NM3_root_s02 - CGN 3421 - Computer Methods Gurley Numerical...

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CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 3 Nonlinear Equations and Root Finding Methods page 68 of 82 Numerical Methods Lecture 3 Nonlinear Equations and Root Finding Methods Lecture covers two things: 1) Solving systems of linear equations symbolically 2) Using Mathcad to solve systems of nonlinear equations 3) Investigating algorithms to find roots of equations Solving Systems of Linear Equations Symbolically Let’s take a look at a very powerful tool in Mathcad that started a revolution in computational analysis. It started in the late 1980’s when I was an undergraduate. A company called Wolfram created a computer program called Mathematica. This was the first computer code that could solve algebraic and calculus equations symbolically . That is, if I had an equation that said x*y = z, Mathematica could tell me that y = z / x, without ever needing me to assign numbers to x, y, or z. It also was able to solve integrals, dif- ferential equations, and derivatives symbolically. This was an incredible advance, and opened the doors to a whole new world of programming, numerical methods, pure mathematics, engineering, and science. Since then, a competing code called Maple was developed and sold itself to other software companies to include in their programs. The end result: Mathcad uses Maple as a solving engine in the background (you don’t see it) to solve problems symbolically. Here we will look at a brief example of how to use this capability in the context of solving a system of linear equations. Example : The structural system below is something you will see in CES 3102 or CES 4141. r1 and r2 are labels that indicate how the ends of the beam are allowed to move. Q and W represent the external loads (a couple and a distributed load, respectively), and material properties are given as E and I. L is the length of the beam. The goal is to solve for the amount or rotation at r1, and the deflection at r2 that occurs for thegiven loads. This would help us to solve for internal stresses, allowing us to design the beam to sur- vive these internal forces. Solution : The way we learn to solve this problem in CES 4141 is using a Matrix-based solution proce- dure. The generic form of the solution is K * r = R K is a 2 x 2 ‘stiffness’ matrix that contains information about the structures shape, boundary conditions and material properties. THe information needed is L, E, and I r is a 2 x 1 vector that contains the unknown rotation and displacement quantities sought. R is a 2 x 1 vector that contains only information about the external loads (W and Q for this problem). r2 r1 W Q L W = 3 K/FT Q = 2 K*FT L = 15 FT E = 4000 ksi I = 1400 in^4
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CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 3 Nonlinear Equations and Root Finding Methods page 69 of 82 Solution continued : So we will have K as a known 2 x 2 stiffness matrix We will have R as a known 2 x 1 load vector We will solve for the unknown displacement vector r We will not go into any detail on HOW we fill in the values for K and R, that’s for another class.
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This note was uploaded on 05/18/2011 for the course CGN 3421 taught by Professor Long during the Spring '08 term at University of Florida.

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NM3_root_s02 - CGN 3421 - Computer Methods Gurley Numerical...

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