## 2 Materials and Methods
### Specim Design  
The design of the vibration test specimen was an iterative process. The vibration specimen geometry was obtained by comparing the first resonance vibration stress pattern with the stress pattern of the beam undergoing three-point bending. The vibration specimen design flowchart is given in Figure 4.  
The stress ratio corresponding to the top face sheet centerline profile of a sandwich beam undergoing three-point bending is shown in Figure 5. Here, the stress ratio is defined \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Face Sheet** & **Foam Core** \\ \hline Density (kg/m\({}^{3}\)) & 1117 & 96 \\ \hline Longitudinal Modulus of Elasticity \(E_{Z}\) (MPa) & 40 \(\times\) 10\({}^{3}\) & 21.507 \\ \hline Transversal Modulus of Elasticity \({}^{\star}\)\(E_{x}\), \(E_{y}\) (MPa) & 40 \(\times\) 10\({}^{3}\) & 21.507 \\ \hline Longitudinal Shear Modulus of Elasticity \(G_{XY}\) (MPa) & 10 \(\times\) 10\({}^{3}\) & 8.994 \\ \hline Transversal Shear Modulus of Elasticity \({}^{\star}\)\(G_{XZ}\), \(G_{YZ}\) (MPa) & 10 \(\times\) 10\({}^{3}\) & 8.626 \\ \hline Poisson’s Ratio \({}^{\star}\)\(v_{xy}\), \(v_{xz}\), \(v_{yz}\) & 0.35 & 0.3 \\ \hline \hline \end{tabular}  
* Predicted FEA vibration natural frequencies are not sensitive to significant changes of these properties.  
\end{table} 
Table 1: Material property values used in FEA. Figure 3: Objective function value as a function of properties iterations. by \(R=(\sigma/\sigma_{all})\), in which \(\sigma\) corresponds to the Von Mises stress and \(\sigma_{ult}\) is the average ultimate stress of the several sandwich beams tested. A beam with dimensions of 16 cm \(\times\) 3.81 cm \(\times\) 1.2 cm underwent fully reversible (\(R=-1\)) 0.75 Hz sinusoidal loading in the three-point bending simulation (see stress profile in Figure 5). This profile was used as the baseline for designing the vibration test specimen. Vibrations occurring at the first resonance frequency usually created mode shapes with the largest displacements. For a cantilever beam, this corresponded to a beam oscillating similar to its static (weight) deflection pattern, thus producing the highest stress level at the cantilevered end during its motion. In order to establish the desired profile, similar to Figure 5, geometric changes needed to be performed. Only core thickness modifications were performed, since the initial analysis demonstrated that increasing the core thickness reduces the stress significantly more than increasing the face sheet thickness. The beam modeled in three-point bending was limited in dimensions due to the jig that was available. The vibration beam was  
Figure 4: Approach used in the optimization design of the vibration specimen. Figure 5: The centerline stress ratio profile that was used in the specimen design. designed so as to optimize the vibration's available experimental space. A rectangular beam with the length of 390 mm and the width of 38.1 mm was chosen. During the geometric modifications, the beam's length and width were held constant. The three-point bending face sheet layout (0-90 carbon face sheets bonded with Epoxy resin, 0.5 mm thick) was adopted and held constant in the design process. In addition, the available amplifier and electromagnet shaker restricted reasonable repeatable experiments to a first resonance of around 40 Hz. The design process of the specimen entailed iterative variations in the numerical values of the parametric variables chosen for the design. In general, this process is not limited to the number used in the research. The values can be increased or decreased. A schematic showing the relative location, with respect to the beam's span of the parametric variables used in the design of the core' geometry, is shown in Figure 6.  
As mentioned before, the width (W) and the total length (L1) were held constant. The thickness at H1, H2, H3, and H4, corresponding to positions L1, L2, L3, and the origin 0, respectively, were varied. In order to reduce the stress at the cantilevered end, H4 was increased.