07-Linear Algebra and Matrices

07-Linear Algebra and Matrices - 7: Linear Algebraic...

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7: Linear Algebraic Equations 7: Linear Algebraic Equations and Matrices and Matrices
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Oct 16, 2006 21:43 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives Linear algebraic equations – matrices – matrix algebra. linear systems of equations – what and where in engineering matrix notation identify types of matrices (identity, diagonal, symmetric, triangular, tridiagonal) matrix multiplication linear equations in matrix form solving linear equations in MATLAB
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Oct 16, 2006 21:43 ECOR2606 -- Hassan & Holtz 3 A Problem Three bungee jumpers in series: determine the steady-state position of each Figure 7.1
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Oct 16, 2006 21:43 ECOR2606 -- Hassan & Holtz 4 A Problem m 1 m 2 m 3 m 1 m 2 m 3 x 2 1 3 length change, chord 1: x 1 tension force, chord 1: k 1 x 1 length change, chord 2: x 2 x 1 tension force, chord 2: k 2 x 2 x 1 length change, chord 3: x 3 x 2 tension force, chord 3: k 3 x 3 x 2 m 1 k 1 x 1 m 1 g k 2 x 2 x 1 m 2 k 2 x 2 x 1 m 2 g k 3 x 3 x 2 m 3 k 3 x 3 x 2 m 3 g Displacements: Forces:
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Oct 16, 2006 21:43 ECOR2606 -- Hassan & Holtz 5 A Problem (contd.) Using F = m a , with a = 0 (steady state): Collecting terms: F = 0 on each free-body diagram m 1 g k 2 x 2 x 1  − k 1 x 1 = 0 m 2 g k 3 x 3 x 2  − k 2 x 2 x 1  = 0 m 3 g k 3 x 3 x 2  = 0 where m i = mass of jumper i (kg) k j = spring constant for chord j (N/m) x i = displacement of jumper i measured from unstretched posn (m) g = gravitational constant (= 9.81 m/s/s) k 1 k 2 x 1 k 2 x 2 = m 1 g k 2 x 1   k 2 k 3 x 2 k 3 x 3 = m 2 g k 3 x 2 k 3 x 3 = m 3 g (7.1) [ k 1 k 2 k 2 0 k 2 k 2 k 3 k 3 0 k 3 k 3 ][ x 1 x 2 x 3 ] = [ m 1 g m 2 g m 3 g ]
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Oct 16, 2006 21:43 ECOR2606 -- Hassan & Holtz 6 7.1: Linear Algebraic Equations General form: a 11 x 1 a 12 x 2  ⋯  a 1 n x n = b 1 a 21 x 1 a 22 x 2  ⋯  a 2 n x n = b 2 a n 1 x 1 a n 2 x 2  ⋯  a nn x n = b n a i j - constant coefficients b i - constants x i - unknowns
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Oct 16, 2006 21:43 ECOR2606 -- Hassan & Holtz 7 7.2: Matrix Algebra 7.2.1: Matrix Notation matrix – rectangular array of elements a ij – an element in row i and column j a matrix having m rows and n columns has dimensions of m x n .
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07-Linear Algebra and Matrices - 7: Linear Algebraic...

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