Skip to main content
SpringerLink
Log in

A Complex Variable Method to Predict an Aerodynamics of Arbitrary Shape Ballistic Projectiles

  • Chapter
  • First Online:
Nonlinear Approaches in Engineering Applications

  • 1292 Accesses

Abstract

Ballistic projectiles (fragments, missiles) thrown due to the explosions can be a greater hazard than the blast wave. Safe distances from the blast can be readily calculated since it falls off as the cube root of distance. Far more complex is predicting how far fragments may be thrown. This work develops an engineering method to predict aerodynamics of explosive ballistic projectiles (EBPs) of arbitrary shapes. Incorporating the numerical solution of the equations of the dynamic motion of projectile with a complex variable method (“linearization of single-bonded area”), the velocities and the trajectories of arbitrary shape EBPs have been determined. The results are compared to previously developed model predictions and the fragment recovery tests results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Anderson, C. E., Predebon, W. W., & Karpp, R. R. (1985). Computational modeling of explosive filled cylinders. International Journal of Impact Engineering, 23, 1317–1330.

    Google Scholar 

  • Baker, W. E., Cox, P. A., Westine, P. S., Kulesz, J. J., & Strehlow, R. A. (1983). Explosion hazards and evaluation. Amsterdam: Elsevier.

    Google Scholar 

  • Bakhtiyarov, S. I. (2017). Complex variable method to predict aerodynamics of arbitrary shape fragmentation. In Proceedings of ISEE 43rd annual conference on explosives & blasting technique, Orlando, 29 January–1 February 2017 (accepted).

    Google Scholar 

  • Bakhtiyarov, S. I. (2013). A complex variable method to predict an aerodynamics of arbitrary shape debris. In Proceedings of 6th IAASS conference “Safety is not an option”, Montreal, 21–23 May 2013, pp. 51–52.

    Google Scholar 

  • Bishop, R. H. (1958). Maximum missile ranges from cased explosive charges. SC-4205 (TR). Albuquerque: Sandia National Laboratories.

    Book  Google Scholar 

  • Bola, M. S., Madan, A. K., Singh, M., & Vasudeva, S. K. (1992). Expansion of metallic cylinders under explosive loading. Defence Science Journal, 42, 157–163.

    Article  Google Scholar 

  • Charron, Y. J. (1979). Estimation of velocity distribution of fragmenting warheads using a modified gurney method. MSc thesis, USA Air Force Inst. of Tech., Ohio.

    Google Scholar 

  • Chou, P. C., Carleone, J., Hirsch, E., Flis, W. J., & Ciccarelli, R. D. (1983). Improved formulas for velocity, acceleration, and projection angle of explosively driven liners. Propellants, Explosives, and Pyrotechnics, 8, 175–183.

    Article  Google Scholar 

  • Cooper, P. W. (1996). Explosives engineering. New York: Wiley-VCH.

    Google Scholar 

  • Cullis, I. G., Dunsmore, P., Harrison, A., Lewtas, I., & Townsley, R. (2014). Numerical simulation of the natural fragmentation of explosively loaded thick walled cylinders. Defense Technology, 10, 198–210.

    Article  Google Scholar 

  • Davis, W. C. (1978). Fragments hazard zone for explosives shots, Los Alamos Internal Report, M-3-78-4. Los Alamos.

    Google Scholar 

  • Davis, W. C. (2003). Introduction to explosives. In J. A. Zukas & W. P. Walters (Eds.), Explosive effects and applications (pp. 1–22). New York: Springer.

    Google Scholar 

  • Elek, P., Jaramaz, S., & Mickovic, D. (2013). Modeling of the metal cylinder acceleration under explosive loading. Scientific and Technical Review, 63, 39–46.

    Google Scholar 

  • Flis, W. J. (1996). Gurney formulas for explosive charges surrounding rigid cores. In: Proceedings of 16th international symposium on ballistics, San Francisco, 23–28 September 1996, pp. 1–11.

    Google Scholar 

  • Grady, D. E. (1981). Fragmentation of solids under impulsive stress loading. Journal of Geophysical Research, 86(B2), 1047–1054.

    Article  Google Scholar 

  • Grisaro, H., & Dancygier, A. N. (2015). Numerical study of velocity distribution of fragments caused by explosion of a cylindrical cased charge. International Journal of Impact Engineering, 86, 1–12.

    Article  Google Scholar 

  • Gurney, R. W. (1943). The initial velocities of fragments from bombs, shells, and grenades, BRL-405. Aberdeen: Ballistic Research Laboratory.

    Google Scholar 

  • Hardesty, D. R., & Kennedy, J. E. (1977). Thermochemical estimation of explosive energy output. Combust and Flame, 28, 45–59.

    Article  Google Scholar 

  • Hirsch, E. (1986). Improved gurney formulas for exploding cylinders and spheres using ‘Hard Core’ approximation. Propellants, Explosives, Pyrotechnics, 11, 81–84.

    Article  Google Scholar 

  • Hirsch, E. (1995). On the inconsistency of the Asymmetric-Sandwich Gurney Formula when used to model thin plate propulsion. Propellants, Explosives, Pyrotechnics, 20(4), 178–181.

    Article  Google Scholar 

  • Huang, G., Li, W., & Feng, S. (2010). Fragment velocity distribution of cylindrical rings under eccentric point initiation. Propellants, Explosives, Pyrotechnics, 35, 1–7.

    Google Scholar 

  • Huang, G. Y., Li, W., & Feng, S. S. (2015). Axial distribution of fragment velocities from cylindrical casing under explosive loading. Internatinal Journal of Impact Engineering, 76, 20–27.

    Article  Google Scholar 

  • Kamlet, M. J., & Fioger, M. (1979). An alternative method for calculating gurney velocities. Combustion and Flame, 34, 213–214.

    Article  Google Scholar 

  • Kamlet, M. J., & Jacobs, S. J. (1968). Chemistry of detonations. I. A simple method for calculating detonation properties of C-H-N-O explosives. The Journal of Chemical Physics, 48, 23–35.

    Article  Google Scholar 

  • Karpp, R. R., & Predebon, W. W. (1975). Calculations of fragment velocities from naturally fragmenting munitions, BRL Report No. 2509. Aberdeen Proving Ground, Maryland: Ballistic Research Laboratories (BRL).

    Book  Google Scholar 

  • Kelleher, J. (2002). Explosive residue: Origin and distribution. Forensic Science Communications, 4(2), 1–11.

    Google Scholar 

  • Kennedy, J. E. (1998). Chapter 7 The gurney model of explosive output for driving metal. In J. A. Zukas & W. P. Walters (Eds.), Explosive effects and applications (pp. 221–257). New York: Springer.

    Chapter  Google Scholar 

  • Kinney, G. F., & Graham, K. J. (1985). Explosive shocks in air (2nd ed.). New York: Springer.

    Book  Google Scholar 

  • Küchemann, D. (2012). The aerodynamic design of aircraft. AIAA Education series. Reston: American Institute of Aeronautics & Astronautics.

    Book  Google Scholar 

  • Lavrentyev, M. A., & Shabat, B. V. (1973). The methods of complex variable functions theory. Moscow: Nauka. 736p (in Russian).

    Google Scholar 

  • Lenz, R. R. (1965). Explosives and bomb disposal guide. Springfield: Thomas Books.

    Google Scholar 

  • Li, W., Huang, G. Y., & Feng, S. S. (2015). Effect of eccentric edge initiation on the fragment velocity distribution of a cylindrical casing filled with charge. International Journal of Impact Engineering, 80, 107–115.

    Article  Google Scholar 

  • Loitsyansky, L. G. (1973). Mechanics of fluids and gases. Moscow: Nauka. 848p.

    Google Scholar 

  • Mock, W., Jr., & Holt, W. H. (1983). Fragmentation behavior of armco iron and HF-1 steel explosive-filled cylinders. Journal of Applied Physics, 54(5), 2344–2351.

    Article  Google Scholar 

  • Mott, N. F. (1947). Fragmentation of shell cases. Proceedings of the Royal Society of London, 300, 300–308.

    Article  Google Scholar 

  • Nystrom, U., & Gylltoft, K. (2009). Numerical studies of the combined effects of blast and fragment loading. International Journal of Impact Engineering, 36, 995–1005.

    Article  Google Scholar 

  • Oxley, J. C., Smith, J. L., Bakhtiyarov, S. I., & Baldovi, P. M. (2017). A complex variable method to predict a range of arbitrary shape ballistic projectiles. Journal of Applied Nonlinear Dynamics, 6(4), 521–530.

    Article  Google Scholar 

  • Oxley, J. C., Smith, J. L., Resende, E., Rogers, E., Strobel, R. A., & Bender, E. C. (2001). Improvised explosive devices: Pipe bombs. Journal of Forensic Sciences, 46(3), 510–534.

    Article  Google Scholar 

  • Oxley, J. C. (2003). The chemistry of explosives. In J. A. Zukas & W. P. Walters (Eds.), Explosive effects and applications (pp. 137–172). New York: Springer.

    Google Scholar 

  • Pearson, J. (1990). Fragmentation model for cylindrical warheads, Report No. NWC TP 7124. China Lake: Naval Weapons Center.

    Google Scholar 

  • Polya, G., & Szego, G. (1962). Izoperimetric inequalities in mathematical physics (p. 336). Moscow: Nauka. (in Russian).

    Google Scholar 

  • Predebon, W. W., Smothers, W. G., & Anderson, C. E. (1977). Missile warhead modeling: Computations and experiments, Report No. 2796. Aberdeen Proving Ground, Maryland: USA Ballistic Research Laboratory.

    Book  Google Scholar 

  • Schiller, L. (1936). Fluids flow in pipes. Moscow: ONTI-NKTp. 230p (in Russian).

    Google Scholar 

  • Smit, G. J. F., & Mostert, F. J. P. D. P. (2001). A novel approach to the multidimensional nature of velocities of fragments originating from convex shaped warheads. In R. C. Iris (Ed.), 19th international symposium of ballistics (pp. 655–661). Thun/Interlaken: RUAG Land Systems.

    Google Scholar 

  • Szmelter, J., Davies, N., & Lee, C. K. (2007). Simulation and measurement of fragment velocity in exploding shells. Journal of Battlefield Technology, 10, 1–7.

    Google Scholar 

  • Taylor, G. I. (1963). Analysis of the explosion of a long cylindrical bomb detonated at the end. In G. Batchelor (Ed.), Scientific papers of G. I. Taylor (Vol. III, pp. 277–286). Cambridge: Cambridge University Press.

    Google Scholar 

  • Thomas, L. H. (1944). Theory of the explosion of cased charges of simple shape, BRL Report No. 475.

    Google Scholar 

  • Wang, M., Lu, F., Li, X., & Cao, L. (2013). A formula for calculating the velocities of fragments from velocity enhanced warhead. Propellants Explosives Pyrotechnics, 38, 232–237.

    Article  Google Scholar 

  • Wesenberg, D. L., & Sagartz, M. J. (1977). Dynamic fracture of 6061-T6 aluminum cylinders. Journal of Applied Mechanics, 44, 643–646.

    Article  Google Scholar 

  • Xiangshao, K., Weiguo, W., Jun, L., Fang, L., Pan, C., & Ying, L. (2013). A numerical investigation on explosive fragmentation of metal casing using smooth particle hydrodynamic method. Materials and Design, 51, 729–741.

    Article  Google Scholar 

  • Zhang, Q., Miao, C. Q., Lin, D. C., & Bai, C. H. (2003). Relation of fragment with air shockwave intensity for explosion in a shell. International Journal of Impact Engineering, 28, 1129–1141.

    Article  Google Scholar 

  • Zulkonski, T. (1976). Development of optimum theoretical warhead design criteria, Report No. ADB015617. China Lake: Naval Weapons Center.

    Book  Google Scholar 

Download references

Acknowledgments

This study was performed under an appointment to the US Department of Homeland Security (DHS) Science & Technology (S&T) Directorate Office of University Programs Summer Research Team Program for Minority Serving Institutions, administered by the Oak Ridge Institute for Science and Education (ORISE) through an interagency agreement between the US Department of Energy (DOE) and DHS. ORISE is managed by ORAU under DOE contract number DE-SC0014664. All opinions expressed in this paper are the author’s and do not necessarily reflect the policies and views of DHS, DOE, or ORAU/ORISE.

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, New Mexico Institute of Mining and Technology, Socorro, NM, USA

    Sayavur I. Bakhtiyarov & Philipp M. Baldovi

  2. Department of Chemistry, DHS Center of Excellence in Explosives, University of Rhode Island, Kingston, RI, USA

    Jimmie C. Oxley & James L. Smith

Authors
  1. Sayavur I. Bakhtiyarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jimmie C. Oxley
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James L. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Philipp M. Baldovi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sayavur I. Bakhtiyarov .

Editor information

Editors and Affiliations

  1. Xiamen University of Technology, Xiamen, China

    Liming Dai

  2. Xiamen University of Technology, Xiamen, China

    Reza N. Jazar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bakhtiyarov, S.I., Oxley, J.C., Smith, J.L., Baldovi, P.M. (2018). A Complex Variable Method to Predict an Aerodynamics of Arbitrary Shape Ballistic Projectiles. In: Dai, L., Jazar, R. (eds) Nonlinear Approaches in Engineering Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-69480-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-69480-1_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-69479-5

  • Online ISBN: 978-3-319-69480-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation