# Calibration and Use of a Strain-Gage-Instrumented Beam: Density Determination and Weight-Flow-Rate Measurement

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1 Chapter 2 Calibration and Use of a Strain-Gage-Instrumented Beam: Density Determination and Weight-Flow-Rate Measurement 2.1 Introduction and Objectives This laboratory exercise involves the static calibration of a system consisting of four strain gages mounted on a cantilever beam. Once calibrated, this system can be used to determine the weight of an object, mass flow rate of a material, or the frequency of a vibrating beam. In this exercise you will see how uncertainties enter into the calibration process and how they subsequently enter into the uncertainty in determining such quantities weight, mass flow rate, and frequency. In this exercise, we will build on the concept that the change of a wire s resistance with strain can be utilized in a practical measurement system. It is possible to take small changes in resistance and, using an electrical circuit, transform the signal into a change in voltage. By making a strain gage one of the resistors in the Wheatstone bridge circuit, you are able to measure a voltage that is proportional to strain. Specifically, in this lab exercise, you will use four strain gages bonded to a cantilever beam. Each of the four gages will serve as one resistor in a leg of a Wheatstone bridge. By using four gages instead of one, the output from the gage effectively is quadrupled. You will perform a static calibration to obtain a mathematical relationship between bridge output voltage and force. Once you know this expression, the measurement system can be used for all sorts of applications, from determining unknown weights to measuring the mass flow rate of a material. This system also can be used for dynamic measurements. Finally, you will utilize this system to measure the natural frequency of a vibrating beam. 4

2 2.2 Instrumentation The following is a list of equipment for this lab: Home-built Wheatstone bridge and operational amplifier measurement system Home-built cantilever load cell Tektronix TDS 210 two-channel digital real-time oscilloscope 4 Micro-Measurement 120 Ω CEA UW-120 strain gages (bonded to the beam) Four unknown materials: circular pipe, rectangular pipe, cylinder and hexagonal cylinder. Plastic bottle with fine-grained sand ml plastic beaker 2.3 Measurements R 1 R 2 E i + - G= var. G R 3 R 4 Figure 2.1: Wheatstone bridge and op amp configuration. Follow the following procedure step by step: Part 1: Static Calibration 1. Record the lab set-up number. The set-up number is located on the top of the cantilever beam. 2. Make sure that the power in ON (the warm-up period for stable readings is about 1 hour). 3. Connect the two strain gages on the tension side of the beam to the Wheatstone bridge as follows: connect one between Bridge Excitation and + Bridge Out, and then other between - Bridge Out and Ground. The tension side gages have yellow end connectors. 5

5 10. Carefully remove the bag from the beam, ensuring that sand does not spill out of the bag. Empty all sand back into the squeeze bottle. 11. Using a dry plastic beaker, measure 100 ml of sand. 12. Place the plastic Zip-Loc bag in the Hunt s can. 13. Place the 100 ml of sand in the Zip-Loc bag lining the Hunt s can and record the Amplifier Out Voltage. 14. Carefully remove the bag from the beam, ensuring that sand does not spill out of the bag. Empty all sand back into the squeeze bottle and return the supplies to your TA. Part 4: Dynamical Measurement 1. Press the Run/Stop button if necessary to view the signal again. Move the vertical position of the signal to the center of the screen using the vertical position knob for CH Now change the SEC/DIV under the Horizontal section of the oscilloscope to 50 ms/div. Gently tap the end of the beam. Wait until the entire signal is visible within the oscilloscope display, then press the Run/Stop button on the oscilloscope to freeze the signal. 3. Press the Cursor button and select Time under the Type menu. Using the vertical knobs, measure and record the time for several periods of the signal. The frequency of vibration (in cycles per second or Hertz) is the inverse of ONE period (in seconds) of the signal. 2.4 What to Report The following must be included in your technical memo (use SI units and 95% confidence throughout): 1. Plot up the Amplifier Out (V) versus Weight (N), where Amplifier Out is on the ordinate and Weight is on the abscissa. Use different symbols for your up calibration sequence (when you added weights), and for your down sequence (when you removed weights). Note that your masses (in g) must be converted into force units (N). Note in your memo any differences between the up and down calibration sequences. 2. Using the m-file caley.m available from the class web site, plot the linear leastsquares regression analysis of the data. 3. Based upon the information provided by caley.m, determine the uncertainty in the voltage that is related to the standard error of the fit (the y estimate ). 8

6 4. Find the uncertainty in determining weight from a voltage measurement (the x-from-y estimate ). To determine this uncertainty use a voltage near the mid-point of your calibration voltages and project it back through the calibration curve to the x-axis to determine the expected x value. Also project back through the appropriate confidence limits to the x-axis to find the minimum and maximum possible x values. Find the differences between these values and the expected x value. Use the greater of the two as the uncertainty. 5. Determine the volume of each unknown material using both the water displacement and caliper measurements. 6. Use both water-displacement and caliper-determined volumes and the weights determined using the calibration, determine the density of each material. For three of the objects you will have two density values (one from each measurement method). 7. In a table report the volume determined by each method and the density for each object. Compare your density values with those found from the literature or on the web. Be sure to cite your source(s). 8. Determine the uncertainty in the volume and density values. This should be done for both types of volume measurement. Which method has the least uncertainty? 9. Assume that you do not have access to calipers and you need a low cost experiment to determine the volume and density of the unknown objects. Design a new water-displacement based experiment that costs less than \$ and has a volume uncertainty less than in 3. Be specific and cite your sources. The best design will receive 2 extra credit additional points. 10. Determine the mass flow rate of the sand using your calibration information and the recorded time difference. 11. Determine the uncertainty in the mass flow rate. 12. Present the weight, volume, density and mass flow rate uncertainties all in one table. 13. Determine the density of the dry sand and calculate the uncertainty of thin the dry sand density. 14. Assume that you want to build an hourglass based on the collected data. How much sand would be needed to measure an hour? 15. Determine the natural frequency of the beam using the data collected from lab (see Supplemental Information). Calculate a theoretical natural frequency for the beam. Compare the experimental and theoretical values and give reasons for any possible differences. 9

7 All of your important experimental results and answers to the posed questions must be presented as a technical memo. Your answers should be contained in the explanation of your results and not listed item-for-item. 2.5 Supplemental Information All solid objects vibrate to some extent when they are hit. Determining the frequency of vibration is very important in design. Vibration can cause wear, reliability problems, and induce unwanted noise. In some instances, vibration is desired and even necessary. A vibrating conveyor belt is a good example of this. The frequency at which an object vibrates depends on both its shape and material properties. We can determine an object s unforced (natural) frequency experimentally by giving the object a short impulse and measuring its frequency of vibration, as is done in this lab. For simple configurations, it is also possible to determine the natural frequency using theoretical approximations. For the cantilever beam configuration used in this lab, the frequency of vibration can be determined by ω r = A r l 2 ( ) 1/2 EI, (2.1) ρa where ω r is the natural frequency, E is the modulus of elasticity, I is the moment of inertia, A is the cross-sectional area, A r is the mode coefficient, l is length, and ρ is the density. The cantilever beams used in this lab are made of 6061-T65 aluminum with a modulus of elasticity of approximately 10 7 psi and a density of slugs/in. 3. The length, l, is the distance for the fixed point to the free end of the beam. For the lab set-up, l is l =(L D). (2.2) where L and D are the distances shown in Figure 2.2. D P L Location of bolts Figure 2.2: Cantilever Beam Diagram Recall, that the moment of inertia for a rectangular cross-section is I = bh3 12 (2.3) 10

8 L l h a b a Strain Gauges d Bolts Figure 2.3: Cantilever beam with four strain gages where b is the width of the beam and h is its height. Figure 2.3 shows a diagram of relevant dimensions. Every beam in this lab has slightly different dimensions; this is true with any engineered object. Table 2.1 contains the measurements for each experimental set-up. When you calculate theoretical values for this lab, make sure you use the appropriate dimensions for your particular station. The accuracy of each measurement is also given. Notice that the dimensions for a and d are not listed in the Beam L (±0.002) l (±0.020) d (±0.0015) b (±0.001) h (±0.0005) Table 2.1: Beam Dimensions (all units are in inches) table. These dimensions are not critical to the calculations and were not measured. The mode coefficient, A r depends upon the mode of vibration, r. The natural frequency is the first mode (i.e., r=1). As the mode of vibration increases so does the frequency of vibration and the mode coefficient. Table 2.2 gives the mode coefficients for the first three vibration modes. Mode 1 A Mode 2 A Mode 3 A Table2.2:ModeCoefficientsforthefirstthreevibrationmodes. You can find more information on vibration modes in any vibrations or dynamics text. 11

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