NUMERICAL EVALUATION OF DISPLACEMENT AND ACCELERATION FOR A MASS, SPRING, DASHPOT SYSTEM
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1 American Society for Engineering Education Salt Lake City, Utah June 24 NUMERICAL EVALUATION OF DISPLACEMENT AND ACCELERATION FOR A MASS, SPRING, DASHPOT SYSTEM ACCELEROMETER MASS SPRING FIXED SUPPORT LVDT Dr. Peter Avitabile, Jeffrey Hodgkins University of Massachusetts Lowell 1 Dr. Peter Avitabile, Assistant Professor
2 Mechanical Engineering Laboratory I & II At UMASS Lowell, the laboratory course is taught in a two semester sequence. The first semester concentrates mainly on basic instrumentation, measuring techniques and processing of data. The second semester begins with the same format as the first semester with more complicated labs and ends with a five week project on designing a measurement system (A companion paper describes that project) 2 Dr. Peter Avitabile, Assistant Professor
3 Mechanical Engineering Laboratory I The first semester concentrates mainly on basic measurement tools (oscilloscopes, multimeters, digital data acquisition, etc), measuring devices (flow meters, manometers, pressure transducers, pitot tubes, strain gages, thermocouples, accelerometers, LVDTs, etc) methods for data collection/reduction (regression analysis, curvefitting, numerical processing) 3 Dr. Peter Avitabile, Assistant Professor
4 Mechanical Engineering Laboratory I One laboratory project requires measurements for the displacement and acceleration of a simple mass-spring-dashpot system. Students acquire digital data using an LVDT and accelerometer to obtain this displacement and acceleration data. 4 Dr. Peter Avitabile, Assistant Professor
5 Mechanical Engineering Laboratory I This final project requires extensive use of all material covered over the semester. The project requires significant numerical manipulation of data. Regression Analysis Data Cleansing Integration Differentiation 5 Dr. Peter Avitabile, Assistant Professor
6 Problem The data acquisition system and transducers are intentionally selected such that the majority of possible errors exist in the data Drift Bias Offset Quantization Noise 6 Dr. Peter Avitabile, Assistant Professor
7 Problem This data is integrated and differentiated to compare displacement measurements to acceleration acceleration measurements to displacement The actual numerical processing of the measured data is performed via spreadsheet calculations. I i y + y 2 ( x x ) = I i 1 i i i+ 1 i 7 Dr. Peter Avitabile, Assistant Professor
8 Problem Students struggle with this less than perfect data to emphasize the importance of good measurements. Students re-visit the project at the end of the semester with better instrumentation and understanding of the problem at hand. A final report is then written covering all aspects of test/analysis and any recommendations to improve process. 8 Dr. Peter Avitabile, Assistant Professor
9 Quantization Errors A 12 bit board with a 1 volt range and no AC coupling is used to measure an accelerometer that has approximately 3 mv peak signal.8.6 Typical Quantization Error Dr. Peter Avitabile, Assistant Professor
10 Noise Effects The measured LVDT signal appears to be acceptable until a closer look is made this has a pronounced effect on differentiation 1.8 Displacement Measured from LVDT.6 Disp (in) Time (sec) Displacement Experimental Zoom in on noise 1 Dr. Peter Avitabile, Assistant Professor
11 Differentiation Effects with Noise Differentiation amplifies any noise which confuses the students until they realize the effects. 4 3 Velocity Comparison 2 1 Velocity (in/sec) -1 Differentiated Velocity Integrated Velocity Time (sec) High Frequency Noise on Signal After the second differentiation, acceleration trend is lost under amplified noise. 11 Dr. Peter Avitabile, Assistant Professor
12 Offset Effects The accelerometer signal conditioner has a DC offset that has a dramatic effect upon integrating the signal to obtain displacement.6.4 Raw Accelerom eter Voltage Dr. Peter Avitabile, Assistant Professor
13 Integration Effect with Offset Integration smooths any noise but the trend is confusing if any offsets exist. 1.8 Plot of Displacement for a Spring-Mass System Displ (inch) c Displ. Meas. Displ. Calc. Time (sec) 13 Dr. Peter Avitabile, Assistant Professor
14 Integration Effect with Offset Integration smooths any noise but the trend is confusing if any offsets exist Comparison of Calculated Veclocity 1. Velocity (ft/sec) First Derivative of Displacement First Integration of Acceleration Time (sec) Some realize the velocity is more reasonable 14 Dr. Peter Avitabile, Assistant Professor
15 Student Ownership of Problem The students generally work hard to try to understand the problem and often do very well in/s/s.2.1 Displacement Accelerometer LVDT 4 2 Velocity seconds -.2 in/s -4 Accelerometer LVDT 4-6 Acceleration 3-8 seconds in/s/s -2 Accelerometer seconds LVDT The professor is just a guide or mentor to help direct them in trying to solve this problem. Beware No answer guide for the Professor either!! 15 Dr. Peter Avitabile, Assistant Professor
16 Student Ownership of Problem Many times very novel and unique approaches result in an attempt to solve this problem. 15 Integration results acceleration Velocity Dr. Peter Avitabile, Assistant Professor
17 Student Ownership of Problem Several students have also resorted to building low pass RC filters in Simulink to remove noise simout Filtered LVDT Data 1 Raw Data from LVDT Input Point 12 1/RC 1 s Integrator xdot Output Point Output Input vs. Output si mout1 Filtered Accelerometer Data 2 Raw Data from Accelerometer Input Point1 12 1/RC1 1 s Integrator1 xdot Output Point1 Output2 Input vs. Output1 17 Dr. Peter Avitabile, Assistant Professor
18 Student Ownership of Problem Several students have also resorted to building low pass RC filters in Simulink to remove noise Magnitude (db) Bode Diagram From: Input Point To: Output Point System: proj4_rc_2 I/O: Input Point to Output Point Frequency (rad/sec): 12 Magnitude (db): Phase (deg) Frequency (rad/sec) 18 Dr. Peter Avitabile, Assistant Professor
19 Student Ownership of Problem LVDT noise filtered Dr. Peter Avitabile, Assistant Professor
20 Student Ownership of Problem Accelerometer noise filtered Dr. Peter Avitabile, Assistant Professor
21 Student Ownership of Problem Comparison of integrated data after Simulink noise removal Comparison of LVDT Data to Displacement via Integration Displacement (in) Final Displacement LVDT Raw Data Time (sec) 21 Dr. Peter Avitabile, Assistant Professor
22 Student Ownership of Problem Comparison of integrated data after Simulink noise removal Displacement (in) Displacm ent Com parisons Time (sec) Displacement(LVDT) Displacement(Accleraometer) (Accel) 22 Dr. Peter Avitabile, Assistant Professor
23 Student Ownership of Problem And of course there are students that persist to the point of success!!! Displacement (in) Displacm ent Comparisons Time (sec) -.2 Displacement(LVDT) Displacement(Accleraometer) Displacement (inches) Displacement Comparison -.6 Measured Displacement Calculated Displacement Time (sec) 23 Dr. Peter Avitabile, Assistant Professor
24 1 1 1 ζ=.1% ζ=1% ζ=2% ζ=5% ζ=1% ζ=2% ζ=2% ζ=1% ζ=5% ζ=2% ζ=1% ζ=.1% Acknowledgements This project is partially supported by NSF Engineering Education Division Grant EEC Multi-Semester Interwoven Project for Teaching Basic Core STEM Material Critical for Solving Dynamic Systems Problems FIRST & SECOND ORDER SYSTEMS SIGNAL CONDITIONER RISE & SETTLING TIME FOURIER TRANSFORM DYNAMIC TESTING PULLS ALL THE PIECES TOGETHER!!! DIGITIAL DATA ACQUISITION ANALOG TO DIGITAL HAMMER TIP CHARACTERIZATION DIFFERENT PULSE SHAPES FOURIER SERIES & FFT x(t) k m c f(t) MODEL APPROXIMATION SDOF DYNAMIC SYSTEM MODEL FREE BODY DIAGRAM 2 d x dx & EQUATION OF MOTION m c kx f (t) dt = dt 1 ζωt h(t) = e sin ωdt mω * d a1 a1 h (s) = + * (s p ) (s p ) 1 1 DISPLACEMENT LVDT IMPACT HAMMER FORCE GAGE HAMMER TIP ACCELEROMETER yi 1 + y I i i = Ii ( xi+ 1 xi ) 2 QUANTIZATION SAMPLING T ALIASING I M LEAKAGE E WINDOWS TIME ACCELERATION NUMERICAL PROCESSING INTEGRATION / DIFFERENTIATION Y TRANSDUCER CALIBRATION REGRESSION ANALYSIS X F R E Q U E N C Y FREQUENCY ω/ω n ω/ω n LAPLACE & TRANSFER FUNCTION Peter Avitabile, John White, Stephen Pennell 24 Dr. Peter Avitabile, Assistant Professor
25 American Society for Engineering Education Salt Lake City, Utah June 24 NUMERICAL EVALUATION OF DISPLACEMENT AND ACCELERATION FOR A MASS, SPRING, DASHPOT SYSTEM ACCELEROMETER MASS SPRING FIXED SUPPORT LVDT Dr. Peter Avitabile, Jeffrey Hodgkins University of Massachusetts Lowell 25 Dr. Peter Avitabile, Assistant Professor
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