Experimentally determine Young’s modulus, shear modulus and Poisson’s ratio of stainless steel rods using magnetostrictive resonance
Abstract
Young’s modulus (E) and the shear modulus (μ) of thin stainless steel rods, as well as Poisson’s ratio (σ), were experimentally found by determining the longitudinal and torsional resonant frequencies for different known lengths of rods using magnetostrictive resonance. Young’s modulus was found to be 140 GPa ±17 and shear modulus 59.2 GPa ±5.7. Poisson’s ratio was found for the rods of varying length and three of these were within right range at: 0.23±0.07 for the 0.417m rod, 0.13±0.04 for the 0.411m rod and 0.11±0.03 for the 0.251m rod.

Introduction

This experiment aimed to determine Young’s modulus (E) and the shear modulus (μ) of thin stainless steel rods, as well as Poisson’s ratio (σ), by finding the longitudinal and torsional resonant frequencies for different known lengths of rods using magnetostrictive resonance. A drive coil connected to a power amplifier was used to vary the driving frequency and excite the steel rods. The vibrations of the steel rods due to the changing magnetic field were measured using a stereo cartridge connected to an oscilloscope.

Theory

Magnetostriction is the effect observed 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 around a ferrous rod, such as the steel rods examined in this experiment, it is possible to induce vibration by the motion of the domains [1].
Young’s modulus and the shear modulus of a material determine the frequency at which it resonates in different modes. Solids can experience three main modes of vibration; longitudinal, torsional, and flexural [2]. The modes examined in this experiment are longitudinal and torsional. Longitudinal vibrations are “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. Young’s modulus determines longitudinal resonance and shear modulus determines torsional resonance. The natural frequencies for longitudinal and torsional vibration of a steel rod are 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 μ are Young’s and shear moduli and ρ is density.
These equations are used to relate f to 1/L and thus find the elastic moduli.
Poisson’s ratio, σ, is the ratio of change 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 γ is given by
[2] . (6)

Method

The method was adapted from that used by Davis and Opat in “Elastic vibrations of rods and Poisson’s 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 cartridge stylus was positioned at the top of the rod, off centre on the flat end, as shown in Figure 1.

Figure 1: Clamp stand with rod clamped in the centre, coil clamped slightly above, and the stereo cartridge positioned above the rod to pick up vibrations. Foam used under clamp stand to attempt to reduce back ground vibrations.
This positioning allowed for the detection of and distinction between longitudinal and torsional resonances. The two outputs of the stereo cartridge each respond to different component of motion of the stylus at 45Ëš to the horizontal. Figure 2 is a diagram of the stereo cartridge stylus and placement on the end of the rod from Davis 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 Opat’s “Elastic vibrations of rods and Poisson’s 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 maximum and record the frequency. This was repeated for rods of different length. Also recorded were the mass and diameter of each rod analysed in order to find the density since each steel rod had slightly different composition.

Results

Figure 3 shows the resonant frequency plotted against the reciprocals of the lengths of steel rods. Also plotted is a line of best fit 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 random 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 enough points for the line of best fit to be very accurate.

Figure 3: Resonant frequencies (kHz) 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 ρsteel =7970 kg m-3 [3, p. 435], or using the average of the densities of steel recorded (ρ =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 μ= 59.2 GPa ±5.7 using ρsteel =7970 kg m-3 [3, p. 435], or using the average of the densities of steel recorded (ρ =8020 kg m-3 ±701) μ=59.6 GPa ±7.8.
Poisson’s Ratio (σ) 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 quantity varies for each rod, again evidence of a large random error in the resonant frequencies. Table 1 shows the different values of σ. The errors for Poisson’s ratio are calculated based on the random error in each of the resonant frequencies.

Length /m±3×10-3

Poisson’s Ratio

0.417

0.23±0.07

0.411

0.13±0.04

0.367

0.7±0.2

0.262

0.04±0.01

0.251

0.11±0.03

Table 1: Poisson’s Ratio for different lengths of rods for mode n=1 from equations (5) and (6)

Discussion

Young’s 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 μ=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 obtain a truly accurate result.
For Poisson’s Ratio, there is a large variation 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].

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One major problem encountered was the fact that no resonant frequencies above 15kHz were observed. Whether this is a limitation of the stereo cartridge or due to the extremely 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 addressed 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 be
In some cases, it was difficult to record data accurately or to detect resonances due to background 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 either side. However, only one strong resonance was detected, which was not close to the predicted resonance of aluminium, 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].

Conclusion

The longitudinal and torsional resonance frequencies for stainless steel rods of varying known length were measured and used to determine Young’s modulus of 140 GPa ±17 and shear modulus of 59.2 GPa ±5.7 using literature values for density of steel. Poisson’s ratio was found for the rods of varying length and three of these were within right range at: 0.23±0.07 for the 0.417m rod, 0.13±0.04 for the 0.411m rod and 0.11±0.03 for the 0.251m rod. The random error in the resonance frequencies was large, which meant that none of the results are very accurate. The accuracy could be improved with more data form more rods.
References

[1]

J. P. Joule, “Mr. J. P. Joule on the Effects of Magnetism upon the Dimensions of Iron and Steel Bars,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. XXX (30) Third Series, no. 199, pp. 76-86, February 1847.

[2]

T. J. Davis and G. I. Opat, “Elastic vibrations of rods and Poisson’s ratio,” American Journal of Physics, vol. 51, no. 2, 1983.

[3]

R. D. Blevins, Formulas for natural frequency and mode shape, New York: Van Nostrand Reinhold Company, 1979.

[4]

“The Engineering ToolBox-Young’s Modulus,” [Online]. Available: http://www.engineeringtoolbox.com/young-modulus-d_417.html. [Accessed 30 March 2014].

[5]

“The Engineering ToolBox- Modulus of Rigidity,” [Online]. Available: http://www.engineeringtoolbox.com/modulus-rigidity-d_946.html. [Accessed 30 March 2014].

[6]

H. F. Meiners, Physics Demonstration Experiments, vol. 1, New York: The Ronald Press Company, 1970.

Acknowledgements
Thanks to collaborator in Data Collection: Bivu Nepaune
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