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Engineering Computations Module 3: Fly at change in systems

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posted on 05.12.2017, 23:28 by Lorena A. BarbaLorena A. Barba, Natalia C. ClementiNatalia C. Clementi
Engineering Computations
—Original material written as Jupyter Notebooks for an undergraduate engineering course, Fall 2017

Module 3: Fly at change in systems

After the two foundation course modules, equipping you with the coding patterns for numerical computing, this module is our launching pad to investigate change, motion, and dynamics, using computational thinking.

Lesson 1: Catch things in motion

Working with images and videos in Python using imageio. Interactive Matplotlib figures in the notebook, and capturing mouse clicks on images for digitizing an object's position. Computing velocity and acceleration from position captures: a falling ball, and projectile motion. Computing numerical derivatives using differences. Free-fall acceleration from real data.

Lesson 2: Step to the future

Computing velocity and position from accelerometer data: a roller-coaster ride. Using the subplot() function go draw more than one plot in the same figure. Euler's method for initial-value problems, and Taylor expansion showing first-order accuracy. The second-order differential model for an object in free fall written as two first-order differential equations, leading to a vector form. General design of a code to solve ordinary differential equations (ODEs). Application to free fall of a tennis ball and comparison with experimental data. Improved model accounting for air resistance.

Lesson 3: Get with the oscillations

Differential model of a spring-mass system without friction: state vector and system in vector form. Amplitude growth with Euler's method on oscillatory systems, and the fix: Euler-Cromer method (semi-implicit Euler). Numerically observed order of accuracy using a convergence plot: numerical error with different time increments, ∆t. Modified Euler's method, and observed order of accuracy.

Lesson 4: Bird's-eye view of mechanical vibrations

General spring-mass systems with damping and a driving force, revealing a variety of behaviors. Presents a powerful new method to study dynamical systems based on visualizing direction fields and trajectories in the phase plane.

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Note—If you have suggestions for changes or improvements to this material, please open an issue on the GitHub repository.

Funding

NSF Award OAC #1730170

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