Music 420 Lecture
Elementary Finite Different Schemes

Julius O. Smith III (jos at ccrma)
Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University

Detailed Contents (and Navigation)

• Two Approaches to Physical Modeling
• Ordinary Differential Equations
  • Equivalent Circuit for a Force-Driven Mass
  • Reference Directions in Equivalent Circuits
  
  
• ODE for a Mass Sliding with Friction
  • Force-Driven Mass with Friction
    Diagram and Equivalent Circuit
  • Mass-Spring ODE
  • Physical diagram:
  • Mass-Spring-Wall System
  • Equivalent Circuit for Mass-Spring-Wall
  • Mass-Spring-Dashpot ODE
  • Physical diagram:
  
  
• Difference Equations
  (Finite Difference Schemes)
  • Backward Euler Finite-Difference Equation for a Force-Driven Mass
  • Accuracy of Backward Euler
  • Summary of Backward Euler
  • Delay-Free Loops
  • Forward-Euler (FE)
  • Centered Finite Difference
  • Trapezoidal Rule for Numerical Integration
  • Filter Design Approach
  • Ideal Differentiator Frequency Response
  • Explicit and Implicit Finite Difference Schemes
  • Semi-Implicit Finite Difference Schemes
  
  
• ODE Laplace Transform Analysis
  • Bilinear Transform
  • Properties of the Bilinear Transform
  • Hybrid Euler-Bilinear Mapping
  • Example: Leaky Integrator
  • Accuracy of Trapezoidal Rule
  • Backward Difference Conformal Map
  • Laplace Analysis of Trapezoidal Rule
  • Trapezoidal Rule Frequency Mapping
  • Summary of Backward Euler vs. Trapezoidal Rule
  • Why Don't We Always Use the Bilinear Transform?
  
  
• Physical Model Formulations
• State-Space Models
  • Digitizing State Space Models (Simplistically)
  • Recommended Related Courses at Stanford
  
  
• About this document ...