Fast calculation of direct fire trajectories taking the earth s rotation into account

Transcription

1 Coputational Ballistics III Fast calculation of direct fire trajectories taking the earth s rotation into account W. Roetel, W. Carnetki & T. Maier Helut-Schidt-Universität / Universität der Bundeswehr Haburg, Gerany Hochschule Esslingen, Gerany Abstract An analytical approach is developed for the subsequent consideration of the Coriolis effect. The target is regarded as a oving target in the star-fied coordinate syste. The stationary atosphere creates a nonunifor cross-wind which reduces the noral oving target deflection of the projectile. The approach is ipleented in a previously developed analytical fast calculation ethod and tested against nuerical calculations with good results. Keyword: Coriolis effect, analytical solution, power law, drag coefficient, Mach nuber. Introduction Several analytical solutions of the point ass equation of otion have been developed for the fast calculation of direct fire trajectories [ 4], which are based on the power law c D C a v * / a v T * T ( Ma) C C () for the drag coefficient variation with Mach nuber. McCoy [] published flat fire solutions for, ½,. The later solutions [ 4] allow for arbitrary values of and any angle of sight β [3, 4]. Wind can be considered using a coordinate transforation [, 4]. For uphill and downhill firing the change of pressure and teperature along the trajectory should be considered. Usually ean values of WIT Transactions on Modelling and Siulation, ol 45, 7 WIT Press ISSN X (on-line) doi:.495/cbal73

2 Coputational Ballistics III pressure and teperature are applied [3]. A ore sophisticated way [4] is the correction of the Mach nuber eponent, which takes not only the changing pressure and teperature but also the curvature of the trajectory approiately into consideration. The above entioned fast calculation ethods do not allow for the earth s rotation. In the present paper a siple analytical ethod is described [5] with which the Coriolis effect can subsequently be considered. The sae coordinate syste and noenclature is used as in the foregoing paper [4]. Nuerical ethod For the precise calculation of the trajectory the coplete equation of otion has to be integrated nuerically. With the previously [,4] defined ballistic coefficient * / π d ² π d ² * * p T ρ a C ( a ) () * D( p, T ) C ρ 8 M 8 M p T the equation of otion with wind velocity w can be written as [] v g D (3) ( v w ) ( v w) b with the Coriolis acceleration b ω v (4) containing the earth s vector ω of rotation cosγ ω ω sinψ. (5) sinγ In eqn (5) the angle ψ is the latitude and the angle γ the aiuth of fire (-ais), easured clockwise fro north. Assuing the standard teperature drop with altitude T/ y -.65 k/ [6] and regarding the atosphere as perfect dry air yields the ballistic coefficient D as function of height y.65 D D y T /, (6) where inde indicates the firing site and origin of the earth-fies coordinate syste. WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

3 Starting at the origin with τ and L {, y, }, the local projectile velocity v can stepwise be calculated as function of tie. Integrating siultaneously over the tie yields L as function of tie. This nuerical calculation is later carried out in order to test the new approach which is derived and described in the following. 3 Analytical approach The equation of otion without the Coriolis acceleration is valid in a star-fied not rotating syste which oves with unifor velocity. Therefore the coordinate syste in which the Coriolis-free solutions [ 4] are valid is considered to ove on uniforly with the fied circuferential (not rotating) velocity u of the firing site (origin of coordinate syste) at the instant of firing τ. For τ this star-fied coordinate syste coincides with the earth-fied syste, for τ > they separate fro each other. 3. The oving centre of gravity In this star-fied syste a flying projectile eperiences an additional gravitational acceleration g in the horiontal direction, as the centre of gravity does not reain eactly perpendicular below the projectile. With the siplifying assuption of constant ean velocity coponents v and v and the radius R of the globe, the additional tie dependent acceleration can be epressed as u + v ω R + / τ g τ g τ. (7) g R R u + v ω R + / τ Substituting ω and ω according to eqn (5) and integrating twice over the tie of flight yields the displaceent of the projectile ω sin γ + / Rτ g 3. (8) L g τ 6 ω cosγ + / Rτ The vertical coponents g y and L gy since cos ωτ. The derivation of eqn (8) is correct under vacuu conditions. The effect of eqn (8) is very week in noral cases and a sufficiently accurate approach. 3. The oving target Coputational Ballistics III 3 The ain effect of the earth s rotation is the fact that in the uniforly oving star-fied syste the earth-fied target appears as a oving target. We look first at the poles of the globe. There the circuferential velocity is ero, but the earth- WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

4 4 Coputational Ballistics III fied coordinate syste rotates relatively to the considered star-fied syste. At the north pole any target will travel fro west to east, i.e. fro right to left. A negative angle of lead α < would be required to hit the oving target. The shift L of the target during the tie of flight ω τ can be epressed as L τ [ ( + )] ω L L τ ( ω L ) + ω L ω g g (9) which leads with eqn (5) to sinψ + y sinγ L. () ω ω τ sinγ cosγ y cosγ sinψ At other latitudes ψ ± 9 an additional target oveent takes place in the star-fied syste which is caused by the growing distance between the origins of both coordinate systes. However, a detailed analysis shows that this additional shift can be oitted. Its effect is copensated by the fact that in the usual standard free-fall acceleration g /s (at sea level) the centrifugal acceleration due to the earth s rotation is included (subtracted fro the ass attraction force). Thus, eqn () is valid for all values 9 < ψ < + 9. The sae shift () with negative sign is found by the twofold integration of the Coriolis acceleration eqns (4,5) using a constant ean projectile velocity v L /τ. The negative sign shows that in the earth-fied syste the projectile is apparently accelerated and displaced in the opposite direction as the target is in the star-fied syste. One could suppose that the final Coriolis deflection of the projectile in the earth-fied syste could be epressed as L L. () g Lω However, this is only a rough approiation under noral shooting conditions which will be discussed later in this paper. Eq () holds true in a vacuu. The contribution L is norally relatively sall. In the special case of the g downward free-fall in a vacuu at the equator ( ψ, L {, L,}) L is g decisive. For γ and with the tie of flight τ ( ) / L / g eqn () yields the correct [7, p.4] east drift The ter / 3 is due to 5%. / / L L L ω L + ω L. () g 3 g 3 L g. Neglecting it would cause an overestiation of WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

5 Coputational Ballistics III The Coriolis wind The oving target proble under consideration differs fro the usual case in so far as the air follows the oving target. If no wind is present, the stationary air in the earth-fied syste rotates in the star-fied syste together with the target. The oving target gets tailwind of equal velocity. The wind is not uniforly distributed but the cross-wind velocity w c is ero at the firing site and grows linearly with the distance to its aiu value w at the target L wc w. (3) τ This Coriolis wind pushes the projectile towards the oving target and reduces the deflection according to eqn (). Since the consideration of unifor wind is no proble, a suitable ean unifor Coriolis wind velocity w is defined and c derived which produces the sae effect as the actual linearly growing Coriolis wind does: w c w L L L ω g f f f. (4) τ τ A horiontal flat shot in the -direction (v v) towards a target at distance with cross-wind of velocity w w c is considered. Gravity is neglected. For w Z << v Fro eqns ( 3) one can derive [] v v v w v. (5) v [ + D( ) v ] τ (6) D and v Dv. (7) v + D( ) v τ Substituting eqns (6) and (7) into eqn (5) yields a first order linear inhoogeneous differential equation for v (τ ). Solving first the hoogeneous differential equation and applying the variation of the constant yields with the boundary condition τ : v the projectile velocity coponent v as function of tie. Integrating v Z fro τ to τ gives the deflection of the projectile at the distance. The sae deflection has to be produced by the WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

6 unifor wind of velocity w c, defined with eqn (4). This deflection can be found by replacing in eqn (5) the variable velocity w / by the constant velocity f w. The solution leads to the known forula of Didion [] ( ) v v w f τ. (8) Equating both deflections and solving for f yields finally the iplicit forulas for the deterination of the correction factor f: [ ] ) )( ( ) ( ), ( η η η f, (9) [ ] ) ( + η η. () For given values of and guessed values of η the diensionless nuber and the factor f can be calculated. Iteratively one can deterine f as function of and. The nuber is defined as, p v τ τ τ. () It is the ratio of the horiontal coponents of the ule velocity and the ean velocity. It can be deterined once the trajectory has been calculated nuerically or analytically (neglecting the Coriolis effect). In a vacuu. The right-hand definition as the ratio of the ties of flight under noral and vacuu conditions is ore appropriate when both v and turn to ero. (e.g. free-fall in a vacuu). In the special case ½ the variable η in eqns (9, ) can be eliinated yielding the eplicit forula + ln / f. () In the special cases and the eqns (9, ) cannot be applied directly and the following iplicit equations have to be used ( ) ) ln(, λ λ λ + f, (3) ( ) µ µ µ e f,. (4) 7 WIT Press WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 6 Coputational Ballistics III

7 Coputational Ballistics III 7 For ½ the deterination of f is inconvenient and tie consuing. Therefore the following epirical eplicit forula has been developed: f +, E.48( + ) +.8. (5) E For the liiting value the eponent E (l +)/ is ore precise. Equation (5) is sufficiently accurate for all direct fire applications. Once the correction factor f is found, the Coriolis deflection of a projectile can subsequently be calculated as follows: First L is calculated fro eqs (8,, ). This vector represents the deflection of the projectile fro the target L neglecting the Coriolis wind. The ean Coriolis wind velocity w is calculated c using eqn (4) with f fro eqn (5). If a real unifor wind is present, the Coriolis wind velocity has siply to be added to the real wind velocity. The su of both winds can be taken into account by a coordinate transforation [, 4] or other known ethods. If no real wind is present the application of Didion s forula is recoended. Cobining the shift L and the Coriolis wind effect according to Didion leads to the following forula for the Coriolis drift of the projectile L c + L + E E. (6) A horiontal flat shot has been assued in the above derivations. For uphill and downhill fire the eponent in eqs. (9 6) has to be corrected for changing air pressure and teperature along the trajectory [4]. The eponent has to be replaced by * n with n according to [4, eq. (6)]: v sin n D β g( ) (.65)( / +. (7) v T ) 4 Test of the analytical approach against nuerical calculations The analytical Coriolis approach has been ipleented into a fast calculation prograe, based on the previously developed eplicit Coriolis-free analytical solution [4, chapter 3. and 4]. The prograe is constructed in such a way that for a given target (, y ) or (L, β) and wind velocity (w, w y, w ) the angle of lead and the gun elevation angle above line of sight is calculated iteratively. The etended prograe has been tested against nuerical calculations described in chapter of this paper. Four typical eaples are presented in the following. WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

8 8 Coputational Ballistics III 4. Eaple The 85 Grain atch projectile Lapua Scenar.3 GB 43, d 7.83, M.988 g is considered with.4855 and C.486 (eq. ()). The projectile is fired with v 93 /s at sea level, p 3.5 bar and t 5 C. The angular velocity of the earth is taken as ω rad/s []. The ule Mach nuber Ma.73. It reduces to Ma.5 (in all eaples) after the fied tie of flight τ.6 s. Table shows calculated Coriolis deflections for flat fire fro the north pole and the equator. The sae results are obtained (with illieter accuracy) fro the analytical and the nuerical ethod. The analytical deflections are calculated using eq. (6) yielding the sae values as the fast calculation prograe. 4. Eaple The sae projectile and data are used as in eaple with the eception of the gun elevation angle φ 46 and the latitude ψ 45. Table shows the results. The values in brackets are analytically calculated deflections which deviate fro the corresponding nuerical values. The largest deviation is in height at the distance (bee-line) of about k. 4.3 Eaple 3 A projectile of identical shape (equal values of and C) and ass density is considered, which has the threefold diaeter d The ass is enlarged by the factor 7, yielding M g. Fro the analytical solution [4] one can see that with the sae ule velocity and threefold tie of flight τ 4,8 s the end velocity will reain the sae at the threefold distance. So Ma.73 to.5. One can also predict that for flat fire the elevation angle above line of sight will increase by the factor 3. Table 3 shows the flat fire results for φ, τ 4.8 s and L The largest deviations are 3. Concerning the superelevation angles: For eaple the Coriolis-free superelevation angle ε ils and for eaple 3 ε ils 3. ε. Table : Flat fire vertical ( y) and horiontal ( ) Coriolis deflections, calculated nuerically and analytically. Projectile Lapua Scenar.3 GB 43, 85 Grain. Gun elevation angle φ. Coriolis-free range 6., hitpoint height y 7.7, distance L 6.. Mule velocity v 93 /s, tie of flight τ.6 s. ψ( ) 9 γ ( ) y () + - () WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

9 Coputational Ballistics III 9 Table : Uphill fire Coriolis deflections, calculated nuerically and analytically (deviating analytical results in brackets). Projectile Lapua of Table, φ 46, 75.45, y 7.58, L 8,4, v 93 /s, τ.6 s. ψ( ) γ ( ) y () +5-5 (-5) () Table 3: Flat fire Coriolis deflections, calculated nuerically and analytically (deviating analytical results in brackets). Projectile of Lapua geoetry and ass density but threefold diaeter (d 3.47, M g), φ, 39., y 6.75, L 39.6, v 93 /s, τ 4.8 s. ψ( ) 9 γ ( ) y () +9-9 () +9-4(-7) +4(+7) Table 4: Uphill fire Coriolis deflections, calculated nuerically and analytically (deviating analytical results in brackets). Projectile of Table 3, φ 47, 6.9, y 78.37, L 337.3, v 93 /s, τ 4.8 s. ψ( ) γ ( ) y () +469(+463) -469(-463) () (-) (+948) Eaple 4 The sae data are used as in eaple 3, however the elevation angle changes to φ 47 and the latitude to ψ 45. The Coriolis-free distance is now L 337.3, which is slightly longer than in eaple 3, although gravity pulls the projectile back. The reason is the decreasing air pressure along the trajectory. Table 4 again shows very good agreeent. The largest error occurs in height with 6 at a distance of about 3 k. As entioned with eq. (6), in the analytical approach the corrected eponent has to be used. In this particular case.4855, n.3574 (eq. (7)) and the corrected eponent *.8. 5 Conclusions The developed analytical approach yields sufficiently accurate vertical and horiontal Coriolis deflections and represents a siple and useful etension of WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press

10 3 Coputational Ballistics III the previously developed [4] analytical fast calculation ethod for direct fire applications. References [] McCoy, R.L., Modern Eterior Ballistics / The Launch and Flight Dynaics of Syetric Projectiles, Schiffer Publishing Ltd: Atglen, PA 93, 999. [] Roetel, W., Analytische Berechnung gestreckter Geschossflugbahnen. Mitteilungen der Schießsport-Arbeitsgeeinschaft an der Universität der Bundeswehr Haburg Nr., Ed. H. Rothe, Helut-Schidt-Universität, Universität der Bundeswehr Haburg, 4. [3] Kuhrt, A., Rothe H., The use of coputer algebra and nonlinear optiiation for realtie coputation of fire orders for direct fire, Coputational Ballistics II, pp , Eds.. Sanche-Galve, C.A. Brebbia, A.A. Motta, C.E. Anderson, WIT Press 5. [4] Roetel, W., Analytical calculation of trajectories using a power law for the drag coefficient variation with Mach nuber. Coputational Ballistics II, pp. 33 3, Eds.. Sanches-Golve, C.A. Brebbia, A.A. Motta, C.E. Anderson, WIT Press 5. [5] Roetel, W., erfahren ur nachträglichen Berücksichtigung von Coriolis- Effekten bei der Bestiung der Flugbahnen von Geschossen, unpublished report, Haburg, 5. [6] Rogers, G.F.C., Mayhew, Y.R., Therodynaic and Transport Properties of Fluids, Basil Blackwell Ltd. Oford, 4 th Edition, 988. [7] Sabó, I., Einführung in die Technische Mechanik, 3. Auflage, Springer- erlag, Berlin / Göttingen / Heidelberg, 958. WIT Transactions on Modelling and Siulation, ol 45, ISSN X (on-line) 7 WIT Press