EXPERIMENT 2: STRAIN GAGE DYNAMIC TESTING Objective: In this experiment you will use the strain gage installation from the prior lab assignment and test the cantilever beam under dynamic loading situations. Theory: Strain will be measured using a resistance strain gage as one resistance arm in a Wheatstone bridge. The gage will be mounted on a cantilever beam and subjected to uniaxial tension or compression. The formula for voltage output is given in Eq. (1) below. For example for = 1000 (microstrain), an input voltage V of 10 volts, gage factor F of 2.005 and an amplifier gain A of 100 the output voltage reading would indicate V o = 500 mv. V O A* F 2 A* F = ( ) Vex 4 6 2+ F * 10 4 Strain V o Output voltage reading (V) V ex Excitation (input) voltage (V) F Gage factor provided by gage manufacturer (typ. ~ 2.0) A Strain gage circuit amplifier gain (external amplifier unit, typ. ~100) (1) Solving Eq. (1) for strain, we obtain: 4V O = V * A* F (2) ex Equipment: Cantilever Beam (Aluminum) Strain gage application chemicals and supplies Beam clamp system (steel sandwich and C-clamp) Precision 120-ohm resisters for bridge circuit Data acquisition system and computer running LabView Strain gage Voltage amplifier Digital voltmeter ProtoBoard Small weights ME 411 last update: 2/9/2016 1
Figure 1. Supplies available for installing a strain gage. These include an installation guide, the gages, cleansing chemicals, abrasive sand paper, adhesive, a soldering iron, and a ProtoBoard that is used to facilitate the creation of a bridge circuit. Precision resistors and the amplifier needed for the bridge circuit are not shown. Procedure: Install and Test Strain Gage Retrieve the instrumented aluminum beam from your prior lab assignment. Reconnect the gage electrical connections and check by first verifying that the resistance across the gage is approximately 120 Ohms. The beam should then be clamped in the steel clamping device as shown in Fig. 2. The center of the gage should be 0.75 from the edge of the clamping fixture. Be careful not to clamp down on the gage itself! Be sure that the cantilever edge is perpendicular to the long edge of the beam. Measure all beam dimensions (thickness, width, length) including distances from cantilever edge to gage center and to loading divot. If you haven t done so already, measure the mass of the beam so that you can estimate the mass per unit length value used in the analysis of beam vibrations. The lead wires should be attached to a bridge circuit constructed with 3 other high-precision 120 Ohm resistors mounted on the ProtoBoard shown in Fig. 1. The circuit should be wired as illustrated in Fig. 3, with the output voltage wired into the input end of the OMNI AMP amplifier and the amplifier output initially measured by a digital voltmeter. Be sure the amplifier is switched on and set to a gain of 25. ME 411 last update: 2/9/2016 2
Figure 2. The completed strain gage installation. The beam is shown mounted to the cantilever structure (steel blocks and clamp). Actual beams in this experiment may be different (e.g., hook). Figure 3. Wiring diagram for bridge circuit. R g is the gage and R 1, R 2, and R 3 are all the same resistance as the nominal gage resistance (120 ). Graphic from Omega.com. Determine beam natural frequency Write a virtual instrument to measure vibration of the beam. Note that the sensor output (a resistance change) is converted into a voltage signal in the transducer stage (bridge circuit) of this experiment. So, the virtual instrument is measuring voltage NOT strain. Set the sampling rate to 500 Hz and set the sample period so that you collect 10 seconds worth of data for each test (5,000 samples). The virtual instrument should store the raw voltage values, but it should also be able to determine and display a frequency analysis for the response (ie the frequencies within which most of the signal power is contained). This can be accomplished in a variety of ways, but will likely include the use of an existing subvi that can be found in the Functions Panel either in the SignalProcessing tool palettes. You will need to explore LabView Help to identify which resource is best suited to your particular need in this experiment. Be sure to record the results of the frequency analysis for every run that you conduct. To start with you should see if there are any undesired vibrations being transmitted into the beam from the table lab floor building. If there appears to be any such noise try to minimize it through isolating the test apparatus (perhaps adjusting the rigidity of the table and cushioning either the table feet or the clamp mechanism). Prior to testing for the natural frequency of the ME 411 last update: 2/9/2016 3
beam record at least one sample that will allow you to quantify the noise in the system. That is, run the vi without touching the beam. Name the resulting file control.txt. To conduct the vibration experiment, run the VI, initiate sampling and wait between 1 and 2 seconds before plucking the beam. Pluck the free end of the beam with a gentle motion of the index finger. Use your fingernail to initiate the beam vibration. The data gathered by the system will provide a measure of strain, and hence deflection vs. time. These data can be analyzed later to determine both the natural frequency of response and the damping ratio of the system. Name the resulting file vibe500_1.txt. Repeat this step 2 additional times (total of 3 replicates), naming subsequent files vibe500_2.txt etc. By conducting the same measurement multiple times you will be able to investigate measurement repeatability in your analysis. To verify that the sampling rate is sufficient run the same expt. at twice the sampling rate (for same total sample period), naming these files vibe1000_1.txt, etc. Analysis and Calculations: The strain can be calculated as a function of beam deflection as predicted by linear beam theory. 3EI P (3) 3 L b where P is the load, E is the modulus of elasticity, and I is the moment of intertia (be sure to calculate this about the correct axis), is the deflection, and L b is the length of the beam (from cantilever point to load). Knowing the load and beam properties/dimensions you can calculate strain at the strain-gage location using beam theory: PLc E (4) I where L is the distance from the point load to the center of the strain gage. From the calibration data and equation 2 the strain gage conditioning circuit gain (G) can be determined. This same equation can then be used with output voltage data to determine the transient strain signal during the vibration tests. As the beam vibrates the output voltage from the strain gage system will oscillate and decay. This oscillation occurs at the natural frequency of the beam. While the natural frequency is the dominant term dictating the beam oscillatory frequency other higher order frequencies of the system may also be evident in the signal. A representative trace of the system response is illustrated in Fig. 4. Output Voltage Time (ms) t Figure 4. Sample trace of the response of the vibrating beam showing the natural frequency and the signal decay associated with damping. ME 411 last update: 2/9/2016 4
The angular frequency is given by = 2/t. To obtain an estimate of natural frequency one can measure t over many (N=5) periods and determine the individual frequency by dividing by N. Assuming there is no damping the natural frequency can be obtained from: EI 2 n (1.875) (5) 4 m' Lb where m is the mass per unit length of the beam and L b is the beam length from the cantilever edge to the free edge. See an appropriate textbook to investigate the theory behind this equation and the determination of the constant (1.875). Note that the natural frequency resulting from eqn (5) is in the units of radians per second. This equation for natural frequency assumes a uniform beam (without the hook at the end). An alternative development found in some vibrations texts allows one to consider the mass of the hook at the end. Such an analysis is optional. Note that in a real cantilever beam system the oscillations are damped. In such a case the response of the beam is sinusoidal with frequency d and an amplitude that decays as: Amplitude n ( t) Ce t (6) To obtain the damping ratio () one can simply take the natural log of the above equation and plot Ln(amplitude) vs. t. The resulting plot should have a slope of - n. The relationship between the natural frequency and the damped frequency that you actually measure is given by: 2 d n 1 (7) ASSIGNMENT: You must write a full report (no more than 6 pages). You should also include an appendix (no more than 4 pages) of raw data & calculations. The following list of questions is intended as a general guide for students as they consider their report write-up. Students should NOT set out to answer this specific set of questions, but rather use them as a guide as they consider the analysis of their experimental data. How do the strain measurement results compare with beam theory predictions for the static loading? How repeatable are the results of the calibration experiment? What factors contribute to any variability? How does the theoretical natural frequency of this beam compare to the results of the transient measurements? What factors contribute to any differences? How does sampling frequency affect your results? What is the damping ratio of the system? ME 411 last update: 2/9/2016 5
References: Figliola, R.S. and Beasley, D.E., Theory and Design for Mechanical Measurements. (chapter 11) 4 th Edition. John Wiley and Sons. 2006. Robert L. Norton, Machine Design - An Integrated Approach, (chapter 4), Prentice Hall, 1997. Shigley, J.E., Mechanical Engineering Design, (3 rd edit.), McGraw-Hill Book Co. 1977. Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill Book Co., 1975. ME 411 last update: 2/9/2016 6