Sreelal Mohanan,Rajesh G

Department of Aerospace Engineering,Indian Institute of Technology Madras,Chennai,Tamilnadu,600036,India

Keywords: Lift separation Sabot Trajectory Shock reflection

ABSTRACT This paper presents a modi fied analytical model to evaluate the trajectories of various lift separation sabot con figurations.The aerodynamic forces acting on the sabot surfaces during a supersonic flight are modeled in the present analytical model by incorporating the pressures on the windward side of the sabot due to the detached/attached shock and its reflections and then integrated using the 3-DoF dynamical equations.The trajectory and the aerodynamic coef ficients were obtained for these con figurations at a projectile Mach number of 3.The sabot con figurations,which include two new designs,are compared with each other and with the conventional free flight trajectory data of the conventional sabots.The mechanical interaction between the sabot and projectile is also addressed in the present work.The comparison shows that the new designs with the aerodynamic surfaces close to the center of gravity,lift-off from the projectile with minimal mechanical interaction compared to a conventional sabot.

1.Introduction

Armor Piercing Fin Stabilized Projectiles(APFSDS)are used as primary ammunition for Main Battle Tanks(MBT).This type of ammunition uses its enormous amount of kinetic energy to penetrate and destroy the armor.An APFSDS system consists of a considerable length to diameter(L/D)ratio projectile,which is stabilized with the fins,multi-segmented sabots,and an obturator,in general.The sabots are the critical components of the system and should be capable of withstanding very high pressures of the order of hundreds of MPa and transmit this pressure effectively to the sub-caliber projectile.Fig.(1)shows the schematic of such a kinetic energy projectile.The momentum transfer is achieved through the interfacial buttress grooves provided on both the sabot and the projectile.The sabot has an after ramp where high-pressure gas due to the combustion of the propellant acts,which pushes the projectile down the barrel.In order to prevent the leakage of hot combustion gases,an obturator(seal)is provided on the rear bourrelet.A forward bourrelet is used to support the projectile during its in-bore travel.A forward ramp is sometimes used to avoid the failure of the sabot during the initial stages of separation.The sabot is divided into three or four components along the axial plane.Three distinguishable flow regimes present in the transit of the projectile and sabot assembly are in-bore gas flow,muzzle blast,and free flight regime.Once the assembly of projectile and sabots exit the gun tube after the in-bore travel,the sabot becomes a parasitic mass and is no longer needed.The discard of the sabot components which needs to be taken place as quickly as possible hence occurs typically in the muzzle blast in the vicinity of the launch tube exit.The most important considerations in the designing of the sabot,which separates rapidly are the trade-off between the dispersion of the projectile and the aerodynamic force required for the quick separation of the sabot without compromising the force transmission ability.Improved dispersion accuracy requires minimizing the magnitude of disturbances acting on the projectile during the launch and the free flight.Improved penetration requires traveling at higher velocities by minimizing the aerodynamic drag.These two objectives can be achieved by a sabot discard with minimal interactions with the projectile.

As it is generally not desirable to change the gun system to enhance the muzzle velocity and the dispersion accuracy,gun system/projectile designers prefer modifying the con figuration of the sabot and choosing a sabot material having very good strength to weight ratio.When it comes to the con figuration changes,the structural integrity of the sabot cannot be compromised,and it becomes a challenging task to design a sabot with less weight and superior structural performance.An ideal con figuration for a sabot can be thought of as one with less weight,and which separates from the projectile without disturbing the trajectory of the projectile.

Fig.1.Components of a kinetic energy projectile.

Conventional sabots employ two bore riders[1,2],i.e.,a forward bourrelet and a rear bourrelet,as shown in Fig.1 to support the highL/Dratio projectiles inside the gun bore and thereby avoiding the balloting motion[3].Out of these two bore riders,the front bore rider acts as an aerodynamic lifting surface as well to facilitate the separation of the sabot.Since most of the lift forces are concentrated on the forward end of the sabot segment,the rotation of the sabot would lead to mechanical contact[4,5]between projectile and sabot.The mechanical contact being generally asymmetric in nature could induce high yaw/pitch to the projectile leading to the loss of dispersion accuracy.Conventional sabots,hence,are known to have this inherent problem of mechanical interaction and high Target Impact Dispersion(TID).

In order to overcome the disadvantages of the conventional sabots,a solution is proposed[6]to avoid the non-uniform detachment caused by the pressure fluctuations in the front bore rider of the sabot petal.In this design,the front bore rider is replaced by a web member to reduce the turbulence of the incoming air flow,which also acts as a support to reduce the balloting of the projectile.Another interesting modi fication[7]is that the aerodynamic lifting surface is placed close to the center of gravity so that the moment arm of the lift force about the CG of the sabot is signi ficantly less compared to the conventional sabot.These modi fications are radically different from a conventional sabot where the separation is mostly through the rotation of the sabot petals,as shown in Fig.2(a).In this case,where the moment arm is smaller,the separation process is partly through the rotation of the petal and partly through the lifting of the sabot,as shown in Fig.2(b).

Fig.2.Different aerodynamic separation strategies.

Determining the sabot trajectories by simple modeling techniques were the main focus of research in the past.A code named AVCO code[5]was developed to determine the trajectories of a simple wedge-like sabot con figuration.This code used a lot of empirical approximations in the modeling.It was later found that the AVCO code becomes inaccurate to predict the sabot trajectories for complicated sabot con figurations.This led to a modi fication of the AVCO code[8-10],and the code was experimentally validated.Later,the formulation was again modi fied[11,12]to suit the particular con figuration of the sabot depending on the application.

There have been many works that propose to design the sabot with minimal Mechanical interactions.A new sabot design is proposed by Cayzac et al.[13],in which an additional aerodynamic surface is provided on the rear end of the sabot.This new design hence consisted of lifting surfaces at the front and rear ends.The idea was to use the muzzle blast flow to initiate the sabot separation by rotating the sabot clockwise first,and the sabot is then rotated counter-clockwise at the leading when the projectile reaches the free flight.The combination of these two would give the sabot a parallel separation,thereby avoiding the mechanical interaction as much as possible.In this work by Cayzac et al.[13],the trajectory of sabots using axisymmetric Euler equations was successfully predicted,and it was qualitatively validated with the ballistic firing data of a 44 mm projectile.

The trajectory prediction of a particular type of sabot is essential for its performance to be assessed.There are various analytical methods that employ the compressible flow equations and the dynamical equations to predict the sabot trajectory.A combination of Computational fluid dynamics and rigid body dynamical equations is a powerful tool to predict trajectories of different sabot con figurations even though they are costly methods.A simulation was carried out[14]to predict the interior,intermediate,and exterior ballistics of the projectile-sabot assembly where the sabot has a rear lifting surface using a combination of 3-D Euler equations and 6-DoF dynamical equations.It has been found that the new sabot with the rear lifting surface has less possibility for mechanical interaction with the penetrator during separation compared to a conventional sabot.However,the new design was not as good due to inadequate structural response owing to the exposure of its aerodynamic lifting surface to a very high-pressure charge detonation.In another work[15],the simulation started from a point where the projectile is outside of the launch tube to reduce the computational time.Appropriate initial velocity will be assigned to each sabot initially,and the trajectories of the sabot have been predicted using the integration of the aerodynamic forces to be included in the 6 DoF dynamical equations and matched well with experimental data on conventional sabots in free flight.

Another interesting work by Nicolas Eches[16]describes how the obturator breakage before sabot separation may affect the Target Impact Dispersion(TID).They used the finite element method coupled with a simple 1-D aerodynamic model predict the event of obturator breakage.Three different types of obturators in notched and un-notched fashion were described,and it was found that the notched obturator breaks in a symmetric fashion imparting less momentum transfer to the sabot before separation.It is clear from this work that asymmetrical obturator breakage initiates an asymmetrical sabot discard and increases the TID.

The effect of exit velocity[17,18]of the projectile on sabot separation in an electromagnetic rail gun has been studied using a combination of CFD and Rigid body dynamics.It has been found that exit velocity affects projectile velocity variation in the separation phase due to shock wave interactions.Another interesting work[19]includes experiments conducted on a shock tunnel using two sabots and a projectile and simulations using a CFD model specially designed for rail gun launches.

Most of the prior research was focused on the design and optimization of conventional drag separation sabots,which has the inherent flaw of mechanical interaction with the projectile.To date,no analytical,computational or experimental data are available to predict the trajectory of the lift separation sabot con figurations,as the mechanism of separation is signi ficantly different from that of the conventional sabots.The transient aerodynamic forces generated on the sabot petals owing to the complicated fluid flow paths in these cases are to be modeled accurately to predict the sabot trajectory.In this work,we attempt to predict the trajectories and the aerodynamic performance of the sabot con figurations,which use the lift force largely to get separated from the penetrator(lift separation sabots).A modi fied analytical method is employed here to estimate the force generated on the sabot petals by modeling the transient shock interactions in the leeward side of the lift separation sabots.We also present two new types of lift separation sabots in this work,which are expected to separate quickly without any mechanical interaction with the projectile when compared to the conventional sabots,as shown in Fig.2(b).

The objective of the present work is hence to analyze the aerodynamic performance and trajectory of the new lift separation sabots compared to the conventional sabots using a modi fied analytical method.Aerodynamic forces and trajectories are computed by first estimating the surface pressure on sabot petals by using the one-dimensional compressible flow equations and integrate the forces and moments.The aerodynamic forces and moments are used in the 3-DoF dynamical equations,which have been solved using the 4th order Runge-Kutta method to obtain the required sabot trajectories.The trajectory of a conventional sabot has been predicted and validated with the experimental and analytical data computed elsewhere.The trajectories of the new lift separation sabots are also analytically predicted and compared with that of the conventional sabots.

2.Formulation of the problem

2.1.General dynamics of motion

Sabot petal trajectories can be predicted by solving the equations of conservation of linear and angular momentum.These equations are written in inertial or non-accelerating frames of reference.However,calculation of aerodynamic forces acting on the sabot petals requires the equations in the body-fixed coordinates,and this enables an invariant moment of inertia tensor.Nonaccelerating coordinate(xf-zf)is aligned with the projectile center of gravity,and body-fixed coordinates(xb-zb)translate as well as rotate about the sabot petal center of gravity as shown in Fig.3.Initially,xfandzfare aligned with each other.The body-fixed six degrees of freedom equations of motion in the scalar form are given(1)through(8).The firing axis relative earth reference frame is xfas shown in Fig.3

Since the geometry of sabot is axisymmetric in nature,evaluation of forces inr-θcoordinates will be convenient.Transformingrθintox-ycoordinate yields:

FrandFθcan be evaluated by multiplying pressure and elemental surface areas of revolution as follows:

WhereFxb,FybandFzbare body fixed components of aerodynamic forces and,p,q，andrare yaw,pitch,and roll rates with respect to the body-fixed coordinates,respectively.This moving coordinate system cannot be easily used for describing the position and orientation of sabot petals.This can be conveniently described in terms of a coordinate system fixed to earth.To analyze the sabot petal trajectories with respect to the earth reference frame,a bodyfixed to earth-fixed transformation is required.The orientation of the sabot petals can be described using Euler angles relative to the earth.This can be achieved through a series of coordinate rotations.The sequence of rotations is(a)rotate(xf,yf,zf)coordinate system about zfaxis through an angleψto(x1,y1,z1),(b)rotate(x1,y1,z1)abouty1axis through an angleθto(x2,y2,z2)and(c)rotate(x2,y2,z2)aboutx2axis through an angleφto(xb,yb,zb).This sequence of rotations is shown in Fig.(4).Whereψ,θandφare Euler angles.

The present analysis assumes that the force in the azimuthal direction(Fyb)is zero.This assumption gives rise to a great deal of simpli fication in Eqs.(1)-(4),and due to this,yaw and roll rates are zero.As a consequence,the problemwill simplify to a 3-DoF motion of the sabot petal or pure longitudinal motion.Variables which are going to zero in Eqs.(1)-(8)are shown in Eq.(9)

The reduced form of equations of motion and appropriate coordinate transformation equations lead to a system of six coupled differential equations.This system of Eq.(10)is solved using the fourth-order Runge-Kutta method.The simpli fied equations are given below.

Fig.3.Coordinate system for predicting sabot trajectories.

Fig.4.Sequence of rotations to reach the body fixed coordinates.

2.2.Aerodynamic model for the sabot and the projectileestimation of forces

2.2.1.Assumptions used in the present aerodynamic modeling

(a)As the projectile exits the muzzle,the projectile will be immediately engulfed in the highly under expanded supersonic jet.Usually,the pressure at the muzzle exit will be on the order of hundreds of bar for a short period,and this gas pressure clamps the sabot components together around the projectile,which may generate a pivoting action at the tail of the sabot and this may further delay the separation of the sabot.However,due to the very high velocity of the kinetic energy projectile,the projectile quickly overtakes the unsteady jet,which suddenly expands to low pressures owing to its very high degree of expansion.The present analysis does not hence consider the effect of muzzle blast on the sabot trajectories due to the highly transient nature of the supersonic jet and it allows the model to be signi ficantly simpli fied.

(b)As soon as the sabot petals separate,a gap is created between the sabot petals and the incoming flow will penetrate into this gap and create series of reflections in this area.This pressure causes the lift component as the side of the sabot is oriented to an angle away from the symmetry as shown in Fig.(5).This lift force might be negligible at the leading edge of the sabot due to the lower side area.However,it becomes signi ficant at the sabot bulkhead due to its greater area.The present analysis does not consider the flow between sabot petals as the treatment of the shock reflections in the passage requires a 3-dimensional modeling of the shock waves.

Once the projectile sabot assembly exits the muzzle,it passes through three different flow regimes.These include the muzzle blast,interaction region,and the free flight regime.The interaction flow regime contains the shock wave interactions between the projectile and the individual sabot petals.The present analysis deals mainly with the interaction regions where strong normal shocks and oblique shocks are present.In the interaction region,the pressure distribution is crucial in mainly two areas of the sabot:the sabot front scoop and the windward side of the sabot.All the models developed to date assumed a simple conical front scoop for the simplicity of the analysis.However,all modern sabots are geometrically complex and require modi fications in the previous aerodynamic models.The following section will describe the modi fied analysis,which takes into consideration the complicated flow features arising out of the complex shapes of the sabot petals,to estimate the forces.

Fig.5.Lift component due to flow between sabot petals.

2.2.2.Transientflow evolution during the conventional sabot separation

A typical shadowgraph flow visualization of the sabot discard process[3]reveals the presence of a detached bow shock at the leading edge of the projectile sabot assembly before separation.Fig.6 shows the shadowgraph visualization of the sabot discard process.As the sabot starts to separate from the projectile,the incoming supersonic flow will enter the annular region between the windward side of the sabot and the projectile.The aerodynamic interactions between the sabot and the projectile when they are in close proximity are critical when it comes to the dispersion accuracy of the projectile.The leading edge sabot shock impinges on the projectile surface and further re flects on the sabot and again reflects on the projectile.The nature of these reflections is decided by the incoming flow Mach number and the instantaneous sabot separation angles.This series of reflections will increase the projectile surface pressure during the interaction period leading to uneven pressure distribution on the projectile surface.If all the sabots separate symmetrically from the projectile,these forces will cancel out without causing any disturbance to the projectile.However,in reality,sabot discard is never symmetrical in nature,and the aerodynamic forces tend to create a moment that will cause projectile pitch-up or down/yaw depending on the asymmetry in the sabot discard process.

Fig.6.Shadowgraph visualization of sabot discard[3].

The flow during this phase is highly transient in nature and is extremely dif ficult to be solved analytically,and the complete flow field is a combination of subsonic and supersonic flows,as shown in Fig.7.Various stages of the sabot separation are sketched through Fig.7(a)-7(e),and the shock structures are sketched based on one-dimensional compressible flow theory.The incoming supersonic flow will be stagnated at the leading edge cavity of the sabot,and this initiates the rotation of the sabot petal because of the formation of a very high-pressure zone in the leading edge cavity as shown in Fig.7(a).During shallow angles of attack of the sabot,flow fromthe high-pressure region could enter the windward side of the sabot leading to another recirculation bubble,as shown in Fig.7(b).The presence of a recirculation bubble may lead to the formation of a throat,and the subsonic flow after the detached shock accelerates to supersonic flow,as shown in Fig.7(b).This supersonic flow further decelerates to subsonic flow through a series of oblique shock reflections.The wind tunnel experiments conducted[1]at low angles of attack reveal a pressure increase towards the trailing edge of the sabot,and this is probably due to the deceleration of the supersonic flow.As the angle of attack increases,multiple shock reflections will continue,and a Mach reflection(MR)is possible on the surface of the projectile,as shown in Fig.7(c).Shock reflection patterns adjust to satisfy the boundary conditions,such as the flow tangency to the wall.The flow process just described is analogous to what happens in a supersonic diffuser.When the sabot angle of attack becomes higher than the critical de flection angle for the incoming Mach number,a normal shock will form in front of the sabot,as shown in Fig.7(e).From here onwards,the flow becomes steady throughout the annular region and will be completely subsonic.

2.2.3.Modeling of aerodynamic forces acting on the primary lifting surfaces

The present analysis considers three different designs of sabots for trajectory analysis.One is the conventional sabot,and the others are the new lift separation sabots with different con figurations.Fig.8(a),(b),and(c)represent the conventional and the two new lift separation sabots,respectively.In the two new designs,the primary aerodynamic lifting surfaces are closer to the CG of the sabot.The sabot in Fig.8(c)is designed with a pressurized cavity closer to the CG to enhance the lift force to separate the sabot,in addition to the shorter moment arm.Fig.8(b)and(c)also consist of the rear lifting surface behind the obturator to utilize the muzzle blast pressure to initiate a parallel separation.However,the effect of muzzle blast is not considered in the present analysis due to the reasons stated in section 2.2.1(a).

Fig.7.Flow features at various orientations of sabot.

A modi fied aerodynamic model incorporating the shock interactions in the windward side of the sabot is used to estimate the forces on the sabots,which,in turn,will be used to predict the trajectories.The trajectories of two new lift separation sabot designs will be compared with the existing conventional sabot design by using the improved trajectory prediction model.Fig.9(a)shows a conventional sabot with the primary aerodynamic.

lifting surfaces at the leading edge and a sabot in the present analysis.In this case,the moment arm is the distance between the leading edge and the CGof the sabot.This could lead to the rotation of sabot about the CG and potentially cause a mechanical interaction/contact between the sabot trailing edge and the projectile.The pressure acting on the annular chamber enclosed by the leading edge is assumed to be the stagnation pressure behind a normal shock corresponding to the free-stream Mach number.This value can be calculated using 1-D gas dynamic relations for a normal shock.The forces and moments acting on the sabot petal are given in equations 11 and 12.The included angle of a single sabot is 120°.The azimuthal variation in pressure is ignored in the present analysis,and the total force and moments are obtained by revolving the cross-section to 120°.

WherePis a pressure andnis the unit vector normal to the surface.The moment arm from a reference point to the aerodynamic surface is represented by a vector r.

Fig.10 shows the leading edge of a conventional sabot where segment k is the primary lifting surface.The forces acting on this segment can be resolved into the lift(Fz)and drag force(Fx)by multiplying with the unit normal to the surface.Equation of line segment k is given by(13).

Expression for the unit normal can be calculated from Eq.(15)and is given by:

Fy=0 due to the symmetry about thex-axis.Eq.(17)through(19)can be integrated over the entire surface to obtain the aerodynamic body forces and moments on each segment as given in Eqs.10-12.Revolution angleζfor a sabot petal is the half of the sector angle.All the sabot con figurations in the present study employ 120°sector angle;hence the limit of integration forθin all Eqs.20-22 is with reference to figure(3)is fromto.

Fig.8.Various con figurations of the sabot for the analysis.

Fig.9.Forces and moments acting on the conventional and new sabots.

WhereMybis the pitching moment about y-axis andlcgis the displacement between center of pressure and the CG of sabot petal when the sabot rotates about the CG.However,during the time when contact happens between the sabot tail and projectilelcgis the distance between the center of pressure and tail end of the sabot.The above formulation is applied to the lift separation sabots as well.For the sabot con figuration with the pressurizing hole,the formulation is as shown below.

Fig.11 shows the schematic of the aerodynamic modeling for the sabot con figuration shown in Fig.8(c).This con figuration has two primary lifting surfaces;one is on the outer circumference of the sabot,and the other is in the form of a cavity in the inner face of the sabot.While formulating the problem,the inner lifting surface pro file is modeled as a sin function shown in Eq.(23).The pressures acting on both the lifting surfaces are assumed to be stagnation pressure behind a normal shock for the incoming free stream Mach number.

Fig.(12)shows a detailed view of the inner surface of the sabot with forces acting on the wall.It is assumed that the surface is symmetric about bothxandzaxes.However,in the real case,symmetry about the z-axis cannot be achieved because of the presence of holes to pressurize the cavity.For simplicity,the present analysis assumes the symmetry about the z-axis,and this will lead to the cancelation of forces in thex-direction(drag),as shown in Fig.(12).The only force which prevails is the lift forceFz.In order to calculate the lift force,the surface normal has to be evaluated using Eq.(15)which results in

Fig.10.Leading edge geometry.

Fig.11.Aerodynamic forces acting on the sabot.

Fig.12.Forces acting on the wall due to pressurization of the cavity.

Eq.(29)is the magnitude of lift force for a 120°sabot petal with a sinusoidal inner surface with constantshandl.

2.3.Modeling of aerodynamic forces acting on the windward side(secondary lifting surface)of the sabot

The initial lift produced by the primary lifting surface leads to the formation of an annular chamber of fluid between projectile and sabot.During the initial stages of sabot flight,this chamber acts as a converging duct,and the supersonic flow decelerates in a converging duct non-isentropically.Pressure measurements conducted on wind tunnels[3]reveal that the deceleration takes place through a series of oblique shock reflections present in the annular chamber.A schematic used for the windward side force modeling is shown in Fig.(13).The present analysis considers all the oblique shock reflections with a terminal Mach reflection(MR).A series of shock reflections will lead to a non-uniform increase of pressure along the sabot surface.The span at which each shock reflection cell occupies is represented using Eqs.30-32 derived using trigonometrical relations.Eq.(32)is used to estimate the nth span.The shock reflection terminates as a Mach reflection(MR)followed by the subsonic flow through the passage between the projectile and the sabot.The subsonic flow parameters in the region downstream of the Mach stem in the MR can be calculated by using the area-Mach number relation.However,since the area change is small,the static pressure is assumed to be constant.The area on which the pressure between the penetrator and the sabot is applied is the projected area of the windward side of the sabot where the buttress grooves are present.

Fig.13.Shock reflections on the windward face of sabot.

The sabot keeps on rotating due to the aerodynamic forces,and the angle of attack of the sabot is increasing when the projectile is in the free flight regime.As the angle of attack reaches the maximum turning angle for the incoming Mach number or the stagnation pressure after the shock reflection under the sabot petal becomes insufficient to push the mass flow through the gap between the sabot and the projectile,a normal shock is formed in front of the sabot.All the shock reflections will disappear when either of these conditions is reached,as shown in Fig.7(e).In this condition,the forces on the windward side of the sabot are modeled from the pressure distribution on the sabot derived from the Area-Mach relations and the isentropic relations.

Fig.(13)shows the schematic of the instantaneous position of the sabot and the shock reflections on the windward side.There is a pressure jump across each shock cell when a shock re flects off from the projectile surface.Pressure across each shock is calculated using 1-D inviscid oblique shock relations.Once the shock angle is determined using the de flection angle and the incoming Mach number,the re flected shock position can be calculated from the geometry as follows:

To calculate thenth shock reflection,the value ofZnis calculated using previous cells’shock angles and flow de flection angles.

This quasi-steady analysis is extended for the entire sabot flight duration,where the angle of attack is constantly changing,leading to the change in pressure distribution due to the characteristic changes in the shock reflection patterns.The pressure distribution and the resulting forces on the sabot petals at each time step are coupled to the 3-DoF equations of motion.Once the flow de flection angle exceeds the flow turning angle for the incoming Mach number,a detached shock is formed in front of the sabot petal.Flow beneath the sabot becomes subsonic in this case,and there is no longer a steep rise in pressure.In this case,the sabot enters the free flight,and there will not be any subsequent shock reflections.However,the bow shock from the sabot petal still can interfere with the projectile,and this may cause asymmetries in the projectile trajectory.It should be noted that the projectile is constrained to move in the axial direction,and it is also assumed that the projectile velocity is constant,and the projectile is in the free flight.Trajectory variation due to muzzle blast is neglected in the present analysis.

3.Results and discussion

3.1.Validation of the numerical method

To prove the accuracy of the numerical analysis presented in this work,the results of the trajectory analysis of a conventional sabot have been compared with existing experimental and numerical data.Present analytical work has been validated with experiments conducted[8-10]in the Institute of Advanced Technologies(IAT),University of Texas,as well as the analytical data of the AVCO code[5]developed for the trajectory predictions of the sabot.The experimental data is available for a sabot named,HVP_016 developed by the IAT Hypervelocity launch facility.Table 1shows the geometrical and mass properties of the HVP_016 sabot used for validation.

Table 1 Mass and geometrical properties of HVP_016 sabot used for validation.

The HVP_016 sabot was launched at a free-stream Mach number of 7.5,and the free-stream pressure was maintained at 0.67 bar.Yaw cards,as well as cameras,were placed downstream to measure the radial and angular trajectories of the sabot.Fig.(14)shows the trajectory comparison of the present analysis with the experimental data,as well as that predicted by AVCO code.The main difference between present analysis and AVCO code is that in AVCO code,shock reflections in the windward side of the sabot are not considered.AVCO code uses empirical relations[5]to represent the flow between sabot and projectile.The present model predicts the radial displacement of the CG reasonably well compared to the experimental data,as shown in Fig.(14)while the AVCO code over predicts the radial displacement,probably due to the inaccurate modeling of the shock reflection pressures.

When it comes to the angular displacement of the CG,the predictions of both the present analysis and AVCO code are not as good as the radial displacement.This may be due to the fact that the present model assumes a complete normal shock stagnation pressure recovery at the leading edge of the sabot,while in reality,the pressure at the leading edge would be less due to the stagnation pressure losses which will lead to reduced moment available leading to reduced rotation of the sabot.Nevertheless,the prediction of the present code is better than that of the AVCO code.It is clear from Fig.(14)that present analysis predicts trajectories of the HVP_016 sabots reasonably well,and this gave them the con fidence to proceed to use the present model for the trajectory predictions of the new lift separation sabots.

Fig.14.Trajectories of HVP_016 sabot at a Mach number 7.5 and at a free stream pressure of 0.67 bar.

3.2.Trajectory analysis of lift separation sabots

The present analysis compares the trajectories of various sabot con figurations discussed in the previous section.Three critical parameters estimated are the axial displacement,radial displacement,and the angular motion of the sabot with respect to the projectile.Before going to the detailed kinematic analysis,it is better for clarity,to outline the sequence of events during the sabot separation as follows.At the very beginning,the sabots rotate about the CG and a contact establishes between the sabot and projectile.At this moment,the instantaneous center of rotation shifts from CG to this contact point for a short time.During this contact period,the moment arm is the distance between the center of pressure and this contact point.When the contact between sabot and projectile ceases,the center of rotation again becomes the CG of the sabot.It is challenging to treat these sequences of events analytically because of the complexity of the mechanical contact between sabot and projectile.In order to simplify this problem,present analysis estimates sabot trajectories in two methods.In the first method,it is assumed that sabots rotate about the CG throughout the time of flight.This assumption will be helpful in identifying the later mechanical interaction between sabot and projectile.The mechanical interaction between projectile and sabot is analyzed by tracing the trailing edge of the sabot when the sabot is rotating about its CG.The past analytical trajectory predictions[5,8,11]were carried out by assuming that the sabots rotate about the trailing edge throughout the course of travel instead of CG,and this assumption is not reasonable unless the sabot trailing edge is in contact with the projectile entire time of flight.In the second method,it is assumed that the sabot rotates about the contact point throughout the time of flight.Even though this assumption is only true for a short period,this would help to identify the kinematic differences between sabot trajectories.Table 2 shows the properties of three types of sabots analyzed in the present work.The material chosen for all three types of sabots is aluminum.To arrive at the dimensions shown in the table,a fixed projectile with anL/Dratio of 27 is selected for all three con figurations.

Fig.15 shows the schematic of a 3-DoF sabot C.G.trajectory where z represents the radial trajectory,andθrepresents the angular displacement of the sabot.Fig.16(a)shows the radial(z)CG displacement of the sabot with respect to the projectile surface when sabot rotates about CG.For a fixed down range displacement,the conventional sabot has the least CG radial displacement,and sabot with a hole has the maximum CG radial displacement.However,the rotation of the conventional sabot about its CG is more compared to the new sabots,as shown in Fig.16(b),and this is due to the fact that the aerodynamic forces will be more signi ficant on the leading edge of the conventional sabot leading to higher pitching moments.It is interesting to note that the sabot with the hole has almost comparable rotation with the conventional sabot.This is due to the presence of the cavity ahead of the CG,and the lifting force generated is mostly rotational in nature due to the hemispherical shape of the cavity.Table 3 shows the estimated aerodynamic coef ficients of the sabots in the free flight at zero angle of attack.It is seen that the conventional sabot has the highest drag coef ficient(CD)and the lowest lift coef ficient(CL).This high drag coef ficient is attributed to the design of the leading edge aerodynamic surfaces of the sabot.The conventional sabot has more area normal to the flow compared to the new design.Owing to the maximum moment coef ficient(CM),conventional sabot rotates faster compared to the new sabots.It is worthwhile noting that the angular rotation of the new sabot with the hole is close to that of the conventional ones because of the less transverse moment of inertia of the sabot.The transverse moment of inertia plays an important role in the rotation of the sabot petal.This reduction in the moment of inertia of the new sabot with the hole is due to its typical con figuration,such as the presence of an inner cavity and pressurizing holes.The new design of the sabot employs less mass and radius of gyration to achieve a less transverse moment of inertia.If the moment of inertia of the new sabot with the hole is equal to the conventional sabot,it would have been rotated to a lesser amount due to the less moment arm available for the new design.In order to prove this,a comparison of angular displacement between conventional and new sabot is presented in Fig.16.Fig.(17)shows the comparison of the angular displacements of both the conventional sabot and the sabot with the hole when transverse moments of inertia of both the sabots are equal.

Table 2 Mass and geometric properties of sabots.

Table 3 Aerodynamic coef ficients of sabots at zero angle of attack.

Fig.18(a)shows the trajectory of the trailing edge calculated when the sabot is rotated about CGusing linear interpolation based on the CG displacement and the rigid body kinematics.It is interesting to note that during the initial stages of the flight up to about 2.3 m down rage,the displacement is negative for the conventional sabot.This means that the sabot intersects with the projectile,and a negative displacement is physically not possible for rigid bodies.A remedy would be a contact de finition between both sabot and the projectile.This can hence be interpreted as a mechanical interaction between the sabot and projectile.As the negative values change to positive values,the interaction will end,and the trailing edge will lift-off from the projectile.From Fig.18(a),it can be seen that the longest interaction time is encountered by conventional sabot.The sabot with a hole clears the projectile quickly with minimal interaction due to its high lift coef ficient.The negative displacement can be used to find the contact force between the sabot and projectile once the stiffness value is known.As long as the sabot separation is symmetric,these contact forces will cancel out and will not perturb the trajectory of the projectile.However,in the case of asymmetric separation,and is mostly the case,these forces will be crucial and affect the trajectory.The new sabots discussed in the present discussion would induce minimal perturbing forces to the projectile.Fig.18(b)shows the axial deceleration of the sabots,and the sabot with no hole has the least deceleration.The other two con figurations rotate quickly to a high drag con figuration,and it will further increase the drag due to the exposure of the windward side to the shock waves from the leading edge of the sabot.

Fig.15.Radial and angular trajectories of sabot with respect to projectile.

Fig.19(a)shows the radial displacement of the trailing edge(contact point)of the sabot as the sabot rotates about the trailing edge.In this case,the trailing edge displacement is always positive making it unable to identify the mechanical interaction unlike Fig.18(a),where a mechanical interaction could be identi fied as a negative displacement.Fig.19(b)shows the angular displacement of the sabot as the sabot rotates about the trailing edge.The magnitude of angular displacement is less when compared to the magnitude of rotation about CG as shown in Fig.16(b).The assumption of the center rotation being CG could reasonably predict the trajectory as can be seen from the validation of the analytical model,as shown in Fig.14,and hence the first method seems more realistic.

Fig.16.Trajectory of the C.G.of the sabot for a projectile with Mach number of 3.

Fig.17.Angular displacements of the conventional and the new sabot with a hole for the same transverse moments of inertia.

4.Conclusions

3-DoF trajectory predictions of different sabots were performed using analytical methods using a modi fied aerodynamic model.Two new lift separation sabot con figurations were discussed in comparison to a conventional sabot.All the projectiles were of 25 mm caliber,and the aerodynamic coef ficients were estimated.It is shown that simple one-dimensional compressible flow relations,along with the 3-DoF equations,are powerful enough to predict the trajectory of the sabot petals.Out of the two new sabots discussed,the sabot with a cavity beneath the obturator and pressurizing holes has the least mass and transverse moment of inertia.This helps the sabot to separate easily from the projectile surface.The symmetrical shape of the cavity beneath the obturator helps to produce extra lift force required during the initial stages of separation.Analytical results show that the new lift separation sabot performs better compared to the conventional sabot in terms of the mechanical interaction and the radial displacement from the projectile.The new design has a very high lift coef ficient compared to the conventional sabot,which makes the lift separation possible within a short downrange distance.

Fig.18.Sabot trailing edge radial displacement and axial velocity(u)of sabots(M=3).

Fig.19.Trajectories of sabot when rotated about the trailing edge.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.