lab three

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650:350 Mechanical Measurements Section 1B 10/23/2023 Lazarus TA: Shai Sabaroche Strain Gauge Measurements Morgan Lazarus , ‘25 Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway New Jersey 08854 Finding the axial and transverse strain for a constant stress beam can help in finding the stresses and the beam’s material properties as well. Using a rectangular strain gauge rosette for the aluminum cantilever beam allows for the flexural stress to be calculated. For Part 1: The Poisson ratios for aluminum, brass, and steel are 0.26, 0.27, and 0.24 respectively. The Young’s moduli for aluminum, brass, and steel are 63.0 GPa, 47.1 GPa, and 148.4 GPa respectively. To check the validity of the strain values found from the Wheatstone bridge voltagemethod, a strain gauge indicator was used. For Part 2: The experimental flexural stress values for the cantilever beam are 0 psi, 1755 psi, 3496 psi, 5251 psi,and 7012 psi for each of the given deflection values, which are very close to the theoretical valuescalculated via the given formulas. Introduction Engineers often use strain measurements to gain an understanding of how certain materials will deform given an applied load or force. These measurements aid us in designing objects as well as engineering tools. To obtain such measurements, we use a strain gauge. A strain gauge is a device that measures the strain on an object by the change in its resistance. Strain gauges are often used in conjunction with a Wheatstone bridge circuit when analyzing the strain while changing the voltage. A Wheatstone bridge circuit consists of four resistances, an input voltage, as well as an output voltage. The following lab experiment uses two different beams to see the effects of a given load on stress and strain. The first type of beam in the experiment is known as a constant stress beam, on which the stress is constant along the axial direction due to the changing width of the beam. The second type of beam used is known as a cantilever beam. A cantilever beam has a load applied on the opposite side of the clamped end. The load amount on the end of the beam can be varied by the equipment for analysis purposes. During the experiment we used two different types of strain gauge sensors. For the constant stress beam, there are two single axis gauges that measure axial and transverse strain. However, when measuring strain in different directions it is best to use a rectangular strain gauge rosette. A rectangular strain gauge consists of three strain gauges, each of which are 45 degrees with respect to one another. The rectangular strain gauge rosette is shown below in Figure 1. Figure 1. Image of a rectangular strain gauge rosette Figure 2. Image of a rectangular strain gauge rosette Since the electrical resistance of a wire in strain gauges is changed when subjected to strain, an equation relating the gauge factor, change in resistance, and strain can be formulated. This is shown in Equation 1. ε = 1 ?? ΔR 𝑅 [1] When a Wheatstone bridge is used in combination with a strain gauge, the output v, and input voltages also come into the equation along

650:350 Mechanical Measurements Section 1B 10/23/2023 Lazarus TA: Shai Sabaroche with the change of resistance. This exact equation is shown in Equation 2. ∆? 𝑂?? ? 𝑖𝑛 = ∆𝑅/𝑅 4+2∆𝑅/𝑅 = ?? 4+2??? ≈ ??? 4 [2] To calculate young’s modulus for a constant stress beam, we must calculate the axial stress as well as the axial strain on the beam. Using equation 2, we can find the Axial strain on the beam. However, the axial stress for a constant stress beam requires knowing the magnitude of the applied load, P, the distance from the strain gauge location to the load, x, the base width at the strain gauge location, b, as well as the thickness at the strain gauge location, h. The equation relating these variables to axial stress is shown below in Equation 3. 𝜎 𝑥 = 6𝑃𝑥 ?ℎ 2 [3] Once axial stress and axial strain are calculated, they can be related to each other to find young’s modulus. the materials from the voltage method and the axial modulus, which would be the slope of the corresponding graph. This relationship is shown in equation 4 below. ? ? = 𝜎 ? [4] The strain gauge rosette that was used for the cantilever beam produced three different strains. Each of the strains can then be used to find the two maximum and minimum principal strains. The relationship between these maximum and minimum strains is shown in Equation 5. ? (? ? +? ? ) 2 ± √ (? ? −? ? ) 2 +(? ? −? ? ) 2 2 ??𝑥,?𝑖? [5] Upon finding the principal strains for the cantilever beam, the stresses can be calculated using young’s modulus, E, and Poisson’s ratio,  . The exact relationship between these values is shown in Equation 6. These calculated stresses can then be compared to the theoretical values. 𝜎 1 = ? 1−𝜈 2 (ℰ 1 + 𝜈ℰ 2 ), 𝜎 2 = ? 1−𝜈 2 (ℰ 2 + 𝜈ℰ 1 ) [6] The theoretical values for the cantilever beam stress can be calculated via the load formula, Equation 7, and the flexural stress formula for a cantilever beam, Equation 8. ? = 3?𝐼? 𝐿 3 [7] 𝜎 𝑥 = ?𝑥? 𝐼 [8] Results and Discussion The first part of the lab involving the constant stress beams required finding the axial and transverse strain values via the voltage method (VM) and the strain gauge indicator (SGI) for each of the beam materials (aluminum, brass, and steel.) The voltage method involved the use of a Wheatstone bridge circuit given certain input and output voltage values, which could then be used to calculate the strain via Equation 2. The strain gauge outputted values for the axial strain of steel. The axial strains for each of the materials from the voltage method and the axial strain from the strain gauge indicator are shown in Figure 3. There is a small discrepancy between the axial strain values for steel from the two different methods. The main source of error that affects this is zero drift because one of the strain values when unloading the beam at 0 is much higher than its counterpart. The percent error between the experimental voltage method (VM) and the theoretical SGI is 11.9%, which was found via the difference in slope values of the plots and the percent error formula.

650:350 Mechanical Measurements Section 1B 10/23/2023 Lazarus TA: Shai Sabaroche Figure 3: Comparison of Axial Strain values for Steel’s constant stress beam from the VM and SGI To calculate the Poisson ratios for all the constant stress beams, the axial and transverse strains must be plotted against each other. The resulting graph depicts a negative slope equivalent to the Poisson’s ratio of the material. The following graph is depicted in Figure 4. The Poisson ratios for aluminum, brass, and steel from the plot are 0.2552, 0.2746, and 0.2361 respectively. There is a huge discrepancy between these values and the theoretical Poisson ratios. This is mainly due to an instrumental systematic offset error in the output voltages, which systematically affects the calculated strain values. This explains why all the Poisson ratio values are offset by 0.06-0.08. Also, we observe the existence of the hysteresis error, which is the behavior of a system where the output variable is dependent on the input variable as well as the previous state of the output variable. In such a system, when the input increases versus when the input decreases, the output variable will assume differing values. The hysteresis error for the aluminum values is noticeable because the data points when unloading and loading are not very close to each other. The hysteresis error is the behavior of a system which This heavily impacts the Poisson ratio for Aluminum. The percent errors for the Poisson ratios for Aluminum, Brass, and Steel are 22.7%, 19.2%, and 17.2% respectively. Figure 4: Plot of Axial Strain vs Transverse Strain to find Poisson’s ratio for Aluminu m, Brass, and Steel To calculate Young’s Modulus for each of the materials, it is necessary to plot axial strain and axial stress against each other. The resulting slope of this plot is Young’s Modulus for the corresponding material. This plot is shown in Figure 4 and is modeled in Equation 4. The experimental Young’s Moduli for Aluminum, Brass, and Steel from the slopes were found to be 9,131,000 psi (63.0 GPa), 6,825,000 psi (47.1 GPa), and 21,520,000 psi (148.4 GPa) respectively. These values are much lower than the theoretical values. The percent errors Aluminum, Brass, and Steel when compared to the theoretical ones are 10%, 52.9%, and 25.8% respectively, which is larger than expected. The main source of these large errors is the existence of an experimental systematic error. Which could have possibly occurred if the clamp’s was not positioned all the way on the edge of the table. Additionally, the clamp needed to be at the point where the base meets the triangular section of the beam. The setup error is believed to be the main contributor to the percent error between the experimental and theoretical values. Such can only be confirmed however by replicating the experiment. Commented [ML1]: Reword

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650:350 Mechanical Measurements Section 1B 10/23/2023 Lazarus TA: Shai Sabaroche Figure 5: Plot of Axial Strain and Axial Stress to find Young’s Moduli for Aluminum, Brass, and Steel The second part of the lab involved the aluminum cantilever beam along with the rectangular strain gauge rosette, which measures three different strain values. The theoretical values were found by using equations 5 and 6. The calculated values were found using Equations 7 and 8. The experimental and theoretical values for the stress in the cantilever beam are very close to each other. For each of the deflections from 0.1 to 0.4, the percent errors are 4.0%, 3.6%, 3.8%, and 3.9% respectively. This small error can be attributed to the resolution error of the tape measure used to measure the dimensions of the cantilever beam. Table 1: Calculated and Measured strains and stresses for the cantilever beam ? ? ??𝑥 𝜎 1 𝜎 2 ? ?𝑖? ? ???𝑡𝑖?𝑒𝑣𝑒𝑟 in in/in psi psi in/in psi meas. calc calc meas. calc 0 0 0 0.1 0.2 0.3 0.4 Conclusion The primary objective of the lab is to help gain a better understanding of how strain gauges work via the use of constant stress and cantilever beams. As mentioned prior, the strain gauge indicator for the constant stress beam gives the theoretical value of the axial stress, which were compared to the experimental value of it using a Wheatstone bridge. Also, there were some large discrepancies in the young’s moduli and the Poisson ratios for the given materials for the constant stress beams. This was majorly due to the systematic error in the equipment and setup, and minorly due to the hysteresis and drift error. However, the experimental and theoretical stress values for the cantilever beam were very close to each other with only a minor percent error mainly due to the resolution uncertainty in the measuring equipment. Collected Data Tables Table 2: ??? Clamped To Load Application (in) Bottom of Strain Gauge to Load App (in) Width (in) Thickness (in) Steel 11-10/16 5-11/16 1-14/16 0.329 Aluminum 11-4/16 7-10/16 2-8/16 0.310 Brass 10-8/16 7-10/16 2-8/16 0.322 Cantilever Table 3: ??

650:350 Mechanical Measurements Section 1B 10/23/2023 Lazarus TA: Shai Sabaroche Weight (lbs) Aluminum Brass Steel Axial (mV) Transverse (mV) Axial (mV) Transverse (mV) Axial (mV) Transverse (mV) SI-Axial (10^-6) 0 0.004 0.002 0.001 0.000 -0.005 0.000 -1 1 0.061 -0.018 -0.033 -0.011 -0.016 -0.006 -7 2 0.117 -0.038 -0.067 -0.022 -0.030 -0.011 -14 3 0.173 -0.057 -0.100 -0.033 -0.045 -0.016 -20 4 0.229 -0.076 -0.134 -0.044 -0.061 -0.021 -27 5 0.284 -0.095 -0.168 -0.055 -0.078 -0.026 -33 6 0.338 -0.113 -0.202 -0.066 -0.093 -0.031 -40 7 0.394 -0.131 -0.235 -0.077 -0.110 -0.036 -46 6 0.338 -0.113 -0.201 -0.066 -0.091 -0.031 -40 5 0.283 -0.094 -0.168 -0.055 -0.074 -0.026 -33 4 0.229 -0.075 -0.134 -0.045 -0.057 -0.021 -27 3 0.174 -0.056 -0.101 -0.034 -0.040 -0.016 -20 2 0.120 -0.036 -0.067 -0.024 -0.023 -0.011 -15 1 0.065 -0.016 -0.033 -0.013 -0.006 -0.006 -8 0 0.010 0.004 0.001 -0.002 0.011 -0.001 -1 Table 4: ?? References [1] Figliola, R. S., & Beasley, D. E. (2019). Theory and design for mechanical measurements. Wiley. [2] Guo, Z. (2022). (tech.). MAE 350 Lab_F22 . [3] Guo, Z. (2022). (tech). Sample Lab Report 1. [4] Hysteris error CITATION!!