## 2 Classical Analytical Formulation for Frequency Calculations
### Existing Closed-Form Solutions  
Assuming the ideal restraint configurations that were schematized in Figure 2c for simply supported (S-S) or clamped (C-C) members, classical theories for slender beams can be taken into account for frequency analyses of LG elements, as far as the dynamic mechanical properties of the interlayer in use are properly described. According to Figure 2, a given LG beam in free vibrations must, in fact, satisfy the well-known Euler-Bernoulli differential equation of motion, that for a given a monolithic (\(A=b\times h\)) \(\times\)\(L_{0}\) member is given by [20]:  
\[\frac{\partial^{2}}{\partial x^{2}}EI(x)\left(\frac{\partial^{2}v(x,t)}{ \partial x^{2}}\right)+\rho A\frac{\partial^{2}v(x,t)}{\partial t^{2}}=0 \tag{1}\]  
Moreover, in Equation (1), \(v(x,t)\) is the vertical displacement at the abscissa \(0\leq x\leq L_{0}\) and time instant \(t\); \(E\) and \(\rho\) the MoE and density of the material in use; and, \(I\) the second moment of area. Thus, the fundamental frequency is conventionally given by the compact expression [20]:  
Figure 2: Double LG beam in free vibrations: (**a**) transversal and (**b**) longitudinal cross-sections, with (**c**) selected ideal restraints. \[f_{n}=\frac{\omega_{n}}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{\beta_{n}^{4}E}{12\overline {m}}}h^{3} \tag{2}\]  
with \(n\) the mode order and \(\beta\) the wavenumber (Table 1). According to several studies of literature, the challenge for a given LG member lies in the estimation of the actual composite stiffness, being strongly related to the shear coupling effect of the bonding interlayer. Besides the availability of simplified analytical approaches that are based on the use of an equivalent, monolithic glass thickness \(h_{ef}=h\) for sandwich sections, according to Figure 2, it was shown in [17] that the \"adjusted dynamic\" effective thickness (adapted from [21] for modal analysis purposes) is able to offer reliable frequency estimates for double LG beams with ideal boundaries. In particular, such an \"adjusted\" thickness is given by [17]:  
\[h_{ef}=\sqrt[3]{\frac{1}{h_{1}^{3}+h_{3}^{2}+12I_{s}}+\frac{1-\eta}{h_{1}^{3} +h_{3}^{3}}} \tag{3}\]  
where \(\eta\) represents the shear coupling of the composite section:  
\[0\leq\eta=\frac{1}{1+\frac{E_{1}h_{2}}{G_{2}b}\frac{I_{1}+I_{3}}{I_{lat}} \frac{A_{1}A_{3}}{A_{1}+A_{3}\cdot\Psi}}\leq 1 \tag{4}\]  
and the other relevant terms are given by:  
\[I_{i}=\frac{bh_{i}^{3}}{12}\ A_{i}=bh_{i} \tag{5}\]  
\[I_{s}=\frac{h_{1}h_{3}}{h_{1}+h_{3}}\cdot[h_{2}+0.5(h_{1}+h_{3})]^{2} \tag{6}\]  
\[I_{tot}=I_{1}+I_{3}+\frac{A_{1}A_{3}}{A_{1}+A_{3}}\cdot[h_{2}+0.5(h_{1}+h_{3}) ]^{2} \tag{7}\]  
The coefficient \(\overline{\Psi}\) in Equation (4) depends on the normalized shape of deflections, for a given homogeneous beam. For basic boundary conditions and several mode orders \(n\), \(\Psi\) can be calculated from Table 1.  
Finally, \(\eta\) in Equation (3) is strictly affected by the shear modulus \(G_{2}=G_{2}(\omega)\) of the bonding layer. Given that the common materials for LG applications have a viscoelastic behaviour that depends on the material composition and its vibration frequency and/or ambient conditions, this turns out in an effective thickness \(h_{ef}=h_{ef}(\omega)\), which explicitly reflects the dynamic response of the interlayer itself, as a part of a composite system it belongs. However, according to Equation (2), it is also \(\omega=\omega(h_{ef})\), and hence an iterative calculation approach is required for accurate thickness/frequency estimates.