random error in the frequency values themselves, but which may also be due in part to the differences in the type of steel used in each rod. They are almost all of the right order of magnitude, and some are very close to the literature value of 0.265 3, p. 435.One major problem encountered was the point that no resonant frequencies above 15kHz were observed. Whether this is a limitation of the stereo cartridge or due to the highly small width of the resonances at these high frequencies, or a combination of both, is unclear. However, it may be possible to detect resonances at higher frequencies with a more sensitive stereo cartridge or a more accurate power amplifier. While the power amplifier used was adjustable to 0.1Hz at low frequencies, above 10kHz this was reduced to 1Hz. Another improvement to the method is to use more and longer rods. This is similar to the problem communicate above of high frequency resonances being difficult to detect. No resonances were found for the shortest rod available because all, including the n=1 mode, were too high. With longer rods, and more data points, a more accurate result could beIn some cases, it was difficult to record data accurately or to detect resonances due to priming vibrations. For example, the movement of a chair 5m away was enough to create a very unstable oscilloscope trace due to the sensitivity of the stereo cartridge to low frequencies. This was the case despite efforts to reduce the background vibrations by placing the clamp stand set up on foam.An extension to non-ferrous materials was attempted by using a small piece of steel with two longer pieces of aluminium attached with screws on any side. However, only one strong resonance was detected, which was not close to the predicted resonance of a luminium, and since the issues mentioned above meant that it was difficult to obtain enough data even for steel rods it was decided not to pursue this. As Davis and Opat put it, Inhomogeneities in the structure of the rod can lead to coupling of the different vibrational modes and the description of the oscillating rod rapidly becomes more complex. 2. A more appropriate method for generating vibrations in rods of non-ferrous materials is outlined by Meiners and may be found in Physics Demonstration Experiments on page 439 6.ConclusionThe longitudinal and torsional resonance frequencies for stainless steel rods of varying known length were measured and used to determine Youngs modulus of 140 GPa 17 and shear modulus of 59.2 GPa 5.7 using literature values for density of steel. Poissons ratio was found for the rods of varying length and three of these were within right range at 0.230.07 for the 0.417m rod, 0.130.04 for the 0.411m rod and 0.110.03 for the 0.251m rod. The random error i n the resonance frequencies was large, which meant that none of the results are very accurate. The accuracy could be improve with more data form more rods.ReferencesAcknowledgementsThanks to collaborator in Data Collection Bivu Nepaune1

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