## 4 Results and Discussion  
### Preliminary Analyses  
The analytical approach results expressed in terms of impactor displacement, contact force, impactor velocity, and kinetic energy versus time are shown respectively in Figures 4-7. The results were obtained by considering an increasing number of modes. Figure 4: Analytical solution—impactor displacement. Figure 3: Experimental set-up. Figures 4 and 5 highlight that lower values of \(m\) and \(n\) results in higher values of stiffness. Indeed, increasing the \(m\) and \(n\) values led to a decrease of the peak force and to an increase in the maximum deflection of the plate central point. Additionally, according to Figures 4 and 5, the panel response in terms of impact force and maximum deflection tended to converge as the values of \(m\) and \(n\) increased. Figure 6 shows the trends of the impact velocity considering different \(m\) and \(n\) values. Although the velocity had the same initial and final values, different \(m\) and \(n\) led to different velocity trends. Moreover, as a direct consequence of the variation of the impact velocity, the kinetic energy, shown in Figure 7, showed different behavior when low values of \(m\) and \(n\) were considered. However, as \(m\) and \(n\) increase, the graphs shown in Figures 4-7 appear to be perfectly overlapped. Figure 5: Analytical solution—contact force history. Figure 6: Analytical solution—impactor velocity. Figure 7: Analytical solution—kinetic energy. Then, the numerical results were compared with the analytical ones. The \(m\),\(n\) = 10 analytical results were chosen as the reference analytical solution. In order to define the numerical model which best-fit the analytical results, several parameters were investigated, such as the element type (reduced integration continuum shell elements SC8R and reduced integration solid elements C3D8R), the in-plane element size (1 mm, 2 mm, and 4 mm), and the number of elements along the thickness direction. In particular, Continuum Shell elements were discretized with 1, 12, and 24 elements in the thickness direction, while Solid elements were discretized with 24 elements in the thickness direction (one element per ply). Hence, more than one orientation is associated to the continuum shell elements belonging to the models characterized by less than 24 elements in the thickness direction. In the numerical analyses, the plate was considered simply supported on the edges, as reported in Figure 8. In the framework of preliminary analyses, aimed to compare the analytical model with the numerical ones to determine the numerical parameters which best represent the mechanical behavior of the plate subjected to low velocity impact, both the inter-laminar and intra-laminar damages were neglected. In Figures 9-13, the numerical and analytical results, in terms of displacement of the impactor and contact force, are compared. In particular, the results shown in Figure 9 were obtained by using a shell element formulation, while the results in Figures 10-12 were obtained by using a continuum shell formulation considering, respectively, 1, 12, and 24 elements in the thickness direction. Finally, in Figure 13, the results obtained by using a solid element formulation, with 24 elements in the thickness direction, are introduced. Figure 11: Continuum shell elements with 12 elements in the thickness direction: analytical–numerical comparison. (**a**) Central point displacement; (**b**) contact force. Figure 12: Continuum shell elements with 24 elements in the thickness direction: analytical–numerical comparison. (**a**) Central point displacement; (**b**) contact force. Figure 10: Continuum shell elements with one element in the thickness direction: analytical–numerical comparison. (**a**) Central point displacement; (**b**) contact force. According to the Figures 9-13, Shell and Solid elements' models were not particularly influenced by the in-plane element size.