Comparing Data from Mathematical Models and Data from Real Experiments

Transcription

1 Comparing Data from Mathematical Models and Data from Real Experiments Building Models with VENSIM Hildegard Urban-Woldron Ogólnopolska konferencja, 28 th of October, 2011, Warsaw, Poland

2 Overview Introduction to VENSIM Building a Model with VENSIM Equations & Initial Values Running the Simulation Graphs Comparing with Data from Real Experiments

3 Example: Mass on a spring Data-Logging Simulation Modelling

4 From learning by using models to learning by making models

5 What is VENSIM? Software VENSIM (Ventana Simulation Environment) Available freely from PLE (Personal Learning Edition) free Integrated framework for o Conceptualizing, o Building, o Simulating, o Analyzing, o Optimizing and o Deploying models of dynamic systems Use of a workbench-toolbox design See the results of simulations superimposed on the model diagrams

6 The Window of VENSIM Sketch Tools Analysis Tools Sketch Drawing Area

7 Building a Model with VENSIM: Mass on a Spring Basic building blocks Variable or Converter Box Variable or Stock Rate of Flow Arrow or Connector

8 Mathematical Model F m a k y ay y k m y Δv v (t + Δt) = v (t) + a y a y = y y y Δt Δy v= y(t + Δt) = y(t) + vy Δt y Δt Δt

9 Defining the variable displacement Click on the rectangular stock icon Position the cursor in the drawing area and Click again Type the stock name in the box and press the Enter key At any instant, the magnitudes of the stock gives a snapshot of the system.

10 Adding v as a rate flow icon Click on the flow icon. Then click to the left of the stock. Without dragging, click the stock. Type the name in the rectangle and press Enter key

11 Adding the variables velocity and a

12 Defining auxiliary variables for mass and spring constant

13 Drawing information arrows ICT for Innovative Science Teachers

14 Entering equations and initial values

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20 Running the simulation ICT for Innovative Science Teachers

21 Time step 0,03125 = 1/32 s, Euler method Time period is ok. But, altogether the result is not satisfying!

22 Time step 0, = 1/128 s, Euler method Time period is ok. But, altogether the result is still not satisfying!

23 Time step 0,03125 = 1/32 s, Runge-Kutta-4 method

24 Studying the behaviour of the system Mass on a spring Changing the mass

25 Studying the behaviour of the system Mass on a spring Changing the spring constant

26 Defining Graphs

27 Defining Graphs

28 Defining Graphs

29 Defining Graphs

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31 Data-logging Simulations Modelling Data-logging automates the process of measurement and data gathering tedious repetitive tasks are removed stronger focus on scientific inquiry Virtual labs and interactive simulations can be powerful tools for learning physics concepts System dynamics tools allow an intuitive approach to the modelling of dynamical systems

32 Comparing experimental data to the sinusoidal function model y t A sin 2 f t What do these parameters represent in the experiment?

33 Further questions max d 2 min d 0.57 Where is the mass when the velocity is zero? Where is the mass when the velocity is greatest? position vs. time- & velocity vs. time graph How are the two graphs the same? How are they different?

34 More investigations Phase diagram (velocity vs. position) acceleration vs. position

35 Getting an intuitive feeling for how oscillators behave How to use the applet? Assign a value for the mass Assign a value for the spring constant Select the amplitude of the motion by grabbing the mass with the mouse

36 How to promote the shift from playing to learning? Teacher decisions Alignment with Curriculum & Learning Goals? What are fitting instructional technologies?

37 Exploring by changing parameters Choose m = 0,125 kg spring constant = 300 N/m and damping factor = 0 Change only the value for the mass Explore Can you find a relationship between mass m and period T?

38 Thank you for your attention!