## 5 Frequency Response Analysis Using TLE
### TLE Modeling
The concentration region of the resonance peaks for the distribution method according to Equation (12) is consistent with the experimental FFT spectrum. In addition, the amplitude variation of the resonance peaks exhibits a similar trend as that obtained in the experiment. Thus, the nodal force distribution method using Equation (12) is more reasonable. Figure 17 shows the FRF curve of the TLE and TIED modeling methods. Compared to the experimental FFT spectrum, it can be seen that the two modeling methods have a similar frequency distribution as the experiment, and only at the frequency of 20,284 Hz are the resonance peaks obtained by the two simulation methods not noticeable. This is because the vibration at the frequency of 20,284 Hz is a non-axial vibration mode. Only the nodal force along the axial direction was applied in the FE model. Therefore, this vibration mode was not completely excited. During penetration, it is impossible to guarantee the complete vertical penetration of the projectile. Thus, the resonance peak (20,284 Hz) appears in the experimental spectrum. Compared with the 'TIED' method, the amplitudes of the FRF curve at each frequency obtained with the TLE method are closer to those of the experimental spectrum. Table 10 lists the resonance frequencies of the TLE method, TIED method, and experiment. Each  
Figure 15: FE model of the projectile–fuse system (source elaborated by authors). vibration response characteristics of the projectile-fuse system. The simulation results were in good agreement with the experimental results for a projectile penetrating a semi-infinite concrete target. The results can be summarized as follows:  
1. Several modal tests of threaded tubes were performed to investigate the nonlinear response characteristics of threaded connections. The results indicate that the frictional damping and contact stiffness had a significant impact on the dynamic characteristics, causing the resonance frequency and resonance peak to decrease. 2. An effective method for determining the TLE material parameters was established based on the thread elastic model and TLE principle. Threaded tubes with different specifications were used to verify the accuracy and reliability of the material parameters. The errors were within 5%, indicating the accuracy of the material parameters. 3. The experimental deceleration signal mainly consisted of the shock response and vibration response. The shock response frequency depended on the frequency of the penetration excitation signal. The vibration response was a damped vibration with the first-order axial vibration frequency as the base frequency. 4. A reasonable force distribution method for a projectile nose node was proposed. Compared with the equal distribution method, the proposed method could better simulate the penetration excitation force. This study thus provides a reference for the reasonable distribution of the penetration excitation force in dynamic analyses of projectile-fuse systems. 5. Frequency response analysis of the projectile-fuse system was performed using the TLE. The maximum error in the resonant frequencies between the TLE method and experiment was 7.8%, whereas the maximum error between the existing 'TIED' constraint method and the experiment was 15.7%. In addition, the resonant peak distribution obtained by the TLE method was in better agreement with the frequency distribution of the experimental deceleration signal. Conceptualization, A.Y. and A.P. ; Methodology, A.Y. ; Software, A.P. ; Validation, A.P., F.H. and X.W. ; Writing--original draft preparation, A.Y. ; Writing--review and editing, A.P. and H.Y. ; Supervision, F.H. and X.W. ; project administration, F.H. and X.W. ; funding acquisition, A.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest.