It is important to evaluate the modal correlation coefficient when the responses to be combined are from modes with closely spaced frequencies. In these cases, the complete quadratic combination (CQC) rule is used instead of the square root of the sum of the squares(SRSS). To evaluate the value of the modal correlation coefficient, it was evaluated as a CQC factor. However, an exception to this approach arises when the modal correlation coefficient in secondary systems is evaluated. When the modal correlation coefficient in secondary systems was evaluated, it was found that the tendency of the modal correlation coefficient in the primary and secondary systems analyzed by time history response was different, with the latter being greater than the former for the modal correlation coefficient when the natural period in the primary system is longer than that in secondary system. Other studies have proposed formulas for the modal correlation coefficient in a primary system. This study aims to propose a formula for the modal correlation coefficient in secondary systems based on primary system formulas reported in these previous studies. Furthermore, the modal correlation coefficient in the secondary system was examined based on our proposed formula. The conclusions of this study are summarized as follows: 1) Comparing of the modal correlation coefficient in secondary systems in single degree of freedom with that in multi-degrees-of-freedom, the tendencies were found to be different. Moreover, they were particularly different from the secondary and tertiary natural periods in multi-degrees-of-freedom. 2) The modal correlation coefficient in multi-degrees-of-freedom is expressed as the sum of the products of a participation vector of primary systems, the Fourier amplitude of ground motion and a factor of the modal correlation coefficient in single-degree-of-freedom primary systems. Additionally, the modal correlation coefficient in the secondary systems calculated by Eq. 10 can be evaluated by adding components around three-order modes. 3) It is confirmed that CQC factors to combine peak values nearly correspond to correlation coefficient not only in primary systems but also in secondary systems.
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