Modal Superposition, Matrix Iteration
This class was divided in two parts,
- modal superposition (mostly a recapitulation),
- matrix iteration procedures, efficient numerical techniques for computing all the eigenvalues and eigenvectors needed in a dynamical analysis.
Revisiting the modal superposition procedure, particular attention was given to damping matrix construction and to modal truncation. In modal truncation, approximation effectiveness is different between displacements and internal forces, the latter requiring more modal contributes to be accurately described
Matrix Iteration is an iterative numerical procedure that permits building a sequence of eigenvalues/eigenvectors, from lower to higher frequencies.
Proof was given for the convergence of the iterative procedure to the first, or fundamental, eigenvector.
Iteration with sweeps and iteration with shifts are two different procedures that can be used to find all the eigenvectors/values of a dynamic system.
Inverse Iteration is a numerical technique, based on LU decomposition of K, that can significantly speed up the matrix iteration procedure.
Rayleigh Quotient can be applied to discrete systems. We have seen what happens when the shape vector is described by a restricted base, known as Ritz base. A new, reduced eigenproblem in Ritz coordinates must be solved. The Ritz eigenvalues are approximations of structural eigenvalues, the Ritz eigenvectors give place to structural shapes that have all the orthogonality properties that are required for the modal superposition procedure.
Subspace Iteration is a crossover of Matrix Iteration and Rayleigh-Ritz procedures, and is the procedure of choice for large dynamical systems.
Material available for this class:
- the slides i used for my lesson,
- the printouts for
- The Python program used to compute the results in the exercise that closed part 0 of today's class,
- An IPython notebook
that gives a demonstration of Subspace Iteration similar to what I will show next week...
You can look at the notebook following the link, if you want to interact with it you can download it from the linked page, clicking in the top-right corner. Of course, to interact you have to install Ipython...