chap08CLASSLECTURE - Chapter 8 Linear Algebraic Equations...

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Chapter 8 Chapter 8 Linear Algebraic Equations Linear Algebraic Equations and Matrices and Matrices
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Linear Algebraic Equations and Matrices Linear Algebraic Equations and Matrices 8.1 Matrix Algebra Overview 8.1.1 Matrix Notation 8.1.2 Matrix Operating Rule 8.1.3 Representing Linear Algebraic Equations in Matrix Form 8.2 Solving Linear Algebraic equations with MATLAB
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Three individuals connected by bungee cords
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Free-body diagrams Free-body diagrams Newton’s Newton’s second law second law = - - = - - - + = - - + 0 x x k g m 0 x x k x x k g m 0 x k x x k g m 2 3 3 3 1 2 2 2 3 3 2 1 1 1 2 2 1 ) ( ) ( ) ( ) ( = + - = - + - = - + g m x k x k g m x k x k k x k g m x k x k k 3 3 3 2 3 2 3 3 2 3 2 1 2 1 2 2 1 2 1 ) ( ) ( Rearrange the equations [ K ] { x } = { b }
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Two types of systems that can be modeled using linear algebraic equations: (a) Lumped variable system that involves coupled finite components (b) Distributed variable system that involves a continuum System of Algebraic Equations System of Algebraic Equations Mass (flow rate) conservation in chemical reactors Spatial variation in a single reactor
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Coupled System Coupled System Mass flowrate = Volume flowrate (Q) * concentration (c) Mass balance: Σ Q i c i = 0 for each reactor
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Newton’s second law – equation of motion Newton’s second law – equation of motion Kirchhoff’s current and voltage rules Mass-spring system (similar to bungee jumpers) Resistor circuits
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Solved single equations previously Now consider more than one variable and more than one equation ( 29 0 x f = ( 29 ( 29 ( 29 0 x ,..., x , x f 0 x ,..., x , x f 0 x ,..., x , x f n 2 1 n n 2 1 2 n 2 1 1 = = = Linear Algebraic Equations Linear Algebraic Equations
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Linear equations and constant coefficients a ij and b i are constants n n nn 2 2 n 1 1 n 2 n n 2 2 22 1 21 1 n n 1 2 12 1 11 b x a ... x a x a b x a ... x a x a b x a ... x a x a = + + + = + + + = + + + Linear Systems Linear Systems
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= + + + = + + + = + + + n n nn 2 2 n 1 1 n 2 n n 2 2 22 1 21 1 n n 1 2 12 1 11 R F a ... F a F a R F a ... F a F a R F a ... F a F a Forces on a Truss Forces on a Truss Most obvious example in Civil Engineering trusses: force balance at joints F 1 2 F 3 R
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} { } ]{ [ ] [ ] ][ [ b x A or b x A = = = n 2 1 n 2 1 nn 2 n 1 n n 2 22 21 n 1 12 11 b b b x x x a a a a a a a a a Mathematical background Mathematical background It is convenient to write system of equations in matrix-vector form
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[ ] = mn 4 m 3 m 2 m 1 m n 4 44 43 42 41 n 3 34 33 32 31 n 2 24 23 22 21 n 1 14 13 12 11 a a a a a a a a a a a a a a a a a a a a a a a a a A Matrix Notations Matrix Notations Column 4 Row 3 (second index) (first index)
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Scalars, Vectors, Matrices Scalars, Vectors, Matrices MATLAB treat variables as “matrices” Matrix (m × n) - a set of numbers arranged in rows (m) and columns (n) Scalar : 1 × 1 matrix
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