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.

rcc02007/CompMech03-IVPs

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