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 $9.8{\mathrm{m}/\mathrm{s}}^2$.

• 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 $9.8{\mathrm{m}/\mathrm{s}}^2$.

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