formulation of fundamental laws ❧ Algebraic ❧ ODE ❧ PDE q y T x T dt x d m dt dv m F E ; ma F = ∂ ∂ + ∂ ∂ = = = = 2 2 2 2 2 2 ε σ X
Mathematical Models Mathematical Models ❧ Modeling is the development of a mathematical representation of a physical/biological/chemical/ economic/etc. system ❧ Putting our understanding of a system into math ❧ Problem Solving Tools : Analytic solutions, statistics, numerical methods, graphics, etc. ❧ Numerical methods are one means by which mathematical models are solved ❧ Computer Mathematics X
Mathematical Models Mathematical Models ❧ It describes a natural process or system in mathematical terms ❧ It represents an idealization and simplification of reality ❧ The model yields reproducible results for predictive purposes ❧ What fundamental laws we use in modeling? ❧ Conservation of mass ❧ Conservation of linear/angular momentum ❧ Conservation of energy ❧ Conservation of charge, etc. X
Bungee Jumper Bungee Jumper You are asked to predict the velocity of a bungee jumper as a function of time during the free-fall part of the jump Use the information to determine the length and required strength of the bungee cord for jumpers of different mass The same analysis can be applied to a falling parachutist or a rain drop
Newton’s Second Law F = ma = F down - F up = mg - c d v 2 (gravity minus air resistance) Observations / Experiments Where does mg come from? Where does -c d v 2 come from? Bungee Jumper / Falling Parachutist Bungee Jumper / Falling Parachutist
Now we have fundamental physical laws, so we combine those with observations to model the system A lot of what you will do is “ canned ” but need to know how to make use of observations How have computers changed problem solving in engineering? Allow us to focus more on the correct description of the problem at hand, rather than worrying about how to solve it.
Newton’s Second Law Exact (Analytic) Solution Exact (Analytic) Solution 2 d 2 d v m c g dt dv v c mg dt dv m - = - = Exact Solution = t m gc c mg t v d d tanh ) (
Numerical Method Numerical Method i 1 i i 1 i 0 t
You've reached the end of your free preview.
Want to read all 29 pages?
- Fall '10
- Numerical Analysis, bungee jumper