Conservatism in summation rules for closely spaced modes

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Abstract
It is shown that the method recommended by the Nuclear Regulatory Commission to be used to combine spectral response in the case of closely spaced modes is unnecessarily conservative for certain systems. Closely spaced modes arise in structures from symmetry and where there is a light appendage with a frequency close to one of the natural frequencies of the structure. In the former case, the closely spaced modes do not involve significant interaction between components of the system and the Nuclear Regulatory Commission Guide is reasonable. The latter case, that is when there are closely spaced modes where interaction of components occurs as in the examples of light appendages and torsionally unbalanced buildings, must be treated by consideration of the interacting components. The approach proposed here is that the modes that are not closely spaced be treated by modal analysis and the closely spaced modes, in the case of two closely spaced modes, be treated as a coupled two‐degree‐of‐freedom system. If this is done, the beat phenomenon, the most important characteristic of the interaction, is evident, as is the associated result that the peak response of the coupled system is developed much later than the peak responses obtained in the individual modes. It is shown that the square root of the sum of the squares procedure underestimates, as expected, the response for undamped and very lightly damped systems, but for damped systems the square root of the sum of the squares method can be extremely conservative. It follows that the other methods specified by the Nuclear Regulatory Commission for closely spaced modes must be even more conservative.

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