Project Numina II UNCW UNCW 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

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

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

Project Numina II UNCW UNCW 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

Project Numina II UNCW UNCW The Cantilever Beam E = Modulus of Elasticity I = Moment of Inertia  = Mass density A = Cross-sectional area L = Length of Beam k = damping Boundary Conditions Initial Conditions

Project Numina II UNCW UNCW NC State Modeling Course  CRSC  Math lab at Centennial Campus  NCSU - Experimental Setup  Main NCSU DAQ Instrumentation  Beam Data  Modeling results

Project Numina II UNCW UNCW 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 Dr. H. T. Banks and Dr. H. T. Tran  Course: Math 573 Level: SMETE Upper division and graduate students Level: SMETE Upper division and graduate students Math Background: ODE’s, Linear Algebra. Math Background: ODE’s, Linear Algebra. Sample Labs Sample Labs  Vibration of Beams  Heat Conduction  Reflection of Acoustic Waves

Project Numina II UNCW UNCW NCSU – Lab at Centennial Campus Beam Lab Heat Conduction Lab

Project Numina II UNCW UNCW Lab at Centennial Campus (Cont.) Acoustic Waves Lab Group Project

Project Numina II UNCW UNCW NCSU - Experimental Setup

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

Project Numina II UNCW UNCW Beam Data  Collect data at one point. PDE  ODE  Consider data after Force has been turned off  Simplified model.

Project Numina II UNCW UNCW Modeling Results  Data Array:  Model Solution:  Cost Function:  Minimize cost function. Matlab: Fmins Matlab: Fmins Mathcad: Minerr Mathcad: Minerr

Project Numina II UNCW UNCW Modeling Results (Cont) Matlab Mcad

Project Numina II UNCW UNCW UNCW Model  Probeware History at UNCW  Course Preparation  Experimental Setup  Experimental Results

Project Numina II UNCW UNCW Probeware History at UNCW  IBM PSL  MBL Explorer  Vernier  Data Harvest Typical Labs =>

Project Numina II UNCW UNCW Newton’s Law of Cooling

Project Numina II UNCW UNCW Uniform Acceleration

Project Numina II UNCW UNCW Simple Harmonic Motion Spring Spring MBL Interface MBL Interface 750g Mass 750g Mass Sonar Probe Sonar Probe

Project Numina II UNCW UNCW Course Preparation  Introduction to Harmonic Oscillators  Pre-project Work Verify form of solutions Verify form of solutions Graphing Typical Solutions Graphing Typical Solutions Fitting Simulated Solutions Fitting Simulated Solutions  Few hints about difference between mass-spring and beam systems

Project Numina II UNCW UNCW 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.Nonlinear Least Squares Curve Fitter 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. Project Description

Project Numina II UNCW UNCW Pre-Project Work

Project Numina II UNCW UNCW Experimental Setup  Handheld Computers  Data Harvest DAQ  Simple Spring and Meter Stick

Project Numina II UNCW UNCW  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 HP Jornada 720 Handheld Computers

Project Numina II UNCW UNCW  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 HP Jornada 568 Pocket PC

Project Numina II UNCW UNCW  12-bit DATAQ system  Numerous probes  Serial interface or CF+ unit  Software runs on HPC, PPC, and desktop computers Data Harvest DATAQ System

Project Numina II UNCW UNCW UNCW - Experimental Setup

Project Numina II UNCW UNCW Experimental Results

Project Numina II UNCW UNCW Modeling Results  Nonlinear Least Squares Curve Fitter was unwieldy  All data taken at one time  Fits done in Excel and some in Maple.  Fits done “by hand” – more pedagogical

Project Numina II UNCW UNCW Modeling Results - Maple

Project Numina II UNCW UNCW Modeling Results - Excel

Project Numina II UNCW UNCW Nonlinear Fit - Maple

Project Numina II UNCW UNCW Data Analysis Functions  Nonlinear Regression  Matlab (bar_lsq2.m) bar_lsq2.m Load – Reads files Load – Reads files Ode23 – Solver (RK) Ode23 – Solver (RK) Fmins – Simplex method Fmins – Simplex method  Mathcad ( Beam1_model ) Beam1_model Readprn – Loads files Readprn – Loads files Odesolve – (RK) Odesolve – (RK) Minerr – Simplex Method Minerr – Simplex Method  Maple (SHOData) SHOData Readdata – Reads files Readdata – Reads files Dsolve Dsolve LeastSquares LeastSquares  Linear vs  Nonlinear (NLFit2.mws) NLFit2.mws  Excel (Beam1_d11a, Harmonic.xls) Beam1_d11aHarmonic.xlsBeam1_d11aHarmonic.xls Fit manually Fit manually  Nonlinear Least Squares Fitter

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

Project Numina II UNCW UNCW Summary  The Problem  NC State Model  UNCW Model  Data Analysis  Comparison of Models

Project Numina II UNCW UNCW Thank you! Russ Herman, Gabriel G. Lugo,