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Introduction to Strain Gage Lab ME 125L: Measurement and Modeling of Dynamic Systems Laboratory 4 Instructor: Professor Silvia Ferrari TA: Chenghui Cai November 5, 2006 ABSTRACT The purpose of this laboratory is to calculate the amount of carbon dioxide put inside a soda via measuring the strain change. Assuming the soda can is a thin-walled pressure vessel, a strain gage is used to measure the strain on the soda can based on its own deformation. The strain value is converted to electrical resistance, outputted through a strain indicator, and used to determine the mass of carbon dioxide inside the can using the ideal gas law. INTRODUCTION During the lab procedure, we will use a strain gage to find the strain due to an increase in pressure within a soda can. This will enable us to determine the amount of carbon dioxide put into the soda can to make it fizz when popped. To do so, one must assume the soda can to be a thin- walled pressure vessel. This indicates that the thickness of the wall is no more than 10% of the vessel’s diameter. The strain gage measures strain based on its own deformation during experimentation [5]. Pressurized Soda in a Can Theory A strain gage, as shown in Figure 1, measures strain based on its own deformation during experimentation and converts strain into electrical resistance. As equation (1) indicates, the relationship between wire resistance, Rw, length of the wire, L, cross-sectional area of the wire, A, and resistivity, , indicates that as the wire is stretched, L increases and A decreases. Figure 1: Strain Gage [4] L (1) strain direction R A etched Backing metal material foil Since also increases, resistance increases linearly with strain as follows: R (2) S connecting wires R where S is the strain gage factor with a value of 2.11 for this experiment. A strain indicator is used to determine these miniscule changes in resistance through the use of an electric circuit, and outputs a visual strain reading in units of microstrain [5]. Output voltage can be determined from Ohm’s Law as follows: R 3 R1 R 4 R 2 Vo Vs ( R 2 R 3)( R1 R 4) where Vs is the supplied voltage, R1=R2=R4=120Ω, and the value of R3 is a sum of the resistance of the strain gage and the internal potentiometer. Adjusting the potentiometer results in an initial R3 resistance (R3i) of 120Ω. When strain occurs, R3i is increased by dR3. Since R1=R2=R3i=R4=120Ω, equation (3) becomes: dR3 S (4) Vo Vs Vs 4 R 3i 4 where dR3/R3i = R/R = S [5]. Figure 2 shows the quarter bridge circuit where we only need connect the strain gage as an active resistance. We also can choose half bridge circuit, where we need make another strain gage on a bar as an active compression and connect the strain gain on the can as R2 (or R3) and connect the strain gain on the bar as R3 (or R2). The principle is the same, i.e., to measure the resistance change and convert to voltage signal. The pressure inside the soda can is calculated by first assuming the soda can is a thin-walled pressure vessel whose wall thickness is no more than 10% of the vessel’s diameter. Strain, , is defined as the change in length over the initial length: L/Lo. Meanwhile, stress is defined by the parameters in Figure (3) as follows: Pidi h h (5a) l (5b) 2t 2 where di is the inner diameter of the soda can, do is the outer diameter, E is the Modulus of Elasticity, Pi is the internal pressure, h is the wall hoop stress, l is the longitudinal stress and t is the wall thickness [5]. Using Hooke’s Law below (according to the coordinate system in Figure 3) for homogeneous isotropic materials, the strain of the material becomes: x y z h l h h , h (1 0.5 ) (6) E E E E E E where is poisson’s ratio and where the stress perpendicular to the surface is neglected. Combining equation (6) with (5a) results in the following formula for Pg: 2tEh (7) Pg (1 0.5 )di where Pi is equivalent to the absolute pressure, Pabs. To find gage pressure, Pg, atmospheric pressure, Patm, must be subtracted from Pi as shown in equation (8). Pg Patm Pabs (8) where Patm is 101.33kPa [5]. Here since h obtained by strain indicator is negative, the pressure change Pg is negative. That means that the initial pressure inside the can, Pabs is greater than the final pressure Patm. The Ideal Gas Law is used to approximate the mass of CO2 released from the soda can. PV nRT (9) where P is absolute pressure [N/m2], V is volume of CO2, n is number of moles of CO2, R is the gas constant with a value of 8.31 J/(mol. K) for CO2, and T is temperature [K] [6]. The mass of CO2 can be determined by solving for n and converting the number of moles to mass as follows: n moles of CO2 · m grams of CO2 = mass of CO2 (10) 1 mole of CO2 Finally, the arithmetic mean and standard deviation for the experimental data are calculated. The mean, χ, is the average of the gathered values and can be calculated as follows: N 1 N X i 1 i (11) where N is the number of gathered values. Finally, the standard deviation, σ, tells us how close the data is gathered around the means and is calculated as follows: N X (12) 1 i N i 1 In addition, the most common value is the mode and when the values are listed from smallest to largest, the median is the value in the center of the list if N is odd or the average of the two middle values if N is even [4]. Materials The procedure for part I was completed through the use of the following materials: stain gage, solder, and a Vishay 3800 Strain Indicator. Figure 4 shows the functional diagram: Figure 4: Instrument Functional Diagram Fluid Strain Gage Strain Gage Wiring Strain Primary Variable Data Measured Sensing Electric Medium Conversion Transmission Element Element Circuit Element Electric Circuit Variable Variable Data Conversion Analog Analog Manipulation Presentation Observer Element Signal Element Signal Element Strain Indicator Strain Indicator Digital Display In addition, Figure 5 shows a visual representation of the strain indicator. Figure 5: Vishay 3800 Strain Indicator [8] Digital display of strain and voltage Wires from strain gage connected to these ports Knob to set voltage to ~0 Procedure A soda can is lightly sanded with 320-grade sandpaper to reveal a sanded surface of approximate 2cm by 2cm. The surface is then cleared running a gauze sponge in one direction along the surface of the can to avoid contamination. Sanding and properly cleaning the surface allows the gage to better attach to the can. The strain gage is then placed on a piece of tape with the copper side (shiniest face) of the gage facing the sticky side of the tape. The opposite side of the gage is then lightly coated with a catalyst to allow it to better attach to the can [5]. Next, the tape is placed vertically on the surface of the can so that the strain gage is accurately positioned on the sanded area with the middle strain gage running along the can’s circumference. One end of the tape is attached to the can and a drop of superglue is added to the interior surface of the hanging tape. The remainder of the tape is pressed onto the surface of the can by running a gauze sponge from bottom to top along the outer surface of the tape. This step must be completed rapidly because the catalyst helps the superglue harden at a faster rate. Pressure is then applied with your thumb to the surface of the strain gage for one minute. Finally, the tape is removed carefully from the surface of the can, leaving the gage firmly attached to the can [5]. When the can is ready for experimentation, two wire leads are soldered onto the gage’s contact pads. If a quarter bridge circuit is employed, these wires are then plugged into the strain indicator by connecting one wire to the top port, and the other to both the third and fifth ports. If a half bridge circuit is used, the connection is different. The connections can be found in the manual of Vishay 3800. The gage factor is set at 2.11 and the displayed strain is set at zero. To conclude this procedure, one of the things is to be done: the top of the soda can is popped or the can is shaken and the strain reading is recorded [5]. Results Results for this laboratory are to be calculated using the material specifications listed in Table A. Table A Variable Value do 6.6040cm di 6.5837cm t 0.01016cm E 72GPa 0.330 Note: (1) This is an introduction to the lab. The only purpose of this handout is to explain the principle of how to calculate the amount of carbon dioxide put inside a soda can. You should understand and derive EVERY equation on your own. (2) In Lab 4, you may just do the experiment once. In other words, in Equation 11, N=1 and ignore equation 12. WORKS CITED [1] Crews, Kelli. T.A. Thermal Anemometer Slides. ME125L, Professor Silvia Ferrari. February 2004. [2] Crews, Kelli. T.A. Voltage, Velocity Conversion Table (Handout). ME125L, Professor Silvia Ferrari. February 2004. [3] Crews, Kelli. T.A. Wind Tunnel Handout. ME125L, Professor Silvia Ferrari. February 2004. [4] Ferrari, Silvia. ME125L: Classnotes. Spring 2004. [5] Jensenius, Mark, T.A. ME125L: Pressurized Soda in a Can (Handout). ME125L, Professor Silvia Ferrari. February 2004. [6] Munson, Bruce R. Fundamentals of Fluid Mechanics (4th Edition). John Wily & Sons, Inc., 2002. pg 14 and inside cover. [7] Olivari, D. & Carbonaro, M. Chapter 5: Hot Wire Measurements (Handout). ME125L, Professor Silvia Ferrari. February 2004. [8] InterTechnology, Inc. Wide Range Strain Indicator Vishay 3800 Model. http://www.intertechnology.com/Vishay/Vishay_Model_3800_Strain_Indicator.html (26 February 2004).