Computational Mechanics 03- Initial Value Problems

Learning to frame engineering equations as numerical methods

Welcome to Computational Mechanics Module #3! In this module we will explore some more data analysis, find better ways to solve differential equations, and learn how to solve engineering problems with Python.

01_Catch_Motion

• Work with images and videos in Python using imageio.
• Get interactive figures using the %matplotlib notebook command.
• Capture mouse clicks with Matplotlib's mpl_connect().
• Observed acceleration of falling bodies is less than $9.8\rm{m/s}^2$.
• Capture mouse clicks on several video frames using widgets!
• Projectile motion is like falling under gravity, plus a horizontal velocity.
• Save our hard work as a numpy .npz file Check the Problems for loading it back into your session
• Compute numerical derivatives using differences via array slicing.
• Real data shows free-fall acceleration decreases in magnitude from $9.8\rm{m/s}^2$.

02_Step_Future

• Integrating an equation of motion numerically.
• Drawing multiple plots in one figure,
• Solving initial-value problems numerically
• Using Euler's method.
• Euler's method is a first-order method.
• Freefall with air resistance is a more realistic model.

03_Get_Oscillations

• vector form of the spring-mass differential equation
• Euler's method produces unphysical amplitude growth in oscillatory systems
• the Euler-Cromer method fixes the amplitude growth (while still being first
• order)
• Euler-Cromer does show a phase lag after a long simulation
• a convergence plot confirms the first-order accuracy of Euler's method
• a convergence plot shows that modified Euler's method, using the derivatives
• evaluated at the midpoint of the time interval, is a second-order method
• How to create an implicit integration method
• The difference between implicit and explicit integration
• The difference between stable and unstable methods

04_Getting_to_the_root

• How to find the 0 of a function, aka root-finding
• The difference between a bracketing and an open methods for finding roots
• Two bracketing methods: incremental search and bisection methods
• Two open methods: Newton-Raphson and modified secant methods
• How to measure relative error
• How to compare root-finding methods
• How to frame an engineering problem as a root-finding problem
• Solve an initial value problem with missing initial conditions (the shooting
• method)
• Bonus: In the Problems you'll consider stability of bracketing and open methods.