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
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:
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
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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