## 3 Similarity Methodology
### The Scaling Law for Perfect Similar Stiffened Plates (Reference Model)
Therefore, using the initial velocity \(v_{0}\), stress \(\sigma_{0}\), and structure mass \(M_{S}\) as fundamental quantities to form a series of dimensionless numbers, we obtain  
\[\underbrace{\left[\frac{M_{0}}{M_{v}v_{0}^{2}}\right]}_{\Pi_{1}},\left[\frac {AM_{I}}{\rho v_{0}^{2}L^{2}}\right]}_{\Pi_{2}},\underbrace{\left[\frac{ \rho v_{0}^{2}}{\sigma_{0}}\right]}_{\Pi_{3}},\underbrace{\left[\frac{\sigma _{0}}{\sigma_{0}}\right]}_{\Pi_{4}},\underbrace{\left[\frac{M_{I}}{M_{S}} \right]}_{\Pi_{5}},\underbrace{\left[\frac{A_{S}}{L^{2}}\right]}_{\Pi_{6}}, \underbrace{\left[\frac{S_{S}}{L^{3}}\right]}_{\Pi_{7}}, \tag{9}\]  
The establishment of similarity laws is based on the principle of equating the terms between a scaled model and a full-scale prototype [6]. These laws provide a structured foundation for establishing correlations between the dimensions of a scaled model designed in proportion to those of the prototype. This alignment of relevant terms within the laws facilitates the analysis and prediction of the prototype behavior through observations made from the scaled model. In the scenario of a perfectly similar scaled model, where the dimensions conform to a uniform scaling factor \(\beta\), specific dimensionless quantities inherently fulfill the conditions of consistency. The pivotal scaling factors governing the attributes of the stiffened plates can be identified using the Buckingham procedure, as computed by Equation (1). Figure 8: Sketch map of stiffened plate section (side view of center section): (**a**) Definition of the calculation segment of the stiffened plate center section; (**b**) deformation mode of the stiffened plate center section. It is worth noting that the strain rate effect is neglected since it is not the primary focus of the proposed methodology. Therefore, consider a reference model with perfectly scaled dimensions and identical material properties, where Equation (1), encompassing \(\Pi_{1}\) to \(\Pi_{7}\), is applied:  
\[\underbrace{\left[\frac{M_{0m}}{M_{Im}v_{0m}^{2}}\right]}_{(\Pi_{1})_{m}}= \underbrace{\left[\frac{M_{0p}}{M_{1p}v_{0p}^{2}}\right]}_{(\Pi_{1})_{p}} \rightarrow\beta_{v_{0}}=\sqrt{\beta_{v_{0}}}=1, \tag{10}\]  
\[\underbrace{\left[\frac{A_{m}M_{Im}}{\rho_{m}v_{0m}^{2}L_{m}^{2}}\right]}_{( \Pi_{2})_{m}}=\underbrace{\left[\frac{A_{p}M_{1p}}{\rho_{p}v_{0p}^{2}L_{p}^{2}} \right]}_{(\Pi_{2})_{p}}\rightarrow\beta_{A}=\frac{\beta_{v_{0}}^{2}}{\beta}= \frac{1}{\beta}, \tag{11}\]  
\[\underbrace{\left[\frac{\rho_{m}v_{0m}^{2}}{\sigma_{0m}}\right]}_{(\Pi_{3})_{ m}}=\underbrace{\left[\frac{\rho_{p}v_{0p}^{2}}{\sigma_{0p}}\right]}_{(\Pi_{3})_{ p}}\rightarrow\beta_{v_{0}}=\sqrt{\beta_{v_{0}}}=1, \tag{12}\]  
\[\underbrace{\left[\frac{\sigma_{m}}{\sigma_{0m}}\right]}_{(\Pi_{4})_{m}}= \underbrace{\left[\frac{\sigma_{p}}{\sigma_{0p}}\right]}_{(\Pi_{4})_{p}} \rightarrow\beta_{v}=\beta_{v_{0}}=1, \tag{13}\]  
\[\underbrace{\left[\frac{M_{\text{Im}}}{M_{Sm}}\right]}_{(\Pi_{5})_{m}}= \underbrace{\left[\frac{M_{1p}}{M_{Sp}}\right]}_{(\Pi_{5})_{p}}\rightarrow\beta _{M_{1}}=\beta_{M_{5}}, \tag{14}\]  
\[\underbrace{\left[\frac{A_{Sm}}{L_{m}^{2}}\right]}_{(\Pi_{6})_{m}}=\underbrace {\left[\frac{A_{Sp}}{L_{p}^{2}}\right]}_{(\Pi_{6})_{p}}\rightarrow\beta_{A_{5 }}=\beta^{2}, \tag{15}\]  
\[\delta=f(L,h_{p},N,\rho,A,M_{0},\sigma,\sigma_{0},v_{0},M_{S},M_{I},A_{S},S_{S }), \tag{8}\]0  
Utilizing these scaling factors facilitates the extrapolation of dynamic responses from a scaled model to a full-sized structure. It is imperative to emphasize that these factors are derived under the assumption that the scaled model adheres to perfect similarity, where the dimensions are proportionally scaled by the same scaling factor \(\beta\). In essence, if geometric attributes, materials, and loading parameters are scaled to maintain identical