## 5 Frequency Response Analysis Using TLE
### Nodal Force Distribution of the Projectile Nose  
To simulate the real penetration excitation environment, the penetration excitation force should be accurately applied to the projectile nose. The projectile has a symmetrical structure. Therefore, the radial forces cancel each other out for vertical penetration, and only the excitation force in the axial direction should be considered. In the frequency response analysis of the projectile-fuse system, the axial penetration resistance was usually evenly distributed to the nodes [31]. However, this cannot accurately represent the excitation of the projectile nose and affects the dominant frequency distribution. Thus, we propose to  
Figure 12: Influence of the thread friction coefficient on the axial stiffness and TLE shear modulus (source elaborated by authors). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Specification** & **Processing Precision** & **Pitch (mm)** & **Length (mm)** & **E (GPa)** & \(\rho\) **(kg/m3)** & \(\
u\) \\ \hline M42 \(\times\) 2-6H/6g-33 & 6 & 2 & 33 & 220 & 7800 & 0.284 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Material parameters (source elaborated by authors). assign the axial penetration resistance to the nodes of the projectile nose according to the node position. Figure 13 shows the force analysis on the oval nose of the projectile. [PERSON] and [PERSON] [32] proposed the relationship between the radial stress on the cavity surface and the cavity expansion velocity based on the cavity expansion theory. The relationship between the radial stress on the projectile surface and the rigid body velocity is expressed as:  
\[\sigma_{n}(V_{z},\phi)=Af_{c}^{\prime}+B\rho(V_{z}\cos\phi)^{2}, \tag{9}\]  
where \(\sigma_{n}\) is the radial stress at the projectile surface, \(f_{c}^{\prime}\) is the unconfined compressive strength of the concrete target, \(V_{z}\) is the rigid body velocity, and \(\rho\) is the target density. \(A\) and \(B\) are target material coefficients, which can be obtained from triaxial tests of the target material. The value of \(B\) is generally within a small range around 1.0 [32; 33]. For concrete materials, \(B\) is approximately 1.0, and \(A\) (\(A=S\)) is related to the compressive strength of the concrete material [34]. Based on many experiments, [PERSON] and [PERSON] [35] obtained the relationship between \(S\) and \(f_{c}^{\prime}\) as \(S=72.0f_{c}^{\prime-0.5}\). For an area of \(ds\) at the projectile nose surface, the axial resistance caused by the radial stress on the projectile surface can be expressed as:  
\[F_{i}=\sigma_{n}ds\sin\theta_{i}=(Af_{c}^{\prime}+B\rho(V_{z}\sin\theta_{i})^{ 2})\sin\theta_{i}ds=(\rho V_{z}^{2}(\sin\theta_{i})^{3}+Sf_{c}^{\prime}\sin \theta_{i})ds \tag{10}\]  
From Equation (10), it can be seen that the axial resistance at any point is a third-order function of \(\sin\theta\). To investigate the relationship between \(\rho V_{z}^{2}(\sin\theta)^{3}\) and \(Sf_{c}^{\prime}\sin\theta\) in Equation (10), the curve of \(\rho V_{z}^{2}(\sin\theta)^{3}/Sf_{c}^{\prime}\sin\theta\) with respect to \(V_{z}\) and \(\theta_{i}\) is shown in Figure 14. Considering that the rigid body velocity \(V_{z}\) gradually decreases to \(0\), \(\rho V_{z}^{2}(\sin\theta)^{3}\) is much smaller than \(Sf_{c}^{\prime}\sin\theta\) for the low-to-medium speed penetration (<500 m/s). Moreover, the closer the location is to the arc-column interface, the smaller \(\theta\) is, and the smaller \(\rho V_{z}^{2}(\sin\theta)^{3}/Sf_{c}^{\prime}\sin\theta\) will be. Therefore, the axial resistance can be regarded as related only to \(Sf_{c}^{\prime}\sin\theta\). The projectile was meshed using the mapped method. The numbers of circumferential and axial nodes are \(s\) and \(n\), respectively (Figure 13). \[\left\{ \begin{array}{c}F_{1}+\cdots F_{i}+\cdots+F_{n}=\frac{F}{s}=\frac{m _{p}a}{s}\\ F_{i}/F_{1}=\frac{Sf_{c}^{\prime}\sin\theta_{i}ds}{Sf_{c}^{\prime}\sin\theta_{ i}ds}=\frac{\sin\theta_{i}}{\sin\theta_{i}}=k_{i}\end{array} \right.