## 2 Materials and Methods  
### Mathematical Models  
Figure 1 shows several mathematical models used to represent viscous damping systems. The pure dashpot (Figure 1a) is an extreme case of the Maxwell system in which the stiffness is infinite. The Maxwell system (Figure 1b) includes both linear stiffness and viscous damping and renders an accurate description of the damper and its mounting elements. The Kelvin model (Figure 1c) presents the spring and damper in parallel and has been proposed by several authors as a simple approximation to the more complex Maxwell system. The standard model (Figure 1d) includes a Maxwell system in parallel with a linear or non-linear spring and can be used to represent the viscous damper inserted within a structural system. In all cases, hysteretic loops for steady state response under harmonic excitation of amplitude \(u_{o}\) and circular frequency \(\omega\) are presented. The constitutive equation of the pure dashpot is:  
\[f=c\big{|}\dot{u}\big{|}^{a}sgn(u) \tag{1}\]  
where,\(f\) indicates force, \(u\) is displacement, \(c\) is the damping coefficient, \(\text{sgn}(\cdot)\) is the sign function and \(a\) is the damper exponent, with \(a=1\) corresponding to linear dampers. For earthquake mitigation, exponents below 1 are common. The constitutive equation for the non-linear Maxwell model is:  
\[\dot{u}=\frac{\dot{f}}{k_{d}}+\frac{|f|^{1/a}}{c^{1/a}}sgn(f), \tag{2}\]  
where, \(k_{d}\) is the damping system stiffness. If \(k_{d}\) takes a very small value, the first term on the right side of Equation (2) prevails. Neglecting the second term and after integration, the equation reduces to that of a pure spring with small stiffness and negligible damping. If \(k_{d}\) takes a very large value, the same term vanishes, and the equation reduces to a pure dashpot. These extreme cases show the behavior dependence of the Maxwell system on its stiffness. For linear dampers, Equation (2) can be expressed as  
\[\dot{f}=k_{d}\dot{u}-\frac{f}{\lambda}; \tag{3}\]  
Figure 1: Mathematical models for viscous dampers and their steady-state hysteresis loops for forced harmonic excitation; (**a**) pure dashpot; (**b**) Maxwell model; (**c**) Kelvin model; (**d**) Standard solid model. where, \(\lambda\) (=\(c/k_{d}\)), expressed in seconds, is referred to as relaxation time. For use in earthquake analysis, the behavior of the models is generally characterized for forced harmonic excitation on the assumption that, under earthquake excitation with broad frequency content, the structure vibrates mainly with its fundamental period. For forced harmonic excitation of circular frequency \(\omega\) and amplitude \(u_{0}\), the steady-state solution to Equation (3) is:  
\[f=\beta_{s}k_{d}u_{0}sin(\omega t)+\beta_{d}ciu_{0}cos(\omega t), \tag{4}\]  
\[\beta_{s}=\frac{\omega^{2}\lambda^{2}}{1+\omega^{2}\lambda^{2}}, \tag{5}\]  
\[\beta_{d}=\frac{1}{1+\omega^{2}\lambda^{2}}. \tag{6}\]  
Equation (4) can be compared to the steady-state solution for the Kelvin system, which adopts the same form with \(\beta_{s}=\beta_{d}=1\). For this reason, the Kelvin system has been proposed as an approximation to the Maxwell system. \(\beta_{s}\) and \(\beta_{d}\) are dependent on excitation frequency and relaxation time and represent the relative magnitudes of spring and damper response, respectively. Figure 2 presents a plot of these coefficients for two different values of relaxation time as a function of the excitation period. For very short periods the predominant behavior is that of a pure spring, whereas for very long periods pure damping dominates the response. In between these two extremes, the behavior is a blend of damper and spring, resulting in the characteristic hysteretic shape of the Maxwell system. It is clear that for MDOF structures, the performance of the added damping system will be different for the fundamental mode and higher modes where spring-like, non-dissipative behavior is to be expected.