2017-04-04

Truncated Sums, Matrix Iteration

This class was divided in two parts,

  1. truncated summation of modal responses (mostly a recapitulation of modal analysis) and
  2. matrix iteration procedures, i.e., 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. At every iteration the Ritz base is changed to have a better agreement between inertial forces and elastic forces.

Material available for this class:



Giacomo Boffi