Mathematics behind System Dynamics An Interactive Qualifying Project Report Submitted to the Faculty Of WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree of Bachelor of Science By ___________________________________ ___________________________________ Thanacha Choopojcharoen Ali Magzari Date: May 23, 2012 Advisor: ___________________________________ Professor Khalid Saeed Abstract The goal of this project is to introduce modeling and representation methods to solve dynamics problems. The intuitive System Dynamics representation is introduced and backed up with advanced mathematical concepts such as differential equations and Control theory techniques. This project attempts to illustrate both abstract and intuitive approaches based on examples arising in social and business systems. The topics include graphical representations, delays, numerical methods, and common behavioral pattern of typical structures. Table of Contents Abstract ...................................................................................................... 2 Preface ....................................................................................................... 5 Organization ..................................................................................................................................5 Motivation behind the Work .........................................................................................................5 Acknowledgements .......................................................................................................................5 Chapter 1: An Introduction to Study ......................................................... 6 Chapter 2: An Introduction to System Dynamics ..................................... 7 Motivation .....................................................................................................................................7 Section 2.1: Definition of System Dynamics ................................................................................7 Section 2.2: Why do we use models? ...........................................................................................7 Section 2.3: Information Links and Causal Loop Diagrams .........................................................8 Section 2.4: Stocks and Flows ......................................................................................................9 Summary .....................................................................................................................................10 Chapter 3: Graphical Representation of System Behavior ..................... 11 Prerequisites ................................................................................................................................11 Motivation ...................................................................................................................................11 Section 3.1: Cartesian graph ......................................................................................................11 Section 3.2: Shape of graph & & first-order and second-order derivative ................................14 Section 3.3: Critical point, Critical value, second-derivative test ..............................................17 Section 3.4: Graphical functions for empirical analysis ............................................................18 Summary .....................................................................................................................................19 Chapter 4: Delays and Relationship to Integration ................................. 20 Prerequisites ................................................................................................................................20 Motivation ...................................................................................................................................20 Section 4.1: First-order and Higher-order Material delay .........................................................20 Section 4.2: Smoothing and information delays: .......................................................................22 Summary .....................................................................................................................................27 Chapter 5: Numerical methods for Solution ........................................... 28 Prerequisites ................................................................................................................................28 Motivation ...................................................................................................................................28 Section 5.1: Euler’s Method ......................................................................................................28 Section5.2: Second-order Runge-Kutta Method ........................................................................31 Summary .....................................................................................................................................33 Chapter 6: Common Behavioral Patterns and their Underlying Structures ........ 34 Prerequisites ................................................................................................................................34 Motivation ...................................................................................................................................34 Section 6.1: Exponential Growth ...............................................................................................35 Detailed Transient Analysis ....................................................................................................... 37 Transfer Function and Frequency Analysis ............................................................................... 40 Section 6.2: Goal Seeking ..........................................................................................................42 More Analysis ............................................................................................................................ 48 Section 6.3: S-Shaped ................................................................................................................50 Section 6.4: Oscillation ..............................................................................................................56 Section 6.5: Overdamped Oscillation ........................................................................................57 Section 6.6: Underdamped Oscillation ......................................................................................59 Section 6.7: Undamped Oscillation ...........................................................................................61 Summary .....................................................................................................................................64 Chapter 7: Conclusion and Future Work ................................................ 65 Appendices .............................................................................................. 66 Appendix A: Differentiation .......................................................................................................66 Average versus Instantaneous Rate of Change .......................................................................... 66 Mathematical Definition of Derivative ...................................................................................... 69 Second and Higher Order Derivative ......................................................................................... 70 Appendix B: Integration .............................................................................................................71 Mathematical definition of integration ...................................................................................... 71 Anti-derivative ........................................................................................................................... 72 Evaluation of an integral ............................................................................................................ 73 Appendix C: Ordinary Differential Equation .............................................................................75 Linearity, and Homogeneity ...................................................................................................... 75 Solutions of Linear Differential Equations ................................................................................ 77 First-order Differential Equation solving technique .................................................................. 79 Second-order Linear Differential Equation solving technique .................................................. 82 Method of Undetermined Coefficients ...................................................................................... 87 Appendix D: The Laplace Transform .........................................................................................94 Inverse Laplace Transform ........................................................................................................ 97 Application of Laplace transformation to Initial-Value Problems .......................................... 100 Bibliography .......................................................................................... 102 Preface In this report, Mathematics behind System Dynamics, we present selected mathematical concepts helpful to understand System Dynamics modeling practice. Selected