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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.

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 (c), 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

$$\begin{equation} m\frac{dv}{dt} = u\frac{dm}{dt} -mg - cv^2.~~~~~~~~(1) \end{equation}$$