Analytical Calculation of Trajectories Using a Power Law for the Drag Coefficient Variation with Mach Number

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

Mach number, the drag coefficient of most projectiles can well be approximated by the power law m −m  a  D = ()= CMaCc   . (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− ad C ρπ ad **2 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 −=− vgv . (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∗ =

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   0 coscos αϕ      v0 = vy0  = v0  sinϕ0  (4)     vz0   0 sincos αϕ  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 = tanϕϑ . (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−= G d σδ  . (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  d ϑϑ . (11) 1 2 ∫   g()cosα  vx0  ϑ0

The side deviation

vzo =α11 tan = xxz 1 . (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.

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): δδ ++= ubub b3 210 . (14) u 0 −= ϑϑ

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,

' b = δ 01 u ' −δδ ' b = ()11 0 > 1. (15) 3 ' 01 −− δδδ 0u1 ' ' 1 −δδ 0 b2 = b3 −1 3ub 1

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 ()mD ()11 ϑϑ 2 2 −+−−== i du i ii 1−m . (16) D ()(.0065 .4 25593+ m ) v 2 12/ +ϑϑ 2 2 − i ()iixi 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 δ 0uG u +++= u  . (17) vx0   12 + 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. Iterations are required because δ1 and δ'1 depend on y (eqn (13)). The coefficients b1, b2, b3 can also be used to calculate points between "0" and "1" or slightly behind "1". If a

WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) 308 Computational Ballistics II strongly curved maximum occurs, the trajectory should be subdivided in the ascending and descending part. At the maximum ϑ = 0, δ = D and δ′ = 0. The polynomial approach can also be applied when D = D0 = const. Then only the first term on the right-hand side of eqn (16) is used. The second term (below term one) is set to zero. No iterations are required.

3.2 Consideration of wind

When the air moves with a uniform wind velocity w, the stationary solution can be used, if a co-ordinate transformation is applied [3]. The gun is moved with wind velocity so that the gun fires in stationary air. In order to reach the same muzzle velocity in the original x, y, z-system, the muzzle velocity has to be changed to ~ 00 −= wvv . (18) The gun fires (τ = 0), when the muzzle passes the origin of the original x, y, z- system. Using the new muzzle velocity eqn (18), the new angle α~ , the time of ~ ~ ~ flight τ1 = τ1, the horizontal downrange distance x1 , the height y1 and side deviation ~z can be calculated in the above described manner. When the 1 ~ ~ ~ projectile has reached the point ( x1 , y1 , z1 ) after the time τ1, the gun has moved away from the origin to a distance of τ1 w. The hit point in the original x, y, z- system must consequently be calculated by the vector addition ~ x1  x1    ~  y1 = y1 τ+ 1w . (19)   ~  z1  z1 

For a given target point (x1, y1, 0) the gun elevation angle ϑ 0 and the side angle α can iteratively be adjusted.

4 Closed form solution for weakly curved trajectories

The general quadrature solution (eqns (8) – (12)) has been used to derive a closed form solution (v, ϑ , τ, y) as functions of x. With variables p, T, ϑ or δ first the downrange velocity ratio has to be determined according to 1 v  δxm  m x1 1−= 1  m 1−m  vx0  v ()cosα   x0  . (20) x 1 1 = δδ dx x ∫ 1 0 For the precise determination of the integral mean value δ , the changing values of D and ϑ along the trajectory must be known. D and ϑ decrease during uphill and increase during downhill firing. For direct fire applications the

WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) Computational Ballistics II 309 following simple iteration-free approximation is recommended. The integral mean value δ is calculated from mean values of D and ϑ . The integral mean value of ϑ is the ratio y1/x1. A suitable mean value of D is the arithmetic mean of D0 and D1. This leads to the formula 1−m 2 1   y   2 DD ()DD 1++=δ  1   . (21) 2 10   x     1  

The velocity ratio is needed for the direct calculation of ϑ 1, τ1, y1 and z1 as functions of x1 and vx1/vx0. The mean value δ in eqn (20) is correct for the calculation of the velocity ratio but not for ϑ , τ, and y. For the analytical solution δ is now regarded as a function of the velocity according to the power law n δ  v   x  = q   . (22) δ 0  vx0 

For uphill firing n > 0, downhill n < 0. The dimensionless coefficient q is close to 1. In the special case D = D0 and m = 1, eqn (22) is exactly fulfilled with q = 1 and n = 0. Thus the derived solutions, based on eqn (22), are exact in this special case. In the course of the derivations the coefficient q is eliminated and the following formulas are found: 2 +− nm  vx0    −1 ()− nmx g cosα vx1 ϑϑ −= 1 ⋅   , (23) 01 −nm   v   2 +− nm v2 1−  x1   x0   v     x0  

1 +− nm  vx0    −1 ()− nmx vx1 τ = 1 ⋅   , (24) 1 −nm x0 ()1 +− nmv  vx1  1−    vx0  x ϑ 2 ()− 2 Fnmgx y 01 += 1 1 2 cosα −nm   v   nm v2 ()nm 12 −+−  x1   x0   v     x0   . (25) −nm ()12 +− nm  v   v  1−  x1   x0  −1  v   v  F =  x0  −  x1  − nm ()12 +− nm

The side deviation z1 can be calculated using eqn (12). For the exponent n the following iteration-free formula has been developed, which is valid for the above mentioned temperature gradient of –.0065 K/m and any measured or calculated Temperature T0.

y vm y  ()− mg ()(.1 0065 .4 25593+ m 2/ ) n = 10  +  . (26) 2 2 v2 T 0 1 + yxD 1  0 0 

For longer distances and stronger curvature of the trajectory an iterative determination of n according to

 δ  ln 1   δ  n =  0  (27)  vx1  ln   vx0  gives better results. Some of the eqns (20, 23, 24, 25) fail numerically if m = 0 , n, n + 1, n + 2. In such cases one can simply add a small amount to m, e. g. ∆m = 10–4. Alternatively one has to form the limiting functions with help of Bernoulli's rule (known as de l'Hospital's rule) or series developments. Logarithmic functions appear in the formulas [3]. Eqn (20) turns into vx1/vxo=exp(- δx1/cosα) if m = 0. In situations with wind the considerations of chapter 3.2 can be applied.

5 Numerical examples

5.1 Subsonic pistol bullet

With the first example it is investigated how eqn (22) is suited to take the curvature of the trajectory into account. The subsonic pistol bullet .45 ACP VMR, M = 14.9 g, v0 = 260 m/s is selected [1, p. 219]. The exponent m is taken as m = 0 and the ballistic coefficient D* is estimated from the velocity data [1] to -1 be D* = D0 = .00118 m . A constant ballistic coefficient D0 is assumed. The results are shown in table 1. First the correct trajectory data are calculated for ϑ 0 = 0,5 and various values of ϑ 1 (column 1), using eqns (8, 10, 11). With m = 0 and D = const the integral in eqn (8)

ϑ D0  2 −1 2 −1  d d =  1 ϑϑσδ sinh 1 ϑϑϑ −+−++ sinh ϑ  . (28) ∫ 2  00 0  ϑ0

In columns 2, 3 of table 1 the true values of x1 and y1 are written. For the comparison the trajectory height y1 (x1) is calculated with the closed form

WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) Computational Ballistics II 311 solution eqn (25). The previously calculated (eqns (8, 28)) velocity ratios are used in eqn (25). The values of n are determined with eqns (26) and (27), respectively, yielding two values of y1. These values (columns 4, 5)show that even for big changes in ϑ eqn (22) is well suited to take the curvature of the trajectory into account. For weakly curved trajectories ( ≤ϑ∆ .1) the iteration- free formula eqn (26) is sufficiently accurate. For stronger curvatures the iterative approach using eqn (27) is the better choice. In practical cases the velocity ratio would have to be calculated from eqns (20, 21) and additional errors arise due to the inaccuracy of eqn (21). For larger values of m (m→ 1) the detrimental angle effect becomes less pronounced and the accuracy improves (no angle effect if m = 1).

Table 1: Calculated trajectory data of pistol bullet .45 ACP, D = .00118m-1,

m = 0, v0 = 260 m/s, ϑ 0 = 0,5, g = 9,80665 m/s² .

Eqns→ (8),(10),(28) (8),(11),(28) (25),(26) (25),(27)

ϑ 1 x1(m) y1(m) y1(m) y1(m) 0,4 341.665 156.240 156.256 156.239 0,25 588.005 238.176 238.522 238.169 0 818.391 269.598 271.831 269.948

5.2 Supersonic rifle bullet

In the second example the fast calculation of ϑ 0 is demonstrated when firing from an altitude of H0 = 425 m uphill and downhill towards a target in the positions (x1 = 850 m /y1 = ± 425 m /z1 = 0). The variation of p and T is taken into account. The rifle bullet Sierra Spitzer Softpoint .277, 110 Grain, is considered, which corresponds to the G8 standard projectile [1, pp. 112/113, tables 6.1/6.11]. In the range 1,8 ≤ Ma ≤ 4, C = .553 and m = .75 [3]. In this example the values are partially used below the validity range Ma< 1,8, which should be allowed for this example. Applying eqns (2) and (13) yields D* = -.25 -.75 -.25 -.75 .1464 m s and D0 = .1400 m s . With the temperature gradient -.0065

K/m one finds T0 = 285.39 K. For the fast determination of ϑ 0 the following quantities (with the eqns used in brackets) are calculated. First from the given data D1(13), δ (21), n(26). Then an approximate value ϑ 0,i is guessed (e.g. y1/x1) and (vx1/vx0)i (20) and y1,i (25) are calculated. The previous approximation

ϑ 0,i is improved according to ϑ 0,i+1 = ϑ 0,i + (y1-y1,i)/xi. With a few iterations one finds for y1 = +425 m (uphill) ϑ 0 = .510651 and vx1/vx0 = .3443. For the downhill target at y1 = -425 m the iteration yields ϑ 0 = -.489357 and vx1/vx0 = .3224. The accuracy of this simple and fast calculation method is now tested against the quadrature solution. For this purpose ϑ 1 has to be calculated using eqn (23). Uphill ϑ 1 = .478310, downhill ϑ 1 = -.522689. Once ϑ 0 and ϑ 1 are known the eqns (8, 10, 11, 14-17) can be applied. No iterations are

WIT Transactions on Modelling and Simulation, Vol 40, © 2005 WIT Press www.witpress.com, ISSN 1743-355X (on-line) 312 Computational Ballistics II necessary. For the uphill situation the coefficients of the polynomial eqn (14) become b1 = -.642390, b2 = +1.208341, b3 = +1.299442 > 1. The downhill calculation yields b1 = +.624847, b2 = -1.166276, b3 = +1.305327 > 1. The numerical integration of eqns (8, 10, 11) containing eqn (17) provide finally for the uphill case: x1 = 850.020 m, y1 = 425.021 m, and vx1/vx0 = 0,3434. At the given distance of x1 = 850 m the quadrature trajectory has a height of 425.012 m. In the downhill case the quadrature solution yields vx1/vx0 = 0,3236, x1 = 850.055 m, and y1 = -425.043 m which changes to y1 = -425.014 m at the given distance of 850 m. The time of flight has also been calculated with both methods. The agreement is excellent as well, the relative errors fall below .04 %.

6 Conclusions

The quadrature solution based on the power law eqn (1) is a useful tool for the calculation of trajectories. A polynomial approach can take the variation of the thermodynamic state of atmosphere into account and simplifies the numerical integration. The closed form solution based on the power law eqn (1) is well suited for the fast calculation of weakly curved trajectories. The approximate consideration of both the variation of angle of flight and thermodynamic state of atmosphere is an improvement over previous flat fire solutions. In the special case m = 1 and D = const the solution is exact. Using a co-ordinate transformation, both analytical solutions can be applied to situations with wind.

References

[1] Kneubuehl, B.P., Geschosse/Ballistik, Treffsicherheit, Wirkungsweise, Verlag Stocker-Schmidt AG: Dietikon-Zürich, 1994. [2] McCoy, R.L., Modern Exterior Ballistics/The Launch and Flight Dynamics of Symmetric Projectiles, Schiffer Publishing Ltd: Atglen, PA 19310, 1999. [3] Roetzel, W., Analytische Berechnung gestreckter Geschossflugbahnen, Mitteilungen aus der Schießsport-Arbeitsgemeinschaft an der Universität der Bundeswehr Hamburg Nr. 1, ed. H. Rothe, Helmut-Schmidt-Universität, Universität der Bundeswehr Hamburg, 2004. [4] Schneider, D., Analytische Berechnung gestreckter Geschossflugbahnen, Studienarbeit bei Prof. Roetzel am Institut für Thermodynamik, Helmut- Schmidt-Universität, Universität der Bundeswehr Hamburg,,2004. [5] Rogers, G.F.C., Mayhew, Y.R., Thermodynamic and Transport Properties of Fluids, Basil Blackwell Ltd: Oxford, Fourth Edition, 1988.