A two degree-of-freedom (2-DOF) model is employed for mid-story isolation buildings or a tuned mass damper mounted on buildings in practical design. These systems are classified into a primary-secondary structural system. A seismic response analysis using this model is an empirical evaluation method in design practice. However, the accuracy of the peak response distribution computed by the 2-DOF model has not been clarified.
The objective of this study is to investigate the error in the inter-story drift calculated by the 2-DOF model. The employed 2-DOF model is derived from its original multi-degree of freedom system (MDOF) based on dynamic reduction technique. Parameters of interest are a tuning ratio and mass ratio. The tuning ratio is a frequency ratio to secondary to the primary system. The mass ratio is a ratio of the secondary mass to the effective mass of the primary system. The mass ratio is set within the parameter of 0.02 to 0.5. The inter-story drifts in primary system are recovered from generalized displacements (modal displacements) of the 2-DOF model and classical modal vectors with respect to the primary system.
A stochastic vibration analysis is conducted in both original MDOF model and corresponding the 2-DOF model. RMS responses are obtained by solving a Lyapunov equation governing the stochastic process. The displacement responses of the MDOF model are decomposed into coupled and decoupled components. The decoupled components are the higher modal responses. The coupled component is the remaining response and corresponds to a controlled response. The controlled mode is limited to the fundamental mode in this study. The response of the controlled mode is compared to the one from the 2-DOF model. The analysis reveals that the 2-DOF model estimates accurate floor displacements. The relative errors are ten percent at most. However, the 2-DOF model underestimates inter-story drifts in the upper story of the primary system. As the mass ratio becomes larger, the error becomes large.
A convenient modal superposition technique is proposed to improve the accuracy of inter-story drift obtained by the 2-DOF model. The method is as follows. The CQC modal combination is applied with modal damping factors evaluated by classical modal vectors. Then, amplitude of the response is slightly corrected to be consistent with the one obtained from the MDOF model. It is confirmed that reasonable accuracy is obtained by the proposed method.
