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Parametric analysis for water ricochet studies of ogive nose-shaped projectile Vijayalakshmi Murali and Smita D Naik The Journal of Defense Modeling and Simulation: Applications, Methodology, Technology published online 23 April 2014 DOI: 10.1177/1548512914530535 The online version of this article can be found at: http://dms.sagepub.com/content/early/2014/04/21/1548512914530535
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JDMS Original Article
Parametric analysis for water ricochet studies of ogive nose-shaped projectile
Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 1–6 Ó 2014 The Society for Modeling and Simulation International DOI: 10.1177/1548512914530535 dms.sagepub.com
Vijayalakshmi Murali1 and Smita D Naik2
Abstract This paper reports important parameters that affect water ricochet studies for artillery projectiles. The factors affecting ricochet are critical angle and critical velocity of impact. This study has developed mathematical models for ricochet studies and derived expressions for critical angle and critical velocity. Simulations have been carried out for standard bullets, the data for which are available in open literature. The effects of mass, diameter, and length of the projectile on critical velocity and angle are studied through two non-dimensional parameters, m and y. The results suggest how ricochet conditions can be avoided by defining initial data at launch.
Keywords Critical angle, critical velocity, mass, diameter, angle of impact
1. Introduction Ricochet studies are important for offensive and defensive modes of actions. In defensive applications, it helps in defining the initial data for launching the projectile to avoid ricochet. The study of the impact of a projectile against its intended target is of great importance in ballistics. Most of the investigations are carried out in ideal collision conditions that are the normal impact of a projectile against its target. In practical situations, it has been observed that sometimes the projectile misses its intended target and hits elsewhere after one or more bounces, causing unexpected damage to its surrounding environment. Ricochet is the rebounding of the projectile from any suitable surface such as wall, water, or armor plating and leads to the round missing its intended target. It is the deviation of the projectile from its normal path after impacting an object. In armor analysis, ricochet occurs when an attacking projectile glances off the sloped armor of an armed forces vehicle (AFV) without penetrating the plate.1 The resultant flight behavior can be unpredictable and dangerous. Impact phenomena on water have been classified by Johnson as penetration,2 bounce, broach, or ricochet based on Richardson’s definitions.3 Bouncing is described as the rebound due to elastic restitution in either projectile or
target material or both. Ricochet is the entry where the projectile undergoes no change but the target is ploughed. Broaching pertains to rebound where there is complete immersion compared to ricochet where the projectile does not get completely immersed. Subsequently, in the ricochet phase, the projectile is partially submerged or ploughed in the medium with little or no deformation on it. This differentiates it from the process of bouncing and broaching, as described by Johnson.2 The nature of impact depends on the angle of impact, the impact velocity, the density of the projectile, and the target. The study of water ricochet has gained momentum after World War II. Early in 1941, Barnes Wallis investigated the deployment of bombs that could destroy large targets, 1
Department of Engineering Mathematics, MESCOE (Wadia), University of Pune, India 2 Department of Applied Mathematics and Reliability, Armament Research and Development Establishment, DRDO Pashan, Pune, India Corresponding author: Vijayalakshmi Murali, Department of Engineering Mathematics, MESCOE (Wadia), University of Pune, 19, Late Principal Jog Road, Wadia Campus, Pune, 411001, India. Email:
[email protected]
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Journal of Defense Modeling and Simulation: Applications, Methodology, Technology
such as dams in the Ruhr in Germany. The need to surmount the protecting nets of the dam led to ricocheting bombs. In military applications, this technique was used to enhance the range of cannons as well as to increase the damage inflicted on the target, where hitting near the water line was more damaging than being hit by a descending shot.4–7 In terms of weapon performance, this knowledge of ricochet is essential for design of attack of projectiles and also for protective stores or structures.5 Ricochet models have been utilized in aerospace structures and applications to reduce the damage caused by space debris to satellites and spacecraft, as well as in trajectory planning for landing space vehicles back on the Earth’s surface, typically known as ‘skip re-entry’.8 Knowledge of ricochet is used in crime investigation to retrace the path of bullets. Most water impact problems extending from missile technology to marine applications and even bio systems refer to the work performed by Von Karman, who analyzed the forces exerted while landing sea planes.9 Recent developments of water ricochet are motivated by the experimental study of a stone skipping over the water surface. The phenomenon of water ricochet is characterized by the underlying physics; potentially, the model can be seen as a water impact scenario. The mystery surrounding the physics of stone skipping is being solved by specially designed laboratory equipment to skip stones and analyze their motion.10,11 The conservation of momentum ensures that as a projectile enters the water, it exerts a downward force on the water surface and the projectile, in turn, is lifted upwards.12 The earliest water ricochet model was empirically found by Birkhoff et al. for non-spinning spherical projectiles.13 The critical angle (measured in degrees from the surface of the water) for ricochet is approximately given by θc = 18 pffiffiffi σ
ð1Þ
where σ is the specific gravity of the projectile. Advances in computational efficiency and software have made simulation of models easier. This helps in predicting the trajectory during the ricochet impact phase of the projectile with greater accuracy. The numerical analysis performed by Park et al. was used to calculate the various forces acting on a disc during ricochet.14 Another approach was attempted by Nagahiro and Hayakawa using smooth particle hydrodynamics to find out the ‘magic angle’ for stone skipping.15
projectile has been analyzed by Wijk.19 This study develops the critical ricochet angle for an ogive-nosed projectile. The effect of projectile velocity is discussed by evaluating the limits for critical velocity using dynamic equations. This is done by approximating the ogive-nosed projectile to a conical-nosed projectile. Validation is carried out for the critical angle and velocity using data from the open literature. As a projectile enters the water surface, the upward force acting on it will be proportional to the cross-sectional area on the water surface and the square of the velocity. We define the impact angle as the angle measured from the normal to the water surface by the projectile. The critical angle of ricochet is defined as the minimum angle of impact for ricochet to take place: projectiles encountering the water surface at angles greater than the critical angle will ricochet. A condition for critical angle of ricochet is obtained by balancing the momentum due to dynamic pressure induced by the medium and the impending projectile body momentum. The following assumptions made in our earlier work for conical projectiles were also applied for the ogive-nosed model while developing the model:20 i) Water is calm and non-wavy. ii) Yaw variation is ignored. iii) The projectile has attained sufficient velocity for it to ricochet and during this phase, which lasts for a fraction of a second, the velocity of the projectile is assumed to be constant.
2.1 Tangent ogive-nosed projectile An ogive-nosed projectile having a mass m, a cylindrical rear end with diameter d, ogive nose length l, and an ogive front end with tip angle 2θ, where tan θ = d/2l, is considered to impact the water surface (Figure 1). The projectile impacts the surface with an initial velocity v which makes an angle α to the normal direction of the water surface. The projectile’s depth in the water at time t is ξ = vt cos (a)
2. Water ricochet model Extensive research has been done on ricochet of spherical projectiles by Hutchings and others.16–18 However, not much analytical work has been done on conical and ogive-nosed projectiles. Water ricochet of a conical-nosed
Figure 1. Schematic view of a tangent ogive projectile.
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ð2Þ
Murali and Naik
3 1 ðsin αÞ2 ξ 2 ξ3 rvR3 ( ) tanðθ + αÞ + 2 R cos α 12R3 2 1ξ ( tan (θ + α))2 cot θ + tanðθ + αÞ( cot θ)2 8 R2 pffiffiffi 1 ξ4 2 2 ξ5=2 tan (α) pffiffiffiffiffiffiffiffiffiffiffiffiffi + ( tan (θ + α))3 + 4 48 R 5 sin (α) pffiffiffi 7=2 2 ξ cos (α) pffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ 14 sin (α)
N=
Figure 2. Projected areas of wetted portion of projectile.
I, the downward momentum due to penetration, is given by As it proceeds in the water, the projectile’s depth when the front tip just touches the water surface can be derived as,20 ξ* =
d cos (α + θ) 2 sin (θ)
ð3Þ
The areas of the projection (Figure 2) of the wetted portion of the projectile on the water surface include the projected area of the conical part of the projectile, B(ξ), and the projected area of the cylindrical part of the projectile, C(ξ). The areas of the projection are calculated by taking into consideration the ogive equation:21 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 2 U y= U ðx l Þ + 2
ð4Þ
where U is the radius of the circle that forms the ogive and is called the ogive radius; it is related to the length and base diameter of the nose cone by the formula 2
U=
( d2 ) + l2 d
ð10Þ
A balance between the momentum N and I is achieved and the corresponding maximum depth of cut ξ# is obtained by solving N = I. The maximum depth of cut, ξ#, of the projectile is obtained when the downward and upward forces are in equilibrium. The critical angle analysis is based on the fact that the projectile will ricochet only if its depth ξ in water is less than ξ*, the depth when the front tip makes contact with water.10,16,19 The lift force is maximum when the front tip of the projectile touches the water surface. If the projectile gets completely immersed, the lift is not sufficient to sustain the weight of the projectile and it will not ricochet.10,19 The condition for ricochet is observed when ξ#, the maximum depth of cut, is less than ξ*, the depth when the front tip just touches the water. A threshold for the critical angle for ricochet α* is obtained by equating ξ# and ξ* for a projectile of given dimensions. Simulation is used to find the critical angle by varying the different parameters.
ð5Þ
2.2. Critical velocity of ricochet
The projected areas on the water surface are obtained by integrating the ogive equation and the corresponding areas of the nose and cylindrical parts are: ξ ξ2 B(ξ) = R 2 tanðθ + αÞ + 2 ( tan (θ + α))2 cot θ R 4R ξ ξ3 2 3 tanðθ + αÞ( cot θ) + + ( tan (θ + α)) 4R 12R3 pffiffiffiffiffiffi ξ3=2 tan (α) 2R ξ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C(ξ) = 4R sin (α) sin (α) 2
ð6Þ ð7Þ
where 2R = d. The total projected area is DðξÞ = BðξÞ + C ðξÞ
I = mv cos (α)
ð8Þ
The momentum N in the upwards direction normal to the water surface due to the lift force corresponding to dynamic pressure is:
The critical velocity of ricochet for the ogive-nosed projectile is derived by approximating its shape to a conicalnosed projectile. We consider the impact phenomena of the projectile when it hits the water surface. The equation of motion is based on the equations derived for stone skipping by Bocquet.10 In this analysis of the two-dimensional motion through the vertical plane, the forces are resolved in two directions; one in the direction of the projectile motion, Iv, and the other in the direction of the lift force, Jv. Drag and lift forces are proportional to the square of the velocity and immersed area of the projectile. The force equation is given by F=
1 1 Clρv2 SimJv + Cf ρv2 SimIv 2 2
ð11Þ
where Sim is the immersed area on the water surface, r is the density of water, and Cl and Cf are the lift and drag coefficients. During the impact process, the impact angle
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Journal of Defense Modeling and Simulation: Applications, Methodology, Technology
α with the normal to the water surface is close to π2 or the angle made with the water surface is quite small. The initial time t = 0 corresponds to the time when the projectile just starts to enter the water surface. The equation of motion for ξ, the depth in water at any time t, is given by m
d2ξ 1 = mg Cρv2 Sim dt2 2
ð12Þ
where C is the coefficient of drag and lift, Sim the immersed area, and v the velocity at the time of impact. Since the incidence angle with respect to water surface is very small, velocity v is approximated to vx0 the initial velocity along the water surface. The equation of motion is made linear by approximating the projected areas of the immersed part of the projectile to triangular areas of length ‘b’ and ‘c’ (Figure 2) and given by m
d2ξ d = mg Cρðvx0 Þ2 ξðtan α tan (α + θ)Þ ð13Þ dt2 10
or d2ξ + ω2 ξ = g dt2
ð14Þ
where ω2 =
d Cρðvx0 Þ2 ξðtan α tan (α + θ)Þ 10m
g vz0 g sin ωt 2 cos ωt + ω2 ω ω
ð16Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 + v2x0 ðcotαÞ2 ω2 g ð17Þ
ω2
Applying the criteria ξ# < ξ* for projectile ricochet, we get the condition for critical velocity Vc as Vc2
≥
16dmg sin θ cot (α + θ) 0:2d 3 ðcotðα + θÞÞ2 Cρðtan αtan (α + θ)Þ
1
16mðsin θÞ2 ðcotαÞ2 0:2d 3 ðcotðα + θÞÞ2 Cρðtan αtan (α + θ)Þ
Mass of the projectile (kg)
Critical angle (°)
6.40 5.70 5.45 5.10 5.00
0.002 0.002 0.002 0.002 0.002
84.6027 85.3132 85.5710 85.9320 86.0351
Table 2. Variation of mass and diameter. Serial number
Mass m of the projectile (kg)
Diameter d (m)
Critical angle (°)
Ratio: m/ρd3
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
0.90846 1.36028 1.81339 2.71957 4.08091 5.44269 8.16499 10.8870 14.5157 29.0306 30.6400 18.2000
0.0604 0.0691 0.0760 0.0871 0.1000 0.1097 0.1256 0.1382 0.1521 0.1917 0.2062 0.1207
83.1818 83.1818 83.1818 83.1818 83.1818 83.1818 83.1818 83.1818 83.1818 83.1818 82.7750 85.2330
4.123 4.123 4.131 4.116 4.081 4.123 4.121 4.125 4.125 4.121 3.493 10.35
3. Simulation study and results 3.1. Validation of the model
The maximum depth in water during this process, ξ#, is obtained as
ξ# =
Diameter (mm)
ð15Þ
Solving the above differential equation we get ξ=
Table 1. Comparison of critical angle for projectiles of different diameters with constant mass.
Simulations were performed by changing the parameters. By taking the mass as 17 kg, diameter d = 10.5 cm, density of water as r = 1 kg/dm3, and the tip angle θ = 15°, the model gives the critical angle α* as 85.87°, which agrees well with the value obtained by Wijk.19 As shown in Table 1 for a projectile of given mass, as the diameter increases, the critical angle decreases or the range of ricochet increases. As mass increases for a projectile of given diameter, the critical angle increases or the possibility of ricochet reduces.
3.2. A non-dimensional parameter ð18Þ
Simulations were done with standard dimensions of projectiles and variations on critical velocity were found by varying mass as well as diameter. Critical velocity increases with increase in mass for constant diameter and when diameter is increased keeping the mass constant the critical velocity reduced.
A non-dimensional parameter, μ = rdm3 , was identified to analyze the effect of mass and diameter on critical angle. For water, taking r = 1000 kg/m3, the result in Table 2 shows that the projectile with different mass and diameter but having the same value μ has the same critical angle α*. An increase in the value of μ increases the critical angle of ricochet α* and hence decreases the ricochet range. The relation between α* and μ is given approximately by the following equation for a tip angle of 15°:
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Murali and Naik
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Figure 3. Non-dimensional parameter μ versus critical angle α*.
Figure 4. Critical angle α* versus tip angle θ.
α * = 0:00079μ3 0:04μ2 + 0:78μ + 81
ð19Þ
The graph in Figure 3 shows the variation of α* versus μ. Simulation studies have been done to study the effect of variations in diameter d, ogive nose length l, and θ, d where tan θ = 2l and θ can be identified as another dimensional parameter of ricochet. Simulations have been done for a projectile of mass 17 kg and diameter 10.5 cm and by varying the tip angle; the resulting graph is shown in Figure 4. As can be seen from the graph, the critical angle is a minimum for a tip angle of 21° and gradually increases.
4. Conclusions Parameters affecting the water ricochet phenomenon of artillery projectiles are expressed in terms of nondimensional parameters μ and θ. The effects of mass and diameter on critical angle and critical velocity have been analyzed. The analysis helps the designer to decide the parameters at launch for avoiding ricochet and hence the unpredictable behavior of the projectile. Ricochet of projectiles can be designed as per requirement for extending the range or drift in the trajectory.
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Journal of Defense Modeling and Simulation: Applications, Methodology, Technology
Declaration of conflicting interest The author declares that there is no conflict of interest. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. References 1. Lakowski P. Armor technology. Justin Kelly 2008. 2. Johnson W. Ricochet of non-spinning projectiles, mainly from water. Part I. Some historical contributions. Int J Impact Eng 1998; 21: 15–24. 3. Richardson EG. The impact of a solid on a liquid surface. Proc Phys Soc 1948; 61: 352–367. 4. Johnson W. The ricochet of spinning and non-spinning spherical projectiles, mainly from water. Part II. An outline of theory and warlike applications. Int J Impact Eng 1998; 21: 25–34. 5. Johnson W and Reid S. Ricochet of spheres off water. J Mech Eng Sci 1975; 17: 71–81. 6. Beauchant TS. The naval gunner. Hurst, Chance, 1829. 7. Stinner A. Physics and the dambusters. Phys Educ 1989; 24: 260–267. 8. Seddon CM and Moatamedi M. Review of water entry with applications to aerospace structures. Int J Impact Eng 2006; 32: 1045–1067. 9. Von Karman T. The impact on seaplane floats during landing. NASA, 1929.
10. Gold RE and Schecter B. Ricochet dynamics for the nine-millimetre parabellum bullet. J Forensic Sci 1992; 37: 90–98. 11. Rosellini L, Hersen F, Clanet C, et al. Skipping stones. J Fluid Mech 2005; 543: 137–146. 12. Bocquet L. The physics of stone skipping. Am J Phys 2003; 71: 150–155. 13. Birkhoff G, Birkhoff GD, Bleick WE, et al. Ricochet off water. A.M.P. Memo 42.4M, 1944. 14. Park MS, Jung YR and Park WG. Numerical study of impact force and ricochet behavior of high speed water-entry bodies. Comput Fluids 2003; 32: 939–951. 15. Nagahiro S and Hayakawa Y. Theoretical and numerical approach to ‘‘magic angle’’ of stone skipping. Phys. Rev. Lett. 2005; 94: 174501. 16. Hutchings I. The ricochet of spheres and cylinders from the surface of water. Int J Mech Sci 1976; 18: 243–247. 17. Goldsmith W. Non-ideal projectile impact on targets. Int J Impact Eng 1999; 22: 95–395. 18. Truscott TT and Techet AH. Water entry of spinning spheres. J Fluid Mech 2009; 625: 135–165. 19. Wijk G. A water ricochet model. Defence Research Establishment, Weapons and Protection Division 98-03-06, 1998. 20. Murali V, Law MG and Naik SD. Study of critical angle for conical nose-shaped projectiles. AIP Conf Proc 2012; 1482: 58–63. 21. Crowell GA Sr. The descriptive geometry of nose cones. Ahmad Nur Shofa Retrieved 2011.
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