球形塑性薄壳在平头圆柱体冲击下贯穿过程的研究

对球形塑性壳在平头圆柱体冲击下的动力贯穿行为进行了分析．通过引入等度量变换,给出了壳体的变形模态．在此基础上对球壳在平头圆柱体撞击下的贯穿机制进行了分析研究,给出了贯穿模型,由此结合Hamilton原理,导出了运动方程组．采用Runge-Kutta方法求解了该方程组,最后分别给出了贯穿时间-弹体初速度;残余窝陷半径-弹体初速度;贯穿后弹体剩余速度-弹体初速度的关系曲线．     

Solution to projectile motion with quadratic drag and graphing the trajectory in spreadsheets

This note gives the analytical solution to projectile motion with quadratic drag by decomposing the velocity vector to x, y coordinate directions. The solution is given by definite integrals. First, the impact angle is estimated from above, then the projectile coordinates are computed, and the trajectory is graphed at various launch angles and speed using an interactive Excel application.     

贫铀合金材料自锻弹丸的数值模拟

利用SPH(Smooth Particle Hydrodynamic)算法对贫铀合金材料自锻弹丸过程进行了数值模拟。其中使用了人造爆轰压力,即给定了爆轰压力的时间分布与空间分布。为了描述材料在高压作用下的变形性能,材料模型采用了Johnson-Cook关系。通过数值计算,给出了自锻弹丸的飞行速度、弹丸几何尺寸及炸高等参数。     

弹侵彻混凝土靶的法向膨胀理论

基于描述材料在高速高压冲击下的动态行为的方程,建立了相应的冲击波前为一般曲面的方程．通过分析,针对混凝土材料,提出了可给出弹体减速度时间历程解析解的法向膨胀理论．它不仅适用于球形弹头和柱形弹头,也适用于其他类型的弹头,如锥形弹头和卵形弹头等;它不但适用于垂直侵彻,也适用于斜侵彻．     

Zur Berechnung der Geschossabweichung unter dem Einfluss eines Seitenwindes

Summary The first order correction which should be made to thez-coordinate of the standard trajectory of a projectile to account for a cross-wind is given by (1). It is shown that this formula is a very good approximation also for angles of departure near 90°.     

一类发烟装液弹装填率的数学模型

装液弹的装填率对弹的安全可靠性极其重要 .如果装液弹的装填率不合理 ,不是造成弹腔容积的浪费 ,就会造成弹内压过大 ,易于破坏密封性能引起渗漏甚至破裂而造成损失和危险 .本文通过对一类发烟装液炮弹弹内压力分析 ,得出了弹内压力计算公式 .在分析压力诸因素时 ,我们运用了固体热膨胀理论 ;考虑了液体的压缩性 .这是本文区别于以往压力计算的两个特点 .通过这类发烟装液炮弹内压压力曲线的分析 ,导出了装液 (炮 )弹弹腔空隙率合理选定最后的数学模型 (PLT方程——可由此确定合理的装填率 ) ,由此所计算的空隙率数据与国外文献值相符 ,并引入了“预极限温度”的概念 .可以相信 ,所谓“预极限温度”,将是装液 (炮 )弹 (或任何装液容器 )设计者必须认真考虑的问题     

Analysis of projectile motion in view of fractional calculus

The projectile motion is examined by means of the fractional calculus. The fractional differential equations of the projectile motion are introduced by generalizing Newton’s second law and Caputo’s fractional derivative is considered to use the physical initial conditions. In the absence of air resistance it is found that at certain conditions, the range and the maximum height of the projectile obtained by using the fractional calculus give the same results of the classical calculus. It is also found that, unlike the classical projectile motion, the launching angle that maximizes the horizontal range is a function of the arbitrary order of the fractional derivative α. Moreover, in a resistant medium, the parametric equations are expressed in terms of Mittag-Leffler function and the results agree with those of the classical projectile as α → 2. Moreover, the trajectories of the projectile are discussed in graphs and compared with those of the classical calculus. In order to explore the validity of modelling the projectile motion by the fractional approach, we compared our results with the experimental data of mortar.     

On a moving boundary problem of transitional ballistics

A hybrid algorithm for calculating the flow around a flat-nosed projectile moving through a fluid is established. At cells of fixed volume away from the projectile, Harten's total variation diminishing (TVD), second-order accurate, shock capturing scheme is utilized. Due to projectile motion, cells adjacent to the projectile can be treated as compressing or expanding in volume. For such cells, a local finite volume approach has been employed to derive a cell update algorithm. Proof of concept for the expanding cell scheme is established through calculating the one-dimensional flow behind a moving piston, whose theoretical solution is well-known. The resulting hybrid scheme is applied to the problem of blast wave simulation in axisymmetric geometries. Flow around a vertical muzzle brake is calculated for the case of an embedded moving projectile.     

Prediction of swerving motion of a dual-spin projectile with lateral pulse jets in atmospheric flight

Using the linear theory for a dual-spin projectile in atmospheric flight, closed form expressions are obtained for swerving motion under the action of lateral pulse jets. Trajectory results generated by the linear theory equations and a fully nonlinear seven degree-of-freedom dual spin projectile model agree favorably. The analytic solution provides a relatively straightforward and computationally efficient means of trajectory estimation which is useful within smart weapon flight control systems. In order to accurately predict the impact point using the analytic solution, the dual-spin projectile linear model must be updated periodically. Terminal impact point prediction degrades rapidly as the linear model update interval is increased beyond a critical value. Control authority, as defined by the change in impact location due to a pulse jet firing, steadily decreases as a function of projectile down range position.     

卵形弹垂直侵彻混凝土介质的量纲分析

针对卵形弹侵彻半无限厚混凝土介质,采用量纲分析的方法建立了相关参数的无量纲化的函数关系,通过质量为2.85kg,弹径为50.8mm的卵形弹垂直侵彻混凝土实验确定撞击混凝土侵深的工程表达式.     

Projectile motion with Mathematica

This note describes how to use the computer algebra system (CAS) Mathematica to analyse projectile motion with and without air resistance. For a projectile fired from ground level with an initial velocity ν ft/s at an angle θ degrees from the horizontal (0 < θ < 90°), it is well known that in the absence of air resistance, the projectile follows a parabolic path. However, this is not true if air resistance is taken into account. In the presence of air resistance, the equations of motion become complicated, thus making traditional handcalculation methods quite ineffective, and a powerful CAS such as Mathematica becomes an invaluable tool to better understand projectile motion. The note discusses how Mathematica can be used to create simulated experiments of projectiles with and without air resistance. These experiments result in several conjectures, leading to theorems.     

Approximate formula for the vertical asymptote of projectile motion in midair

The classic problem of the motion of a point mass (projectile) thrown at an angle to the horizon is reviewed. The air drag force is taken into account with the drag factor assumed to be constant. An analytical approach is used for the investigation. An approximate formula is obtained for one of the characteristics of the motion – the vertical asymptote. The value of an asymptote is determined directly by the initial conditions of throwing. Analytically derived values of asymptotes in comparison with numerical values obtained by integrating the equations of motion are given. The motion of a baseball is presented as an example.     

Motion of a circular viscoplastic plate subject to projectile impact

A method is presented for the solution of problems involving the impact on circular plates of projectiles moving with velocities high enough to produce plastic deformations. The plate material is assumed to obey the Mises-Huber yield condition and its associated flow rule for static deformations and behave as a viscoplastic solid for dynamic deformations. The nonlinear constitutive equations of the material are partially linearized in such a way as to allow the required result to be obtained by superposition of the static rigid perfectly-plastic solution and the dynamical elastic solution of equivalent problems. The method is illustrated by application to the case of a clamped plate struck at the center by a projectile of negligible radius.In addition a method for the rapid estimation of the motion of the projectile after impact is given and the result for the time required to bring the projectile to rest and the consequent central deflection of the plate are obtained.

---

Zusammenfassung Es wird eine Methode zur Lösung von Problemen beschrieben, die sich im Zusammenhang mit dem Aufschlag von Geschossen auf kreisförmige Platten ergeben; dabei soll die Geschossgeschwindigkeit so hoch sein, dass plastische Deformationen auftreten. Das Plattenmaterial soll der Mises-Huberschen Fliessbedingung und dem zugehörigen Fliessgesetz für statische Deformationen genügen und sich für dynamische Deformation wie ein viskoplastischer Festkörper verhalten. Die nicht linearen Stoffgleichungen des Materials werden teilweise linearisiert, so dass das gesuchte Resultat als Superposition der statischen starr-ideal-plastischen Lösung und der dynamisch-elastischen Lösung erhalten werden kann. Die Methode wird durch Anwendung auf den Fall einer eingespannten Platte veranschaulicht, auf deren Mittelpunkt ein Geschoss von vernachlässigbarem Radius auftrifft.Ausserdem wird eine Methode zur Abschätzung der Geschossbewegung nach dem Einschlag gegeben und die Zeit vom Aufprall bis zum Stillstand des Geschosses sowie die resultierende Auslenkung des Scheibenmittelpunktes berechnet.

     

膛口流场动力学过程数值研究

采用基于ALE方法的动网格及嵌入网格技术,运用有限体积方法,结合二阶精度Roe格式,对弹丸由高压气体驱动从静止状态加速至超音速,射出膛口到完全飞离初始流场的整个过程进行了数值模拟．根据数值结果,详细讨论了初始流场、火药燃气流场的形成与发展以及与弹丸的耦合和相互作用过程,揭示了在这一变化过程中激波与激波、激波与漩涡、激波与弹丸等的相互作用以及激波衍射、聚焦等对弹丸加速的影响．     

On the motion of a projectile in the atmosphere

The equation of rectilinear motion of a projectile which moves in an atmosphere, of which the density decreases exponentially with the altitude, is solved. It is found that the velocity of the projectile can be expressed explicitly in terms of confluent hypergeometric functions. This theory is applied to treat two specific problems: (1) the flight analysis of a sounding rocket during the freeflight period and (2) the calculation of ambient temperature from the trajectory of a spherical projectile.

---

Zusammenfassung Die Bewegungsgleichung eines vertikal ansteigenden Geschosses wird gelöst für den Fall, dass die Dichte der Atmosphäre exponentiell mit der Höhe abnimmt. Die Geschwindigkeit des Projektils kann dabei in konfluenten hypergeometrischen Funktionen ausgedrückt werden. Die Theorie wird hierauf auf zwei spezifische Probleme angewendet: 1. Fluganalyse einer Vertikalrakete während der Zeit des schubfreien Fluges und 2. Berechnung der Umgebungstemperatur der Bahn eines sphärischen Projektils.

     

Some Real Analysis behind Optimization of Projectile Motion

This article addresses the problem of maximizing the lateral displacement of a projectile that is launched from atop a tower and is subject only to gravity. Here, the surface to which we launch is represented by a continuous function whose intersection with every flight path is a single point. In this general setting, we are forced to access tools from real analysis to gain detailed information about the shape of the graph and the maximum of the distance function. In particular, we derive manageable formulas for the maximal lateral displacement of the projectile and the corresponding optimal launch angle.     

An analytic approach to projectile motion in a linear resisting medium

The time of flight, range and the angle which maximizes the range of a projectile in a linear resisting medium are expressed in analytic form in terms of the recently defined Lambert W function. From the closed-form solutions a number of results characteristic to the motion of the projectile in a linear resisting medium are analytically confirmed, and asymptotic and approximate expressions valid within the weak and strong damping limits are developed. The problem provides an accessible account of the increasingly applicable Lambert W function and highlights much of the important mathematics associated with this simple yet rapidly emerging ‘implicitly’ elementary function.     

陶瓷/金属复合靶板受可变形弹体撞击的理论分析模型

针对弹体撞击陶瓷/金属复合靶板的问题,将弹体的变形、陶瓷面板的碎裂和金属背板的变形结合起来,建立了新的可变形弹体垂直撞击陶瓷/金属靶板的理论分析模型．模型中计入了弹体刚性区长度和运动速度、塑性变形区长度、横截面积和运动速度的变化以及弹体对靶板的侵入速度和深度;对陶瓷面板考虑了陶瓷锥体积和抗压强度的变化;对金属背板的变形,根据其塑性变形功、外力功及其动能守恒原理,得到金属背板的运动方程．最后对具体算例进行了分析,得到了各物理量随时间的变化,给出了一些有价值的规律．结果表明,模型能较好地描述撞击过程中的有关规律;与实验结果和数值模拟结果进行对比,吻合较好,说明了模型的有效性．     

Ballistic performance of a Kevlar-29 woven fibre composite under varied temperatures

Armours are usually manufactured from polymer matrix composites and used for both military and non-military purposes in different seasons, climates, and regions. The mechanical properties of the composites depend on temperature, which also affects their ballistic characteristics. The armour is used to absorb the kinetic energy of a projectile without any major injury to a person. Therefore, besides a high strength and lightness, a high damping capacity is required to absorb the impact energy transferred by the projectile. The ballistic properties of a Kevlar 29/polyvinyl butyral composite are investigated under varied temperatures in this study. The elastic modulus of the composite is determined from the natural frequency of composite specimens at different temperatures by using a damping monitoring method. Then, the backside deformation of composite plates is analysed experimentally and numerically employing the finite-element program Abaqus. The experimental and numeric results obtained are in good agreement.