Calibrating a Pressure Transducer, Accelerometer, and Spring System

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1 Laboratory Experiment 6: Calibrating a Pressure Transducer, Accelerometer, and Spring System Presented to the University of California, San Diego Department of Mechanical and Aerospace Engineering MAE 170 Prepared by Kimberly Nguyen, A05 Grace Victorine, A05 5/15/15

2 Abstract The purpose of this lab was to calibrate a pressure transducer and accelerometer, determine the constant of a spring system, and explore the capabilities of the wheatstone bridge, which was involved in both mentioned transducers. In the first part of this experiment, the sensitivity of the pressure transducer was determined to be mv/cm. Compared with the listed value of mv/cm, this was a % error. In the second part of this experiment, the sensitivity of the accelerometer was determined to be mv/g, resulting in a % error from the listed value of 500 mv/g. The third part of this experiment provided spring constants of N/m with the static method and N/m with the dynamic method. Finally, the last part of this experiment, 350 Ω was determined to be exactly where the bridge voltage was zero. The value resulted in a 0 % error from the theoretical value of 350 Ω. 1

3 I. Introduction Transducers are instruments that convert physical parameters into electrical outputs. 1 This experiment explored a pressure transducer and an accelerometer. In addition, the wheatstone bridge was evaluated for its capabilities. The first two parts of this experiment focused on calibrating a pressure transducer and accelerometer as well as discerning levels of hysteresis in the instruments. The pressure transducer was calibrated by incrementally increasing and then decreasing the height of the water volume while recording the corresponding voltage output values. The accelerometer was calibrated by incrementally adjusting the angle of inclination on the tilt table while recording the corresponding voltage output values. Hysteresis was then evaluated in the two calibrations by comparing ascending trial values to the corresponding descending trial values. The third part of this experiment aimed to calculate the spring constant of a system through a static method and dynamic method. The static method determined the spring constant through measuring the final displacement given an object of mass m. The dynamic method determined the spring constant by using an accelerometer to measure the frequency of oscillation for an object of mass m. In the final part of this experiment, a Wheatstone Bridge was used to plot a resistance vs. bridge voltage graph. The graph was then used to determine at what value the voltage was zero. II. Theory The strain gauge in this experiment was a piezoresistive gage which used the resistances in a wheatstone bridge circuit to respond to changes in pressure-force and provide a corresponding change in voltage. Pressure, as detected by the piezoresistive gauge, was given by, F/A = P = Po + gh (1) 1 where F was Force, A was cross-sectional area, P was the total pressure, P o was the atmospheric, or starting, pressure, was the density of the fluid, g was gravity, and h was the height of the fluid. Hysteresis, the time-dependence quality of a system, was evaluated for by conducting two trials, one in which the height of the water was decreasing and one in which the height of the water was increasing. 1 The difference between the two outputs demonstrated the experimental hysteresis in the system. The second part of this experiment focused on calibrating an accelerometer, which responded to changes in accelerative-force and subsequently provided a corresponding output voltage. The angle was related to the change in acceleration through the equation, F = ma = gcos θ (2) 1 in which F was Force, m was mass of the object, a was acceleration, g was gravity, and theta was the angle of inclination. The third part of this experiment used Hooke s Law to calculate for the spring constant k, with x as displacement, m as mass, ω as angular acceleration, and F as force. The first method of determining the spring constant was the Static Method: F = kx (3) 1 The second method for determining the spring constant was the Dynamic Method: 2 F = ω m (4) 1 Finally, the wheatstone bridge analysis was constructed around the following derivation, where Rx was the unknown resistance in an R1, R2, Rx, to R3 counter-clockwise setup. R 1 R 2 R = 3 Rx (5) 1 2

4 III. Experimental Procedures Part I: Calibration of Pressure Transducer The transducer was set up as in Figure 1. Once configured, the height of both thistle tubes were set to an equivalent height of one meter. Voltage on the DMM was recorded as one of the thistle tubes was systematically incremented 5 cm downwards. Once a reasonable bottom value was obtained, the same thistle tube was systematically incremented upwards while voltage values on the DMM were likewise recorded. The values were plotted and compared for hysteresis. The listed sensitivity of the pressure transducer was then recorded. Figure (1) Pressure Experiment Arrangement 1 Figure (2) Accelerometer Wiring 1 Figure (3) Spring System Arrangement 1 Part II: Calibration of Accelerometer The accelerometer was set up as in Figure 2. Once configured, the accelerometer was incrementally adjusted to be at the 90 degree angle of inclination on the tilt table. After recording the voltage value at that point, the accelerometer was systematically adjusted 10 degrees down, and voltage values were recorded correspondingly. Once an angle of 0 degrees was reached, the accelerometer was, as in the first experiment, systematically increased, and corresponding voltage values were recorded. The descending and ascending plots were then used to determine experimental hysteresis in the system. Part III: Determination of Spring Constants The third part of this experiment focused on determining the spring constant of a system through the static and dynamic methods. Under the static method, four displacement measurements were made corresponding to the mass values of the four objects causing the displacement. Under the dynamic method, four frequency values provided by the oscilloscope were recorded corresponding to the respective mass values of the four objects causing the oscillations. Part IV: Evaluation of the Wheatstone Bridge The final part of this experiment double-checked the accuracy of a Wheatstone Bridge by taking down bridge voltage values and plotting them in line with the resistance that was set using the Decade Resistance Box. After finding the close to zero point using the DMM output, a null function was used to set the zero point. Then the dial was adjusted in increments of two and five values were recorded above the zero point and five values were recorded below the zero point. The resistance value which balanced the bridge was then found. 3

5 IV. Data and Results Part I: Calibration of Pressure Transducer Voltage can be seen plotted against calculated pressure from eq (1) and sensitivity was determined through slope. Figure (4) Calibration Curve for Pressure Transducer (Blue: Descending; Yellow: Ascending) Listed Sensitivity Without Gain = mv/cm Calculated Sensitivity With Gain = mv/cm Voltage Gain (given) = 100 dimensionless Calculated Sensitivity Without Gain = mv/cm Percent Error = % Table 1: Pressure Transducer Data Part II: Calibration of Accelerometer Gravitational acceleration from eqn (2) was plotted against generated voltage and slope determined sensitivity. Figure (5a) and Figure (5b) Calibration Curves for Accelerometer Slope = V / gcosθ = dimensionless Sensitivity = V / cos(θ) = mv / g Listed Sensitivity = mv / g 100%* [listed-exp]/listed = % Table 2: Accelerometer Data 4

6 Part III: Determination of Spring Constants The spring constant k was determined in the static method through from eqn (3) using the slope of Δh vs. m plot. IV. Data and Results (Continued) Spring constant k is determined in the dynamic method through average of results from eqn (4). Figure (6) Spring System Calibration Curve for Static Method Static Method Dynamic Method h_o (m) Weights (kgm/s2) Height (m) Δh (m) k (N/m) Mass (kg) w^2 (rad) slope avg Table 3: Static & Dynamic Method Data Part IV: Evaluation of the Wheatstone Bridge Recorded values of bridge voltage and conductance were plotted such that the inverse of Rx+350 was used. Figure (7) Relationship Curve between Voltage and Conductance (Inverse Resistance). For Resistance vs. Voltage m b y= 3.30E E+05 At y = 0 x = 3.50E+02 Theoretical Value = 350 Error =100%* (theor.-exp.)/theor. = 0% Table 4: Static & Dynamic Method Data 5

7 V. Discussion and Error Analysis The sensitivity of the pressure transducer was experimentally determined to be mv/cm, which was in % error of the listed sensitivity of mv/cm. The error was not significant as it was much less than 1%. Hysteresis was indeed present in the system, as shown in Table 5 (Appendix). The sensitivity of the accelerometer was experimentally determined to be mv/g, resulting in a sizeable error of % from the listed value of 500 mv/g. A multitude of factors could have contributed to the sizeable error. Movement around the table could have easily affected the readings, since movement would distort the level of tilt. Lastly, there was a chance that the listed MEM value for the accelerometer was not correct for the accelerometer used. At the same time, several factors may be ruled out. From the plot in Figure 5a, hysteresis values were either very small or nonexistent, deeming them negligible for scope and aims of this experiment. In addition, any noise by the DMM was nulled out, evidenced by the zero voltage value at the 90 degree angle measurement. The spring constants determined by the static and dynamic methods were N/m and N/m, respectively. The static method yielded a value that was in 36.02% error to the dynamic method value. The large error is likely due to the systematic error caused by damping, leading to a lower frequency than ideal oscillation. With headnod to the data, it would most likely be that the dynamic method was more accurate and precise, if at least only more precise. The spring constant data from the static method, when manually calculated from each of the mass and displacement trials, were largely incongruent with one another, as displayed in Table 3. The spring constant of N/m was only derived through a linear fit that hugely compensated for the scattered data (greater than 30% difference). The dynamic method data, on the other hand, were all within 10% of one another. Although the end results were relatively close, with the 5.3% difference, the static method seemed inefficient due to its need for multiple mass and displacement trials and unreliable data scatter. As well, eye-balling the displacement in the static method may have been a large point of possible error. However, with the dynamic method, an accelerometer, which measures frequency by capturing angular acceleration, was used. The use of an accelerometer likely produced more precise data than a human measurement. A method of improving the static method would be to have a precise device measure the displacement. An accelerometer allowed the determination of frequency because it transduced the changes in acceleration into changes in voltage. Since these voltage changes manifested in the form of sine waves present on the oscilloscope, the time between cycles was inverted to determine the frequency of the signal. This information was used in conjunction with equation (4) to determine spring constant in the dynamic method. For the accelerometer calibration plot, voltage was plotted against cos(θ) as opposed to just θ because the weight had a constant mass and moved in the same direction as the constant force of gravity. The variance was observed in terms of the two-dimensional plane of the tilt table and allowed for recording of the voltage changes in the accelerometer transducer. Conversely, using only theta did not allow for such a direct comparison. The last part of this experiment was less about determining a value as it was checking the precision, accuracy, and functionality of the Wheatstone Bridge. The resistor value, 350Ω, interpolated from a plot constructed from variable resistance and corresponding voltage values was 0% in error from the theoretical resistor value of 350Ω. Indeed, the Wheatstone Bridge was certainly a very precise and accurate tool. It was very unlikely, then, that the errors in the pressure transducer and accelerometer experiments were due to the built-in wheatstone bridge. 6

8 VI. Conclusions The overarching purpose of this week s experiment was to explore the calibration methods of pressure transducers, accelerometers, and spring systems. Specifically, the experiments focused on constructing calibration curves for these transducers and calculating sensitivity values from the curves. The deliberate juxtaposition of the static and dynamic method for determining spring constant demonstrated the value of the oscilloscope and cosine wave functions in capturing characteristic oscillation. Finally, the wheatstone portion of this experiment aided us in understanding how the pressure transducers and accelerometers that use resistors function and how precise the bridge can be. 7

9 VII. References 1. Nicholas Busan, Steve Roberts, and Rahul Kapadia. Experiment 6A/B: Measurement of Pressure and Acceleration. (2014). UCSD MAE 170. Web. 10 May < > VIII. Appendix I: Raw Data Descending Ascending Hysteresis Δh (cm) Uncertainty (cm) DC voltage (mv) Uncertainty (mv) Δh (cm) Uncertainty (cm) DC voltage (mv) Uncertainty (mv) (cm) Table 5. Height and Voltage Data for Pressure Transducer Calibration Descending Ascending Acceleration vs. Voltage Angle ( ) Voltage (mv) Uncertainty (mv) Angle ( ) Voltage (mv) Uncertainty (mv) θ (rad) g*cosθ (m/s2) Average (mv) Table 6. Angle of Inclination and Voltage Data for Accelerometer Calibration 8

10 VIII. Appendix I: Raw Data (Continued) Resistance (ohms) Error - 5% (ohms) 1/(Rx+350) Voltage (mv) Error - std (mv) Δ R / Δ V E E E E E E E E E E-09 Average = Table 7. Resistance and Voltage Data for evaluation of Wheatstone Bridge -6.18E-09 9

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