## Ritz analysis and Modal Eigen Analysis Methods

Modal Ritz analysis and Modal Eigen analysis are two different methods used in structural dynamics to determine the natural frequencies, mode shapes, and dynamic response of structures.

Modal Ritz analysis is a numerical method used to determine the natural frequencies and mode shapes of structures by approximating the mode shapes using a trial function. The trial function is chosen based on the physical characteristics of the structure, and the natural frequencies and mode shapes are obtained by solving a generalized eigenvalue problem. Modal Ritz analysis is commonly used when analytical solutions are not available or when the structure has complex geometries.

Modal Eigen analysis, on the other hand, is a numerical method that directly solves the eigenvalue problem to obtain the natural frequencies and mode shapes of a structure. The method involves solving the matrix equation of motion and the corresponding boundary conditions to determine the eigenvalues (natural frequencies) and eigenvectors (mode shapes) of the structure. Modal Eigen analysis is commonly used in linear structural analysis and is a widely accepted method for determining the dynamic response of structures.

Furthermore, In many seismic analyses, the choice of Ritz vs eigenvectors is not clear beforehand. You may need to compare both methods and choose the most suitable one. A >90-95% of mass participation is a common threshold for evaluating if the analysis of modes is ok.

Ritz analysis is load dependent, captures coupled modes better, and is usually quicker. Unfortunately, when there are no clear or known dominant modes, Ritz analysis can be a too rough approach, for example, too broad for discovering unexpected local resonance problems. Unlike eigenvectors, Ritz does not always produce the same modes. For example, with limits of 10 and 100 eigenvectors, the first 10 of each set will be the same. However, with the Ritz option, the modes are not exactly the same when increasing the set.

Eigenvector analysis determines the undamped natural modes which is a good way to know about the behavior of the structure when you have no clue. And it is load independent for linear analyses. Ok, you will obtain many meaningless modes but after some trial and error, it is possible to adjust the cut-off and shift frequencies and run faster.

In summary, Modal Ritz analysis is an approximate method that uses a trial function to determine natural frequencies and mode shapes, while Modal Eigen analysis is a direct method that solves the eigenvalue problem to determine the natural frequencies and mode shapes of a structure.

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