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Analysis of Rocket Flights Section 4, Team 4 Student 1, Student 2, Student 3, Student 4
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Temperature Predictions Decrease in temperature upon ascent Rise in temperature upon descent Further increase after landing
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Thermistor 3 (middle of rocket body, on surface)
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Thermistor 2 (on fin)
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Altimeters Expect decrease in pressure with increase in altitude, and vice versa Used barometric equation to find altitude Calibrated sensors in lab using vacuum chamber
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Altimeter vs. Models for Flight
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Flight Modeling (2-D) CGCG CPCP mg D T y z r Wind T CGCG CPCP mg r Wind D y z θ
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Euler’s Integration Method for numerical integration Iterative For given a(t) and initial conditions for x and v: v(t+Δt)=v(t)+a(t)*t x(t+Δt)=x(t)+v(t)*t
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IMU Analysis: Mudd IIIC (Large) Rocket Rotation from local to global axes Euler integration of rotation matrix azaz axax ayay AyAy AzAz AxAx y azaz ayay axax
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Processing Algorithm (Matlab) Raw RDAS Data (counts) Local Acceleration (m/s 2 ) Calibrations Global Acceleration (m/s 2 ) Global Velocity (m/s) and Position (m) Local Rotation Rate (radians/sec) Rotation Matrix (radians/sec) Calibrations Euler integration Local Rotation Angle (degrees) Filtered Global Acceleration (m/s 2 ) Acceleration Filtering (optional)
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Principle Axis Rotation: Plot vs. Video
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1-D Model Comparison (launch 2) day 1 2 nd launch (13-20 mph winds) IMU dataR-DAS pressure altimeter % difference from altimeter Student model % difference from model Rocksim model % difference from model Apogee height 160.3390 meters 154.53 m+3.759%147.3- 182.3 m (lies within range) 165.3 m-3.001% Apogee time 6.38 sec4.645- 6.145 sec +3.824%5.763- 6.293 sec +1.382%5.888 sec+8.356% Max z vel 54.35 m/sN/A 52.46- 58.88 m/s (lies within range) 57.24 m/s-5.049% Max z accel 206.4 m/s 2 N/A 180.8- 198.4 m/s 2 +4.032%202.5 m/s 2 +1.926%
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Acceleration Filtering (Before)
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Acceleration Filtering (After)
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Acceleration Filtering (Descent Plot) E80 teams wind z
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Bad Data Mudd IIIA IMU rocket Failure to eject parachute Flat spin crash after apogee WHY? http://www.tribuneindia.com/2002/20020715/world.htm
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Principle Axis Rotation Plot vs. Video, Round 2
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Acceleration Data… Very pronounced 0.2149 Hz oscillations Possible causes: camera interference, camera overpowering Band-stop filter might be able to retrieve data
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Vibration Analysis
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Tap tests on hollow tube are inaccurate Mass spring damper system Theoretical Analysis
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Spring-Mass-Damper Model Rocket can be modeled as a single degree of freedom spring- mass-damper system. Effective mass, m Spring constant, k Damping Half-Power Bandwidth Predicted Resonance Frequency
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Analysis No control variables! Treat Sensor 10 as input. Create FRFs of other sensors to see relative peaks Sensor 10
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FRF plots Removed DC offset fdomain.m used to generate Fourier Coefficients Relative Amplitudes First set of data is not trustworthy Second set of data has more coherent peaks Used 1 st second of data, short motor burn time
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1 st Set of Data Results Peak around 60 or 70 Hz Other peaks are inconsistent Sensor 15 seems to be malfunctioning Locally, 3 sensors show local peaks between 60-80 No video
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2 nd Set of Data Results Consistent peaks at 64 Hz Possible peaks around 30 Hz, but not consistent Sensors 1, 3, and 8 are 13 show peak frequencies Sensor 13 farther away from the input source
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Noises Only 64 Hz showed in every FRF Others are jumbled by the noise Running averages smoothes out the data too much. Too little data during the 1 st second of input Ineffective way of removing noise
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Mode Shapes Absolute magnitude of Fourier Coefficients vs Relative Sensor Distances Sensor 10 was normalized as “0” point.
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Results from FRF Not enough frequencies to test all 3 mode shapes Does not deal well with noise, especially with highly aliased data
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Problems with FFT Using just FFT coefficients to calculate Frequency Response Functions assumes a clean periodic signal. The rocket data is neither. A better estimator is Power Spectral Density (PSD).
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Power Spectral Density Auto power spectral density Cross power spectral density Frequency Response Function
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PSD and Noise H(j x(t)v(t)y(t) n(t) Assume n(t) is unrelated to v(t) 0
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Hamming Window Time DomainFrequency Domain
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Averaging Overlap Overlapping windowed segments by 50% minimizes attenuation of time domain signal near the end of segment
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Frequency Response Function
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Waterfall Analysis freq (Hz) time (.1 sec) magnitude (dB) FRF of Sensor 5 over time
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Conclusions Thermistor behavior depends on location Euler Integration Method not sufficient to model whole flight path Spring-Mass-Damper model can simplify system to find theoretical resonance FFT method of finding FRF is not consistent due to large noise component PSD method gives much sharper peaks in FRF
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Interesting Precautions... Check battery…sensors are sensitive! Wait until last moment to turn on R- DAS and video camera…otherwise, ejection charge could go off early! Don’t try to catch rocket…it may have a chute, but it’s still falling fast!
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Extra: Altimeter Plots
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Extra: Why We Didn’t Do 2-D Model Comparison
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Acknowledgements The professors and proctors who helped to make this beta-test a success. All of our classmates for their infinite support and advice during this semester Student A for a discussion on the causes of small rocket IMU corruption Student B for his help with setting up the Single Degree of Freedom model
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References E80 The Next Generation Spring 2008, http://www.eng.hmc.edu/New E80/index.html. R. Wang, http://fourier.eng.hmc.edu/e80/inertialnavigation/ Q. Yang, http://www.eng.hmc.edu/NewE80/PDFs/Lecture_PressureSensor Thermistors.ppt H. Buchholdt, Structural Dynamics for Engineers (Telford, 1997), pp. 17-22. The Hanning Window, http://www.dliengineering.com/vibman/thehanning window.htm
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