Martin Pickett
Michele Meo
Strain Gauge Martin Pickett Wednesday 23rd April 2008
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Martin Pickett
Michele Meo
Summary We used a beam which in a structure so that we could easily strain the beam. On the beam there was a strain gauge that was attached to three different circuits. The first was a simple potentiometer circuit with the strain gauge acting as the variable resistor. The second was a Wheatstone Bridge and the third a Wheatstone Bridge followed by an operational amplifier. In all three circumstance we strained the beam in increments and recorded the voltage difference in an effort to find the best circuit. Of the three set ups, the third was the best as it was able to resolve a strain of 0.27μstrain where as the first set up was the worst with a resolution of only 0.36 mili strain. The second experiment’s results were between the first and last. The main conclusion was that in order to measure strain electronically, a strain gauge should be placed in a Wheatstone Bridge and then the resultant signal amplified using an operational amplifier.
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Martin Pickett
Michele Meo
Table Of Contents
Introduction
Page 4
Theory
Page 4
Method
Page 4
Results
Page 6
Discussion
Page 11
Conclusion
Page 12
Acknowledgments and References
Page 13
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Martin Pickett
Michele Meo
Introduction This laboratory exercise is designed to demonstrate the limitations in the measurements of strain when done by way of a Strain Gauge. It will then go on to suggest two improvements that can be made to the circuitry that will increase the accuracy of measurements made using a strain gauge. I will also outline the limitations of these new methods in a quantitative way. This is very important as in industry the measurement of strain is one of the fundamental measurements which allow engineers to analyse a structure so as to insure it ’s safety. Without accurate ways to measure strain an engineers life would be a lot more difficult.
Theory In order to analyse the effectiveness of the strain gauge on a bent beam, we require a way of calculating the strain on the beam at the point where the strain gauge is positioned. From the work we have done in Solid Mechanics class ’s we are able to calculate the strain on a cantilevered beam close to it ’s gripped end. Therefore by placing the strain gauge at that end of the beam and measuring the beam thickness and deflection when applied we can calculate the strain which the gauge is recording using the following formulae:
3"d ! =
2L
Where:
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- Strain δ - Deflection of the end of the beam d - Beam thickness L - Beam length ε
Method The initial set up for this experiment is fairly simple as we have a beam suspended and supported at one end with a metric screws at the other for accurate deflection measurement. This set up can be seen in figure 1. This allows strain to be calculated using the formulae outlined in the Theory section above. There is also a strain gauge placed very close to the supported end of the cantilevered beam for reasons explained in the theory section. This strain gauge is then connected in series with a 1.5KΩ resistor. The strain is then measured in teams of the voltage across the strain gauge. The circuit diagram for this can be seen in figure 2. With this set up the minimum strain resolution can be measured by bending the beam until the smallest measurable voltage change occurs and then calculate the strain using the formulae in the Theory section.
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Martin Pickett
Michele Meo
Figure 1 - Strain Gauge apparatus.
Figure 2 - Circuit diagram for initial measurements.
I’ll then refine the circuitry by implementing a Wheatstone Bridge into the measurement circuit as well as changing the 1.5KΩ resistor into a 680KΩ resistor. Apart from that rest of the experiment set up remains the same. So as to keep the Wheatstone bridge as simple and accurate as possible two of the other three resistors required will also be strain gauges but ones that are not being strained. The fourth resistor will be a variable resistor which will be altered to ‘balance’ the bridge, i.e to insure a zero voltage is measured when there is no strain. The circuit diagram for this can be seen in figure 3. From this the same resolution measurement can be made and this time an out-of-balance voltage versus calculated strain graph can also be drawn and compared to a theoretical one using the Wheatstone Bridge balance voltage calculation.
Figure 3 - Circuit diagram for Wheatstone Bridge set-up.
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Martin Pickett
Michele Meo
Finally, I will add an operational amplifier to the circuitry to amplify the out-of-balance voltage created by the Wheatstone Bridge to a more easily read value. The circuit diagram for this can be seen in figure 4. I will also be able to calculate the resolution of this circuit as well as compare voltage-strain graphs from this circuit and the theoretical values of this circuit when gain has been accounted for.
Figure 4 - Circuit diagram for amplifier set-up.
Results If we start with the first experiment there are only a few values obtained experimentally, however there are quite a few that will be useful that have to be calculated using the formulae outlined in the Theory section. Below is the recorded data: Deflection / mm
Voltage / mV
Strain / mε
0
1.11
o
18.9
1.12
[0.36]
Table 1 - Experiment 1 data.
From this I can calculate the strain that an 18.9mm deflection reflects. (This value is also shown in the above table in brackets.) I can also calculate the resistance of the strain gauge used and the resolution of the circuit: Strain / mε
Resistance / Ω
Resolution / mε
0.36
119.9
0.36
Table 2 - Derived data from experiment 1.
Finally, using the formulae outlined in the Theory section I can produce a calibration graph for strain against deflection, the data for which is shown in the following table:
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Martin Pickett
Michele Meo
Deflection / mm
Strain / mε
0
0
5
0.096
10
0.192
15
0.288
20
0.384
25
0.48
30
0.576
35
0.672
40
0.768
Table 3 - Calculated strains from deflections.
Using the above data I can produce the following graph: 0.8
0.6
0.4
0.2
0 0
10
20
30
40
Graph 1 - strain vs. deflection.
Moving on to the second experiment, in which a range of deflections were used and voltages recorded. The raw data from this is provided below:
Deflection / mm
Voltage / mV
1.71
0
3.43
0.1
5.14
0.1
6.86
0.2 7
Martin Pickett
Michele Meo
Deflection / mm
Voltage / mV
8.57
0.2
10.3
0.3
12
0.3
13.7
0.4
15.4
0.4
17.1
0.5
Table 4 - Deflection and out-of-balance voltages from Wheatstone Bridge
From this raw data the strain can be calculated. Also by using the Wheatstone Bridge equation for out-of-balance voltage the theoretical out-of-balance voltages can be calculated. All both of these pieces of data can be found in the following table: Deflection / mm
Strain / mε
Theoretical Voltage / mV
1.71
0.033
0.08
3.43
0.066
0.16
5.14
0.099
0.24
6.86
0.132
0.32
8.57
0.165
0.39
10.3
0.198
0.47
12.0
0.230
0.55
13.7
0.263
0.63
15.4
0.296
0.71
17.1
0.328
0.79
Table 5 - Calculated strain and voltage for Wheatstone Bridge experiment.
Using the recorded voltages and the calculated strains you can draw a graph shown below and a further graph can be drawn using the calculated strain and the theorised voltage.
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Martin Pickett
Michele Meo 0.500
0.375
0.250
0.125
0 0
0.1
0.2
0.3
0.4
Graph 2 - Actual voltage vs. Strain.
0.8
0.6
0.4
0.2
0 0
0.1
0.2
0.3
0.4
Graph 3 - Theorised voltage vs. Strain.
There are two other pieces of data that were recorded that should be presented. The first is the resolution of the voltmeter used and the second (closely related) piece is the overall resolution of the circuit measured in terms of strain. Voltmeter Resolution / mV
Strain Resolution / m ε
0.1
0.071
Table 6 - Extra data for Wheatstone Bridge experiment.
Finally, the last experiment included an amplifier on the end of the circuit. For this experiment we recorded the resistors in the amplifier circuit R 1 and R2 so that the gain of the amplifier could be calculated. We also measured the voltages out of the amplifier at a range of deflection the same way as we did when there was just a Wheatstone Bridge. The results are shown in the tables below: 9
Martin Pickett
Michele Meo
R1 / Ω
R2 / Ω
Gain
100
100000
1000
Deflection / mm
Strain / mε
Voltage / V
1.71
0.033
0.05
3.43
0.066
0.09
5.14
0.099
0.14
6.86
0.132
0.17
8.57
0.165
0.22
10.3
0.198
0.26
12.0
0.230
0.3
13.7
0.263
0.35
15.4
0.296
0.39
17.1
0.328
0.44
Table 7 - Amplifier data
Table 8 - Strain vs. Amplified voltage.
Using the data above the following graph was plotted: 0.60
0.45
0.30
0.15
0 0
0.1
0.2
0.3
0.4
Graph 4 - Strain vs. Amplified voltage.
Again, the resolution of the of the circuitry is required to complete my analysis, and in order to attain this we first need the resolution of the voltmeter. Below is a table containing both of these pieces of information:
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Martin Pickett
Michele Meo
Voltmeter Resolution / mV
Strain Resolution / με
0.1
0.27
Table 9 - Amplifier experiment extra data.
Discussion Lets start with the first experiment. The first thing is to explain why there is a 1.5KΩ resistor in series with the strain gauge. The strain gauge is effectively a variable resistor so by placing another resistor in series you can create a potential divider. This allows voltage to change as the resistance of the strain gauge changes thus producing an electrical readout for the strain. This is the basic principle behind all three different measurement circuits and the improvements made on this basic voltage measurement system will be explained soon. This set up relies on a voltmeter to measure the voltage over the strain gauge which mean the accuracy of the strain derived from the voltage measured will only be as good as the accuracy of the voltmeter. This accuracy is measured in terms of its resolution. As expressed in tables 6 and 9 the resolution of the voltmeter we were using was 0.1mV. This means the smallest voltage change the voltmeter could measure was 0.1mV. In order to convert this into a strain we strained the beam until there was 0.1mV change on the reading of the voltmeter and then noted the deflection we caused to the beam and from that calculated the strain on the beam. As shown in table 2 for the first experiment we conducted the resolution was 0.36 mili strain. What this means is that the smallest possible change in strain that our equipment could measure would be 0.36 mili strain. What none of the above answers is the question as to what the relationship between deflection, strain and voltage is. In order to do that we can utilise the equation outlined in the theory section of this report. Utilising the equation I was able to produce a table of data seen in table 3 and from that graph 1 was then drawn. As you can see from graph 1 the relationship between deflection and strain is a linear, and that should come as a surprise as the equation linking the two boils down to the fact that deflection is proportional to strain. In order to further the understanding of the relationship between deflection, strain and voltage we need to move onto the second experiment. In experiment two we used a Wheatstone Bridge which can be seen in figure 3. This provided us with a larger range of measurable voltage values which enabled us to recored out-of-balance voltages and deflections, as can be seen in table 4 from which the strains could be calculated. Also, using information from the first experiment the theoretical outof-balance voltages could be calculated and this information is all shown in table 5. From this data two graphs could be drawn both with the same axis, strain vs. out-of-balance voltage. The difference between the two is which out-of-balance voltage was used. in the first the recorded out-of-balance voltage was used where as in the second the calculated out-of-balance voltage was used. These two graphs are graphs 2 and 3 and both show straight lines. This shows one thing very strongly and that is that the relationship between strain and out-of-balance voltage is linear. As the relationship between strain a deflection is also linear this shows that all three of the quantities are related to each other in a linear means. This is important as it shows that measuring the out-of-balance voltage can be considered as measuring strain as the relationship between the two is linear. Now we just need the required resolution to make strain gauges an effective way of measuring strain. 11
Martin Pickett
Michele Meo
The Wheatstone Bridge is an electric circuit which allows you to measure the change in a voltage as opposed to the actual voltage. Due to the extra resistors the voltage change for a given change in resistance is higher then that used in the first experiment. This results in the same resolution of the voltmeter, still 0.1mV but a better strain resolution as seen in table 6, of 0.071 mili strain. However, small voltage changes still require a sensitive voltmeter to measure or they need to be amplified. In the third experiment we added an amplifier to the circuit to boost the out-of-balance voltage generated from the Wheatstone Bridge. In order to control the gain we set the resistance of two ‘control’ resistors, which in turn limited the gain of the operational amplifier. The two values of the resistors and the limited gain can be found in table 7. This shows that all the out-of-balance voltages should have been multiplied by 1000 making all the voltage measurement a lot more accurate, especially when you consider that the voltmeter still has a resolution of 0.1mV. This results in a resolution of 0.27 μstrain as seen in table 9. It’s important to realise the differences in the resolution the different techniques have achieved. A basic potentiometer design for a strain gauge results in a resolution of 0.36 mili strain where as using an operational amplifier and a Wheatstone bridge you can achieve a resolution of 0.27 μstrain over a thousand times more detailed. Not to mention the advantage that you get the change in voltage rather then the actual voltage so you don’t have to do any further calculations. There are three main things which affect the resolution, the first of which is the change in resistance produced by the strain gauge. As this is what we are hoping to measure I will ignore this. The first thing which affects the resolution is the voltage which powers the amplifier as this is the maximum voltage to which any signal could get amplified to. The second is the ratio of ‘control’ resistors used. In the third experiment we used a ratio of 1000 between the two resistors and thus we had a gain of 1000. If we had used a different selection of resistors we would have achieved a different gain and therefore different outof-balance voltages.
Conclusion The first (and arguably obvious) conclusion to this report is that strain gauges are an affective way of measuring strain in a beam as the resistance and therefore voltage changes linearly with strain making all the calculations to convert changes of voltage into mechanical strain relatively simple. However there are some more important conclusions. The first of which is that just using a strain gauge in a potentiometer is in affective as the resolution is too low to be of any real value. This statement also goes for experiment two where a Wheatstone Bridge is used also. However there are certain advantages with using the Wheatstone Bridge such as you get a slightly better resolution and you only measure the voltage change which makes calculating the strain a lot easier. Finally, by use of a Wheatstone Bridge and an amplifier you achieve the required affect. You are only measuring the change in voltage due to the Wheatstone Bridge and you can then amplify this signal to greatly improve the resolution at which the strain can be determined. 12
Martin Pickett
Michele Meo
This means with the correct selection of some fairly basic electrical components and a strain gauge very small strains can be measured with high degrees of accuracy.
Acknowledgments and References Thank you to: Matthew Watson Andrea Pedrazzini The Lab. Technician whose help was invaluable (feel guilty as i didn ’t ask him his name)
… and references: http://www.sensorland.com/HowPage002.html diadem.jp/english/products/gages/pdf/howsgw.pdf http://www.customsensorsolutions.com/ap-opamp.htm
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