LectureEigenanalysis.ppt

Lecture 12 - Eigen-analysis CVEN 302 June 22, 2001 Lecture’s Goals What is an eigenvalue and eigenvector? Direct computation of the l’s and eigenvectors. Power Method Shift technique Eigenvalue Problems In solving homogeneous linear differential equations, an acceptable form of solution is determined by an eigenvalue problem. To solve ODE, one must find an eigenvalue/eigenfunction for which in general are infinite in numbers. These are different than eigen problems. Eigen-Analysis Matrix eigenvalues arise from discrete models of physical systems Discrete models Finite number of degrees of freedom result in a finite number of eigenvalues and eigenvectors. Eigenvalues Computing eigenvalues of a matrix is important in numerous applications In numerical analysis, the convergence of an iterative sequence involving matrices is determined by the size of the eigenvalues of the iterative matrix. In dynamic systems, the eigenvalues indicate whether a system is oscillatory, stable decaying oscillations or unstable growing oscillation Eigenvalues Oscillator system, the eigenvalues of differential equations or the coefficient matrix of a finite element model are directly related to natural frequencies of the system Regression analysis, eigenvectors of correlation matrix are used to select new predictor variables that are linear combinations of the original predictor variables. Physical Examples Natural vibration of systems of mass springs Flutter of the airplane wings vibration of membranes oscillation of a suspension bridge torsional vibration of multi-cylindrical engine structural response of earthquakes General form of the equations The general form of the equations Example For a set of equations Example The equations can be Example Expand the equation Examples The equation can be factored: Example The eigenvector for l 3, can be determined by plug-in the equation Example Example Eigen-analysis Eigen-analysis Power Method Power Method Power Method Power Method