An overview of **Ritz and Eigen vectors**, taken from the <=
a href=3D"http://www.csiamerica.com" class=3D"external-link" rel=3D"nofollo=
w">CSI *=
Analysis Reference Manual* (Modal Analysis > Overview, page 323), is =
given as follows:

- Eigenvector analysis determines the und= amped free-vibration mode sh= apes and frequencies of the system. These natural modes provide an excellen= t insight into the behavior of the structure.

- Ritz-vector analysis seeks to find modes that are excited by a particul=
ar loading. Ritz vectors can provide a better basis than do Eigenvectors wh=
en used for response-spe=
ctrum or time-history=
analyses that are based on modal superposition.

The user should determine the type= of modes which are the most appropriate.

Eigen modes are most suitable for determining response from horizontal g= round 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 whe= ther or not there are enough modes, though it does not provide information = about localized response.

Eigen analysis is useful for checking behavior and locating problems wit= hin the model. Another benefit is that natural frequencies indicate when re= sonance should be expected under different loading conditions. Users may co= ntrol the convergence tolerance. Orthogonality is strictly maintained to wi= thin the accuracy of the machine (15 decimal digits). Sturm sequence checks= are performed and reported to avoid missing Eigen vectors when using shift= s. Internal accuracy checks are performed and used to automatically control= the solution. Ill-conditioned systems are detected and reported, then stil= l produce Eigen vectors which may be used to trace the source of the modeli= ng problem.

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

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

Sources of documentation on Load-dependent Ritz vectors include:

- Structural-analysis textbooks (Wilson, 2004)

- Finite-element textbooks (Cook et al., 2001)

- Structural-dynamics textbooks (Chopra, 2000)

- Both Eigen and Ritz modes may be calculated simultaneously, in the same= model, and in the same analysis run, such that their behavior may be compa= red.

- Sources of flexibility available during calculation of Eigen and Ritz m=
odes include:

- Consideration of P-Delta and= other nonlinear effects
- Modes at different stages o= f construction
- Frequency shifts for specialized loading