Ritz and Eigen.

Ritz, not Rizz and Eigen Vector are two ways of solving a Modal Analysis problem. It is what ETABS use while doing the Modal Analysis. Today's blog will be about the difference between these two.

EIGEN-VECTOR :

It determines the undamped free-vibration mode shapes and frequencies of the system. What free-vibration means in our term is that there is no external loads acting on it for any period of time, just the initial conditions. The idea of bending a ruler and allowing it to reverberate. The natural modes obtained through this provide an excellent insight into the behaviour of the structure. 

But remember, it's the free-vibration response while the structures will be under various different loads. What should be done then? Here comes the Ritz Vector.

RITZ-VECTOR :

Research has indicated that the natural free-vibration mode shapes are not the best basis for a mode-superposition analysis of structures subjected to dynamic loading : Wilson, Yuan and Dickens, 1982. I stole this line from the ETABS documentation. Hence Ritz vectors are preferred. It seeks to find modes that are excited by a particular load. The loads can be any load you defined in the analysis. The Ritz vector takes into account the distribution of the dynamic loading across the entire structure while the Eigen-vector completely ignores it.

The takeaway here is: Ritz-Vector modes do not represent the actual characteristics of a structure, Eigen does. The modes obtained by Ritz are biased by the starting load vectors.


Calculation:

While both the Analysis can be solved using standard Eigen-solution techniques, there is a difference in the final output results. 

2-DOF free vibration.

As you now know, Eigen calculates only the free vibration. So it misses some modes which will react to a particular loading condition. Fret not, ETABS/SAP2000 have the solution for this too. 

Come Static-Correction Modes:
A Static-Correction Mode is the solution to that portion of the specified load that was missed out by the Eigen-Vector. It can be specified for any load patterns. When applied to acceleration loads, the Seismic Loads, it is called as missing-mass or residual-mass modes. 
It is to be noted that Ritz Vectors always include the residual-mass effect for all loading vectors.

Conclusion: Eigen is better for understanding the intrinsic behaviour of the structure. Ritz is preferred for the actual response. 

PS: There will be updates on this topic later on as I gain more insights with experience. The mathematics on this two vectors will be uploaded too.

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