mase201sp09_Jan27_printable

mase201sp09_Jan27_printable - MASE201: Systems of Linear...

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MASE201: Systems of Linear Equations Systems of linear equations occur frequently in engineering. Some examples include: – Truss structure equilibrium – Electrical networks – Elastic deformation – Chemical reaction equilibrium 1/27/2009 1 System of two equations: 2*x + 3*y = 7 4*x – 2*y = 6 Systems are often written in matrix form b x A v v = = ] [ 6 7 2 4 3 2 y x MASE201 Spring 2009 Example • Eight-member truss Balance forces in x and y direction at pins A, B, C, D – results in eight equations in terms of the forces in each truss, e.g. at pin A: ) 25 sin( 4000 sin sin ) 25 cos( 4000 cos cos A pin A pin F F F F F F AC AC BA AB y AC AC BA AB x θ ° + + = ° + + = 1/27/2009 2 . cos cos cos B pin etc F F F F BD BD BC BC BA AB x + + = ° ° = 0 0 0 0 0 0 ) 25 sin( * 4000 ) 25 cos( * 4000 0 0 0 0 cos cos 0 cos 0 0 0 0 0 0 sin sin 0 0 0 0 0 0 cos cos etc F F F F etc BD BC AC AB BD BC BA C A BA AC BA MASE201 Spring 2009 Solving Systems of Linear Equations using MATLAB MATLAB has built-in functions for solving systems of linear equations s These functions solve a system of linear equations without explicitly calculating a matrix inverse : • Gauss elimination method (with or without pivoting) • Gauss-Jordan method s We will: 1/27/2009 3 s Review how these methods and functions work s Apply them to engineering problems s Learn how they scale with problem size (i.e. the number of equations we want to solve) s Learn how to calculate a matrix inverse s Discuss limitations and caveats of the methods and functions MASE201 Spring 2009 Objective: “Look under the hood” and become intelligent users of MATLAB’s tools Gauss Elimination Method • We start with a system of equations: • Step 1: Forward elimination – The rows of the equation are manipulated to put matrix a in an upper diagonal form, e.g. for 1/27/2009 4 a 4 x 4 matrix shown here • Step 2: Back-substitution – Use the 4 th equation to solve for x 4 = ( b 4 / a 44 ) . Then solve the 3 rd equation for x 3 =( b 3 a 34 * x 4 )/ a 33 etc. back to x 1 MASE201 Spring 2009
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Step 1: Forward Elimination Procedure • Define the matrix [ a ] and vector b • For convenience, combine these into a single rectangular matrix (called the augmented matrix) • Starting with the second row, multiply the first row by the ratio of 1 a 1 nd subtract from the = = 4
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mase201sp09_Jan27_printable - MASE201: Systems of Linear...

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