Rigid body dynamics modeling,experimental characterization,and performance analysis of a howitzer

Nchiket TIWARI,Mukund PATIL,Rvi SHANKAR,Ahishek SARASWAT, Riturj DWIVEDI

aDepartment of Mechanical Engineering,Indian Institute of Technology,Kanpur,India

bOrdnance Factory,Kanpur,India

Rigid body dynamics modeling,experimental characterization,and performance analysis of a howitzer

Nachiketa TIWARIa,*,Mukund PATILa,Ravi SHANKARa,Abhishek SARASWATa, Rituraj DWIVEDIb

aDepartment of Mechanical Engineering,Indian Institute of Technology,Kanpur,India

bOrdnance Factory,Kanpur,India

A large caliber howitzer is a complex and cumbersome assembly.Understanding its dynamics and performance attributes’sensitivity to changes in its design parameters can be a very time-consuming and expensive exercise,as such an effort requires highly sophisticated test rigs and platforms.However,the need of such an understanding is crucially important for system designers,users,and evaluators.Some of the key performance attributes of such a system are its vertical jump,forward motion,recoil displacement,and force transmitted to ground through tires and trail after the gun has been fired.In this work,we have developed a rigid body dynamics model for a representative howitzer system,and used relatively simple experimental procedures to estimate its principal design parameters.Such procedures can help in obviating the need of expensive experimental rigs,especially in early stages of the design cycle.These parameters were subsequently incorporated into our simulation model, which was then used to predict gun performance.Finally,we conducted several sensitivity studies to understand the influence of changes in various design parameters on system performance.Their results provide useful insights in our understanding of the functioning of the overall system. ?2016 Production and hosting by Elsevier B.V.on behalf of China Ordnance Society.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Howitzer gun;CAD modeling;Dynamic simulation;Recoil assembly parameters;Sensitivity analysis

1.Introduction

Performance prediction and analysis of artillery weapons have been going on across the world for a long time.Different approaches,analytical,experimental,and numerical,have been deployed to achieve these objectives.Walton et al.[1]have analyzed the performance of hydraulic gun buffers by building a test facility to simulate the reaction loads imposed on the recoil absorbers.They used these facilities to understand the sensitivity of the buffer performance with respect to changes in recoil mass,velocity,fluid viscosity and density.Eksergian’s work[2]is a fairly well known reference used for study of recoil systems.Seah and Ooi[3]have performed FE simulations on an artillery system.They used their model to predict the recoil displacement,pressure and force as functions of time and angle of elevation.Ozmen et al.[4]have conducted static, dynamic and fatigue analyses for a semi-automatic gun locking device,with theaim to reducerecoil forcesacting onthe device. Letherwood and Gunter[5]have simulated the dynamic behavior of wheeled and tracked ground vehicles over at different points of time in their life cycle.To determine the stiffness and damping characteristicsofshock absorbers,Rao andGruenberg [6]developed an electrodynamic shaker based test rig.In this work,we have developed a detailed rigid body model for a typical howitzer gun.This included developing a 3D CAD model and incorporating it into the rigid body dynamics simulation model for the gun.Next,we developed experimental and analytical tools to estimate important design parameters affecting gun performance.Specifically,these tools determined stiffness and damping of recoil mechanism,tire stiffness,and friction coefficient between cradle and guide rails.The estimated design parameters were then incorporated into our simulation model,and the gun’s performance was evaluated.The model was also used to conduct numerous sensitivity studies to understand the influence of variations in key design parameters on system’s performance.

Fig.1.Gun assembly.

1.1.Construction of a gun

Fig.1 provides an overview of the gun assembly.The gun is assembled around a chassis,which is connected to the ground through two tires and a trail during the firing operation.Thus, all the reaction forces are transmitted to the ground through tires and trail.However,when the gun is moving,the trail no longer touches the ground,and instead two extra tires are attached to the chassis.The chassis is the“foundation”for all other important gun sub-assemblies.On top of it is where the saddle is mounted,which provides two supporting arms for placement of the cradle through two trunnion eyes.It is in these eyes that the trunnions of cradle are seated.The cradle is free to rotate around the axis of trunnions.Such an arrangement facilitates changing of the elevation angle(or azimuth)of the gun’s firing direction with the help of an elevating gear mechanism, which is not shown in Fig.1.The cradle houses the recoil system.On its upper side,there are two longish slots,one on each side,which run along its entire length.These slots provide a seat for guide-rail slide mechanism.The assembly is crafted in such a way that the slide mechanism can move back and forth in these slots.

Fig.2 shows the details of guide rails and guide ring subassembly,which is mounted in the cradle.The sub-assembly has two guide rings,front and rear,used for seating the barrel which is shown in Fig.3.The barrel is the most important part of the entire system.Its rear and front ends are termed as breech and muzzle ends.At the breech end,attached are a breech ring and a breech block.The latter component acts as a door,which is firmly shut once the projectile and charge are inserted in the barrel for firing.The muzzle end is connected to the muzzle break.The barrel with its breech assembly and the muzzle brake fits firmly into the guide rings.

Fig.2.Guide rails and guide rings.

Fig.3.Barrel.

The gun also has a recoil and counter-recoil mechanism, which serves two important functions.Firstly,such a system absorbs extremely high recoil forces which are generated during gun firing.Secondly,the system also ensures that the barrel gets back to its original position post firing of the projectile.Its details are shown in Fig.4.

The recoil and counter-recoil mechanism is made up of a recoil brake assembly,and a recuperator assembly[7].While the former element absorbs recoil energy so that only a small fraction of it is transmitted to the chassis,the latter element ensures that the gun barrel returns back to its original position after the recoil period.A system with an inefficient recoil system will require very heavy chassis.Similarly,a system with a suboptimal counter-recoil system will not bring back the barrel to its original position post firing of a projectile.Both these conditions are not desirable from a standpoint of operational efficiency.The recoil brake system is essentially a damper and has a hydraulic piston–cylinder arrangement.Its cylinder is bolted to the cradle while its piston is bolted to guide rails.The cylinder is filled with viscous oil,which is forced through a large number of orifices by compressive forces generated because of recoil motion of the piston.The recuperator is essentially a spring which is made up of one or two piston–cylinder arrangements.Similar to the recoil system,this assembly is connected on one side to the cradle,and on the other sideto the guide rail.Many recuperators are pneumatic systems filled with pressurized nitrogen.During the recoil movement, this gas stores some of the kinetic energy of the recoiling barrel in a form of potential energy.Once the recoil motion is over, this potential energy brings back the barrel to its original position.The recuperator system also serves one more important purpose.It keeps the barrel in its reference position regardless of the elevation angle when there is no firing going on.This is accomplished through a static force generated by compressed nitrogen which keeps the barrel flush against a constraint.The barrel recoils only when recoiling forces exceed this static force.

Fig.4.Recoil and counter recoil mechanism.

2.Rigid body dynamics model

To determine the performance of the gun,a rigid body dynamics(RBD)model was created and analyzed.The model was built upon a detailed CAD model for the entire assembly. All the parts of the model,unless specified,were rigidly connected to each other.Special care was taken to prescribe appropriate constraints and degrees of freedom(DOF)for those assembly components,which have the freedom to move relative to components they are attached to.These inter-component mobility constraints are given below.

1)Frictional contact was specified between ground and trail, and ground and tires.

2)Translational DOF with friction was specified for so that these components and all other assemblies attached to the rails could move with respect to cradle along the length direction of rails.

3)Translational DOF with friction was specified for the projectile so that it could move along the axis of barrel.

Additionally,some of the connections between specific component pairs were idealized as springs and dampers.The response of these elements was described through prescription of force–displacement and force–velocity relationships which were determined experimentally.Specifically,these elements were located at interface of following component pairs.

1)Piston–cylinder interface in the recuperator assembly was modeled as a nonlinear spring element.

2)Piston–cylinder interface in the recoil cylinder assembly was modeled as a damper element.

3)During firing,the wheels are in locked position,and thus they cannot rotate around their axes.However,reaction forces can drive them to jump upwards and skid on ground.Furthermore,tires have a finite stiffness.Thus, tires were modeled as springs,with one end attached to the axis of the tire,and the other end in frictional contact with the ground.

All these details are also shown in Fig.5,which is a schematic representation of the gun.

When the projectile is fired,it moves outwards of the barrel due to high ballistic pressures built behind it.Also,the same pressure pushes the barrel in the direction opposite to motion of projectile.This motion is opposed by recoil elements having a gas-spring of stiffnessKRand damper with a damping coefficientCR.The recoil as well as counter-recoil motion of the barrel is also resisted due to friction between cradle and rails.μsand μkrepresent the static and dynamic friction coefficients between guide rails and cradle.Furthermore,as the system recoils,it also causes the tires to be pushed against ground,and also to roll backwards.The downward motion of the tires is resisted by tire stiffness(which could be modeled as a springbetween tire axle and ground).This stiffness element(KT)is modeled such that it opposes vertical motion,but does not inhibit tire’s rolling motion.The rolling motion is resisted by frictional contact between tire’s surface and ground.The coefficient of this dynamic friction between the tire and ground is designated as μg.Also,the recoil of barrel also pushes the entire gun system backwards.This horizontal motion is also resisted by the trail.The coefficient of this dynamic friction between the trail and ground is also designated as μg.

Fig.5.Schematic model of rigid body dynamics.

The values of all the stiffness and damping elements were determined experimentally.Experimental procedures were also used to estimate values of dynamic and static coefficients of friction between guide rails and the cradle.Sensitivity studies showed that friction coefficients corresponding to the contact between projectile and barrel’s internal surface have negligible effectonprojectile motion,as well as dynamics of theassembly. This is because the energy dissipated due to such friction is significantly less than the kinetic energy of projectile and the energy contained in rapidly expanding gases which push the barrel rearwards.Hence,the value of this friction coefficient (μb)was arbitrarily set at 0.4.Finally,the friction coefficient between tire and ground as well as between trail and ground (μg)was also set at 0.4.

3.Experimental determination of RBD model parameters

The principal parameters governing the performance of a howitzer are its mass distribution in space,mobility constraints, forces acting on projectile and barrel,stiffness and damping of recoil and counter-recoil mechanisms,friction between cradle and rails,and tire stiffness.In our study,the role of assembly’s overall mass distribution on assembly dynamics was automatically taken care of by incorporating an accurate CAD model of the system in the RBD model of the gun.This model also included appropriate mobility constraints as explained earlier. Also,the value of force acting on the projectile was computed by multiplying the pressure curve,p(t),specific to the charge which when burnt drives the projectile outwards with barrel’s cross-sectional area.Such a pressure curve was provided by the manufacturer.The remaining RBD model parameters,which are stiffness of the recoil mechanism,damping of the recoil mechanism,coefficient of friction between cradle and rails, and tire stiffness,were determined experimentally.Here,we describe experimental procedures used to determine these parameters.

3.1.Friction coefficient between cradle and guide rails

If the recoil and counter-recoil mechanisms are removed from the gun assembly,then Fig.6(a)and(b)represents freebody diagrams of all the mass mounted on cradle in static and dynamic states,respectively.In these figures,the sliding surface,the rectangular block,and angle θ represent the contact surface between cradle and guide rails,the sliding mass on guide rails,and the angle of elevation,respectively.

In the configuration as represented by Fig.6(a),the motion of sliding parts is restricted only by friction,which can be overcome by increasing the elevation angle.For a specific angle of elevation,θs,static frictional force would be just sufficient to negate the pull of gravitational force along the inclined plane. For such a configuration,the following equation can be used to compute coefficient of static friction,μs

Such a condition was simulated experimentally by slowly raising the angle of elevation of the cradle till the mass mounted on it barely started to slide downwards.The angle of elevation corresponding to such a configuration was measured as 30.27°. Thus μswas calculated to be 0.58.

For calculating the coefficient of kinetic friction we have to consider Fig.6(b),which depicts forces acting on the system when the block just starts sliding down with some acceleration. For such a system,the expression for coefficient of kinetic friction μkcan be expressed as

Thus,to estimate coefficient of kinetic friction,the acceleration of the sliding mass over an inclined surface has to be determined.This was done through the experimental setup as shown in Fig.7.

Fig.7.Experimental setup used for measuring kinetic friction coefficient.

Fig.8.Acceleration plot for mass sliding down the cradle slots.

As shown in Fig.7,an accelerometer was attached to front guide rings.Also,arrangements were made to stop the sliding mass when it slid down the cradle and reached its extreme end. With the proper set-up done,the cradle was elevated gradually till the guide rails and the attached barrel barely started moving down the cradle.For such a configuration,the angles of elevation and acceleration readings were recorded.Acceleration was recorded using accelerometer B&K 4517 with sensitivity of 10 mV/g and data acquisition was done using 24-bit resolution NI 9234 DAQ card with voltage range of-5 V to+5 V.Using the sensitivity of the accelerometer and resolution of DAQ card, the overall sensitivity of the system comes out to be 0.58 mm/s2.

Fig.8 shows the acceleration plot for the barrel moving down the inclined surface.Its average value was calculated to be 1.55 m/s2.Substituting this in Eq.(2),gave the value of coefficient of kinetic friction μkas 0.40.

3.2.Stiffness of counter-recoil mechanism

The stiffness of the counter-recoil mechanism is attributable to compressed nitrogen gas in the recuperator cylinder.We computed this stiffness,using a pull-back apparatus as shown in Fig.9.

Fig.9.Experimental setup for measuring recuperator stiffness.

Fig.10.D-bolt with four strain gauges mounted on it.

Here,the barrel with guide rings and breech apparatus was pulled back from the rest position by a known distance using a screw-based mechanism.While the front end of this mechanism was attached to the breech end of the barrel through a series of very stiff chain links,and a machined D-bolt,its rear end was attached to the trail end of the chassis using a series of equally stiff chain links.Also,one of the rear end links was designed in such a way that it is easily and instantly disengaged with other links if needed.In such a system,when the pull back rod is pulled back with the help of the screw mechanism,the air spring in recuperator gets compressed thereby generating a spring force.This force is a function of barrel position,and it was measured through four strain gauges mounted on flattened surfaces of the D-bolt.Fig.10 shows details of the D-bolt which was machined for providing flat mounting surfaces for strain gauges.In this figure,only two strain gauges are visible as the other two strain gauges lie on underside of the bolt arm.The presence of four gauges helped cancel out the effect of bending strains generated in the bolt during the pull-back process.

Strain readings were converted into tensile force by multiplying strain values with the cross-sectional area of the bolt at the location of strain gauges,and the Young’s modulus of the bolt material.In this way,recuperator force corresponding to several positions was calculated.Fig.11 shows the average force–displacement relation for the recuperator.During calibration process,the cross-section of the U-bolt was tuned such that each microstrain in the U-bolt corresponded to 53 N.The experimentally measured value for this parameter was in very good agreement with the theoretically computed value.This parameter was later used to generate the force–displacement curve for the recuperator.Corresponding to a noise level in strain data to being the range of±10 μ?,the amount of inaccuracy in load measurement would be±530 N.Thiscorresponds to an error of 5.3%at the low end of Fig.11,i.e. when the load level was 10,000 N,and 2%at the high-end of the Fig.11,i.e.when the load level was 27,000 N.Finally,we note that the displacement was measured using a simple ruler with a least count of 1 mm.As data for load-displacement curve for the recuperator were recorded in steps of 5 cm,such a least count would introduce negligible errors in our estimates for displacement.

Fig.11.Force–displacement graph of recuperator from experiment.

It is seen in this figure that the force–displacement relationship is somewhat nonlinear.Least square curve fitting method was used to generate an equation for this force–displacement relationship for the recuperator,which was used as an input for our RBD model.

3.3.Damping parameter for the recoil mechanism

Fig.12 is a schematic representation of the moving parts mounted on the cradle corresponding to an elevation angle of θ.

For such a spring–mass–damper system,the value of damping coefficientc,may be computed as

Here,mis the mass of recoiling parts,andK(x)is the position dependent secant stiffness of the system,which can be computed from Eq.(3).For such a system,damping parameterccan be determined if displacement,velocity and acceleration of recoiling parts are known.This was accomplished by initially pulling back the recoiling parts by 31 cm using the pull-back apparatus,and then disengaging the chain link.Such a sudden release of tension in the chain link renders the system out of equilibrium,thereby generating a rapid counter-recoil motion. The acceleration of the counter-recoiling barrel was measured through an accelerometer.These acceleration data were then integrated in time to get velocity and position as a function of time.Fig.13(a)–(c)shows the acceleration,velocity and displacement curves for counter-recoiling parts.Acceleration data were recorded at a rate of 25.6 kS/s using a 24-bit DAQ with a measurement range of±5 V and an accelerometer with a sensitivity of 10 mV/g.Thus the sensitivity of such a system comes out to be 0.58 mm/s2.The noise level observed in the accelerometer data was found to be of the order of 0.03 m/s2.Such a noise level is not significant given that measured values of acceleration as per Figs.8 and 13 were between 1.5 and 70 m/s2. Data in Fig.13(a)–(c)were subsequently used in Eq.(3)to calculate the value of damping coefficient as a function of barrel position.Result for such calculations is shown in Fig.13(d).It is seen in this figure that the damping coefficient remains more or less constant over barrel’s position,and it has average value of 10,810 N-s/m.

Fig.12.Spring–mass–damper system on an inclined plane with friction.

Fig.13.(a)Barrel acceleration data from pull-back experiment(b)Barrel velocity during counter recoil motion(c)Barrel displacement of counter recoiling part(d)Damping coefficient for the recoil system.

3.4.Tire stiffness

Fig.14.Tire deflection due to loadF.

In this analysis,we have assumed that tires behave as linear springs.This assumption was based on the understanding that bulk of the stiffness of tires comes from compression of air,and the change in its air volume is relatively small with respect to the original volume of air inside the tire.Fig.14 depicts the deformed shape of a tire in loaded condition.A first approximation of tire stiffness can be computed by physically measuring parametersRandr,as depicted in the figure,corresponding to loadFon the tire,and then plottingFagainst deflection (R-r).The slope of such a curve will then give us the value of tire stiffness.In our experiment,we measured deflection(R-r) corresponding to three different load conditions.In the first case,cradle and all the components over it were removed from the gun assembly.In the second case,only cradle was attached to the gun assembly.And,in the third case,the barrel assembly was mounted to the gun assembly as well.For each load case, the value of force exerted on the tire was computed through a rigid body static analysis of the system after accounting for the weight distribution of the entire system,and the reaction forces exerted by the tires to balance out these forces.Fig.15 shows the load-deflection plot for the tire.From this graph,average tire stiffness was found to be 692.3 N/mm.

Table 1 lists the values of dynamic parameters used in the model,and which have been calculated using experimental data.

3.5.Dynamic simulation and model verification

Fig.15.Force on tire vs.tire deflection.

Table 1 Dynamic Parameters used in the Model.

Next,trial RBD simulations were run and their results were compared with analytical results to ensure that our model was set up accurately.In such an analysis,the forcing functions for the projectile and the breech end were chosen to be as defined in Fig.16(a).Fig.16(b)and(c)shows that results from analytical calculations,and RBD model simulation as predicted by software agree with each other for different angles of elevation. For a given angle of elevation(θ),the equation used to analytically predict motion of recoiling parts mounted on a stationary chassis can be written as

Fig.16.(a)Force profile for projectile and breech block(b)Analytical and simulation acceleration results of projectile(c)Analytical and simulation acceleration results of gun.

Here,positivex-axis corresponds to axis of barrel pointing outwards,andFpis time-varying force on barrel due to pressureof expanding gases in barrel.This force pushes the barrel and projectile in negative and positivexdirections,respectively. This force was assumed to vary with time as shown in Fig.16(a).

4.Results and discussion

Post setting up of the rigid body dynamics model for the gun, simulation studies were conducted to understand the influence of changes in various design parameters on gun’s performance. Specifically,we conducted such studies by varying recuperator stiffness,damping of recoil brakes,stiffness of tires,and friction coefficient between cradle and barrel.Simulation studies were conducted to understand the effect of these variations on several of gun’s performance parameters.Here we discuss the influence of these parameters on only three parameters:recoil displacement,tire jump,and gun’s forward movement.

4.1.Effect on recoil displacement

The length of recoil mechanism should be sufficient to accommodate for recoil and counter-recoil motions of the barrel.Furthermore,the recoil and counter-recoil systems should be such designed that the barrel reaches back to its original position once the counter-recoil motion is complete. Here,we explore the effect of four design parameters; recuperator stiffness,recoil damping,tire stiffness,and cradle–guide rail friction coefficient on the amplitude of recoil displacement through Fig.17(a)–(d),respectively.In each figure, the design parameter has been varied around its normal value. From these figures,we make the following observations.

1)Recoil damping and cradle–guide friction coefficient strongly influence recoil displacement magnitude.Recoil displacement decreases with increasing values of these parameters.In contrast,changes in recuperator stiffness, and tire stiffness influence have a moderate and marginal influence,on recoil displacement,respectively.

2)A more compliant recuperator system not only tends to increase recoil displacement,but also does not ensure that the barrel returns to its original position.This occurs because the elastic energy stored in such systems is not sufficient to overcome effects of damping and frictional forces beyond a certain position during the counter-recoil stage.Thus the barrel stops short of its original,i.e.in battery position.In contrast,stiffer recuperators not only reduce the overall recoil displacement,but also ensure that the barrel reaches back to the in-battery position. However,such systems require thicker and stronger cylinders for storing high pressure nitrogen gas.Additionally,stiffer recuperators tend to increase the force transmitted to the chassis.This will necessitate either a heavier chassis,or reduced life for the chassis.Either of these consequences is not desirable.

Fig.17.(a)Effect of recuperator stiffness on recoil displacement(b)Effect of recoil damping on recoil displacement(c)Effect of tire stiffness on recoil displacement(d)Effect of friction coefficient between cradle and guide rails on recoil displacement.

3)As mentioned earlier,increasing the system’s damping coefficient or friction coefficient also leads to reduced recoil length.Here,the energy of recoiling parts transforms into heat,and hence it is not transmitted to the ground through the chassis.However,excessive damping has its own implications.Firstly,we see through Fig.17(b)and(d)that excessive damping causes the counter-recoiling barrel to stop mid-way,which is away from the in-battery position.This occurs because the spring forces in recoil energy are not sufficiently high to counter the effects of viscous or damping forces,especially during the counter-recoil portion of barrel movement.Secondly,we note that even though increased friction coefficient drives recoil amplitude to significantly lower values,such a strategy may not be necessarily desirable.This is because excessive friction at the interface ofcradle slots and guide rails generates excessive heat and shear forces,thereby causing increased wear and tear of the cradle–guide interface over gun’s operational life.

4.2.Effect on gun’s forward motion

Post firing of a projectile,the interplay of forces on projectile and reaction forces tends to drag the gun forward.Such a motion is not desirable as it shifts the gun away from its original position thereby necessitating recalibration of gun’s firing parameters to ensure firing accuracy.Fig.18(a)–(d)depicts effects of changes in four design parameters;recuperator stiffness,recoil damping,tire stiffness,and cradle–guide rail friction coefficient on the gun’s forward motion.As seen in the figures,these four design parameters have a significant influence on gun’s forward motion.We note that such a motion increases with increasing recuperator stiffness,and with decreasing tire stiffness.Similarly,increasing of recoil damping coefficient,as well as increasing of cradle–guide rail friction coefficient leads to lesser forward motion because in such situations a larger fraction of system’s kinetic energy gets converted into heat and thus,is not available to induce motion in the gun.Comparing Fig.18(a)–(d),we see that the gun’s forward motion is most sensitive to changes in cradle–guide rail friction coefficient.An excessive friction coefficient can significantly reduce the forward gun motion,but as discussed earlier,it can also cause damage to the cradle–guide rail interface.Furthermore,we have limited flexibility in terms of altering tire stiffness,as it is dominated by the tire’s air pressure. Increasing air pressure in recuperator springs is not also a preferred solution as it increases the load on the chassis,and it also necessitates more sturdy gas cylinder designs.Given these constrains,perhaps the best option maybe is to opt for higher damping coefficients for recoil system.

4.3.Effect on gun’s upwards jump

Post firing of a projectile,the gun not only moves in the forward direction,but it also has a tendency to move upwards as well.Such a motion is undesirable for two reasons.First,such a motion shifts the gun away from its original position thereby necessitating recalibration of gun’s firing parameters to ensure firing accuracy.Second,the gun post its upwards motion falls down.Such a free-fall may damage the gun.Such motion is a strong function of angle of elevation.Typically,the gun has a tendency to move upwards especially for low angles of elevation.For higher angles of elevation,the reaction forces have a significant downwards component,which do not permit the gun to move upwards.With reduced angles of elevation,this downwards component of reaction forces become significantly weaker and the gun develops a significant tendency to move upwards due to the moment of horizontal component of reaction force computed about the point where trail hits the ground. This moment tries to rotate the gun counter-clockwise(when barrel is pointing rightwards in the plane of paper).Such a propensity to exhibit an upwards jump at reduced angle of elevation is seen in Fig.19.As seen in the figure,the magnitude of vertical jump is significantly large when the angle of elevation is 0°,and it becomes negative corresponding to even moderate increases in angle of elevation.For larger angles of elevation,the gun’s tire moves in the negative direction,therebyindicating that it gets compressed beyond its static equilibrium position,and then slowly settles down to its final steady state position.

Fig.19.Effect of angle of elevation on upwards jump.

Fig.20.(a)Effect of recuperator stiffness on upwards jump(b)Effect of recoil damping on upwards jump(c)Effect of tire stiffness on upwards jump(d) Effect of friction coefficient between cradle and guide rails on upwards jump.

Fig.20(a)–(d)depicts the effect of changes in recuperator stiffness,recoil damping,tire stiffness,and cradle–guide rail friction coefficient on the gun’s vertical jump,corresponding to angle of elevation of 30°.These figures show that for such an angle of elevation even significant variations in any of the four key design parameters are not sufficient to induce a vertical upwards jump of the gun.We also see that in general,such a vertical jump is more-or-less insensitive to significant changes in recuperator stiffness,recoil damping,and cradle–guide rail friction coefficient.However,more compliant tires tend to increase the downward motion of tires.Such analysis may be used to reduce the magnitude of positive tire jump corresponding to lesser angles of elevation.

5.Conclusions

Rigid body dynamics simulation is critical to design of a howitzer gun.Such a simulation can help us determine optimal values of principal design parameters of the gun and improve gun’s performance.These parameters could be geometric,as well as dynamic in nature.While geometric parameters influence gun’s dynamics by impacting the system’s mass distribution,the latter category parameters affect gun’s behavior by influencing relative magnitudes of inertial,elastic,and dissipative forces present in the system.In this work,we have developed an accurate Rigid body dynamics model of the system by building upon a detailed and accurate CAD model of the gun, and by using relatively simple methods for estimating key dynamical parameters of a howitzer gun.Specifically,we have devised these methods to estimate tire stiffness,recoil damping, recuperator stiffness,and friction between cradle and guide rails.Our parameter measurement methods help us obviate the need for expensive test rigs,and also reduce the time needed for determining these parameters.The model developed was used to conduct a limited number of sensitivity studies for understanding the influence of changes in four design parameters on recoil displacement,gun’s forward motion,and gun’s vertical jump.The results show that maximizing recoil damping coefficient may be the best way to reduce recoil displacement,and the gun’s forward motion.Other methods,which include increasing recuperator stiffness and increasing cradle–guide rail friction,are also effective methods in this regard.However, such changes can lead to undesirable consequences which are a heavier gun system,a gun with shorter life cycle,and excessive damage to cradle–guide rail interface.Our analysis also shows that the gun has a strong propensity to jump vertically primarily at lower angles of elevation.In general,we have been able to develop an effective Rigid body dynamics model for design of howitzer guns,and an inexpensive suite of procedures for measuring its dynamical parameters.

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Received 16 June 2016;revised 29 September 2016;accepted 10 October 2016 Available online 20 October 2016

Peer review under responsibility of China Ordnance Society.

*Corresponding author.Tel.:+919559270306.

E-mail address:ntiwari@iitk.ac.in(N.TIWARI).

http://dx.doi.org/10.1016/j.dt.2016.10.001

2214-9147/?2016 Production and hosting by Elsevier B.V.on behalf of China Ordnance Society.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).