## 2 Materials and Methods
### Establishing the Face Sheet Properties Values  
Face sheet material properties were determined by numerically varying their property values in the FEA of a rectangular cantilever beam until an adequate match with the first three natural frequencies from experimental vibration testing of the same beam was obtained. Six sandwich composite rectangular beams with a length of 40 cm, a width of 3.81 cm, and a core height of 1.27 cm were manufactured. Experimental vibration values were acquired by vibrating a double back-to-back cantilever beam configuration with random white noise. The averages of the first three natural frequencies were 27.5 Hz, 245 Hz, and 665 Hz, respectively. Both the face sheet and core were modeled using solid brick elements. The interface between the face sheet and core was assumed to be perfectly bonded, which means that no delamination was allowed. A schematic of the FEA mesh is shown in Figure 2.  
The face sheet material properties needed in the FEA are the mass density \(\rho\), the modulus of elasticity, \(E_{x}\), \(E_{y}\), \(E_{z}\), the shear modulus of elasticity, \(G_{xy}\), \(G_{xz}\), \(G_{yz}\), and the Poisson ratio, \(v_{xy}\), \(v_{zz}\), \(v_{yz}\). The influence these material properties have in predicting the natural frequencies was examined. A sensitivity analysis was performed in which each individual face sheet material property value was significantly increased and decreased (\(\pm\)60%) while the other values were held constant [39]. Core material property values available from the manufacturer were not varied during this process. The results showed that the face sheet properties \(E_{x}\), \(E_{y}\), \(G_{xz}\), \(G_{yz}\), and \(v_{xy}\), \(v_{xz}\), \(v_{yz}\) had little or no influence in predicting the natural frequencies. Thus, the need to accurately obtain ten variables for FEA was reduced to three (\(\rho\), \(E_{z}\), and \(G_{xy}\)).  
These three face sheet values were determined using a quadratic interpolation method [40]. Once all three face sheet property values were obtained, FEA vibration frequencies were calculated and the accuracy of the interpolations was examined with the following objective functions:  
\[f(\omega_{1}^{n},\ \omega_{2}^{n},\ \omega_{3}^{n})=\left(\frac{\omega_{1}^{ \epsilon}-\omega_{1}^{n}}{\omega_{1}^{\epsilon}}\right)^{2}+\left(\frac{ \omega_{2}^{\epsilon}-\omega_{2}^{n}}{\omega_{2}^{\epsilon}}\right)^{2}+ \left(\frac{\omega_{3}^{\epsilon}-\omega_{3}^{n}}{\omega_{3}^{\epsilon}} \right)^{2} \tag{1}\]  
Figure 2: Finite Element mesh used in FEA for the rectangular sandwich cantilever beam.  
Here \(\omega_{1}^{n}\), \(\omega_{2}^{n}\) and \(\omega_{3}^{n}\) are the first three natural frequencies predicted via FEA, and \(\omega_{1}^{e}\), \(\omega_{2}^{e}\) and \(\omega_{3}^{e}\) are the first three natural frequencies measured through experimental testing. This procedure was repeated until the objective function value fell within a desired tolerance. The initial values of the density, \(\rho\), the longitudinal modulus of elasticity, \(E_{x}\), and the transverse modulus of elasticity, \(G_{xy}\), were 1500 kg/m\({}^{3}\), 8000 MPa, and 5500 MPa, respectively, yielding an initial objective function value of 0.02. This process was terminated after six iterations, when the objective function yielded a value of 6.27 \(\times\) 10\({}^{-4}\). The minimizing behavior of the objective function using this method is shown in Figure 3.  
The determined properties of the face sheet allowed accurate prediction of the first three experimental natural frequencies, with the maximum error observed being 4%. The determined material properties, along with the face sheet properties that did not influence the natural frequency calculations in FEA, are given in Table 1.