Static and Dynamic Response of a Cantilever Beam Scale

In this experiment you will calibrate and experimentally determine the natural frequency and damping ratio of a cantilever beam scale. We expect a cantilever beam to act as a second order system because a second order differential equation (based on Newton’s second law) governs its transient behavior. The cantilever beam is fitted with a strain gauge mounted on top of the beam and it completes a Wheatstone bridge circuit when connected to other three resistors that are contained in a “bridge completion module” attached to the base set-up. The excitation to the bridge is supplied by an electronic module mounted on the ISO-RACK-08 interface board. This module also amplifies the bridge output so that it can be read by the 12 bit A/D board installed inside the PC. The strain gage actually measures the local strain on the cantilever beam surface, and this strain and the related stress could be calculated from the amplifier and strain gage specifications. However, in this experiment, we will perform a calibration to relate the output voltage to the amount of weight placed on the end of the beam. The calibration of the beam turns this set-up in to a weight scale, thus after your calibration of the set-up the LabVIEW program should report the weight value at the tip. This can be turned in to a force by multiplying it by gravity (Weight in kg * g (= 9.8 m/s2) give you force in Newton ? 1 N = 1 Kg?m/s2). Subsequent to the calibration, the transient characteristics of the beam will be explored by placing a weight on the beam and then causing the system to oscillate by giving the beam an initial deflection with your finger. This data will be recorded versus time and an FFT analysis will be used to establish the ringing frequency. The decay of oscillation amplitude will be used to estimate the damping ratio. From these two parameters, the natural frequency can be determined. Procedure 1) Open your Calibration LabVIEW Program 2) Collect the weights for your station (A through G). If any of the weights for your station are missing, borrow from your neighboring group. 3) Measure the weights in the laboratory scale (keep track of units) 4) Use four of these weights (with approximate weights of ¼, ½, ¾, and one pound) to perform the calibration of the beam scale. 5) Modify a LabVIEW program to display the Force at the tip of the beam. 6) The other three weights will be used to test verify the calibration. The difference between measured and actual values should be less than the error in the linear fit, which is t3,95*Syx/41/2. Transient Part of the Experiment 7) Add a “Tare” numeric control to allow you to “zero” or tare the scale. 8) Make sure your LabVIEW program saves time and force in a text file 9) Place the 1/4 pound weight on the beam and make sure it is secured in place. 10) Tare the readout. This is done because the beam and weight together is threated as the system, and thus the output should be zero with the weight in place. 11) Press the beam a little bit with your finger. Note that the bottom of the beam should, at most, lightly touch the mechanical stop which is mounted to the scale base. Then remove your finger, and the beam/weight system will oscillate. Try to make a rough estimate of the frequency of oscillations – this will guide your choice of sampling rate. Now, repeat the oscillations and record the data to a file. Note the constraints on sampling rate and number of samples – you need to sample at least twice as fast as the frequency of oscillations, and you need to have 2n points to use in an FFT analysis. Actually the required sampling frequency is significantly higher than the minimum of twice the oscillating frequency, because we want to accurately capture the peaks in the waveform. About fs = 75 to 100 Hz is a good place to start. Take at least 300 points so you can choose 256 points for the FFT analysis. You may also consider taking 600 points so you can use 512 for the FFT. 12) After successfully collecting a data set, minimize LabView and open your data file in Excel. If you plot the output signal versus time to observe the overall behavior, you should find that it contains a sine wave of decaying amplitude. Now make a plot of the signal versus time and save it so it can be included in your report. Using the signal values, perform the calculations required to generate an amplitude-frequency plot, and note the dominant frequency (which will be the ringing frequency). Save the data in an Excel file because this will be used later to determine the damping coefficient. If the data does not appear to be adequate for the frequency analysis, redo Step #6. 13) Redo Step #7-12, but replace the1/4 pound weight with the, ½ and the 1 pound weight. 14) You’ll have to play with the Sampling rate and the number of points to allow you to capture at least 10 points per period. 15) Using the method discussed in class, determine the damping coefficient for the two cases (i.e. using the1/4 pound and 1 pound weights). Now, from the ringing frequency (found in Step #7) and the damping coefficient, find the natural frequency for each case. 16) Your results should address the following points: a) Include a calibration plot for the beam, showing the linear fit and calibration equation. Also show a table comparing the actual and measured weights from Step #4. Comment on any significant differences – are the deviations within the calculated uncertainty band? b) Include the signal-time and amplitude-frequency plots for the two cases examined in Step #6 and #7 (and include the data files in the report appendix). Note the ringing frequency values. c) Include the plots used to determine the damping coefficients for each case. Are the values for the two cases similar? d) Calculate the natural frequencies for the two cases. We would expect the natural frequency to be proportional to (W)-1/2, where W is the weight on the beam. Are the two natural frequencies consistent with this prediction? (The value for the1/4 pound case should be a factor of (4)1/2 larger than the value for the one pound case.)