Analysis On The Use Of A Strain Gauge, Accelerometer, And .

Analysis on the use of a Strain Gauge,Accelerometer, and GyroscopeFor calculating the displacement at the end of a cantilever beam .By – Tyler ConeFor – Dr. David TurcicME – 410Due 5/15/2014Portland State UniversityMechanical and Materials Engineering Department

Cone 1Contents1 – INTRODUCTION . 22 – THEORY . 22.1 – Strain Gauge. 22.2 – Accelerometer . 32.3 – Gyroscope . 42.4 – Natural Frequency And Damping Ratio . 43 – EXPERIMENTAL SETUP . 53.1 – Strain Gauge. 63.2 – Accelerometer & Gyroscope. 73.3 – Experimental Procedure . 74 – RESULTS & DISCUISSION . 84.1 – Natural Frequency and Damping Ratio . 95 – CONCLUSION . 10APPENDIX A – MATLAB CODE . 11APPENDIX B – ZERODATA CODE . 13APPENDIX C – SAMPLE CALCULATIONS . 14APPENDIX D – DISPLACEMENT GRAPHS FOR EACH SENSOR. 16

Cone 21 – INTRODUCTIONThe purpose of this experiment was to analyze the accuracy and effectiveness of using a straingauge, accelerometer, and gyroscope for measuring the displacement at the end of a cantileverbeam. This was done by applying a step input to the end of the beam, and letting the beamoscillate freely. The natural frequency and damping ratio was also calculated for the beam.2 – THEORYEach of the instruments used operate on a different theory.2.1 – Strain GaugeA strain gauge consists of a length of wire with several loops that have been mounted on a pieceof flexible backing. The backing is then mounted to a beam and will deform with the beam andwill exert a compressive or tensile force on the wire. As the wire is put under tensile andcompressive stress, its resistance will change accordingly. This relationship is expressed in Eq. 1Δ𝑅 (𝐺𝐹)𝜖𝑅Eq. 1Where 𝜖 is the strain in the gauge, 𝑅 is the nominal resistance of the gauge, Δ𝑅 is the change inresistance, and GF is the Gauge Factor that isprovided with the strain gauge.Due to the usual small values for strain, Δ𝑅 is𝑅𝑆𝐺usually very small. Thus, the best way tomeasure Δ𝑅 is to use a wheatstone bridge𝑉𝑚which will output a measureable voltage witha minimal change in resistance. Figure 1 is aschematic of a wheatstone bridge where 𝑅𝑆𝐺 isthe strain gauge, 𝑅2 , 𝑅3 , & 𝑅4 are resistors,𝑉𝐸 is the excitation voltage, and 𝑉𝑚 is themeasured voltage. Equation 2 gives the𝑉𝑒Figure 1- Wheatstone bridge

Cone 3relationship for the measured voltage in terms of the excitation voltage𝑉𝑚 [𝑅2 𝑅4 𝑅1 𝑅3]𝑉(𝑅1 𝑅2 )(𝑅3 𝑅4 ) 𝑒Eq. 2It can be seen, that if 𝑅1 𝑅2 𝑅3 𝑅4 , then no voltage will pass through to the voltmeter at𝑉𝑚 . At this point, the bridge is considered to be “balanced”. If one of the resistances were tochange, then there would be a measured voltage. It was then passed through a DifferentialAmplifier with a gain of G where G is defined as𝐺 1 50,000 Ω𝑅𝑔Eq. 3And 𝑅𝑔 is a resistor that can be changed to give the desired amplification.According to Appendix A, in Lab 5: Position Estimator using a Strain Gauge, anAccelerometer, and a Gyroscope, the equations for strain and voltage can be rewritten as𝛿𝑚𝑎𝑥4𝑉𝑚 𝐿3 []3 ((𝐺)(𝐺𝐹)(𝐿 𝑥)(𝑉𝑒 )(𝑐)]Eq. 4Where L is the length of the cantilever beam, x is the distance from the Strain Gauge to thesupport, and c is the distance from the neutral axis to the outermost fiber in the cantilever beam.2.2 – AccelerometerAn accelerometer outputs a voltage proportional to the acceleration. The voltage is thenconverted to acceleration using Eq. 5𝑎 𝑆𝑎𝑐𝑐 𝑉𝑎𝑐𝑐Eq. 5Where 𝑆𝑎𝑐𝑐 is the sensitivity in 𝑚𝑉/𝑔 as found in the Data Sheet and 𝑉𝑎𝑐𝑐 is the voltage from theaccelerometer.Since displacement is the double integral of acceleration, the displacement can be found.𝛿 𝑎 𝑑𝑡Eq. 6

Cone 42.3 – GyroscopeA gyroscope works by converting angular velocity to a voltage. Like Eq. 5, the value for theangular velocity can be calculated as𝜃 𝑆𝑔𝑦𝑟 𝑉𝑔𝑦𝑟Eq. 7Where 𝑆𝑔𝑦𝑟 is the sensitivity in 𝑚𝑉/𝑐𝑦𝑐𝑙𝑒. The value for the angular velocity was thenconverted to 𝑟𝑎𝑑/𝑠𝑒𝑐. Then, the angular displacement is the integral of the angular velocity.Eq. 8𝜃 𝜔 𝑑𝑡Where 𝜃 is the angular displacement and 𝜔 is the angular velocity. Then, using Appendix B fromLab 5: Position Estimator using a Strain Gauge, an Accelerometer, and a Gyroscope, thedisplacement can be written as𝛿𝑚𝑎𝑥 2𝜃𝐿3Eq. 92.4 – Natural Frequency And Damping RatioTo calculate the ringing frequency, the period of the wave was inverted and than multiplied by2𝜋 such as in Eq. 91 2𝜋 (𝑟𝑎𝑑/𝑠)𝑇Where T is the period of the oscillation.𝜔𝑑 Eq. 10To calculate the damping ratio, the logarithmic decrement was used.𝛿′ 1𝑦1ln ( )𝑛 1𝑦𝑛Eq. 11Where 𝑦1 is the height of the first peak, 𝑦1 𝑛 is the height of another peak n peaks away. Thedamping ratio could then be found𝜁 1 1 (2𝜋′ )𝛿2Eq. 12

Cone 5The natural frequency is𝜔𝑛 𝜔𝑑Eq. 13 1 𝜁 23 – EXPERIMENTAL SETUPAn aluminum beam was obtained and mounted to a table using a clamp and the dimensions ofthe beam were recorded. The strain gauge, accelerometer, and gyroscope were mounted. Anexperimental schematic can be found in Figure 2 and the dimensions are located in Table 1Strain GaugeAccelerometerTGyroscopexLFigure 2 - Experimental Setup SchematicTable 1 - Dimensional Qualities for a cantilever beam.ItemVariableValueLengthL9.825 inchesDistance from the table to the strain gaugeX1.5 inchesThicknessT0.0661 inches

Cone 63.1 – Strain GaugeThe Wheatstone bridge was constructed using three 120Ω resistors, the strain gauge, and a10 𝑘Ω potentiometer to balance the bridge. The output of the Wheatstone bridge was then passedthrough a INA105E differential amplifier to remove the noise. The wheat stone bridge waspowered by a constant 8V and the differential amplifier was powered by 8V and -8V. The ideaoutput of the amplifier was a maximum of 2V, so a resistor of 180 Ω was chosen to be the gainFigure 3 - Wheatstone bridge with the differential amplifier.resistor which gave an amplification of 278 according to Eq. 3. An oscilloscope was then used tomeasure the output of the differential amplifier and referenced to ground.Figure 3 is aschematic showing the setup.Once the Wheatstone bridge and differential amplifier were built, the power was turned on andafter approximately 5 minutes, the bridge was balanced using the potentiometer. The fiveminutes was allowed to elapse to allow the resistors to heat up and reach a constant resistance.

Cone 73.2 – Accelerometer & GyroscopeThe accelerometer was attached tothe end of the cantilever beamalong the x-axis. It was then wiredusing the schematic in Figure 4. Apower supply of 8V was passedthrough a 3.3V Voltage Regulatorand the output was run to the VCCpin of the accelerometer. The GNDpin was wired to ground and theoscilloscope was attached to the XFigure 4 - Schematic of the accelerometer &gyroscope wiringpin.The gyroscope was attached to the end of the cantilever beam in the X direction. The 3.3V waswired to the output of the Voltage Regulator, and GND was attached to ground, and the Xx4 wasattached to the Oscilloscope. Later, it was realized that the Oscilloscope had been hooked up tothe four times amplification and the Eq. 8 was changed to𝛿𝑚𝑎𝑥2𝜃𝐿 34Eq. 14to compensate for the amplification.3.3 – Experimental ProcedureAfter the experiment was setup, the beam was depressed by approximately 1 inch and allowed tovibrate. The oscilloscope was set up to capture first ten cycles of the beam. The data was thensaved to a thumb drive in Comma Separated Values (.csv) format. Three runs were conducted forthe purposes of this experiment.

Cone 84 – RESULTS & DISCUISSIONTwo sets of runs were conducted. The first set was conducted in the middle of the table and thesecond set was conducted on the side by a supporting member. Figure 5 is a graph of thedisplacement as recorded by the strain gauge and the accelerometer. It can be seen that the straingauge adequately verpeculiaroutput.When the cantilever beamwasdepressed,thensuddenly released, part ofthe force was transmitted tothetablecausingittooscillate to. The vibrationscaused by this were thentransmittedtotheaccelerometer located at theFigure 5 - The displacement for the Strain Gauge and Accelerometer for a cantileverbeam mounted in the middle of the table.end of the beam. That iswhy the accelerometer data is peculiar looking while the strain gauge looks normal.Theproblem was then rectified by mounting the beam close to a table leg which. This decreased theoscillations in the table. Figure 6 is a plot of the displacement measured by the strain gauge,accelerometer, and the gyroscope. It can be seen that the strain gauge has a very consistentdecrease in amplitude. The accelerometer and the gyroscope though, appear to lag behind theresponse of the strain gauge.For each test, the beam was displaced approximately one inch. This was accomplished byholding a ruler up next to the beam. Thus, it would appear that the strain gauge gives the mostaccurate, initial displacement. For this purpose, it can be argued that the strain gauge is the bestmeasurement of the displacement of the beam.

Cone 9Figure 6 - Displacement of a cantilever beam according to a Strain Gauge, Accelerometer, and Gyroscope.4.1 – Natural Frequency and Damping RatioThe natural frequency and damping ratio was calculated for all three sensors. They werecalculated using plots generated by the m-code and equations 9-12. Sample calculations can befound in Appendix C. Table 2 lists the results for each sensor as an average between two runsTable 2 - Damping Ratio and Natural Frequency for each sensor.Strain GaugeAccelerometerGyroscopeDamping Ratio.0114.0229.00353Natural Frequency (rad/s)92.2192.5492.62It can be seen, that there is a significant discrepancy between the damping ratio for the sensors.This is due to the nature of the data recorded by the accelerometer and the gyroscope. Appendix

C o n e 10D contains graphs of the displacement for the strain gauge, accelerometer, and gyroscope. It canbe seen that the strain gauge has a smooth decrease of the amplitude, while the accelerometerand the gyroscope does not. This is why there is such a discrepancy for the damping ratio for thesensors. It is to be noted, that with the accelerometer and gyroscope mounted at the end of thebeam could have changed the natural frequency and the damping ratio. The natural frequencythough, is reasonable for each sensor.5 – CONCLUSIONThe purpose of the experiment was to analyze the displacement recorded for a cantilever beamusing a strain gauge, accelerometer, and gyroscope. It was found that the strain gauge produced anice, reasonable plot similar to what was expected. The accelerometer and the gyroscope though,produced graphs that did not show the characteristic oscillation expected of underdampedoscillations. Therefore, the conclusion is that the strain gauge is the optimal sensor for thispurpose.