Computational Ballistics II 303 Analytical calculation of trajectories using a power law for the drag coefficient variation with Mach number W. Roetzel Institute of Thermodynamics, Helmut Schmidt University, University of the Federal Armed Forces, Hamburg, Germany Abstract For hand calculations and for the fast computation of direct fire trajectories two analytical solutions of the point mass equation of motion in stationary air are presented, which are based on the power law for the drag coefficient variation -m with Mach number cD = CMa . The first quadrature solution provides time of flight τ, horizontal downrange distance x and height y as functions of the angle of flight ϕ. The second closed from solution yields ϕ (x), τ (x), y (x) and is applicable to weakly curved trajectories. Using a moving co-ordinate system, both analytical solutions can also be applied to situations with wind. Keywords: point mass trajectory, analytical solution, direct fire, drag coefficient, Mach number, wind. 1 Introduction For the precise computation of trajectories numerical methods are required and available, which take the variation of drag coefficient with Mach number into account as well as the change of air pressure and temperature along the trajectory. Wind can also be considered in the equation of motion and other side effects may be accounted for. There seams to exist no longer any need for analytical solutions. However, for direct fire applications with short distances and moving targets, highly sophisticated computer programmes are prohibitively time consuming. Not extremely but sufficiently accurate analytical solutions are more appropriate in such cases. Also for hand calculations for small arms using a pocket computer, analytical solutions are very useful. In wide regions of the WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) 304 Computational Ballistics II Mach number, the drag coefficient of most projectiles can well be approximated by the power law m −m a cD = C() Ma= C . (1) v In the supersonic region (Ma > 1.1) the exponent m may assume values –0.2 ≤ m ≤ 1.0 depending on the shape of the projectile and the range of Mach number. In the subsonic region (Ma < 0.8) the exponent is usually close to zero. The power law has widely been used and several analytical solutions, based on this law, were published. Kneubuehl [1, pp 89-91] recommended a value of m = 0 and provided the flat fire solution together with deceleration coefficients of various small arms projectiles, which depend on altitude and muzzle velocity. McCoy [2, pp 88-97] presented the flat fire solutions for the special cases m = 0, m = ½, and m = 1. The author of this paper developed recently [3] the flat fire solution for arbitrary values of m and suggested approximations [3, 4] to account for changing air pressure and temperature during uphill and downhill firing. In the present paper two analytical solutions are presented, which are based on the power law eqn (1) with arbitrary values of m. The first solution is a quadrature solution. The second solution is in principle the previously published [3] closed form flat fire solution which is modified and refined in order to improve the accuracy and to simplify the consideration of wind. 2 Equation of motion It is assumed that the dimensionless drag coefficient cD depends on the Mach number alone, although it is also affected by the Reynolds number. The dependence on the Mach number is described by the power law eqn (1). This equation is incorporated in the non-dimensionless ballistic coefficient m m 1− Cπ d2 ρ * a * p T * 2 D = ()⋅ , (2) * 8M p T with which the point mass equation of motion for stationary air can be written as Dv1−m v− g = − v . (3) The first term represents the aerodynamic forces which vary due to variations in projectile velocity v as well as air pressure p and thermodynamic temperature T along the trajectory. Eqn (2) is derived for a perfect gas and takes the dependence of air density ρ on p and T, and the variation of air sound velocity a with T into account. The superscript ( )∗ denotes a standard state of the atmosphere which can be defined arbitrarily. Usually the thermodynamic state of the International Standard Atmosphere [5] at sea level is defined as the reference standard state: p∗ = 1.01325 bar, T∗ = 288.15 Κ, ρ∗ = 1.2250 kg/m3, a∗ = WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) Computational Ballistics II 305 340.3 m/s, (g* = 9,80665 m/s²). In eqn (2) C and m are the dimensionless constant and exponent of eqn (1), d is the diameter and M the mass of the projectile. The unit of D is (m)m-1 . (s)-m. For p = p∗ and T = T∗ the ballistic coefficient D turns into the standard ballistic coefficient D∗. Fig. 1 shows the co- ordinate system used. The muzzle velocity vx0 cosϕ0 cos α v0 = vy0 = v0 sinϕ0 (4) vz0 cosϕ0 sin α must not fall into the x, y-plane, a side deviation by the angle α is allowed. Finite values α ≠ 0 occur, if wind has to be considered. This will be discussed later in this paper. First the simpler case of stationary air is considered. The boundary conditions for the equation of motion eqn (3) are: v = v0, (x, y, z) = 0 for τ = 0. The target point is denoted with index "1", it lies usually in the x, y-plane (z1 = 0). Figure 1: Co-ordinate system. 3 Quadrature solution An analytical quadrature solution has been developed which provides τ, x and y as functions of the angle of flight ϕ. The tangens of the angle ϕ is introduced as the independent variable ϑ= tan ϕ . (5) Defining the function 1−m δ =ϑD()1+ 2 2 (6) and the parameter 2−m 2 − m v G = x0 (7) g cosα WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) 306 Computational Ballistics II one finds for the downrange velocity ratio as function of ϑ 1 − ϑ 2−m vx =1 − Gδ d σ . (8) vx0 ∫ ϑ0 The time of flight τ1, the downrange distance x1, and the height y1 can be determined from the integrals: ϑ1 vx0 vx τ1 = − d ϑ , (9) g cosα ∫ vx0 ϑ0 ϑ 2 v 2 1 v x0 x x1 = − ∫ dϑ , (10) g cosα vx0 ϑ0 ϑ 2 v 2 1 v y = − x0 x ϑ d ϑ . (11) 1 2 ∫ g()cosα vx0 ϑ0 The side deviation vzo z1=α x 1 tan = x1 . (12) vxo The variable velocity ratio according to eqn (8) has to be introduced into eqns (9) – (11) leading to twofold integrals which have generally to be evaluated numerically. If the ballistic coefficient D = D0 = const, the numerical integration can be performed for any given value of m. In the special case m = 0 the internal integral (eqn (8)) can be solved analytically (eqn (28)) and only single definite integrals have to be evaluated numerically. In the special case m = 1 (and D = D0) the integrand δ = D0 and the twofold integrals can be solved analytically. This leads to the known flat fire solution [2, 3] for m = 1 and reveals that the flat fire solution (m = 1, D = D0) is exact also for strong changes in angle ϕ, provided the ballistic coefficient D can be regarded as constant. This will be confirmed later in this paper. 3.1 Procedure for variable ballistic coefficient For uphill and downhill firing at longer distances the dependence of D on y has to be taken into account. For the standard atmosphere up to an altitude of H =11000 m, a linear temperature drop of ∆T/∆H = -.0065 K/m is presumed. WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) Computational Ballistics II 307 With this value the variation of the ballistic coefficient D with altitude H = H0 + y can directly be calculated according to 4.25593+m/ 2 D HH− j =1 − () .0065 . (13) D j T j The index "j" refers to any reference altitude. One could incorporate eqn (13) into the process of numerical integration of eqn (11), which comes to a stepwise calculation of y, τ and x. The following method is faster and more convenient. A polynomial approximation is introduced for the integrand ϑ in eqn (8): δ= δ +b u + b ub3 0 1 2 . (14) u =ϑ0 − ϑ The coefficients b1, b2, b3 are determined such that at the terminal points "0" and "1" the values of ϑ and their derivatives correspond with the true values, ' b1= δ 0 u δ' − δ ' b = 1() 1 0 > 1. (15) 3 ' δ1− δ 0 − δ 0u1 ' ' δ1 − δ 0 b2 = b3 −1 b3 u1 The necessary condition b3 > 1 is usually fulfilled. If b3 ≤ 1, other polynomials would have to be applied. The derivatives at the points i = 1, 2 can be determined from the equation 1+m dδ − δ ' =i = −D()1 − m ϑ() 1 + ϑ 2 2 − i du i i i 1−m . (16) D ()(.0065 4.25593+ m/ 2 ) v2 ϑ 1+ ϑ 2 2 − i xi i() i 2 gTi ()cosα With the coefficients b1, b2, b3 eqn (8) turns into 1 − v b b 2−m x 1 2 2 1+b3 =1 +Gδ 0 u +u + u . (17) vx0 2 1+ b3 Substituting eqn (17) into eqns (9, 10, 11) with ϑ = ϑ 0 - u and d ϑ = - du yields single definite integrals for the determination of τ1, x1 and y1.