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 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 \(9.8\rm{m/s}^2\).
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.
HW_03#
Project_03 - Engineering Design of rocket flights#
You are going to end this module with a bang by looking at the flight path of a firework. Shown above is the initial condition of a firework, the Freedom Flyer in (a), its final height where it detonates in (b), the applied forces in the Free Body Diagram (FBD) in ©, and the momentum of the firework \(m\mathbf{v}\) and the propellent \(dm \mathbf{u}\) in (d).
The resulting equation of motion is that the acceleration is proportional to the speed of the propellent and the mass rate change \(\frac{dm}{dt}\) as such