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.
Work with images and videos in Python using
imageio.Get interactive figures using the
%matplotlib notebookcommand.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 \(9.8\rm{m/s}^2\).
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.
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
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.