## 3 Similarity Methodology
### Correction for Thickness Distortion (Distorted Model I)  
Distortion occurs when scaled models deviate from scaling according to the same factor, which is a common issue often observed in small-scale models owing to various constraints [6]. Extensive discussions on the characteristics of distorted scaled models can be found in the existing research [11; 22]. Geometric distortions, especially in terms of plate thickness, as illustrated in Figure 9, are frequently encountered in scaled models, and these distortions can lead to performance deviations between the scaled model and the prototype structure. In the proposed methodology, the scaled model, which exhibits the same overall scaling factor as the reference model in all dimensions except for the plate thickness, is designated as distorted model I. Consequently, building upon the similarity law introduced for the reference model and prototype in Section 3.1, the concept of distorted similarity in this section aims to compensate for the influence of the existing plate thickness distortion of distorted model I compared to that of the reference model. Here, the Buckingham procedure is applied to relate the distorted model I and the reference model using the same dimensionless number in Equation (9), where the subscript d1 refers to distorted model I. As detailed in Section 3.1, similarity is achieved when the dimensionless numbers in both systems are identical. The distortion scaling factor for any variable \(X\) between distorted model I and the reference model is defined as follows:  
\[\lambda_{X}=X_{d1}/X_{m}, \tag{17}\]  
In the context of distorted model I, despite the variations in the attached plate thickness, the geometric dimensions remain consistent with those of the reference model, which are expressed by the initial condition \(X_{d1}/X_{m}=1\), except for \((h_{p})_{d1}/(h_{p})_{m}\
eq 1\). This consistency inherently satisfies the major dimensionless numbers in Equation (9). Nonetheless, the deviation stemming from the difference in the attached plate thickness leads to distorted sectional geometric properties compared to the reference model. It is worth noting that the impact of the distorted plate thickness on the sectional geometric attributes is deemed negligible within this section, owing to practical experimental constraints that limit the extent of plate thickness distortion. Importantly, such a distortion is unlikely to significantly alter the predominant membrane behavior of the thin plate within the stiffened structure. Furthermore, the strain rate approximation closely aligns distorted model I with the reference model, ensuring consistency between the initial velocities. This alignment is crucial for maintaining similarity between the two models. Consequently, the corrective approach for the thickness distortion focuses on the impact mass coupled with the dimen  
Figure 9: Schematics of reference model and distorted model I.  
sionless number \(\Pi_{5}\). Leveraging the Buckingham-\(\Pi\) theorem, the corrected impact mass is interconnected with the distorted geometry, as illustrated below:  
\[\left(\Pi_{5}\right)_{m}=\left(\Pi_{5}\right)_{d1}\rightarrow\left(\frac{M_{I}} {M_{S}}\right)_{m}=\left(\frac{M_{I}}{M_{S}}\right)_{d1}\rightarrow\lambda_{M_ {I}}=\lambda_{M_{S}}, \tag{18}\]  
\[\left(M_{I}\right)_{d1}=\lambda_{M_{I}}(M_{I})_{m}=\lambda_{M_{S}}(M_{I})_{m}, \tag{19}\]  
Equation (18) serves as the fundamental equation employed to address the discrepancy in the dynamic responses arising from plate thickness distortion, by adjusting the impact mass according to the distorted geometry. As a result, the dynamic responses of distorted model I closely resemble those of the prototype. In addition, several other scaling factors related to impact events can be derived using dimensionless numbers. Table 5 provides a comprehensive summary of the relationships between distorted model I and the reference model, where the length scaling factor is \(\lambda_{L}=\lambda=1\).