ME 368 Laboratory 10 2nd-order system response

Laboratory10

Cantilever Beam / 2nd order system response

Warning: this is a long lab. You may have to work efficiently or work overtime to finish it.

Equipment Needed – Day 1

  • Cantileverbeaminstrumented with 4 strain gages,connected into asolderless-breadboard-based full Wheatstone bridge circuit(no potentiometer) and energized by a benchtop (e.g. Tenma) power supply maintained at 15.0 V. The beam is oriented parallel to the floor and contains an eyescrew at the end.
  • 3-kg electromagnetic mass connected to 24VDC power supply through logarithmic LED dimmer
  • Upside-down trashcan with foam on top of it to catch the electromagnetic mass
  • Function generator  oscilloscope wallplug amplifier  tactile transducer assembly. The setup includes a 2260 Ω / 1 MΩ voltage divider. The tactile transducer is free (not attached to the beam).
  • myDAQ / LabVIEW

Equipment Needed – Day 2

  • You must use the same exact cantileverbeamsetup you did on day 1 so the calibration remains valid. The only difference will be that the beam is oriented parallel to the wall with a tactile transducer attached instead of the eyescrew.
  • Function generator  oscilloscope wallplug amplifier  tactile transducer assembly. The setup includes a 2260 Ω / 1 MΩ voltage divider. The tactile transducer is free (not attached to the beam).
  • myDAQ / LabVIEW

Goals and Objectives

  • Measure the step response of a cantilever beam and infer its natural frequency and damping ratio
  • Measure the frequency response of a cantilever beam throughout the 5 – 140 Hz range. Determine how the magnitude ratio and phase shift vary with frequency.
  • Compare the above to theoretical responses

Background resources useful for understanding this lab include Dunn Chapter 4 and the Labs_9-10_background.docx file on the course website.

1.0 Calibrating the beam to measure weight

Make sure the bridge is supplied with 15 V. Check this periodically throughout the entire lab. Setup the myDAQ to measure the bridge output voltage, so that:

  • you get a 2-second record each time you run your LabVIEW code
  • you include sample compression but not so much that you compromise your ability to observe a 140 Hz sine wave (a reasonable target would be to have about 10 points in a 140 Hz sine wave)

completion a: Explain the ai0 voltage range, ai0 sample rate, ai0 number of samples, and sample compression factor you chose[STS1].

Run your vi and look for noise, particularly 60 Hz noise. Take steps to reduce the effects of noise such as:

  • jumper AGND to ai0-
  • unplug your laptop power supply when you are taking data
  • include a 10th order Butterworth bandstop filter with cutoff frequencies of 55 and 65 Hz

Recall from lab5 that the bridge output voltage should be proportional to the weight hung from the eyescrew. Perform a 2-point calibration to make your cantilever beam into a scale that reports the weight [N] hung from the end as follows:

  • Include a “software zero” control to “balance” the bridge output so that with no mass hung on the eyescrew, you measure 0 V.
  • Hang a heavy mass (at least 1.5 kg but not more than 5 kg) from the eyescrew to determine the aforementioned constant of proportionality.

Check another mass or 2 on your scale to make sure your scale always reads within 10% of the correct weight. As you know from lab 5, much better than 10% is possible, but 10% will be good enough for this lab since we are mostly concerned with dynamic errors.

2.0 Measuring the response of the cantilever beam to a step input

In lab 5, we measured masses hung from the end of the beam. The measured mass was not perfect, due to various uncertainties such as the temperature sensitivity of the strain gages. In other words, the measurement had static uncertainty. In this lab, we will study an additional uncertainty, the dynamic uncertainty. We will remove (approximately instantaneously to simulate a step change) a large mass from the end of the ruler and look at the dynamic approach to the new measured mass, which will ideally be zero mass. If you haven’t already, configure your VI to record and display weight [N] versus time [s].

Make sure the eyescrew is screwed all the way into the beam, and make sure a piece of packing tape is stuck to the top of the electromagnet. Place the inverted trashcan under the beam. Put foam on top of it. The purpose of the trashcan/foam is to catch the magnetic weight when it falls from the beam. Put the magnetic weight on top of the foam, electromagnet facing up and positioned entirely below steel and adjacent to the eyescrew. Electrically connect the free leads from the LED dimmer to ao0. With a separate .vi, prove that you can make the electromagnet hang from the beam when ao0 supplies 10 V, and fall when ao0 supplies 0 V.

Back to your original (scale) VI, configure it so that this one VI communicates both with the magnet and the Wheatstone bridge. Specifically, use DAQ Assistant Express blocks so that the electromagnet is de-energized (released from the beam) and subsequently re-energized (so it is ready for you to stick it back on again), and strain data recording initiates just before the magnet releases. It may be helpful to wire error inputs/outputs from one DAQ Assistant to another to control timing.

When you get a nice-looking step-response trace with an initial value within 5% of 29.5 N and a stabilized final value within 1.5 N of 0 N, replace the constant in the homework problem you did with your data, and use the guess-and-check method to determine the natural frequency and damping ratio for your beam.

mini-report 1: Provide a clear, leglibleLabVIEW screenshot (or better, a plot generated with software designed to make plots) of your measured and simulated step response curves. List the natural frequency and damping ratio you inferred on the plot or in the caption.

Plot the FFT of your measured weight-versus-time signal. It should have a peak at a frequency near the natural frequency, and its shape should resemble the lowest damping ratio case of Figure 4.8 Dunn:

This result shows that it might be possible to infer the cantilever scale’s frequency response directly from a single, simple step response experiment. This possibility exists because the step effectively provides frequency-white excitation of the beam (it excites a broad range of frequencies).

mini-report 2: Compare n,meas [rad/s] to the theoretical prediction you made for homework. Provide possible reasons for any discrepancies over 5%.

mini-report 3: Imagine your cantilever beam was to be used for dynamic mass measurements. Specifically, imagine that an unknown weight was to be instantaneously added to the end of your beam. Based on your above experiments, how long do you expect to wait [s] before you’re the dynamic error will remain below 5%.

From the edit menu, select “make current values default”, then save your code.

3.0 Measuring mass added to the beam

The smallest weight students were able to reliably measure when added to the end of the beam in static mode (lab 5) was ~ 0.2 N (0.7 ounces), or about 7 pennies. Affix < 7 pennies (e.g., 4 pennies with tape or a binder clip) to the end of the beam. The pennies should move rigidly with the beam. Note that such a small weight would not have been detectable in lab 5.

Measure the step response function as above.

Repeat the measurement of n and  using the guess-and-check fitting method. Using your original n,meas (without the added mass), the new measured n,meas,mass(with the added mass), and lab10_added_mass.vi, determine the added mass. Compare it to the mass measured by the yellow digital scale in the lab.

completion_b: Quantify the accuracy (in %) of your measurement of a small mass.

======YOU MUST AT LEAST GET TO THIS POINT BEFORE LEAVING DAY 1 ======

(however, push ahead if at all possible, to increase your chances of finishing on time on day 2)

4.0 Sinusoidal forcing

4.1 Preparing to use the tactile transducer: operation and associated measurements

The tactile transducer is basically a loudspeaker, but instead of having a goal of moving air (creating sound), it has a goal of moving a mass (creating force). It is intended to be attached to the bottom of couches, etc. to enhance home theater experiences and the like.

Prof. Sanders determined that the mass within the tactile transducer you will use is 0.155± 0.009 kg. It is basically a magnet with a foam pad stuck to each end and is pictured below:

The mass is driven by a coil within a housing, pictured below:

  • Make sure the Buttkicker amplifier is off. Turn on the function generator and oscilloscope. Use oscilloscope readings or measurements to set the function generator to produce a 140 Hz sine wave with an amplitude as small as you can make it, leaving the AMPL knob on the function generator pushed in. It should be about 4.4 Volts RMS.
  • Before turning on the amplifier make sure that its filters are off (buttons out). The amplifier output should already be connected to the transducer. You need to monitor this same signal on your myDAQ, channel ai1. The wires from the voltage divider allow you to do this. Because of the voltage divider, the actual signal being fed to the transducer is 443 x bigger than what you will measure on ai1. When connecting the wires from the voltage divider to ai1, be sure to get the polarity correct.
  • Start a new LabVIEW code. Setup a single input DAQ Assistant as follows:

channel / range [V] / mode / samples / rate [Hz] / (measurement)
ai0 / -2 to 2 / continuous samples / 100k / 200k / strain (Wheats. brdg.)
ai1 / -2 to 2 / continuous samples / 100k / 200k / transducer excitation

Put this in a while loop. Sample-compress the output of the DAQ Assistant by a factor of 100. Add the same bandstop filter as above. Feed the resulting signal to each of the following:

  • a 1x2 signal splitter allowing independent access to each of the channel
  • a graph

Stretch the graph horizontally so it fills the width of your front panel. Check that the code runs.Make sure that the strain signal and transducer excitation vary sinusoidally.

  • If it’s day 1, have someone hold the transducer. If they get sick of holding it, place the foam from the trashcan on the floor and place the transducer on that. We don’t want them vibrating off the table and falling to the floor. If it’s day 2, expect the lab station to be shaken side-to-side, so shore up items on the shelves at your lab station, e.g., moving equipment on the shelves to the center.
  • You may wish to temporarily power-off the amplifier to stop vibrations and to prevent the transducer from overheating (it overheats in about 20 minutes when running constantly at 10 V peak drive).
  • While observing the ai1 signal on your looping LabVIEW code, turn on the amplifier. For the remainder of this experiment, the peak voltage at ai1 should be kept at or below 22.6 mV (or peak-to-peak voltage below 45.2 mV. This corresponds to an actual transducer drive signal of 10 V peak (20 V peak-to-peak). You can control this amplitude using the function generator as well as the volume buttons on the amplifier.
  • Next, go to the splitter output in the block diagram, and wire the bottom terminal (the ai1 terminal) to math that accounts for the voltage divider (divide the signal by 0.00226), so that you know the actual voltage being fed to the transducer. Feed this result to a tone measurements Express block (Signal Analysis Palette); use this block to measure the Amplitude, Frequency, and Phase of the ai1 signal. Run the code again and make sure all 3 values seem correct. Adjust the function generator to various frequencies in the 5 – 140 Hz range and make sure all 3 values seem correct throughout the range. When adjusting frequency, remember to keep the peak transducer voltage at or below 10 V. At low frequencies, the foam pads on the magnet can audibly whack the cage of the tactile transducer; when this occurs you must decrease the transducer drive amplitude below 10 V (peak) until the physical contact stops. When you get back to higher frequencies you are encouraged to go back to the 10 V limit. In summary, you should try to run the tactile transducer always near but below either the physical contact limit (low frequencies) or the 10 V limit (high frequencies).
  • Modify your DAQ Assistant so that you record 200k points rather than 100k points on each channel.
  • The tactile transducer has been characterized using a commercial laser vibrometer to determine its frequency response: how the amplitude and phase of the actual force it delivers depends on the frequency of the drive signal. Download lab10_tactile_transfer_function.vi from the course website. If you save this file in a known location, you can right-click within the block diagram of the VI you are building, choose “select a vi…”, navigate to the lab10_tactile_transfer_function.vi file, and insert it as a sub-vi into your code. Now you can supply to this sub-vi the frequency obtained from the tone measurement, and the VI will provide 2 useful outputs. Multiply the ai1 signal (bottom output of the splitter) by the top output of the sub-vi. The result is scaled to be a Force [N] rather than a drive voltage [V]. Next we have to adjust the phase; we will do this by deleting initial points from the signal using an Extract Portion of Signal express block. In that block, select the top radio button “Begin at offset…”. Wire the phase output of the sub-vi to the to the “begin offset” of the express block. The output of this block is now Force [N] vs. time and has the correct phase.
  • The top output of the splitter also needs to be scaled so it is ready to provide measured weight [N] rather than bridge output voltage. Do it.
  • Now your split, processed signals are ready for final analysis and plotting. Make 2 copies of your existing tone measurements block and feed each signal to one. Arrange the now 9 tone measurement indicators neatly on the front panel. Finally, merge the signals back into 1 and feed it to a graph indicator. This second graph should be as wide as your first one. On this second graph, right click, and select “ignore time stamp”, to make the phase shift work out as desired.
  • We are interested in the phase shift [degrees] between the two signals in your second graph. Compute this quantity and display it in an indicator on the front panel. We are also interested in the magnitude ratio (force measured divided by force applied). Divide the two amplitude measurements from the associated tone measurements block and create a magnitude ratio indicator on the front panel.
  • Run your code, vary frequency, and make sure that everything seems to be working.

======YOU CANNOT GO PAST THIS POINT UNTIL DAY 2 ======

(because you must have the tactile transducer mounted to the beam, and the beam rotated, to continue)

4.2 Determining the frequency responseof the cantilever beam

The next part will shake the lab station side-to-side. Shore up items on the shelves at your lab station, e.g., moving equipment on the shelves to the center.

Use your setup to measure the magnitude ratio and phase shift for many different frequencies spanning the 5 – 140 Hz range. Suggested target frequencies are: 5, 5.5, 6, 6.5, 7, 8, 9, 9.5, 9.75, 10, 10.25, 10.5, 10.75,11, 11.5, 12, 13, 15, 20, 25, 35, 50, 70, 90, 110, 140 Hz, but it is not important to hit integer values. Log your data (3 columns) in Excel or similar. Plotted, it should look roughly like the ξ = 0.1 curves of Figures 4.8 and 4.9 Dunn. You may need to add or subtract 360 degrees from some phase data points to get a smooth graph.

mini-report 4: Prepare two high-quality plots (1 magnitude, 1 phase) from your data. Explain the key features of the plots in the caption(s).

mini-report 5:Determine the frequency [Hz] below which the measured weight from your cantilever scale will remainwithin 5% of the correct value.

mini-report 6: Determine the frequency [Hz] below which the phase shift of the measured weight will always be < 10 degrees.

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[STS1]GOOD ANSWERS: +-2V, 200 kS/s, 400 kS, sample compression factor = 140