## 2 Theory
### From Homogeneous to Sandwich Partitions  
Switching from a homogeneous to a sandwich partition changes the dynamic behavior of the wall considerably, as demonstrated by several extensions on the original Euler-Bernoulli theory aiming at making it applicable to thick, more complex partitions. Reviews of some of these pioneering studies can be found for instance in [23; 24]. In 1990, a sophisticated model describing the lateral displacement of sandwich beams featuring thick core and thin laminates has been proposed by [PERSON] [25], featuring a complex set of ill-conditioned equations that could be solved only with considerable numerical effort. When such a fine level of detail is not required, as in most practical applications, the use of simpler methods can be investigated. In particular, the present work uses the model introduced in 2002 by [PERSON] and [PERSON] [26], who proposed a different approach based on the formulation of the differential equation governing the motion of a sandwich beam according to [PERSON]'s principle. The derivation of the model is summarized in the following. [PERSON]'s principle states that the action integral of a mechanical system, expressed as the integral of the difference between kinetic and potential energy, is stationary. In order to write the founding equation of the model, the laminates are assumed to move in phase. Moreover, pure bending of the complete beam and core shear effects are considered (Figure 3b). The potential energy results as the sum of five contributions: pure bending of the complete beam, shear of the core, bending of the laminates induced by core shear effects and external loads \(F^{\prime}\). Kinetic energy is the sum of two contributions, deriving from lateral and rotational motion of the beam. The resulting formulation of the variational problem leads to a set of differential equations that can be rearranged in a single sixth-order equation in the variable \(\zeta\) and the relative boundary conditions. For the beam represented in Figure 3a, the governing equation is:  
\[ \begin{array}{l}-2D_{1}^{\prime}D_{2}^{\prime}\frac{\partial^{2}\zeta_{c}^{ \prime}}{\partial x^{6}}+2D_{2}^{\prime}I_{\omega}^{\prime}\frac{\partial^{2} \zeta}{\partial x^{4}\partial t^{2}}+G_{c}SD_{1}^{\prime}\frac{\partial^{4} \zeta}{\partial x^{4}}-\left[\left(D_{1}^{\prime}+2D_{2}^{\prime}\right)m^{ \prime}+G_{c}SI_{\omega}^{\prime}\right]\frac{\partial^{4}\zeta}{\partial x^{2 }\partial t^{2}}+\\ +G_{c}Sm^{\prime}\frac{\partial^{2}\zeta_{c}^{\prime}}{\partial t^{2}}+m^{ \prime}I_{\omega}^{\prime}\frac{\partial^{4}\zeta}{\partial t^{4}}=G_{c}SF^{ \prime}-\left(D_{1}^{\prime}+2D_{2}^{\prime}\right)\frac{\partial^{2}F^{ \prime}}{\partial x^{2}}+I_{\omega}^{\prime}\frac{\partial^{2}F^{\prime}}{ \partial t^{2}}.\end{array}  \tag{10}\]  
Here, \(G_{c}\) is the effective shear modulus of the core, \(I_{\omega}^{\prime}\) is the mass moment of inertia of the beam and \(m^{\prime}\) is its mass per unit length, which for a given width \(b\), is calculated as \(\mu\cdot b\). \(D_{2}^{\prime}\) and \(D_{1}^{\prime}\) are the bending stiffnesses of the beam and of one laminate, respectively:  
\[D_{1}^{\prime}=b\left[\frac{E_{c}h_{c}^{3}}{12}+E_{l}\left(\frac{h_{c}^{2}h_{ l}}{2}+h_{c}h_{l}^{2}+2\frac{h_{l}^{3}}{3}\right)\right], \tag{11}\]  
\[D_{2}^{\prime}=b\frac{E_{l}h_{l}^{3}}{12}. \tag{12}\]  
Assuming a solution in the form:  
\[\zeta=\exp\left[\mathrm{i}\left(\omega t-k_{x}x\right)\right] \tag{13}\]  
and setting the external load \(F^{\prime}\) to zero, the dispersion equation relating the wavenumber \(k_{x}\) and the angular frequency \(\omega\) is obtained:  
\[ \begin{array}{l}2D_{1}^{\prime}D_{2}^{\prime}k_{x}^{6}-2D_{2}^{\prime}I_{ \omega}^{\prime}k_{x}^{4}\omega^{2}-\left[m^{\prime}\left(D_{1}^{\prime}+2D_{2 }^{\prime}\right)+I_{\omega}^{\prime}G_{c}S\right]k_{x}^{2}\omega^{2}+\\ +G_{c}S\left[D_{1}^{\prime}k_{x}^{4}-m^{\prime}\omega^{2}\right]+m^{\prime}I_{ \omega}^{\prime}\omega^{4}=0.\end{array}