## 2 Theory
### Determination of the Apparent Bending Stiffness  
The frequency-dependent apparent bending stiffness \(D^{\prime}_{a}\) introduced in Section 2.3 can be used in Equations (7)-(9) to obtain the sound transmission coefficient for a sandwich structure. For a finite Bernoulli-Euler beam of length \(L\), the eigenfrequency \(f_{n}\) of the bending wave equation for mode \(n\) can be expressed as:  
\[\omega_{n}=2\pi f_{n}=\frac{(k_{n}L)^{2}}{L^{2}}\sqrt{\frac{D^{\prime}}{m^{ \prime}}} \tag{20}\]  
where the eigenvalues \((k_{n}L)=a_{n}\) depend on the specific boundary conditions. If Equation (20) is rewritten as a function of the apparent bending stiffness, the following expression is obtained:  
\[D^{\prime}_{x,n}=\frac{(2\pi f_{n})^{2}\;m^{\prime}L^{4}}{a_{n}^{4}}. \tag{21}\]  
Approximate values for \(a_{n}\) are reported in Table 1 for the free-free boundary condition [28].  
The natural frequencies of a free-free beam can be measured in a relatively simple way. The beam can be suspended by strings, so to simulate free-free conditions, and excited at one end for example by an impedance hammer (Figure 5). The response of the beam can be measured by means of an accelerometer, whose mass must be negligible with respect to the mass of the tested specimen. The accelerometer has preferably to be mounted close to the opposite endpoint. The resulting signal, usually acceleration or velocity as a function of frequency, displays local maxima corresponding to the system eigenfrequencies, which can be used to obtain \(D^{\prime}_{x,n}\) through Equation (21).  
The \(A\), \(B\) and \(C\) coefficients of Equation (18) can be estimated by applying the least-square (LS) method to the set of \(\left(f_{i},D^{\prime}_{x,i}\right)\) points obtained experimentally for each eigenfrequency, thus allowing the reconstruction of the apparent bending stiffness function along the entire frequency range of interest (usually 50 Hz-5 kHz). These coefficients can be computed once the eigenfrequencies and some simple geometric and mass data of the tested specimen are known. They can be used together with the definitions (19) to estimate some parameters that may be difficult to know in advance, such as the shear modulus \(G_{c}\), the static bending stiffness of the beam \(D^{\prime}_{1}\) and the bending stiffness of the laminates \(D^{\prime}_{2}\).  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \(n\) & **1** & **2** & **3** & **4** & \(\geq\)**5** \\ \hline \(a_{n}\) & 4.73 & 7.85 & 11.0 & 14.14 & \(n\pi+\pi/2\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Approximate \(a_{n}\) value for increasing vibration modes, free-free boundary condition.  
Since many materials have an orthotropic behavior, in order to obtain information along the two main directions, two beams should be cut, in orthogonal directions, from a panel and tested separately. The composed bending stiffness can then be computed with a general formula that is applicable to different types of materials [29].  
It is worth noting that a similar approach as the one developed for beams has been successfully applied to panels from which no beam could be cut, due to the large size of the products [30] or because they had been already installed [31]. In this case, the apparent bending stiffness function can be estimated from point mobility measurements.