Unifying causal model of rate-independent linear damping for effectively reducing seismic response in low-frequency structures

Rate-independent linear damping (RILD), which is also known as structural damping or hysteretic damping, refers to a type of damping having a constant imaginary part in its complex stiffness that generates damping forces independent of the excitation frequency. In contrast, linear viscous damping (LVD) is another type of damping with a frequency-proportional imaginary part in its complex stiffness, resulting in damping forces proportional to frequency. RILD demonstrates similar performance to that of LVD for the same loss factor when incorporated in a structure to control seismic response displacement. Nevertheless, the damping force generated by the RILD is relatively low in frequency ranges higher than the natural frequency of the primary structure. This leads to efficient displacement control with low damping force when RILD is integrated with low-frequency structures. However, the noncausality of RILD hinders its practical applications, and thus, causal models are widely studied to mimic the RILD behavior. This paper proposes a causal model of RILD using Maxwell elements whose damping force is generated according to the fractional derivative of displacement. The proposed model, further represented by a fractional-order damping function, is found to be a unifying model that includes existing causal RILD models from the literature, thereby providing further insights to better understand the nature of RILD. Furthermore, numerical examples using linear as well as nonlinear structural models illustrate the benefit of physical realization of the proposed methods in improving the seismic performance of low-frequency structures.