## 4 Experimental Results
### Validation of Ricochet Model  
The data presented in Figures 9 and 10 clearly show that the penetration behaviour of the ricocheting projectiles can be quite different with the increasing velocity and obliquity. This kind of change in the ballistic behaviour of the impacting plate is basically related to the ricochet angle of the projectile. To validate the ricochet model, the ricochet points indicated in Figure 9 are used to calculate the deflection angle of the projectile, where the striking velocities are, respectively, 398 m/s and 411 m/s. The corresponding experimental results are presented in Figure 11. The average distance between the impacting point and the impacting point of the ricochet is equal to 51.25 cm. The measured deflection angles of the projectile are, respectively, 34.22\({}^{\circ}\) and 33.94\({}^{\circ}\), and the deflection angle is calculated, where the material restitution coefficient \(e\) is 0.2 and the incidence angle \(\theta\) is 60\({}^{\circ}\). The graphical result of the analytical computation is displayed in Figure 12, where \(\omega_{2}=\)103.6 rad/s, \(\beta=-\)28.56\({}^{\circ}\) and \(\omega=61.44^{\circ}\), and the equation of \(PG\) is defined as:  
\[y=1.84x-1.16 \tag{13}\]  
Figure 11: Experimental results of the ricocheting projectile: (**a**) 398 m/s, 30\({}^{\circ}\) obliquity; (**b**) 411 m/s, 30\({}^{\circ}\) obliquity.  
The coordinate of impacting point \(B\) is (0.67, 0.07); thus, the deflection angle (\(\angle\)NAB) is 34.66\({}^{\circ}\), which is the sum of \(\beta\) and the angle between line AB and the \(X\) axis, with an arc tangent of 0.07/0.67. Compared with the experimental results, the prediction error of the deflection angle is no larger than 2.1%.  
To further verify the ricochet model, the experimental data from the ricochet tests conducted by [PERSON] and [PERSON] et al. was utilized to compute the deflection angle, as cited in Figure 220 of [PERSON]'s review paper [7], where the 12.7-mm-diameter, 26.5 g cylindrical monobloc, spin-stabilized hardened steel projectiles with half-cone angles of 30\({}^{\circ}\) projectiles, were striking a 6.35-mm-thick armour plate of 230 BHN at 30\({}^{\circ}\) obliquity (namely \(\theta=60^{\circ}\)) and an impact velocity of 343 m/s. As the ricochet obliquity \(\beta_{f}=54.44^{\circ}\), the deflection angle was the complementary angle of \(\beta_{f}\) and equal to 35.56\({}^{\circ}\). Correspondingly, the analytical calculating result was 37.85\({}^{\circ}\), with \(\omega_{2}=371.76\) rad/s, \(\beta=-41.9^{\circ}\). The prediction error of the deflection angle was 6.44%. The above comparative results demonstrate that the proposed model is accurate and can be an efficient tool to predict the trajectory and range of ricochet and optimize the design of bulletproof structures in engineering.  
It should be noted that the ricochet model is not universal enough to explain all the problems related to the ricochet complex phenomenon. Additional data should be obtained to extend this model. At the present stage, a few assumptions involving rigid body movements are made, and the warhead shape and material properties are not fully considered, but this will be investigated thoroughly in the subsequent study. The model is established based on a momentum analysis, and it is assumed that the deformation of the projectile and target are both elastic plastic. This can be utilized to predict the movement trajectory of a ricochet projectile reasonably, which lays the foundation for optimizing the bulletproof structures. Compared with other existing complex analytical models, the advantage of this model is convenient for application and time-saving for computations.