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Inverse Problems for Vibrating Beams ICTCM – 2002 Orlando, FL.

Inverse Problems for Vibrating Beams ICTCM – 2002 Orlando, FL. Russ Herman, Mathematics and Statistics Gabriel G. Lugo, Mathematics and Statistics University of North Carolina at Wilmington. Outline of Presentation. Mathematical Modeling Pedagogical Gains

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Inverse Problems for Vibrating Beams ICTCM – 2002 Orlando, FL.

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  1. Inverse Problems for Vibrating BeamsICTCM – 2002 Orlando, FL. Russ Herman, Mathematics and Statistics Gabriel G. Lugo, Mathematics and Statistics University of North Carolina at Wilmington

  2. Outline of Presentation • Mathematical Modeling • Pedagogical Gains • The Cantilever Beam • NC State Modeling Course • Centennial Campus Lab • Experimental Setup • Modeling Results • UNCW Project • Course Preparation • Experimental Setup • Experimental Results • Data Analysis • Summary

  3. Mathematical Modeling • Objective • To develop a quantitative description of a physical problem. • Inverse Problem • Make observations and Acquire Data. • Develop equations from basic principles. • Make assumptions to simplify equations. • Solve the equations and run simulations. • Test how well the model fits the data. • Revise model if necessary.

  4. Pedagogical Gains • Students learn how to apply mathematical concepts to solve real problems. • Solutions of differential equations is more meaningful when students collect their own data. • Group interaction. • Inter-departmental applications. • Practice in writing reports

  5. Boundary Conditions Initial Conditions The Cantilever Beam E = Modulus of Elasticity I = Moment of Inertia r = Mass density A = Cross-sectional area L = Length of Beam k = damping

  6. NC State Modeling Course CRSC Math lab at Centennial Campus NCSU - Experimental Setup Main NCSU DAQ Instrumentation Beam Data Modeling results

  7. NC State Modeling Course • Center for Research and Scientific Computation. Director: Dr. H. T. Banks • Math Instructional and Research Lab • Dr. H. T. Banks and Dr. H. T. Tran • Course: Math 573 • Level: SMETE Upper division and graduate students • Math Background: ODE’s, Linear Algebra. • Sample Labs • Vibration of Beams • Heat Conduction • Reflection of Acoustic Waves

  8. NCSU – Lab at Centennial Campus Heat Conduction Lab Beam Lab

  9. Lab at Centennial Campus (Cont.) Acoustic Waves Lab Group Project

  10. NCSU - Experimental Setup

  11. Main NCSU DAQ Instrumentation • HP Dynamic spectrum Analyzer • Cost prohibitive • Piezoceramic Actuator • Hard to install • Great to control Force • Accelerometer • Fast response. Proximity meter would be better • Electronic Impulse Hammer • A hammer really worth $800

  12. Beam Data • Collect data at one point. PDE  ODE • Consider data after Force has been turned off • Simplified model.

  13. Modeling Results • Data Array: • Model Solution: • Cost Function: • Minimize cost function. • Matlab: Fmins • Mathcad: Minerr

  14. Modeling Results (Cont) Matlab Mcad

  15. UNCW Model • Probeware History at UNCW • Course Preparation • Experimental Setup • Experimental Results

  16. Probeware History at UNCW • IBM PSL • MBL Explorer • Vernier • Data Harvest Typical Labs =>

  17. Newton’s Law of Cooling

  18. Uniform Acceleration

  19. Simple Harmonic Motion • Spring • MBL Interface • 750g Mass • Sonar Probe

  20. Course Preparation • Introduction to Harmonic Oscillators • Pre-project Work • Verify form of solutions • Graphing Typical Solutions • Fitting Simulated Solutions • Few hints about difference between mass-spring and beam systems

  21. Project Description Differential Equations Project Part 1: Analysis of the Damped Harmonic Oscillator and Simulated Data In this part you will explore the behavior of solutions to the damped harmonic oscillator by answering questions given on the Part 1 handout. You will also test the Nonlinear Least Squares Curve Fitter listed at the course Links page. This will prepare you for doing the other two parts with real data. Part 2:Data Collection and Analysis for Spring-Mass Oscillations In this part of the lab you will collect data with the Data Acquisition Equipment for at least three different mass-spring combinations. Using the techniques from Part 1, you will determine the system parameters from your data (b/m and k/m) and the frequency of oscillation. Part of your report should describe your setup and any relevant observations you made during the experiment. You should provide plots of the data, the fit based upon the parameters you determined and a discussion of your results. Part 3:Data Collection and Analysis for Beam Oscillations In this last part you will study the behavior of a vibrating beam, namely a meter stick. The meter stick will be clamped at several points and data taken for the oscillation of a point on the beam. The data will be analyzed similar to that of the system in Part 2 and a similar report written. Note differences and similarities between the systems and support any differences with data analysis.

  22. Pre-Project Work

  23. Experimental Setup • Handheld Computers • Data Harvest DAQ • Simple Spring and Meter Stick

  24. HP Jornada 720 Handheld Computers • StrongArm CPU (206 MHz) • 32 MB RAM • 640 x 240 Color Display • Compact Flash Type I & PC Card Type II Slots • RS232C Serial & IrDA ports, 115Kbps • 56k Modem • 9-Hour Li-ion Battery • MS HPC2000 v. 3 OS

  25. HP Jornada 568 Pocket PC • Intel StrongArm CPU (206 MHz) • 64 MB RAM • 240 x 360 color display • Compact Flash slot • 14-hour Li-ion battery • Pocket PC 2002 OS • MS Pocket Office suite • Internet Explorer

  26. Data Harvest DATAQ System • 12-bit DATAQ system • Numerous probes • Serial interface or CF+ unit • Software runs on HPC, PPC, and desktop computers

  27. UNCW - Experimental Setup

  28. Experimental Results

  29. Modeling Results • Nonlinear Least Squares Curve Fitterwas unwieldy • http://members.aol.com/johnp71/nonlin.html • All data taken at one time • Fits done in Excel and some in Maple. • Fits done “by hand” – more pedagogical

  30. Modeling Results - Maple

  31. Modeling Results - Excel

  32. Nonlinear Fit - Maple

  33. Data Analysis Functions • Nonlinear Regression • Matlab (bar_lsq2.m) • Load – Reads files • Ode23 – Solver (RK) • Fmins – Simplex method • Mathcad (Beam1_model) • Readprn – Loads files • Odesolve – (RK) • Minerr – Simplex Method • Maple (SHOData) • Readdata – Reads files • Dsolve • LeastSquares • Linear vs • Nonlinear (NLFit2.mws) • Excel (Beam1_d11a, Harmonic.xls) • Fit manually • Nonlinear Least Squares Fitter

  34. Comparison of Projects • Student Demographics • NCSU - Senior/Grads • UNCW – First course in ODE’s • Common Problems • Students not comfortable with CAS and data. • Weak backgrounds in Physics • Common Solutions • Working in groups • Help files on Web • Project Differences • UNCW – Rougher experiment, portable DAQ, classroom only once.

  35. Summary • The Problem • NC State Model • UNCW Model • Data Analysis • Comparison of Models

  36. http://aa.uncwil.edu/numina Thank you! Russ Herman, hermanr@uncw.edu Gabriel G. Lugo, lugo@uncw.edu

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