Ballistic Equations Research Articles

Solution of the Equations of Internal Ballistics for the Non Homogeneous Linear Law of Burning

In this paper, the fundamental differential equation of internal ballistics for the non-homogeneous linear law of burning for tubular and cord charges has been solved numerically and results presented graphically. The results for the usual homogeneous linear law of burning have been deduced as a particular case. Tables for maximum pressure and position at all burnt have been prepared. Finally conclusions have been drawn to show the effect of non-homogeneous term in the law of burning, on the maximum pressure and position at all burnt have been prepared. Finally conclusions have been drawn to show the effect of non-homogeneous term in the law of burning, on the maximum pressure and on the all burnt positions.

A Novel Interior Ballistic Prediction of Gun Propellants Based on Experimental Pressure-Apparent Burning Rate Model in Closed Vessel

A new experimental pressure-apparent burning rate model was established based on the recorded pressure-time data in the closed vessel to study the actual burning gas generate rate of a propellant whose form function is not available. The new burning model can replace the Vieille’s law and form function of propellants in the interior ballistic equations to predict the interior ballistic performance of gun. As distinct from the traditional geometric form function based burning rate model, the new model introduced the concept of relative pressure impulse related to the actual burnt web thickness of propellant statistically. The apparent burning rate was expressed by the propellant mass fraction burnt vs. relative pressure impulse curve. The interior ballistic simulation of selected gun propellants was conducted using the new burning rate model. The results predicted by the new model show a good agreement with the classic interior ballistic predication results though the agreement with the experimental results is still to be improved.

Range Prediction and Trajectory Correction of Long Range Rocket with Attitude Stabilization

The dispersion of long range rocket is significantly suppressed by the attitude stabilization in the boost phase. However, the attitude stabilization system cannot govern the propellant impulse error and mass error, which induces large dispersion in the longitudinal direction. Therefore, it is necessary to conduct range correction in the post-boost phase of flight trajectory. A range prediction method based on the elliptic ballistic theory is proposed in this paper. Elliptic ballistic equations and range estimation equations are derived and the modified range prediction method with correction factor is presented. It is verified that the proposed method can predict the residual flight range accurately. After that, the lateral pulse jets control is presented and utilized to correct the trajectory and eliminate the range error. A unique control law is reported that combines elliptic ballistic theory and control mode for lateral pulse jets. The impact point is directly controlled and rocket dispersion in the longitudinal direction is efficiently reduced.

ChemInform Abstract: Discrete Molecular Dynamics

AbstractReview: 56 refs.

Prediction of flyrock trajectories for forensic applications using ballistic flight equations

The paper presents a method for determination of the maximum throw of flyrock fragments and the estimation of safe distances. The method is based upon formulation and solution of differential equations of ballistic flight of the flyrock fragments. The equations are formulated according to Newton's law of motion. Two possible solutions are presented, an approximate numerical solution and the application of the Runge–Kutta algorithm of the fourth order. As an illustration of the presented method a post-accidental forensic analysis case study is given describing the procedure for determination of the input parameters (especially the launch velocity).

Mirrors and smoke: A. V. Hill, his Brigands, and the science of anti-aircraft gunnery in World War I

In 1916 Captain A. V. Hill was transferred from the infantry to the Ministry of Munitions to work on anti-aircraft gunnery. He determined the three-dimensional coordinates of flying objects by placing two mirrors far apart. The mirrors were viewed from a fixed distance above them and observers simultaneously marked the position of the object. He gathered brilliant men, most too old or too young for conscription, who became known as Hill's Brigands. They determined the coordinates of the explosions of shots fired with different fuse settings and fitted them with the ballistic equations to construct accurate gunnery tables. They solved the puzzle of erratic fuse timing at high altitudes. They developed apparatus to locate aircraft by sound. Travelling groups of Brigands worked with anti-aircraft gunners, which Hill regarded as the dawn of operations research. Hill was as adept at leading scientists as he was at doing science.

Generalized Ballistic-Diffusive Formulation and Hybrid SN-PN Solution of the Boltzmann Transport Equation for Phonons for Nonequilibrium Heat Conduction

A generalized form of the ballistic-diffusive equations (BDEs) for approximate solution of the Boltzmann Transport equation (BTE) for phonons is formulated. The formulation presented here is new and general in the sense that, unlike previously published formulations of the BDE, it does not require a priori knowledge of the specific heat capacity of the material. Furthermore, it does not introduce artifacts such as media and ballistic temperatures. As a consequence, the boundary conditions have clear physical meaning. In formulating the BDE, the phonon intensity is split into two components: ballistic and diffusive. The ballistic component is traditionally determined using a viewfactor formulation, while the diffusive component is solved by invoking spherical harmonics expansions. Use of the viewfactor approach for the ballistic component is prohibitive for complex large-scale geometries. Instead, in this work, the ballistic equation is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM) and the control angle discrete ordinates method (CADOM). Results of each method for solving the BDE are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two different two-dimensional transient heat conduction problems at various Knudsen numbers. It is found that standalone CADOM (for BTE) and hybrid CADOM-P1 (for BDE) yield the best accuracy. The hybrid CADOM-P1 is found to be the best method in terms of computational efficiency.

Optimal Design of Cathode’ Working Tooth in ECM for Rifle Based Visual C++

To design an electrochemical machining (ECM) cathode for machining the mixed rifle with large spiral angle and the experiments are difficult, it takes a long period. Visual C++ was adopted to simulate the process for machining rifle as cathode moves according to the Ballistic equation. The models of 3D working tooth on cathode and the processing environment were built and the arithmetic was established. The length of working tooth was initialized. Having compared above points with design parameter, the parameter of cathode’ working tooth will be modified till it be close to design parameter. At last the result for appropriate parameter of cathode’ working tooth was proposed.

Discrete molecular dynamics

AbstractDiscrete molecular dynamics (DMD) has emerged as a simplification of traditional molecular dynamics (MD). DMD employs discrete step function potentials in place of the continuous potential used in traditional MD. As a result, the simulation engine solves the ballistic equations of motion for only those particles participating in a collision, instead of solving Newton's equations of motion for every particle in the system. Because fewer calculations are performed per time step, the DMD technique allows for longer time and length scales to become accessible in the simulation of large biomolecules. The use of coarse‐grained models extends the computational advantage of this method. Although some accuracy is sacrificed to speed, because of the usefulness of DMD to the simulation of many particles at longer timescales, the technique has seen application to diverse molecular systems. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 80–92 DOI: 10.1002/wcms.4This article is categorized under: Structure and Mechanism > Molecular Structures Structure and Mechanism > Computational Biochemistry and Biophysics Molecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods

Jumper Kinematics on Terrain Park Jumps: Relationship between Takeoff Speed and Distance Traveled

Abstract The performance of individuals jumping tabletop features in terrain parks has not been widely studied. A field study was conducted to measure the takeoff speed and horizontal distance achieved by jumpers at two tabletop style jump features: A smaller jump at Snow Summit, CA and a larger jump at Mammoth Mountain, CA, USA. Analyses were also conducted to determine the effects of equipment type (skis versus snowboard) on jumper kinematics. Before each data collection session, the physical dimensions of the jump were measured and recorded. For consecutive jumpers, the speed parallel to the ramp was measured at the end of the takeoff by using a laser speed trap accurate to 0.11 m/s. The landing zone was marked at 3.0 m intervals with colored dye, beginning just past the flat deck portion of the jump. A high-definition video camera was used to record the landing of each jumper, and the landing point of the projected center of mass was determined by using photogrammetry, accurate to within 5 cm. The actual landing distances measured in the field were compared to predicted landing distances by using ordinary ballistic equations. A total of 280 jumps was observed on the two features: 105 on skis and 175 on snowboards. The correlation coefficients between the square of the takeoff speed and the landing distance for the smaller and larger jumps were 0.75 (R2=0.56) and 0.41 (R2=0.16), respectively. The measured landing distances differed significantly from those predicted by models using the ballistic equations of motion. The average landing distance beyond the knuckle was 2.4 m on the smaller jump and 2.1 m on the larger jump. For both features, skier and snowboarder jumpers did not differ in average landing distance.

Optimal Sliding-Mode Terminal Guidance Law Design of Airborne Boost-Phase Ballistic Missile Interception

Aiming at terminal guidance law design of airborne boost-phase ballistic missile interception under acceleration of target without availability, we study on optimal sliding-mode terminal guidance law design of Kinetic-Kill Vehicle. Motion characteristic and infrared characteristic of ballistic missile are analyzed basing on founding ballistic missile boost-phase motion equations. Optimal terminal guidance law of Kinetic-Kill Vehicle is designed basing on undershoot quantity least and energy minimum. Optimal guidance law and augmented proportional navigation law having the same form is proved in theory. Optimal sliding-mode terminal guidance law is designed under acceleration of target without availability, using optimal control theory and sliding-mode control theory. Simulation show that optimal sliding-mode terminal guidance law satisfies required index, reaching the aim to direct hit the target. The correctness and effectiveness of the optimal sliding-mode guidance law are proved.

Numerical Analysis for Dynamic Characteristics of New Launcher with Coupled Rigid and Flexible Motion

This paper is concerned with an analysis of the dynamic characteristics of the high performance launcher—rarefaction wave gun(RAVEN) by numerical simulation. Based on its launch mechanism and launch structure, a rigid-flexible coupling dynamic model which considered the coupling effect between the flexible virbation of the launch barrel and the motion behaviors of the other parts of the RAVEN is established via a subsystems method. The actual motion of the projectile and inertial breech during the lauching are described by the interior ballistic equations of the RAVEN. The dynamic characterisitcs of RAVEN is illustrated by the numerical simulation about a small caliber launcher, and the interaction between launch barrel and the other parts is also studied.

Scalings for a Ballistic Aggregation Equation

We consider ballistic aggregation equation for gases in which each particle is iden- ti?ed either by its mass and impulsion or by its sole impulsion. For the constant aggregation rate we prove existence of self-similar solutions as well as convergence to the self-similarity for generic solutions. For some classes of mass and/or impulsion dependent rates we are also able to estimate the large time decay of some moments of generic solutions or to build some new classes of self-similar solutions.

A simplified quantum mechanical model for nanowire transistors based on non-linear variational calculus

A simplified quantum mechanical model is developed to investigate quantum transport features such as the electron concentration and the current flowing through a silicon nanowire metal-oxide-semiconductor field-effect transistor (MOSFET). In particular, the electron concentration is extracted from a self-consistent solution of the Schrödinger and Poisson equations as well as the ballistic Boltzmann equation which have been solved by exploiting a nonlinear variational principle within the framework of the generalized local density approximation. A suitable action functional has been minimized and details of the implementation and its numerical minimization are given. The current density and its related current-voltage characteristics are calculated from the one-dimensional ballistic steady-state Boltzmann transport equation which is solved analytically by using the method of characteristic curves. The straightforward implementation, the computational speed and the good qualitative behavior of the transport characteristics observed in our approach make it a promising simulation method for modeling quantum transport in nanowire MOSFETs.

Analysis of Dynamic Characteristics for Rarefaction Wave Gun During the Launching

Rarefaction wave gun (RAVEN) propulsion has renewed interest in the fundamental limits of recoil reductions attainable by redirecting propellant gases rearward from a gun without compromising the projectile propulsion. Compared with a conventional gun there is a great difference in the launch process and launch structure. This paper is concerned with an analysis of the dynamic characteristics of this high performance weapon system by numerical simulation. Based on its launch mechanism and launch structure, the vibration equation describing the vibration characteristics of RAVEN was established by vibration theory, which considered the actual movement of the projectile and inertial breech by coupling the interior ballistic equations of the rarefaction wave gun. A rigid-flexible dynamic model, which considered the coupling effect between the elastic vibration of the launch barrel and the dynamic behaviors of the other parts of the RAVEN, is established via a subsystem method. The vibration response of RAVEN during the launch is analyzed by numerical simulation. Comparisons are presented based on the conventional gun, as well as the rules of how the different parameters affect the vibration response. During the launching of RAVEN, the launch barrel shows significant vibration due to the effect of the propellant gases, the inertial breech, and the projectile, and there is some reduction in the vibration amplitude compared with that observed in a conventional closed chamber gun. The vibration amplitude and duration of the launch barrel, which increased with a decrease in the loading density, an increase in the mass of the inertial breech and projectile, and a delay of the venting time, is affected in a more significant manner by changes in loading density and the mass of projectile. The coupled effect between the launch barrel and the other parts of RAVEN are most prevalent in the z-direction. The vibration amplitude along the z-direction is higher than that of the y-direction. When the coupled effect is considered, the transverse vibration response of the flexible barrel has some reduction compared with the one that does not exhibit the coupling effect.

Numerical–analytical model of penetration of long elastically deformable projectiles into semi-infinite targets

A new numerical–analytical model of penetration of long axisymmetric elastically deformable projectiles in semi-infinite targets is presented. A background of this model is the integral–differential equation of ballistics for non-deformable projectile. This equation is obtained on the basis of the Lagrange–Cauchy integral for non-stationary irrotational motion of an incompressible fluid, as well as the solutions for the quasi-static spherical cavity expansion problem in an infinite medium. The velocity field in a target is defined by actual projectile shape. The functional dependence of penetration velocity is determined for both elastic and rigid projectiles. The effect of forced elastic longitudinal oscillations on penetration velocity is estimated. An estimate is made for the critical impact velocity at which point the projectile plastically deforms causing irreversible changes in its shape, and also leads to instability of its trajectory in the target. This velocity depends on both elastic and strength characteristics of the projectile and target, their densities and projectile shape. Results from our penetration modeling are compared with existing experimental and calculated data.

Device Characteristics of Nanoscale Metal/Insulator Tunnel Transistors in the Ballistic Quantum Transport Regime

The device characteristics of the nanoscale metal/insulator tunnel transistor are investigated by solving the ballistic quantum transport equation in the device. The device performance in terms of the transfer characteristics, the drain output, and the threshold voltage change is assessed as the channel length is gradually reduced down to a few nanometer. We have found that the device characteristics remain almost the same if the channel length is reduced to around 10 nanometer, but below it, the device performance becomes drastically degraded. Effects of other device parameters such as the channel depth, tunnel barrier height, and the gate dielectric constant are also discussed.

MODELING LEAF DISPERSAL IN MIXED HARDWOOD FORESTS USING A BALLISTIC APPROACH

In mixed-species stands, modeling leaf litter dispersal is important to predict the physical and chemical characteristics of the forest floor, which plays a major role in nutrient cycling and in plant population dynamics. In this study, a spatially explicit model of leaf litterfall was developed and compared with two other models. These three models were calibrated for a mixed forest of oak and beech using litterfall data from mapped forest plots. All models assumed that an allometric equation described individual leaf litter production, but they strongly differed in the modeling of the probability density of leaf shedding with distance from source trees. Two models used a negative exponential function to account for leaf dispersal with distance, and this function was allowed to vary according to wind direction in one of them. In contrast, our approach was based on a simple ballistic equation considering release height, wind speed, wind direction, and leaf fall velocity; the distributions of wind speeds and wind directions were modeled according to a Weibull and a Von Mises distribution, respectively. Using an independent validation data set, all three models provided predictions well correlated to measurements (r > 0.83); however, the two models with a direction-dependent component were slightly more accurate. In addition, parameter estimates of the ballistic model were in close agreement with a foliar litter production equation derived from the literature for beech and with wind characteristics measured during leaf litterfall for both species. Because of its mechanistic background, such a spatially explicit model might be incorporated as a litterfall module in larger models (nutrient cycling, plant population dynamics) or used to determine the manner in which patch size in mixed-species stands influences litter mixture.

Experimental determination of drag coefficient for volcanic materials: Calibration and application of a model to Popocatépetl volcano (Mexico) ballistic projectiles

This study shows the results of experiments performed in a subsonic wind tunnel to measure drag coefficients (Cd) for volcanic particles emitted by Popocatépetl volcano (Mexico). The results indicate that volcanic projectiles move in a range of Reynolds number (Re) values where Cd is independent of Re. Drag coefficients of volcanic fragments were found between the values of spheres and cubes with low values. In this study, initial conditions of the ballistic equations were calibrated with the “launching” kinetic energy from ballistics expelled by Popocatépetl volcano during the eruptive event of December 17, 1998. The maximum range of the projectiles calculated with the experimentally measured drag coefficient is in better agreement with the field and observational data reported here than using the values of geometrical bodies as in previous studies. Our study can be useful for definition of safety zones around volcanoes to protect people and infrastructure adequately.

Lagrange–Euler–Euler CFD approach for modeling deep-bed filtration in trickle flow reactors

A Lagrange–Euler–Euler computational fluid dynamic approach was developed to represent the evolution of two-phase pressure gradients in trickle-bed reactors undergoing deposition of colloidal/non-colloidal fines under deep-bed filtration conditions. The ballistic trajectory equation to analyze fines interception was extended to two-phase flows using the inclined slit as the basic collector geometry. New equations for the collection efficiencies and filter coefficients were established for the various filtration stages encountered in single-phase flow and trickle flow regime both for monolayer and multilayer collection mechanisms. These slit-based expressions were established for non-polar and non-electrolytic petroleum-like liquids where there is a need for the refining industry to model deep-bed filtration in multiphase flow. By embedding the new collection efficiency and filter coefficient expressions in an unsteady-state multidimensional computational fluid dynamics code, the deep-bed filtration process in trickle flow reactors was simulated and the increase of pressure drop during plugging was explained by increasing local specific surface area and decreasing local porosity due to fines deposition. The simulations were benchmarked using the experimental pressure drop data and observations of [M.R. Gray, N. Srinivasan, J.H. Masliyah, Pressure build-up in gas–liquid flow through packed beds due to deposition of fine particles, Can. J. Chem. Eng. 80 (2002) 346] in their study of kaolinite–kerosene–air flows in trickle-bed reactors.