## 2 Materials and Methods
### Mathematical Models
The energy dissipated in a cycle of harmonic motion for a pure damper is [26]:  
\[W_{d}=\pi bc\omega^{a}u_{0}^{1+a}, \tag{7}\]  
where, \(b\) is a function of \(a\) representing the relationship between dissipated energy in the non-linear damper and a linear damper with the same peak force response and amplitude. \(b\) can be found as  
\[b=\frac{2^{2+a}}{\pi}\frac{\Gamma^{2}(1+a/2)}{\Gamma(2+a)}; \tag{8}\]  
in this expression \(\Gamma(\cdot)\) represents the gamma function. For any non-linear damper, energy equivalence can be used to propose an equivalent linear damper with damping coefficient:  
\[c_{1,eq}=bc\omega^{a-1}u_{0}^{a-1}, \tag{9}\]  
\[\dot{u}=\frac{\dot{f}}{k_{d}}+\frac{|f|^{1/a}}{c^{1/a}}sgn(f), \tag{2}\]0 Coefficients \(\beta_{s}\) and \(\beta_{d}\) in Equation (4) as a function of harmonic excitation period \(T\): (**a**) \(\lambda\) = 0.02 s; (**b**) \(\lambda\) = 0.1 s.  
and the relaxation time \(\lambda_{a}\) for the non-linear Maxwell system can be defined as \(c_{1,a/b}/k_{d}\). For the standard solid model presented in Figure 1d, the viscous damping ratio \(\xi\) can then be expressed using [PERSON]'s equation:  
\[\dot{u}=\frac{\dot{f}}{k_{d}}+\frac{|f|^{1/a}}{c^{1/a}}sgn(f), \tag{2}\]1  
where, \(W_{s}=k_{s}\cdot{u_{0}}^{2}/2\) is the strain energy at peak displacement. Combining the previous equations, and recalling that \(\omega=2\pi/T\):  
\[\dot{u}=\frac{\dot{f}}{k_{d}}+\frac{|f|^{1/a}}{c^{1/a}}sgn(f), \tag{2}\]2  
Some authors recommend a maximum value of \(k_{s}/k_{d}\) in order to keep the ratio \(\lambda_{a}/T\) low, thus ensuring prevalence of the damper behavior. For instance, [PERSON] and [PERSON] (2005) suggested \(k_{s}/k_{d}\leq 0.2\); with a 30% added damping ratio, this condition is transformed approximately into \(\lambda/T\leq 0.02\), and the authors conclude that, if this condition is met, the influence of brace stiffness on response is negligible regardless of the damper exponent. [PERSON] and [PERSON] (2016) suggested a more restrictive limit \(k_{s}/k_{d}\leq 0.1\); for the same viscous damping ratio, this limit results in \(\lambda/T\leq 0.01\). The large difference between both limits indicates a certain ambiguity in this condition. Moreover, Equation (11) applies only to SDOF systems. Despite that, its use has been extended to MDOF systems. For this type of systems, however, it has been proposed that a Maxwell stiffness for all storeys \(k_{d,n}=k_{s1}\) (first storey stiffness) is enough for a reasonable performance of the added damping system ([PERSON] and [PERSON], 2016). Verification of the validity of these conditions is of interest. Figure 3 presents hysteresis loops for Maxwell models with different values of \(k_{d}\), for the linear (\(a=1\)) and non-linear (\(a=0.5\)) cases, both subjected to the same forced harmonic excitation and energy-equivalent damping coefficients as per Equation (9). The figure shows the features of behavior with limited stiffness: (i) large reduction of dissipated energy expressed as area enclosed within the loop; (ii) appearance of forces at maximum displacement, i.e., increase in storage stiffness; (iii) reduction of forces at maximum velocity, i.e., decrease in loss stiffness; and (iv) moderate reduction of maximum forces. \[\dot{u}=\frac{\dot{f}}{k_{d}}+\frac{|f|^{1/a}}{c^{1/a}}sgn(f), \tag{2}\]3 Hysteretic force-displacement loops for Maxwell systems subjected to forced harmonic vibration with different values of \(k_{d}=10,000\) kN\(/\)mm, \(20\) kN\(/\)mm, \(10\) kN\(/\)mm, \(5\) kN\(/\)mm; **(a)** linear case; **(b)** non-linear case.