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# 201-Lect 9 - 5 EIGENVALUE PROBLEMS 5.1 The engenvalue and...

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MECH201 Engineering Analysis (2009) p. 21 5 EIGENVALUE PROBLEMS 5.1 The engenvalue and eigenvectors Section 27.2.1 Eigenvalue or characteristic value problems are used in a wide variety of engineering contexts involving vibrations, elasticity, and oscillating systems. Let us consider a set of non-homogenous linear algebraic equations of the form [ ] { } { } A x b = where [A] is a square matrix of n rows and columns, {x} and {b} are column vector of n elements. If the determinant of [A] is not zero, [ ] 0 A |, then it is possible to construct the inverse of [A], that is [ ] 1 A - , and a set of unique solution existed and is given by: { } [ ] { } 1 x A b - = On the other hand, the inverse for [A] does not exist if [ ] 0 A = . The result is that there will be no unique solution for {x} For example, [ ] 3 4 1 6 10 2 9 7 3 A - = - the determinant of [A] is zero. The solution for [ ] { } { } 0 A x = is { } 0 3 x α α = - where α is any values. This means that there is not unique solution for the above system of equations. Eigenvalue problems associated with engineering are of the form [ ] { } { } A x x λ = Where λ is an unknown parameter called the eigenvalue or characteristic value. Let [I] be a unit matrix of the same size as [A]. The above equation can be rewritten as [ ] [ ] { } 0 A I x λ - = The determinant [ ] [ ] A I λ - must be zero to determine λ , giving a solution {x} as an eigenvector .

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MECH201 Engineering Analysis (2009) p. 22 5.2 A Mass-Spring System
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