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Module 1 - Dynamic Fracture Mechanics with Analog Electronics

Module 1 - Dynamic Fracture Mechanics with Analog Electronics

For this expeirment, we will be measuring the crack speed as a function of length along a brittle polymer material (PMMA). We will do this following the work of Fineberg et. al., which you will find a copy of in the `references' section below. Essentially, we will coat the PMMA with a thin layer of aluminum, which will form one resistor in a Wheatstone Bridge configuration. We will use a Universal Testing Machine to load the sample in tension, and monitor the output of an instrumentation amplifier circuit with an oscilloscope to record the test. This module will last 6 weeks, over which time you should become familiar with the key apparatus for the experiment.

Readings

Week 1

A simple website introduction to the oscilloscope: How to Use an Oscilloscope - learn.sparkfun.com

I also encourage you to read the appendix of the Art of Electronics (AoE) on the oscilloscope. This is available in both the 2nd and 3rd (most recent) editions of the book.

Week 2

Read about transistors (AoE 2nd Ch. 2, through 2.05) and op-amps (AoE 2nd Ch. 4, through 4.22 - stop at comparators)

Week 3

Read chapters 4, 7 and 8 in the HBM handout on Wheatstone Bridge circuits (wikipedia Wheatstone bridge if you've never seen it before) *posted 5.10 - for those whose thirst for knowledge about how to practically drive bridge circuits isn't yet sated, I emphatically recommend this article by Jim Williams*

Read this short handout on instrumentation amplifiers. It should be clear after reading this why one would use an instrumentation amplifier - particularly in a Wheatstone Bridge circuit.

If you can find the 2nd edition of AoE, read 15.03 on measuring strain and displacement - it covers almost any type of tranducer that you'd ever be interested to use in measurement applications (!)

Week 4

For a general discussion of error analysis and general error propagation, this website has a great, concise summary - from Werner Boeglin. You should understand how error in measured quantities used to derive another quantity can be used to determine the error bounds on the derived quantity.

From the Keithley low level measurements handbook, 7. ed (link below in resources): section 1.2, 1.4, all of section 3.

Week 5

No reading for this week.

Week 6

This week I encourage you to re-read the manuscript from Fineberg et. al. You might also read the manuscript from Goldman on acquisition of inertia by a moving crack, in references. These papers, and citations therein, can form a basis for theory against which to evaluate your measurements.

Exercises

Week 1

Familiarize yourself with the oscilloscope and other bench-top instrumentation at your desk. A suggested starting point is to feed a sine wave from the function generator into the oscilloscope input, and monitor the wave on the scope. Set up a proper trigger for the experiment. Measure the frequency - does it exactly correspond to the frequency of the function generator?

Propose and conduct an experiment to verify that a resistor satisfies Ohm's law
Make a voltage divider, and explain why the output voltage takes on the value that you measure
Drive a resistive load with your voltage divider, and continue to monitor the output voltage. Does this value change when you attach the load? Why or why not? Try this for a few values of R, varying from less than to greater than the resistor value you used in the original divider.
Make a low-pass filter using a resistor and a capacitor
Make a high-pass filter using another resistor and capacitor pair (they can have the same values as the components you used in the low-pass filter)
Propose and conduct an experiment to measure the 3 dB frequency for each of these filters (hint: the function generator has a sweep feature). Does the value agree with your expectation based on the capacitance and resistance values you used?
Bonus: make a bandpass filter

Week 2

Use an npn transistor to make an emitter-follower using ~ 1kOhm on the emitter leg to ground, and between 10 and 15 V on the collector; use 100 Ohm on the base leg. What happens when you drive the base at 5 V peak-to-peak?
- Now connect the 1kOhm on the emitter leg to -15V, leaving everything else as it was. What changes about the output signal?
Plug in & power (with the correct polarity) the op-amp (LF411 in lab). Monitor the output of the amplifier with an oscilloscope probe. If you short the inverting and non-inverting inputs, what do you observe on the oscilloscope? Do you see oscillations, or a steady signal? Is the signal anywhere near the power-supply voltage?
Construct an op-amp follwer. How does this circuit behave when compared to the emitter follower you made in exercise 1 when driven with a 5Vpp sinusoidal input?
Build a non-inverting amplifier with a 10k in the feedback loop, and a 1k to ground. Drive the input with a sinusoid, and observe the magnitude of the output swing by varying the input magnitude. What sets the limit of the output swing at 1kHz? Now, increase the driving frequency by a lot (hint ~ 1 MHz) - do you observe a change in the amplifier's response? How did the response change? Is there a value from the data sheet can account for the observed change?
Build an inverting amplifier using the same resistor values as the previous exercise, and measure it's input impedance by inserting a 1k between the function generator and the 1k leg of the amplifier. What is the input impedance? What sets the input impedance value of the inverting amplifier?

Week 3

Take a measurement of the resistance of the coating on a `typical sample.' Bear in mind that you'll have to carefully account for lead resistance (look up a 4-wire meausrement method - you can use your multimeter) First convince yourself why you'll need the 4-wire measurement for this task. Record the resistivity of the coating.
Take a measurement of the resistance of the artifically `cracked' coating (cut with a razor blade) to get a sense for the `minimal' value of resistivity that you should anticipate. Record this value of the resistivity.
These measurements provide a range of typical resistance values you should anticipate, within about 10%. Design a Wheatstone bridge architecture, specificying the values of each of the resistors. If you consider the above measured values as the upper-bound and lower-bound for the coating during the fracture experiment, you might obtain a useful range for your bridge design.
- Bear in mind that precision resistors might be required, depending on your design.
- Account for thermal loading - if too much current passes through the bridge, you can be in trouble, because you'll get Joule heating of the resistors, and they'll drift from their design values. If you stay below 1/8 W for each resistor, you should be fine.
Once you have converged on a bridge structure, build a voltage source for the bridge, and build the bridge itself. Test the output, and develop a relationship between resistance and voltage. Now that you have the transfer function of the bridge, and given what you know about the resistivity of the `cracked' sample, can you come up with a relationship between output of the bridge and crack length?
Consider circuits that you might used to measure crack length & velocity. If you choose to measure the crack position as a function of time, and then digitally differentiate the recorded signal, you can do this; this is what the original paper describes. Can you think of another way to measure crack tip velocity, which is our quantity of interest? (hint: an passive high-pass filter, or even an active high-pass filter, are also called differentiator circuits). Build one of these circuits (use an INA to deal with common-mode for the position measurement), and measure its bandwidth. If you build the differentiator, compare it with the passive-version constructed using a high-pass filter.