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Unformatted text preview: Lecture 9 Introduction to Linear Systems How linear systems occur Linear systems of equations naturally occur in many places in engineering, such as structural anal- ysis, dynamics and electric circuits. Computers have made it possible to quickly and accurately solve larger and larger systems of equations. Not only has this allowed engineers to handle more and more complex problems where linear systems naturally occur, but has also prompted engineers to use linear systems to solve problems where they do not naturally occur such as thermodynamics, internal stress-strain analysis, fluids and chemical processes. It has become standard practice in many areas to analyze a problem by transforming it into a linear systems of equations and then solving those equation by computer. In this way, computers have made linear systems of equations the most frequently used tool in modern engineering. In Figure 9.1 we show a truss with equilateral triangles. In Statics you may use the method of joints to write equations for each node of the truss 1 . This set of equations is an example of a linear system. Making the approximation 3 / 2 . 8660, the equations for this truss are: . 5 T 1 + T 2 = R 1 = f 1 . 866 T 1 = R 2 = . 433 f 1 . 5 f 2 . 5 T 1 + . 5 T 3 + T 4 = f 1 . 866 T 1 + . 866 T 3 = 0 T 2 . 5 T 3 + . 5 T 5 + T 6 = 0 . 866 T 3 + . 866 T 5 = f 2 T 4 . 5 T 5 + . 5 T 7 = 0 , (9.1) where T i represents the tension in the i-th member of the truss. You could solve this system by hand with a little time and patience; systematically eliminating variables and substituting. Obviously, it would be a lot better to put the equations on a computer and let the computer solve it. In the next few lectures we will learn how to use a computerand let the computer solve it....
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