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SAP2000An hasoverview always offeredof *both*Ritz exactand Eigen vectors*, andtaken Load-dependentfrom Ritz (LDR vectors). The the {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} [_Analysis Reference Manual_|doc:CSI Analysis Reference Manual] (Modal Analysis > Overview, page included323), withis SAP2000,given statesas thatfollows:

* Eigen-vectorEigenvector analysis determines the [undamped|kb:Damping] free-vibration modes [mode|kb:Modal analysis] shapes and frequencies of the system. These natural modes provide an excellent insight into the behavior of the structure.

* Ritz-vector analysis seeks to find modes that are excited by a particular loading. Ritz vectors can provide a better basis thatthan do eigenvectorsEigenvectors when used for [response-spectrum|kb:Response-spectrum analysis] or [time-history|kb:Time-history analysis] analyses that are based on modal superposition.

We always recommend performing an Eigen analysis of the structure, since it can help to check behavior and find modeling problems. The natural frequency information can also be important for understanding where resonance can be expected with different types of loads.

For determining the response to\\
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You, the engineer, may determine the type of modes which you feel are the most appropriate.


h2. Eigenvectors

Eigen modes are most suitable for determining response from horizontal ground acceleration, eigen modes are generally quite efficient, although it may be necessary to include though a missing-mass (residual-mass) mode may need to be included to account for missing high-frequency effects that are missed by the Eigen modes. Most common structural analysis programs have this capability, as does SAP2000.

For vertical ground acceleration, or localized loading such as by machine vibration, eigen modes may be much less efficient, and determining whether or not you have enough modes can be difficult. Load-dependent Ritz vectors are very effective for this purpose, and have the advantage of always including the missing-mass modes automatically. Even for horizontal ground excitation, Ritz vectors are more efficient and widely used.

Mass participation is a common measure for determining whether or not you have enough modes. However, it is only useful for ground acceleration, and does not give information about localized response. SAP2000 provides static and dynamic participation measures for other types of loading as well. When determining convergence of localized response with respect to the number of modes, Ritz vectors converge much faster and more uniformly than do Eigen vectors.

Load dependent Ritz vectors are well documented in the open literature (e.g., Wilson et al), in standard finite-element textbooks (e.g., Cook et al), and in standard structural dynamics textbooks (e.g., Chopra). Please let us know if you need detailed citations.

SAP2000 lets you, the engineer, determine the type of modes you feel are most appropriate. *In fact, you can calculate both types of modes simultaneously, in the same model, even in the same run, and compare their behavior.* SAP2000 offers much flexibility in the calculation of Eigen or Ritz modes, including the consideration of P-Delta and other nonlinear effects, modes at different stages of construction, and frequency shifts for specialized loading.

When using Eigen vectors, you 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 any Eigen vectors when using shifts. Internal accuracy checks are performed and used to control the solution automatically. SAP2000 will detect and report ill-conditioned systems, but will still produce eigen vectors that you can use to trace the source of the modeling problem. Ritz vectors are not subject to convergence questions, but strict orthogonality of the vectors is maintained as it is for eigen vectors.. [Mass|kb:Mass] participation is a common measure for determining whether 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 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 Eigen vectors 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 Eigen vectors which may be used to trace the source of the modeling problem.


h2. Ritz vectors

Load-dependent Ritz vectors are most suitable for analyses involving vertical ground acceleration, localized machine vibration, and the [nonlinear|kb:Nonlinear] [FNA|kb:Comparison of FNA and direct-integration time-history analysis] 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 or not 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 Eigen vectors. Ritz vectors are not subject to convergence questions, though strict orthogonality 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)


h2. Tips

* 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 compared.

* Sources of flexibility available during calculation of Eigen and Ritz modes include:
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** Consideration of [P-Delta|kb:Geometric nonlinearity] and other [nonlinear|kb:Nonlinear] effects
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** Modes at different [stages of construction|kb:Staged construction]
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** Frequency shifts for specialized loading