## 2 Classical Analytical Formulation for Frequency Calculations
### Existing Closed-Form Solutions
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Beam Restraints**} & \multicolumn{3}{c}{\(\beta_{n}\)} & \multicolumn{3}{c}{\(\overline{\Psi}\)} \\ \cline{2-7}  & \multicolumn{3}{c}{**Mode Order \(n\)**} & \multicolumn{3}{c}{**Mode Order \(n\)**} \\ \cline{2-7}  & **1** & **2** & **3** & **1** & **2** & **3** \\ \hline Simply supports (S-S) & \(\pi\)/\(L_{0}\) & \(2\pi\)/\(L_{0}\) & \(3\pi\)/\(L_{0}\) & \(\pi\)/\(L_{0}\)\({}^{2}\) & \((2\pi)^{2}\)/\(L_{0}\) & \((3\pi)^{2}\)/\(L_{0}\) \\ \hline Clamps (C-C) & \(4.73\)/\(L_{0}\) & \(7.8532\)/\(L_{0}\) & \(10.996\)/\(L_{0}\) & \(40.7\)/\(L_{0}\)\({}^{2}\) & \(82.6\)/\(L_{0}\)\({}^{2}\) & \(148\)/\(L_{0}\)\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reference wavenumbers \(\beta_{n}\) and shape coefficients \(\Psi\) for beams with simple restraints (Figure 2c) and bending span \(L_{0}\). Both the real, frequency-independent term (storage modulus \(G_{2,0}\)) and the imaginary part (loss modulus, \(G_{2,\omega}\)), are in fact involved in the frequency domain, where:  
\[G_{2}(\omega)=G_{2,0}+G_{2,\omega}(\omega) \tag{8}\]  
and their typically high sensitivity to frequency is shown in Figure 3 (selected examples reproduced from [7; 17; 22]). In this regard, literature projects have been dedicated to the mechanical characterization of interlayers in use for LG systems, under the assumption of various severe conditions of temperature or time loading (i.e., [23; 24; 25; 26; 27; 28; 29; 30]). As far as the interlayer composition and the test method both modify; however, different mechanical properties can be derived for a given interlayer material [31]. The study reported in [32] also emphasized that the mechanical properties of the interlayer samples (i.e., material test), or interlayer foils belonging to small portions of LG sections (i.e., section test), can result in markedly different stiffness estimates, due to variation of the actual boundary conditions. Finally, for in-service glass structures, it is generally recognized that the degradation of interlayers can affect several material features, including the shear stiffness, but also the adhesion properties, and other thermo-mechanical parameters that could indirectly affect the overall structural performance of a given LG section (see [33; 34; 35]). In other words, Equation (3) represents a practical tool for design, being able to simplify the original dynamic problem of composite beams with flexible, viscoelastic connection and ideal boundaries. On the other side, the reference \(G_{2}=G_{2}(\omega)\) value for dynamic estimates should be properly assessed, including possible delaminations (Section 4). In Figure 4, frequency calculations are proposed for selected LG beams in the S-S or C-C conditions, as a function of \(G_{2}\). The collected frequency values are derived from Equation (2), while assuming that \(h_{ef}\) (Equation (3)) modifies with \(G_{2}\), and \(10^{-4}\) MPa \(<G_{2}<10^{5}\) MPa. In the figures, both the limit \"_layered_\" and \"_monolithic_\" conditions can be easily detected. In addition, the grey regions represent all the possible frequency values that could characterize the dynamic performance of a given LG beam geometry, as far as its end restraints are characterized by a certain translational/rotational stiffness that can be comprised within the limit conditions of ideal simple supports (S-S) or clamps (C-C). Figure 3: Examples of (**a**) storage and (**b**) loss moduli variation with frequency, for Polyvinyl Butyral (PVB) or SentryGlas Plus (SGP) interlayers at 25 \({}^{\circ}\)C, according to [7; 17; 22]. As far as higher vibration modes are taken into account in Figure 4, it is possible to notice that the grey region progressively minimizes for \(n=2\) and \(n=3\). As such, major uncertainties for simplified analytical calculations can be expected, especially for \(n=1\), that in most of the cases is a key parameter for design purposes.