# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 3 Numerical Model of Viscoelastic Dampers
### Generalized Maxwell Model  
In this study, the VE material is represented by means of a Generalized Maxwell Model with four Maxwell elements. The optimal number of Maxwell elements has been established based on the best trade-off between the number of parameters of the model and its effectiveness in reproducing the hysteresis cycles of the selected VE material at different frequencies. A schematic representation of the Generalized Maxwell Model with n = 4 (GMM4) is shown in Figure 5. The constitutive equation of the GMM4 is given as follows:  
\[\tau_{\text{GMM4}}(t)=\gamma_{0}\big{[}G^{\prime}_{\text{GMM4}}\sin(\omega t) +G^{\prime\prime}_{\text{GMM4}}\cos(\omega t)\big{]} \tag{22}\]  
where \(G^{\prime}_{\text{GMM4}}(\omega)\) and \(G^{\prime\prime}_{\text{GMM4}}(\omega)\) are the shear storage modulus and the shear loss modulus, respectively. These parameters are obtained by means of the following expressions  
\[G^{\prime}(\omega)=G_{\text{E,0}}+\sum_{i\,=\,1}^{4}\frac{G_{\text{E,i}}\,( \omega\,\lambda_{\text{i}})^{2}}{1+(\omega\,\lambda_{\text{i}})^{2}} \tag{23}\]  
\[G^{\prime\prime}(\omega)=\omega\,\,G_{\text{C,0}}+\sum_{i\,=\,1}^{4}\frac{G_{ \text{E,i}}\,\omega\,\lambda_{\text{i}}}{1+(\omega\,\lambda_{\text{i}})^{2}} \tag{24}\]  
where  
\[\lambda_{\text{i}}=\frac{G_{\text{C,i}}}{G_{\text{E,i}}}i=1\text{ to }4 \tag{25}\]  
In the above equations, \(\lambda_{\text{i}}\) is the \"relaxation time\" of the material, \(G_{\text{E,i}}\) and \(G_{\text{C,i}}\) are the elastic modulus of the spring and the viscous coefficient of the dashpot of the \(i\)-th Maxwell element, \(G_{\text{E,0}}\) and \(G_{\text{C,0}}\) are the elastic modulus of the spring and the viscous coefficient of the dashpot of the Kelvin-Voigt element, and \(\omega\) is the circular frequency of excitation of the structure (to be taken equal to the oscillation frequency of the structure in the fundamental mode of vibration). Details about the derivation of Equations (22)-(24) are reported in Appendix A. Note that both the storage and the loss moduli depend on the frequency of oscillation. Based on experimental values of \(G^{\prime}\) and \(G^{\prime\prime}\) at various frequencies and fixed ambient temperatures, it is possible to calibrate the storage and loss moduli of the GMM4 model in order to reproduce the hysteretic behavior of the selected VE material. In this study, the elastomer ISD-111H tested by [PERSON] [52] is considered. Experimental values of \(G^{\prime}\) and \(G^{\prime\prime}\) obtained through cyclic tests at an ambient temperature of 20 \({}^{\circ}\)C and at frequencies of 0.1 Hz, 0.3 Hz, 1.0 Hz, and 2.0 Hz are shown in Table 1, together with the loss factor \(\eta\).  
Calibration of the coefficients in Equations (23) and (24) is obtained through least square regression analysis performed on the data reported in Table 1 by means of the MS-Excel solver. The values obtained for the ten constants \(G_{\text{E,0}}\), \(G_{\text{C,0}}\) and \(G_{\text{E,i}}\), \(G_{\text{C,i}}\) (with \(i=1\) to 4) of the GMM4 model are reported in Table 2.  
Equation (22) is expressed in terms of stress and strain. However, for design purposes, it is convenient to consider a force-displacement formulation in order to take account of the geometric characteristics of the VED  
\[F_{\text{GMM4}}(t)=u_{0}\big{[}K^{\prime}_{\text{GMM4}}\sin(\omega t)+K^{\prime \prime}_{\text{GMM4}}\cos(\omega t)\big{]} \tag{26}\]  
where \(K^{\prime}_{\text{GMM4}}\) and \(K^{\prime\prime}_{\text{GMM4}}\) are called the \"in-phase stiffness\" and the \"out-of-phase stiffness\" of the VED, respectively. These parameters are obtained by means of the following relations  
\[K^{\prime}_{\text{GMM4}}=n_{1}\frac{A_{\text{I}}}{t_{\text{I}}}G^{\prime}_{ \text{GMM4}} \tag{27}\]  
\[K^{\prime\prime}_{\text{GMM4}}=n_{1}\frac{A_{\text{I}}}{t_{\text{I}}}G^{\prime \prime}_{\text{GMM4}} \tag{28}\]  
where \(A_{\text{I}}\) and \(t_{\text{I}}\) are the area and the thickness of a single elastomeric layer of the VED and \(n_{\text{I}}\) is the number of layers.