Eigen Vector and Ritz Vector

Discuss Eigen Vector and Ritz Vector?

Eigenvector analysis determines the undamped free-vibration mode shapes and frequencies of the system. These natural modes provide an excellent insight into the behaviour of the structure.

Ritz-vector the analysis seeks to find modes that are excited by a loading. Ritz vectors can provide a better basis than do Eigenvectors when used for response-spectrum or time-history analyses that are based on modal superposition.

Eigenvectors

Eigenmodes are most suitable for determining response from horizontal ground acceleration, though a missing-mass (residual-mass) mode may need to be included to account for missing high-frequency effects. Mass participation is a common measure for determining whether there are enough modes, though it does not provide information about the localized response.

Eigen analysis is useful for checking behaviour and locating problems within the model. Another benefit is that natural frequencies indicate when resonance should be expected under different loading conditions. Users may control the convergence tolerance. Orthogonality is strictly maintained to within the accuracy of the machine (15 decimal digits). Sturm sequence checks are performed and reported to avoid missing Eigenvectors when using shifts. Internal accuracy checks are performed and used to automatically control the solution. Ill-conditioned systems are detected and reported, then still produce Eigenvectors which may be used to trace the source of the modelling problem.

 Ritzvectors

Load-dependent Ritz vectors are most suitable for analyses involving vertical ground acceleration, localized machine vibration, and the nonlinear FNA method. Ritz vectors are also efficient and widely used for dynamic analyses involving horizontal ground motion. Their benefit here is that, for the same number of modes, Ritz vectors provide a better participation factor, which enables the analysis to run faster, with the same level of accuracy.

Further, missing-mass modes are automatically included, there is no need to determine whether there are enough modes, and when determining convergence of localized response with respect to the number of modes, Ritz vectors converge much faster and more uniformly than do Eigenvectors. Ritz vectors are not subject to convergence questions, though strict orthogonality of vectors is maintained, like Eigenvectors.