Computational Mechanics 5 - Boundary Value Problems

Computational Mechanics 5 - Boundary Value Problems#

01_Revisiting_derivatives#

  • How to approximate error in approximate derivatives

  • How to approximate derivatives using forward difference methods

  • How to approximate derivatives using backward difference methods

  • How to approximate derivatives using central difference methods

  • How to approximate higher order derivatives with forward, backward, and

  • central differences

02_Keep_it_steady#

  • The difference between a PDE and an ODE

  • How to approximate differential equations with boundary conditions

  • Solve steady-state heat transfer problem

  • Solve static deflection of elastic beam

  • Demonstrate convergence of finite difference solutions

03_Good_Vibrations#

  • How to set up and solve an eigenvalue problem

  • How eigenvalues and eigenvectors relate to natural frequencies and mode shapes

  • How to turn a PDE into a finite difference ODE

  • How to use eigenvalues to calculate natural frequencies of a vibrating string

  • How to solve a coupled set of ODEs

  • Visualize solutions to finite difference equations with 3D plots and

  • animations

  • How to listen to vibration solutions

HW_05#

Project 05 - String vibrations of guitars#

6-string guitar diagram

In this final project, you will consider all six strings of a guitar and the deflection of the neck of the guitar. Here are the inputs for each of the strings, all L=0.64 m:

string

density (g/m)

tension (kg)

E

0.401

7.28

B

0.708

7.22

G

1.140

7.32

D

2.333

8.41

A

4.466

9.03

E

6.790

7.71