To study the flight dynamics of fragments, the following input data are considered: initial coordinates, velocities, air density, the fragment’s exposed area, drag coefficient, and mass. The parameters for the fragmentation process are determined through experimental studies and finite element analysis of the natural fragmentation of a high-explosive projectile. The simulation of the natural fragmentation of an explosive projectile shell leverages the finite element method, and stochastic failure theory is applied using the Ansys Autodyn software. The point mass trajectory model is employed to predict the trajectory of a fragment moving under the impact of drag and gravitational force. In the current study, the main focus was on the development of the methods and tools for implementing trajectory models with varying drag coefficients for different flow speeds. Different approaches for determining drag coefficient are discussed. The nonlinear trajectory model was converted to a linear system of differential equations by employing quasi-linearization. The linear system of differential equations was solved using the Haar wavelet method. The fragment trajectory model with improved accuracy can be considered as the final result of the study.
1. Angel, J. Methodology for dynamic characterization of fragmenting warheads. Presented at Guns and Missiles Symposium, New Orleans, LA, 22 April 2008.
https://doi.org/10.21236/ADA506417
2. Ahmed, K., Malik, A. Q., Hussain, A., Ahmad, I. R. and Ahmad, I. Blast and fragmentation studies of a scaled down artillery shell-simulation and experimental approaches. Int. J. Multiphys., 2021, 15(1), 49–72.
https://doi.org/10.21152/1750-9548.15.1.49
3. Arnold, W. and Rottenkolber, E. Fragment mass distribution of metal cased explosive charges. Int. J. Impact Eng., 2008, 35(12), 1393–1398.
https://doi.org/10.1016/j.ijimpeng.2008.07.049
4. Zecevic, B., Terzic, J., Catovic, A. and Serdarevic-Kadic, S. Characterization of distribution parameters of fragment mass and number for conventional projectiles. In New Trends in Research of Energetic Materials, Czech Republic, 2011.
5. Djelosevic, M. and Tepic, G. Probabilistic simulation model of fragmentation risk. J. Loss Prev. Process Ind., 2019, 60, 53–75.
https://doi.org/10.1016/j.jlp.2019.04.003
6. Ugrčić, M. and Ivanišević, M. Characterization of the natural fragmentation of explosive ordnance using the numerical techniques based on the FEM. Sci. Tech. Rev., 2015, 65(4), 16–27.
https://doi.org/10.5937/STR1504016U
7. Kljuno, E. and Catovic, A. Trajectory estimation model for a solid body with an irregular shape undergoing extremely high aerodynamic forces. Period. Eng. Nat. Sci., 2021, 9(2), 545–568.
https://doi.org/10.21533/pen.v9i2.1694
8. Szmelter, J. and Lee, C. K. Prediction of fragment distribution and trajectories of exploding shells. J. Battlef. Technol., 2007, 10(3), 1–6.
9. Zhang, F., He, Y., Xie, W., Wei, N., Li, J., Wang, S. et al. Drag coefficients for elongated/flattened irregular particles based on particle-resolved direct numerical simulation. Powder Technol., 2023, 418, 118290.
https://doi.org/10.1016/j.powtec.2023.118290
10. McCleskey, F. Drag coefficients for irregular fragments. Naval Surface Warfare Center, 1988, 87.
https://doi.org/10.21236/ADA201943
11. Mott, N. F. Fragmentation of shell cases. Proc. R. Soc. Lond. A, 1947, 189, 300–308.
https://doi.org/10.1098/rspa.1947.0042
12. Grady, D. E. and Kipp, M. E. Mechanisms of dynamic fragmentation: factors governing fragment size. Mech. Mater., 1985, 4(3–4), 311–320.
https://doi.org/10.1016/0167-6636(85)90028-6
13. Moxnes, J. F., Frøyland, Ø., Øye, I. J., Brate, T. I., Friis, E., Ødegårdstuen, G. et al. Projected area and drag coefficient of high velocity irregular fragments that rotate or tumble. Def. Technol., 2017, 13(4), 269–280.
https://doi.org/10.1016/j.dt.2017.03.008
14. Catovic, A., Zečević, B., Kadic, S. S. and Terzic, J. Numerical simulations for prediction of aerodynamic drag on high velocity fragments from naturally fragmenting high explosive warheads. In New Trends in Research of Energetic Materials, Pardubice, Czech Republic, 18–20 April 2012.
15. Li, Y., Yu, Q., Gao, S. and Flemming, B. W. Settling velocity and drag coefficient of platy shell fragments. Sedimentology, 2020, 67(4), 2095–2110.
https://doi.org/10.1111/sed.12696
16. Gubinelli, G. and Cozzani, V. Assessment of missile hazards: evaluation of the fragment number and drag factors. J. Hazard. Mater., 2009, 161(1), 439–449.
https://doi.org/10.1016/j.jhaz mat.2008.03.116
17. Hu, J., Chen, H., Yu, Y., Xue, X., Feng, Z. and Chen, X. Experimental study on motion law of the fragment at hypersonic speed. Processes, 2023, 11(4), 1078.
https://doi.org/10.3390/pr11041078
18. Seltner, P. M., Willems, S., Gülhan, A., Stern, E. C., Brock, J. M. and Aftosmis, M. J. Aerodynamics of inclined cylindrical bodies free-flying in a hypersonic flowfield. Exp. Fluids, 2021, 62(9), 182.
https://doi.org/10.1007/s00348-021-03269-6
19. Kivistik, L., Eerme, M., Majak, J. and Kirs, M. Numerical analysis of dynamics of flight of the fragments. AIP Conf. Proc., 2024, 2989(1), 030016.
https://doi.org/10.1063/5.0189331
20. Kivistik, L., Mehrparvar, M., Eerme, M. and Majak, J. Dynamics of flight of the fragments with higher order Haar wavelet method. Proc. Estonian Acad. Sci., 2024, 73(2), 108–115.
https://doi.org/10.3176/proc.2024.2.02
21. Arda, M., Majak, J. and Mehrparvar, M. Longitudinal wave propagation in axially graded Raylegh–Bishop nanorods. Mech. Compos. Mater., 2024, 59(6), 1109–1128.
https://doi.org/10.1007/s11029-023-10160-4
22. Ratas, M., Majak, J. and Salupere, A. Solving nonlinear boundary value problems using the higher order Haar wavelet method. Mathematics, 2021, 9(21), 2809.
https://doi.org/10.3390/math9212809
23. Mehrparvar, M., Majak, J., Karjust, K. and Arda, M. Free vibration analysis of tapered Timoshenko beam with higher order Haar wavelet method. Proc. Estonian Acad. Sci., 2022, 71(1), 77–83.
https://doi.org/10.3176/proc.2022.1.07
24. Abouzaid, K., Bassir, D., Guessasma, S. and Yue, H. Modelling the process of fused deposition modelling and the effect of temperature on the mechanical, roughness, and porosity properties of resulting composite products. Mech. Compos. Mater., 2021, 56(6), 805–816.
https://doi.org/10.1007/s11029-021-09 925-6
25. Lee, J. K. and Lee, B. K. Coupled flexural-torsional free vibration of an axially functionally graded circular curved beam. Mech. Compos. Mater., 2022, 57(6), 833–846.
https://doi.org/10.1007/s11029-022-10003-8
26. Teng, S., Chen, G., Yan, Z., Cheng, L. and Bassir, D. Vibration-based structural damage detection using 1-D convolutional neural network and transfer learning. Struct. Health Monit., 2023, 22(4), 2888–2909.
https://doi.org/10.1177/14759217221137931
27. Bassir, D., Lodge, H., Chang, H., Majak, J. and Chen, G. Application of artificial intelligence and machine learning for BIM: review. Int. J. Simul. Multidisci. Des. Optim., 2023, 14, 5.
https://doi.org/10.1051/smdo/2023005
28. Xiao, Y., Li, Z., Liu, Z., Zang, M. and Zhu, Y. Effect of material design and weak link setting on the energy absorption of composite thin-walled beams under transverse loading. Mech. Compos. Mater., 2021, 57(3), 401–414.
https://doi.org/10.1007/s11029-021-09963-0
29. Yan, Z., Jin, Z., Teng, S., Chen, G. and Bassir, D. Measurement of bridge vibration by UAVs combined with CNN and KLT optical-flow method. Appl. Sci., 2022, 12(10), 5181.
https://doi.org/10.3390/app12105181
