Large Eigenvalue Problems in Dynamic Analysis

An effective solution technique is presented to calculate the p lowest eigenvalues and corresponding vectors in the problem KΦ = ω²MΦ, when the order and bandwidth of the matrices is large. The eigenvalue problem is solved directly without a transformation to the standard form. The mass matrix, M, may be diagonal with zero elements as in a lumped mass analysis or may be banded as in a consistent mass formulation. The algorithm establishes q starting vectors, q > p, from the elements in M and K and iterates with all vectors simultaneously. This iteration is described as a subspace iteration, where best eigenvalue and eigenvector approximations can be calculated in each iteration. Operation counts are given which show the cost effectiveness of the algorithm when the bandwidth of the system is large. A program is described to solve the eigenvalue problem when the system has practically any order and bandwidth. Two example analyses are presented.