## Saturday, September 25, 2010

### Labview Temperature Measurement and First-Order Dynamic Response

Summary: In this lab exercise, you will determine how various temperature measurement systems respond to different inputs. This analysis will include a thermometer, a thermocouple, and a thermistor.

## Introduction:

In this lab exercise, you will determine how various temperature measurement systems respond to different inputs. This analysis will include a thermometer, a thermocouple, and a thermistor.

## Teaching Objectives:

The purpose of this laboratory is for each student to understand and do the following:
1. Instrument calibration.
2. Measurements with thermocouples and thermistors.
3. Determination of time constants and comparisons of the dynamic responses of three measuring systems.
4. Application of first-order dynamic behavior to dynamic temperature measurement problems.

Figliola and Beasley
Measurement System Behavior, First Order Systems: pp. 72–87
Thermistors: p. 295
Thermoelectric Temperature Measurement: pp. 300-313

## Overview:

In this lab, we will use first-order models to approximate the response of a thermometer, a thermocouple, and a thermistor to temperature inputs. Each of these temperature sensors measure temperature in a different way.
• A thermometer senses a change in temperature as a change in the density of a fluid.
• A thermocouple consists of two wires of different metals joined at one end (the junction). When a voltage is applied across the free ends of the two wires, the differing properties of the wires create an induced voltage that it proportional to the temperature change at the junction.
• A thermistor is a thermally sensitive resistor. When the temperature of the thermistor changes, a proportional change in resistance occurs.

## Procedure:

### Part 1: Modify a VI for temperature measurements

In this lab, we will build on the skills that were developed in Introduction to Benchtop Equipment and Data Acquisition. It is assumed that the student understands the concepts of a Front Panel, a Block Diagram, the Control Palette, and the Functions Palette. It is also assumed that the student can search the control and functions palette for necessary features.
1. Connect a thermocouple to channel 0 of the SCXI-1112 module.
2. Open LabVIEW 8.0.
3. Open the VI that you developed for Introduction to Benchtop Equipment and Data Acquisition.
4. Click File>>Save As…>>Substitute Copy for Original to save your VI with a new name.
5. On the Block Diagram, double click the DAQ Assistant to open the configuration dialogue box.
6. Configure the DAQ Assistant as shown above.

1. Click Voltage and press Delete to delete the channel you used in the first lab.
2. A dialog box will ask if you want to delete the selected channel. Click Yes.
3. Add a new measurement channel.
2. Select the channel that the thermocouple is connected to (SC1Mod2>>ai0).
3. Click OK.
4. The input range should be 80mV max to -80mV min. (Thermocouples have a low voltage output.)
5. The Terminal Configuration should be Differential. (The ground terminal is not used.)
6. Change Acquisition Mode to Continuous.
7. Click OK.
8. A dialog box will open asking if you would like a While Loop to be made around the DAQ Assistant. Click No. (You will place a While Loop in the next step that includes the subVIs that are already on the Block Diagram.
7. Place a While Loop around the entire Block Diagram as shown below. LabVIEW While Loops are similar to While Loops in any text-based programming language. The routine inside the loop will run repeatedly until a true value is passed to the loop condition, represented by a red button at the bottom right corner of the loop.
8. A toggle button can be used to end the While Loop routine. Right click on the loop condition icon and click Create>>Control.
9. Double click the Write to Measurement File ExpressVI to open the configuration box. Under If a file already exists, select Append to File. Click OK. (Because the Write to File function now occurs in a loop, the data must be appended to the end of the file, rather than overwriting the data from the previous loop.)
10. On the front panel, place a Thermometer.
11. Right Click the Thermometer. Click Visible Items>>Digital Display.
12. On the Block Diagram, inside the While Loop, place a Time Delay Express VI. When the Configuration Box opens, change the Time Delay (seconds) to 0.001. (Without a delay in the While Loop, the program will repeat as quickly as it can and consume processor power.)
13. Your Block Diagram should resemble Figure 3.
14. Click File>>Save As…>>Copy>>Create Unopened Disk Copy. Save the VI with “Lab3” in the file name. (You will use the VI as it is now as a starting point in Operational Amplifiers and Signal Conditioning.)
15. Save a second copy with “Lab4” in the file name. (You will also use the VI as it is now as a starting point in Strain Gages and Force Measurement.)

### Part 2: Calibration Curve

You will create a static calibration curve to correlate voltage measurements with temperature readings. We will use the thermometer as the standard. (Note: We use the thermometer as a standard for convenience. Do not assume that an alcohol-in-glass thermometer is a temperature measurement device with superior accuracy.)

#### 2.1 Calibrate the thermocouple

1. Take temperature readings with the thermometer in ice water and boiling water. Be careful not to allow the devices to come into contact with the hot plate or the bottom of the beaker.
2. Record the temperatures measured.
3. Use the thermocouple to measure both temperatures. Again, be careful not to allow the devices to come in contact with the hot plate or the bottom of the beaker.
4. In Excel:

1. Plot the temperatures measured with the known standard values on the y-axis and the ADC output (voltage) resulting from the thermocouple input on the x-axis.
2. Create a linear curve fit using the two data points.
The linear fit results in an equation that relates the ADC output obtained from the thermocouple to the known temperature from the thermometer:
where “T” is the temperature, “k” is the gain (°C/V), “Vadc” is the voltage read in to the computer, and “b” is the zero-voltage offset. The gain “k” is the inverse of the static sensitivity, which has units of V/°C.
5. Repeat the calibration procedure for the thermistor.

#### 2.2 Modify your VI for calibration

In this section, you will multiply the data by the conversion scalar “k” that you found experimentally. You will also add an offset “b”.
1. Delete the Tone Measurement ExpressVI. Delete the Frequency and Amplitude indicators.
2. Delete the wires that connect the DAQ Assistant to the thermometer, the Waveform Graph, and the Write to File ExpressVI. Press Ctrl+b to remove any broken wire remnants.
3. Place a Multiply and an Add on the block diagram.
4. Create a control and connect it to the y terminal of the Multiply icon. Rename the control “k”.
5. Create a control and connect it to the y terminal of the Add icon. Rename the control “b”.
6. Wire your block diagram as shown in Figure 4.
7. On the front panel, enter the values you found experimentally for the “k” and “b” values.
8. When you run the VI, your measurements should now be displayed as temperatures.

### Part 3: Time Constants

The calibration curves you found represent static calibrations. In the following section, you will consider the dynamic behavior of the sensors. The dynamics of all the temperature sensors used in this lab can be modeled accurately as first-order systems. The natural response of a first-order system can be characterized by a single parameter — the time constant.
Time constants can be found by observing the response of the instrument to a step input in temperature. Note that the second medium governs the physics of the transient response. For each of the sensors, you will determine the time constant for transfers from ambient air to ice water and from hot air to ambient air.
An ice water bath will be provided for your tests. A heat gun will be used to generate hot air. Caution: Keep all temperature sensors (particularly the thermometer and the thermistor) at least 9 inches away from the heat gun. Holding them closer to the heat gun will result in temperatures outside the operating range of the sensors.
Four methods will be used to calculate various time constants. Table 1 summarizes the tests to be performed. The testing methods are described below.
Table 1. Time constant calculation methods.
 thermometer thermocouple thermistor ambient air to ice water 1, 2 3, 4 3, 5 hot air to ambient air 1 3 3 ice water to ambient air - 3 -

#### Method 1- Time Elapsed at 63.2% Sensor Accuracy

Thermometer: Perform this test for ambient air to ice water. Repeat for hot air to ambient air.
1. Record the temperature of the initial medium, T0, and the temperature of the final medium, T_inf.
2. Calculate the intermediate reference temperature, Tr that is 63.2 percent of the way from T0 to T_inf.
3. With the instrument at steady state (reading T0) in the initial medium, move it quickly to the final medium.
4. The time from when the step in temperature began to when the instrument reads Tr is the time constant, . This time can be measured using a stopwatch.

#### Method 2 - Mathematic Definition of Tau

Thermometer: Perform this test moving from ambient air to ice water.
1. Record the temperature of the initial medium, T0, and the temperature of the final medium, T_inf.
2. Choose an arbitrary reference temperature, Tr that is between 60 and 80 percent of the way from T0 to T_inf.
3. With the instrument at steady state (reading T0) in the initial medium, move it quickly to the final medium.
4. Record the time trthat it takes to reach the reference temperature Tr.
5. The following expression can be used to calculate the time constant tau:

#### Method 3 – Analyze Acquired Data

Thermocouple: Perform this test for ambient air-to-ice water, hot air-to-ambient air, and ice water-to-ambient air.
Thermistor: Perform this test for ambient air-to-ice water, and hot air- to-ambient air.
1. Run the VI.
2. Insert the thermocouple into the initial medium.
3. Click the Enable toggle control as you transfer the thermocouple to the final medium.
4. When the sensor comes to equilibrium, stop the VI.
5. Open your measurement file in Excel.
6. Determine the time required for the sensor to complete 63.2 percent of the transient from the initial temperature to the final temperature.
7. Change the file path to prevent overwriting data. (You can either change the name of the file after it is written, or you can change the name of the path that the next file will be written to.) You will use the saved data for subsequent calculations.

#### Method 4- Error Fraction

Thermocouple: Complete this step using the data from the ambient air-to-ice water transfer.
Thermistor: Repeat this step using the data from the ambient air-to-ice water transfer.
Equation 2 below gives an expression for the error fraction.
1. Calculate the error fraction at each instant of time for the data you have gathered.
2. Take the natural log of the error fraction data.
3. Plot the natural log of the error fraction versus time.
4. Find the negative reciprocal of the slope of this curve. This is the time constant.
The error fraction versus time curve provides an easy way to judge the accuracy of a first-order model for the data gathered. If the error fraction plot is linear, then the first-order model is a good approximation of the dynamic behavior of the system. (See pages 78–79 of Figliola and Beasley for a more detailed discussion of the error fraction and its use to calculate the time constant.) Are the thermocouple and thermistor in water adequately modeled by the first-order system approximation?

#### Method 5- Least Squared Error

Thermistor: Complete this step using the ambient air-to-ice water data.
Equation (3) below gives the theoretical expression for the temperature, T(t), as a function of time when the temperature of the environment is suddenly changed from T0 to T∞. The thermocouple is assumed to be at temperature T0 when the change occurs at t = 0.
1. Using Excel, compute a value for tau so that the square of the error between the theory and your experimental result is minimized. This is called a least square error approach and is very common in engineering.
2. Plot both the test data and the results from Equation (3) on the same axes to compare how well this model fits the data.

Once you have performed all of the tests outlined above, consider the following tests:
•Transfer the thermocouple from ice water to room temperature air.
How does the time constant compare with the one obtained from the air-to-ice-water transition? Is it different? Why or why not?
•When calculating the time constant for the thermocouple in transition from air to ice water, you measured the time to go 63.2 percent of the way from the initial temperature (around 21 C) to the final temperature (around 0 C). Using the same graph, recalculate the time constant using14 C as the starting temperature (i.e., what is the time required to go 63.2 percent of the way from 14 C to 0 C?). Is this time constant value significantly different from the initial value you computed? Why or why not?

### Part 5: Thought Questions

1. How do the time constants of the thermometer, thermocouple, and thermistor compare? Why is the thermometer “slow” compared to the other devices?
2. How do the time constants obtained for air and water compare? Are they significantly different? If so, explain why.
3. Does the instrument need to be at steady state before transferring it to the final medium?
4. How accurate (qualitatively) is the first-order model approximation for the different sensors and media?

### Part 6: Frequency Response

From the measured time responses of the thermometer, the thermocouple, and the thermistor, it can be seen that the response time of these sensors is not instantaneous. From this we can infer that these measurement systems may have some difficulty following inputs that change quickly. Frequency response is a means of characterizing the dynamic response capabilities of a system.
The frequency response of a system can be determined by calculating the response of a system to sinusoidal inputs at different frequencies. Two important characteristics are:
• the magnitude ratio between the output and input waveforms
• the phase difference between the output and input waveforms
For a first-order system, we can create the frequency response plot using only our knowledge of the system time constant. The magnitude ratio can be calculated from the following expression.
The phase difference is determined from expression 5.
In Excel, create a frequency response plot for the thermometer, thermocouple, and thermistor using the time constant from the ice water tests. Plot the magnitude ratio on a log-log scale and the phase difference on a semilog scale (log scale of frequency axis). Consider frequencies from 0.1 rad/s to 1000 rad/s. Your plots will look nice if you use ten points per decade of frequency (i.e., 0.1, 0.2, 0.3, … ,1, 2, 3, … , 10, 20, 30, …). Overlay the three magnitude responses on one plot and the three phase responses on another. How do the frequency response characteristics of the thermometer, thermocouple, and thermistor compare?

### Part 7: Dynamic Temperature Measurement Problem

Include the analysis for this problem in your lab book.
Assume that the thermocouple you used is placed in a water stream at a point where the temperature varies sinusoidally between 60 C and 80 C at a frequency of 40 rad/s. Assuming that the thermocouple time constant is the same as the value you determined in the ice water case, what will be the range of sinusoidal variation in the temperature readings? Repeat the calculations for a temperature frequency of 2 rad/s. Now assume the thermocouple is in air instead of water. Determine the range of the variation using input frequencies of 10 rad/s and 1 rad/s.

## Clean Up:

Please straighten up your lab work area. Turn off the data acquisition hardware and leave your work area clean. Your help in keeping the lab neat is appreciated.

## Report:

For this lab, you are to write up the Method section of a full formal report. Guidelines are given in the example report in the Undergraduate Guide. Avoid presenting results in the Methods section. Your writing assignment is relatively short. Take the time to do a good job. Since you will not be submitting Results or Discussion of Results in the report, you will need to document your results in your lab book. Also, be sure to answer the questions from the lab handout in your lab book.

## Content actions

### labview Strain Gages and Force Measurement

Summary: In this lab, you will learn about strain gages and the Wheatstone bridge circuit. You will see how they can be used for strain and force measurement. You will modify an existing program to measure the dynamic characteristics of a second-order system.

## Introduction

In this lab, you will learn about strain gages and the Wheatstone bridge circuit. You will see how they can be used for strain and force measurement. You will modify an existing program to measure the dynamic characteristics of a second-order system.

## Teaching Objectives

• Gain practical experience with resistance strain-measurement techniques.
• Learn about the Wheatstone bridge and how it is used in strain measurement.
• Use a beam instrumented with strain gages as a force measurement device.
• Use strain gages to measure the natural frequency and damping in a beam.
• Design a force transducer for measuring thrust from a model rocket motor.

Figliola and Beasley
Strain Measurement: pp. 425–446

## Procedure

### Part 1: Strain Gages and the Wheatstone Bridge

The metal foil strain gages used in this lab are resistors with a nominal (unstrained) resistance of 120 ohms. As they are put in tension, their resistance increases; as they are compressed, their resistance decreases. The Wheatstone bridge provides a way to convert these changes in resistance to changes in voltage, which are easy to work with. These voltages can be conditioned, transmitted, or stored digitally.
Figure 1 shows a Wheatstone bridge configuration.
• Four resistors are connected in an end-to-end fashion.
• The input or excitation voltage is connected to the bridge between top and bottom nodes of the circuit.
• The output is the difference between the voltage at the left node and the voltage at the right node.
• An excitation voltage is required to convert the change in resistance (in the legs of the bridge) to a change in voltage at the output of the bridge.
For the bridge shown, the output voltage is expressed as
When building a Wheatstone bridge with strain gages, all four resistors have the same nominal value. Bridges can be built in the following configurations:
• Quarter Bridge-One strain gage and three fixed resistors
• Half Bridge- Two strain gages and two fixed resistors
• Full Bridge- Four strain gages

### Quarter Bridges

Figure 3 illustrates a quarter bridge configuration. The quarter bridge has one active leg, i.e., one leg with a changing resistance. From equation (1) above we can derive an expression for the output voltage as a function of the resistance change ∆R:

### Half and Full Bridges

Figure 5 and Figure 6 show half-bridge and full-bridge configurations respectively.
• Half bridge: two active legs, one in tension and one in compression. These legs are adjacent legs in the bridge.
• Full bridge: four active legs, two in tension and two in compression. The gages in tension are on opposite legs of the bridge.
Using equation 1 and Figure 1 as a guide, derive expressions for the output voltage of the half-bridge and full-bridge circuits.
Thought Question
A half bridge could be made with two gages in tension on opposite legs. When would this be useful? What would be the main problem with doing this?

### Part 2: Calibration of the Strain-gaged Cantilever Beam

Your TA will provide an aluminum beam instrumented with strain gages.
1. Clamp the beam securely to the edge of the lab table with the gaged portion of the beam cantilevered beyond the edge of the bench.
2. Using the junction box provided, connect a single strain gage in a quarter bridge configuration. (Refer to Figure 3 as a guide.)
3. Connect the junction box output cable to Channel 2 of the SCXI-1520 strain gage module. The SCXI-1520 provides an excitation voltage to the bridge and amplifies the output voltage from the bridge to ranges that are easily observable or acceptable for data acquisition. The excitation voltage Vi is 5 V and the gain is 500. When connected to the signal conditioning board, the amplified bridge output can be read by the data acquisition software.
4. Open the “Lab4” VI that you created in Temperature Measurement and First-Order Dynamic Response.
5. Hang weights from the end of the beam.
6. Record the voltage measured with LabVIEW in your lab book.
7. Repeat steps 5 and 6 for several different weights.
8. In Excel, plot the voltage (input) versus weight (output).
9. Find the best fit linear relationship for the data. The resulting equation can be used to calibrate the voltage output of the strain gages.
10. Apply several loads not used for calibration to test the validity of the linear curve fit.

### Part 3: Comparison of Theoretical Strain to Measured Strain

In this step, we will compare a theoretically based estimate of strain for a given load to that which was measured earlier. First we will determine the strain corresponding to the voltage measurements of step 2.
1. Measure the excitation voltage for bridge circuit, Vi.
The circuit analyses from step one showed that the bridge output Vo is related to the excitation voltage by the following relationship
where K equals 1/4 for a quarter bridge, 1/2 for a half bridge, and 1 for a full bridge. The voltage measured in LabVIEW is related to the bridge output by
where Kamp is the gain of the signal conditioning board. Because the bridge resistances are not balanced exactly and the weight of the beam itself produces some strain, you will observe a nonzero output voltage when there is no load applied.
2. Measure the output voltage with no load. Call this voltage Voffset.
3. To determine the strain induced by the applied loads, measure the changes in the display voltages relative to this offset voltage (Vdisplay–Voffset).
For a strain gage, the gage factor is defined as
where epsilon is the strain experienced by the gage. The gages used in this lab have a gage factor of 2.12±0.8%.
4. Derive an expression for the strain in the beam using equations (2) through (4). Your empirical strain estimate should be in the range of 0 to 2000 microstrains.
A theoretical estimate of the strain can be obtained by drawing on topics from CEEn 203. The stress on the surface of a beam in bending is given by
where M is the applied moment at the location of interest, y is the distance from the neutral axis (in this case, the half height of the beam cross section), I is the area moment of inertia of the cross section with respect to the neutral axis. Recall that for a rectangular cross section,
Recall also that stress and strain are related by Young’s modulus:
For aluminum, E = 10.4x10^6 psi.
5. Using equations (5) through (7), estimate the strain where the gages are bonded to the beam. How does the theoretically obtained strain compare to the value determined from measurements? If they are different, what are some possible reasons?

### Part 4: Experiments with Half and Full Bridges

Repeat parts 2 and 3 for the half bridge and the full bridge configurations.

### Part 5: Measurement of an Unknown Load

Based upon the calibration determined in step 2, use your beam to determine the weight of an arbitrary object. You may use the quarter bridge, the half bridge, or full bridge configuration for this test. Measure the actual weight using a precision scale. How does the weight determined with your beam compare to the object's true weight? How certain is your measurement? What are some possible sources of uncertainty?

### Part 6: Sensitivity to Extraneous Loads and Temperature Variations

Strain gages are often used as transducers to measure force in a structure or mechanical device. A perfect sensor is sensitive only to changes in the variable of interest and is completely insensitive to changes in extraneous parameters. (Such a sensor is impossible to create.) In practice, every effort should be made to minimize sensitivity to extraneous variables. In the case of the cantilever-beam scale, the objective is to measure the weight of objects or alternatively, to measure forces in the vertical plane. As such, a good design will be much less sensitive to loads in other directions (axial, lateral, torsional, etc.). Also a good design will not be sensitive to strains induced by other means, such as thermal expansion.
Your TA will supervise in the following two demonstrations.

#### 6.1 Lateral Sensitivity

Use the instrumented beam of square cross section provided by your TA.
1. Clamp the beam in a cantilever fashion so that it will measure vertically applied loads.
2. Using your hand, gently apply lateral and axial loads to the beam. Do your best to not apply vertical bending loads to the beam.
3. Compare the sensitivity of the quarter bridge to that of either the half bridge or full bridge to these extraneous loads. Which configuration seems to work the best overall? Explain why this is so. Which extraneous loads are "rejected" by which configurations? Explain why. (Note: The steps of this paragraph are more easily said than done. The point is that the quarter bridge is more sensitive to off-axis loads than either the half bridge or full bridge. If you have difficulty demonstrating this, don't stress out.)

#### 6.2 Temperature Sensitivity

Using a quarter bridge configuration, apply a load to the beam and measure the voltage output.
Apply heat uniformly to the beam (140 F max) near the gages and observe any changes in the voltage output. Do not apply heat to the gages directly! Why does the voltage (which corresponds to the measured strain) change? Repeat the procedure using a full-bridge configuration. Compare the sensitivity of the quarter bridge and full bridge configurations to temperature variations. How would you expect the half bridge to behave?

### Part 7: Dynamic Characteristics

Up to this part of the lab, you have examined the static characteristics of the strain-gage bridge. Now you will modify the VI that you developed for Temperature Measurement and First-Order Dynamic Response to measure the dynamic characteristics of a signal from the strain-gage bridge. You will excite the dynamics of the cantilever beam by plucking it. You will use your VI to display the response of the vibrating beam.

#### 7.1 Differential Model

The first mode of vibration of the cantilever beam can be modeled using a simple mass-spring-damper model. This model results in a second-order differential equation that describes the dynamics of the system. You will observe that the response of the strain –gages to an initial deflection is a damped sinusoid. This is the expected response for a second-order system. Using acquired data; you will compute and display values for the damping ratio and the natural frequency of the first vibrational mode of the beam.

#### 7.2 Damping Ratio and Damped Natural Frequency

Figure 5 shows the response of a second-order system to an initial condition. This response plot will be used to define the damping ratio and the natural frequency for this system. From the amplitudes x1and xn, the damping ratio can be calculated using the following expression:
The damped natural frequency can be determined by measuring the period of the damped oscillations. An accurate measurement of the period T is obtained by considering several periods:
Once and are known, the natural frequency of oscillation can be calculated by

#### 7.3 Programming Exercise

You will modify your VI to automate the calculation of and for the cantilever beam.
1. Your TA will direct you to a subVI named “Damping Ratio” that is has been started for you.
2. To place the subVI on your block diagram, open the functions palette and select Select a VI…
3. Connect the Data wire from the DAQ Assistant to the Measured Data input of the Damping Ratio subVI.
4. Open the block diagram by double-clicking the subVI icon. (The block diagram is shown in Figure 18.)
5. You will need to modify the block diagram for the VI to work. Refer to equations 8-11 as you design a block diagram that will calculate the Damping Ratio of your signal.
6. You will notice that the necessary inputs for the subVI are placed along the left of the block diagram. The outputs are placed along the right of the block diagram. The Peak Detector VI is used to locate the peaks (or valleys) of the signal. The locations of the peaks are given as indexed data points. For example, if the first peak occurs in between the sixth and seventh data points, the location will be 6.5. The time at which this peak occurs can be calculated using the sample rate.
7. For more information about the Peak Detector or other VIs, press Ctrl+H to activate the context help window.
(Hint: you may want to use the index array function to pull values such as x1, xn, t1 and tn out of the amplitude and location arrays.)
This exercise is meant to give you a feel for the possibilities for data acquisition, processing, and display using computer software tools. Be sure to save a copy of your data for plotting. You will need it for your writing assignment.

### Part 8: Design of a Force Transducer

For your course project, you will design a force transducer for measuring the thrust of a model rocket motor. This exercise is intended to help you initiate the design process. You may assume that the maximum thrust that will be sensed by your transducer is 5 lb.

#### Cantilever-Beam Force Transducer

Assume that you will use a cantilever-beam configuration similar to what was used in this lab. A full strain-gage bridge will be used. The width of two gages side-by-side is 1/2 inch. A good design will result in 2000 microstrains under the maximum load and will have a natural frequency above 100 Hz. Meeting both of these objectives might not be possible with a cantilever-beam design. You are to determine the thickness of the cross section where the gages are to be mounted and the distance from where the motor is mounted to the center of the gages (i.e., the length of the cantilever).

Develop a spreadsheet to perform your calculations. A spreadsheet will simplify the evaluation of different designs. This analysis will be a good starting point for the analysis of your project. Note: Your project will involve different loads and different dimensions than the transducer used in this lab. The calculations, however, are the same.

## Lab Report

For this lab you will write the Results and Discussion of Results sections of a full report. As you perform the lab, think about what data should be saved or recorded for presentation and why these data are important. For some of your data, tabulation is sufficient (e.g., calculated strain vs. measured strain). Other data should be recorded using LabVIEW (e.g., dynamic response of strain when the beam is plucked). Make sure that you include your thoughts about the results you obtained and why they are important. Discuss their agreement with your expectations.

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