Finite difference methods for vibration problems

Contents:

• Finite difference methods for vibration problems
• Finite difference discretization
  • A basic model for vibrations
  • A centered finite difference scheme
    • Step 1: Discretizing the domain
    • Step 2: Fulfilling the equation at discrete time points
    • Step 3: Replacing derivatives by finite differences
    • Step 4: Formulating a recursive algorithm
    • Computing the first step
    • The computational algorithm
    • Operator notation
• Implementation (1)
  • Making a solver function
    • Computing \(u^{\prime}\)
  • Verification (1)
    • Manual calculation
    • Testing very simple solutions
    • Checking convergence rates
• Long time simulations
  • Using a moving plot window
  • Making a video
  • Using a line-by-line ascii plotter
  • Empirical analysis of the solution
• Analysis of the numerical scheme
  • Deriving a solution of the numerical scheme
  • Exact discrete solution
  • The global error
  • Stability
  • About the accuracy at the stability limit
• Alternative schemes based on 1st-order equations
• Standard methods for 1st-order ODE systems
  • The Forward Euler scheme
  • The Backward Euler scheme
  • The Crank-Nicolson scheme
  • Comparison of schemes
  • Runge-Kutta methods
  • Analysis of the Forward Euler scheme
• Energy considerations
  • Derivation of the energy expression
    • Energy of the exact solution
  • An error measure based on total energy
• The Euler-Cromer method
  • Forward-backward discretization
  • Equivalence with the scheme for the second-order ODE
  • Implementation (2)
  • The velocity Verlet algorithm
• Generalization: damping, nonlinear spring, and external excitation
  • A centered scheme for linear damping
  • A centered scheme for quadratic damping
  • A forward-backward discretization of the quadratic damping term
  • Implementation (3)
  • Verification (2)
    • Constant solution
    • Linear solution
    • Quadratic solution
  • Visualization
  • User interface
  • The Euler-Cromer scheme for the generalized model
• Exercises and Problems
  • Problem 1: Use linear/quadratic functions for verification
  • Exercise 2: Show linear growth of the phase with time
  • Exercise 3: Improve the accuracy by adjusting the frequency
  • Exercise 4: See if adaptive methods improve the phase error
  • Exercise 5: Use a Taylor polynomial to compute \(u^1\)
  • Exercise 6: Find the minimal resolution of an oscillatory function
  • Exercise 7: Visualize the accuracy of finite differences for a cosine function
  • Exercise 8: Verify convergence rates of the error in energy
  • Exercise 9: Use linear/quadratic functions for verification
  • Exercise 10: Use an exact discrete solution for verification
  • Exercise 11: Use analytical solution for convergence rate tests
  • Exercise 12: Investigate the amplitude errors of many solvers
  • Exercise 13: Minimize memory usage of a vibration solver
  • Exercise 14: Implement the solver via classes
  • Exercise 15: Interpret \([D_tD_t u]^n\) as a forward-backward difference
  • Exercise 16: Use the forward-backward scheme with quadratic damping
  • Exercise 17: Use a backward difference for the damping term
  • Exercise 18: Simulate a bouncing ball
  • Exercise 19: Simulate an elastic pendulum
  • Exercise 20: Analysis of the Euler-Cromer scheme
• References

Indices and tables

• Index
• Module Index
• Search Page