1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 2, pp , Warsaw 2010 DYNAMICS OF A CONTROLLED ANTI-AIRCRAFT MISSILE LAUNCHER MOUNTED ON A MOVEABLE BASE Zbigniew Koruba Zbigniew Dziopa Izabela Krzysztofik Kielce University of Technology, Faculty of Mechatronics and Machine Building, Kielce, Poland In the work, dynamics of a controlled anti-aircraft missile launcher mounted on a moveable base(wheeled vehicle) is analysed. Pre-programmed controls were used for basic motion of the launcher during target interception, and stabilizing controls were applied to counteract disturbances resulting from the operation of the anti-aircraft system. This type of control allows setting the longitudinal axis of the missile with respect to the target line of sight irrespective of the carrier vehicle motion. The system vibrations are due to road-induced excitations and they act directly on the vehicle, the missile launch and the control of launcher basic motion. Selected results of computer simulation were presented graphically. Key words: dynamics, control, launcher, missile 1. Introduction Modern self-propelled anti-aircraft missile systems with short-range selfguided missiles should be able to detect, identify and track aerial targets when the launcher-carrier vehicles are in motion. To improve the system accuracy, it is necessary to apply pre-programmed controls for target detection and corrective controls for target tracking(dziopa, 2004, 2005, 2006a; Koruba, 2001; Koruba and Osiecki, 1999). Synthesis of the launcher control involves determination the impact that road-induced kinematic excitations may have on the launcher performance. It is desirable that any transitional process accompanying terrain unevenness(a bump) be optimally attenuated. In this paper, motion of a launcher-missile system was considered in a three-dimensional Euclidean space and Earth s gravitational field. There were
2 280 Z. Koruba et al. sixdegreesoffreedominthediscretemodelofthesystemdescribedbyanalytical relations in the form of equations of motion, kinematic relations and three equations of equilibrium(dziopa, 2006b,c). In the general case, a launcher-missile system is not symmetric about the longitudinal vertical plane going through the centre of the system mass(dziopa, 2008; Mishin, 1990; Mitschke, 1972; Svietlitskiy, 1963). The symmetry refers to selected geometrical dimensions and properties of flexible elements. In the general case, the inertial characteristic departs from this symmetry. The launcher turret can rotate with respect to the carrier together with the guide rail and the missile. The turret rotates in accordance with the azimuth angleψ pv,whichistheturretyawangle.theturretandtheguiderailmountedonitconstitutearotarykinematicpair.theguiderailcanrotatewith respecttotheturretinaccordancewiththeelevationangleϑ pv,whichisthe guide rail pitch angle. This leads to an asymmetric distribution of masses. The system is reduced to a structural discrete model describing the phenomena that are mechanical excitations in character. The basic motion of the turretisreducedtobasicmotionofthecarrier.theturretisanobjectwhose inertial characteristic is dependent on the position of the target with respect to the anti-aircraft system. The mass of the turret remains constant, but its moments of inertia and moments of deviation change. The launcher was modelled as two basic masses, four deformable elements and a control system, as shown schematically in Fig. 1. Fig. 1. Physical model of the controlled launcher
3 Dynamics of a controlled anti-aircraft missile Thelauncherisaperfectlystiffbodywithmassm v,momentsofinertia I vx,i vy and I vz andamomentofdeviation I vxz.thelauncherismounted on the vehicle using four passive elastic-attenuating elements with linear parametersk v11 andc v11,k v12 andc v12,k v13 andc v13 aswellask v14 andc v14, respectively. Theturretisaperfectlystiffbodywithmassm w andmaincentralmomentsofinertiai wξ v,i wη v,i wζ v.theguiderailisalsoaperfectlystiffbody withmassm pr andmaincentralmomentsofinertiai prξpv,i prηpv,i prζpv. Ifthebasicmotionofthelauncherisnotdisturbed,thenthe 0 v x v y v z v, S v x v y v z v ands v ξ v η v ζ v coordinatesystemscoincideatanymoment.theturret model as an element of the spatially vibrating system performs complex motion withrespecttothe0 v x v y v z v referencesystemconsistingofstraightlinemotion ofthemasscentres v,inaccordancewithachangeinthey v coordinate,rotary motionaboutthes v z v axisinaccordancewithachangeinthepitchangleϑ v androtarymotionaboutthes v x v axisinaccordancewithachangeinthetilt angleϕ v. Prior to the launch, the missile is rigidly connected with the guide rail. The mounting prevents the missile from moving along the guide rail. Once the missile motor is activated, the missile moves along the rail. The guide rail-missile system used in the analysis ensures collinearity of pointsonthemissileandtheguiderail. It is assumed that the longitudinal axis of the missile coincides with the longitudinalaxisoftheguiderailatanymomentandthatthemissileisastiff body with an unchangeable characteristic of inertia. The missile is a perfectly stiffbodywithmassm p andmaincentralmomentsofinertiai pxp,i pyp,i pzp. The geometric characteristic of the guide rail-missile system shown in Fig. 2a andfig.2bcanbeusedtoanalyzethedynamicsofthecontrolledsystem. The main view of the turret-missile system model is presented in Fig. 2a. It includes an instantaneous position of the missile while it moves along the guide rail. Figure 2b shows the top view of the turret-missile system model. It includes the instantaneous position of the missile while it moves along the guide rail. The missile model performs straightline motion with respect to the S v ξ pv η pv ζ pv referencesystemaccordingtoachangeintheξ pv coordinate. The model of the launcher-missile system has six degrees of freedom, which results from the structure of the formulated model. The positions of the system were determined at any moment assuming four independent generalized coordinates:
4 282 Z. Koruba et al. y v verticaldisplacementoftheturretmasscentres v, ϕ v angleofrotationoftheturretaboutthes v x v axis, ϑ v angleofrotationoftheturretaboutthes v z v axis, ξ pv straightlinedisplacementofthemissilemasscentres p alongthes v ξ pv axis. Fig.2.(a)Sideviewoftheturret-and-missilemodel;(b)topviewofthe turret-and-missile model 2. Modelofmotionofthelauncheronamoveablebase The mathematical model of the system was developed basing on the physical model. Six independent generalized coordinates were selected to determine the kinetic and potential energy of the model and the distribution of generalized forces basing on the considerations for the physical model. By using the second order Lagrange equations, it was possible to derive equations of motion of the analysed system. The system was reduced to a structural discrete model, which required applying differential equations with ordinary derivatives. Six independent coordinates were used to determine motion of the launcher-missile system model: turretvibrations:y v,ϑ v,ϕ v, basicmotionoftheturretandtheguiderailψ pv andϑ pv,respectively, missilemotionwithrespecttotheguiderailξ pv.
5 Dynamics of a controlled anti-aircraft missile Motion of the controlled launcher mounted on a moveable base can be illustrated by means of the following equations ϑ v = 1 b 7 [z 2 (b 2 ÿ v +b 8 ϕ v +b 9 ξpv +b 10 ψpv +b 11 ϑpv )]+M ϑ ϕ v = 1 b 12 [z 3 (b 3 ÿ v +b 8 ϑv +b 13 ξpv +b 14 ψpv +b 15 ϑpv )]+M ϕ where ÿ v = 1 [z 1 (b 2 ϑv +b 3 ϕ v +b 4 ξpv +b 5 ψpv +b 6 ϑpv )]+F y b 1 (2.1) ψ pv = 1 [z 5 (b 5 ÿ v +b 10 ϑv +b 14 ϕ v +b 17 ξpv +b 20 ϑpv )]+M p ψ b 19 ϑ pv = 1 [z 6 (b 6 ÿ v +b 11 ϑv +b 15 ϕ v +b 18 ξpv +b 20 ψpv )]+M p ϑ b 21 ξ pv = 1 [z 4 (b 4 ÿ v +b 9 ϑv +b 13 ϕ v +b 17 ψpv +b 18 ϑpv )] b 16 y v verticaldisplacementoftheturretmasscentres v, ϕ v,ϑ v angleofrotationoftheturretaboutthes v x v ands v z v axis,respectively, ξ pv straightlinedisplacementofthemissilemasscentres p alongthes v ξ pv axis, ψ pv turretyawangle, ϑ pv guiderailpitchangle, M p ϑ,mp ψ preprogrammedmomentofcontroloftheturretpitchangleand the guide rail yaw angle, respectively, F y forcestabilizingverticalprogressivemotionofthelauncher, M ϑ,m ϕ momentsstabilizingangularmotionsofthelauncheraboutthes v z v ands v x v axes, b i (wherei=1,...,21) coordinatefunctionsy v,ϑ v,ϕ v,ψ pv,ϑ pv,ξ pv, z i (wherei=1,...,6) coordinatefunctionsy v,ϑ v,ϕ v,ψ pv,ϑ pv,ξ pv and theirderivativeswithrespecttotimeẏ v, ϑv, ϕ v, ψ pv, ϑpv, ξ pv. Explicit forms of the functions b i (y v,ϑ v,ϕ v,ψ pv,ϑ pv,ξ pv ) and z i (y v,ϑ v,vp v,ψ pv,ϑ pv,ξ pv,ẏ v, ϑ v, ϕ v, ψ pv, ϑ pv, ξ pv ) are long mathematical expressions, which developed analytically are available at the authors of the article.
6 284 Z. Koruba et al. 3. Controlofmotionofthelauncheronamoveablebase The launcher under consideration is mounted on a moveable base(wheeled vehicle).itcanbeputintorotarymotionbyangleψ pv aboutthes v y v axisand rotarymotionbyangleϑ pv aboutthes v z v axis.whenthemissilelongitudinal axis S v x p isset,forinstance,tocoincidewiththetargetlineofsight,it is necessary to apply the following pre-programmed control moments of the launcher M p ψ =kψ pv(ψ pv ψ p pv)+c ψ pv( ψ pv ψ p pv) M p ϑ =kϑ pv(ϑ pv ϑ p pv)+c ϑ pv( ϑ pv ϑ p pv) (3.1) where k ψ pv,k ϑ pv gaincoefficientsofthecontrolsystem, c ψ pv,cϑ pv attenuationcoefficientsofthecontrolsystem. The principle of operation of the launcher control system is presented in a schematic diagram in Fig. 3. Fig.3.Diagramofthesystemforautomaticcontrolofthelauncheronthe moveable base
7 Dynamics of a controlled anti-aircraft missile The quantities ψpv p and ϑp pv present in the above formulas are preprogrammed angles changing in time according to the following laws ψ p pv =ωψ pv t ψ p pv =ωψ pv ϑp pv =ϑ pv0sin(ω ϑ pv t+ϕ pv0) ϑ p pv =ϑ pv0ω ϑ pv cos(ωϑ pv t+ϕ pv0) (3.2) where ω ψ pv= 2π T ψ ω ϑ pv= 2π T ϑ Once the launcher is in the pre-determined final state, the longitudinal axis oftheguiderailneedstocoincidewiththetargetlineofsightirrespectiveof motions of the base or external disturbances. The stabilizing(corrective) controls used for the automatic control of the launcher need to have the following form M ϑ =k ϑ (ϑ v ϑ p v)+c ϑ ( ϑ v ϑ p v) M ϕ =k ϕ (ϕ ν ϕ p v)+c ϕ ( ϕ ν ϕ p v) (3.3) F y =k y (y ν y p v)+c ϕ (ẏ ν ẏ p v) where ϑ p v,ϕp v,yp v quantitiesdefiningthepre-determinedpositionofthelauncherin space during angular motions of the base, k ϑ,k ϕ,k y gaincoefficientsoftheregulator, c ϑ,c ϕ,c y attenuationcoefficientsoftheregulator. 4. Results 4.1. Systemparameters Motion of the hypothetical controlled anti-aircraft missile launcher was described using the system of equations denoted by(2.1). The launcher system was assumed to have the following parameters: Parameters of the launcher(turret and guide rail) m v =m w +m pr I vη pv =7kgm 2 I vx =(I wξ v +I prξpv cos 2 ϑ pv +I prηpv sin 2 ϑ pv )cos 2 ψ pv + +(I wζ v +I prζpv )sin 2 ψ pv I vy =I vη pv +I prξpv cos 2 ϑ pv +I prηpv sin 2 ϑ pv
8 286 Z. Koruba et al. I vz =(I wξ v +I prξpv cos 2 ϑ pv +I prηpv sin 2 ϑ pv )sin 2 ψ pv + +(I wζ v +I prζpv )cos 2 ψ pv I vxz =(I wξ v +I prξpv cos 2 ϑ pv +I prηpv sin 2 ϑ pv I wζ v I prζpv ) cosψ pv sinψ pv m w =50kg I wξ v =10kgm 2 I wη v =7kgm 2 I wζ v =12kgm 2 m pr =30kg I prξpv =0.6kgm 2 I prηpv =4kgm 2 I prζpv =3.5kgm 2 Launcher suspension parameters k v11 =30000N/m k v12 =30000N/m k v13 =30000N/m k v14 =30000N/m c v11 =150Ns/m c v12 =150Ns/m c v13 =150Ns/m c v14 =150Ns/m Missile parameters m p =12kg I pxp =0.01kgm 2 I pyp =2kgm 2 I pzp =2kgm 2 Geometric characteristics l 1 =0.3m d 1 =0.2m l p =1.6m l 2 =0.3m d 2 =0.2m l sp =0.8m Thrust and operation time of the missile starter motor P ss =4000N t ss =0.07s 4.2. Kinematicexcitations Theterrainunevenness,i.e.abump,wasassumedtobeakinematicexcitation. The basic motion of the vehicle carrying the launcher was defined as where s n =V n (t t gb )
9 Dynamics of a controlled anti-aircraft missile V n =8.3m/s velocityofthevehiclewiththelauncher, t gb =0.5s timeinwhichthevehiclecarryingthelaunchergoesoverthe bump with the front wheels. Inthecaseconsideredhere,allthevehiclewheelsclimbasinglebump. The excitations have the following form y 01 =y 0 sin 2 (ω 0 s n ) ẏ 01 =y 0 ω 0 V n sin(2ω 0 s n ) y 02 =y 0 sin 2 (ω 0 s n ) ẏ 02 =y 0 ω 0 V n sin(2ω 0 s n ) y 03 =y 0 sin 2 [ω 0 (s n l wn )] ẏ 03 =y 0 ω 0 V n sin[2ω 0 (s n l wn )] y 04 =y 0 sin 2 [ω 0 (s n l wn )] ẏ 04 =y 0 ω 0 V n sin[2ω 0 (s n l wn )] wherey 0 =0.05m,l 0 =0.35m,ω 0 =π/l 0,l wn =l 1 +l 2. Control parameters T ψ =1s T ϑ =1s ϑ pv0 = π 2 ϕ pv0 = π or ϕ pv0 =0 4 k ϑ =50000 k ϕ =20000 k y =50000 c ϑ =5000 c ϕ =2000 c y =5000 low values of the coefficients of the pre-programmed controls k ψ pv =2000 kϑ pv =2000 c ψ pv =200 cϑ pv =200 high values of the coefficients of the pre-programmed controls k ψ pv =10000 kϑ pv =10000 c ψ pv=2000 c ϑ pv=2000 Figures 4-21 show selected results of computer simulation conducted for the hypothetical controlled missile launcher. Figures 5-13 illustrate the performance of the launcher with no or some correction applied to the automatic control system(figures a and b, respectively) during motion of the vehicle over the uneven terrain(a single road bump). By analogy, the effects of the pre-programmed controls are presented infigs thelowandhighcoefficientsofthecontrolsystemareshown in Figs a and b, respectively.
10 288 Z. Koruba et al. Fig. 4.(a) Time-dependent profile of the terrain unevenness(bump); (b) time-dependent translatory displacements of the rocket with regard to the guide rail Fig. 5. Time-dependent vertical displacement y of the launcher without(a) and with(b) correction Fig.6.Time-dependentangulardisplacementϑ v ofthelauncherturretwithout(a) and with(b) correction
11 Dynamics of a controlled anti-aircraft missile Fig. 7. Time-dependent angular displacements of the launcher without(a) and with(b)correction,ϑ pv andψ pv,respectively Fig. 8. Time-dependent vertical velocity of the launcher without(a) and with(b) correction, ẏ Fig. 9. Time-dependent angular velocity of the launcher turret without(a) and with(b)correction, ϑv
12 290 Z. Koruba et al. Fig a. Time-dependent angular velocities of the launcher without(a) and with(b)correction, ϑpv and ψ pv,respectively Fig. 11. Time-dependent vertical acceleration of the launcher without(a) and with(b) correction, ÿ Fig. 12. Time-dependent angular acceleration of the launcher turret without(a) and with(b)correction, ϑv
13 Dynamics of a controlled anti-aircraft missile Fig. 13. Time-dependent angular accelerations of the launcher without(a) and with(b)correction, ϑpv and ψ pv,respectively Fig. 14. Time-dependent angular displacements of the launcher turret during pre-programmedcontrolwithout(a)andwith(b)correction,ϑ v andϕ v, respectively Fig. 15. Time-dependent angular velocities of the launcher turret during pre-programmedcontrolwithout(a)andwith(b)correction, ϑv and ϕ v, respectively
14 292 Z. Koruba et al. Fig. 16. Time-dependent angular velocities of the launcher turret during pre-programmedcontrolwithout(a)andwith(b)correction, ϑv and ϕ v, respectively Fig.17.Time-dependentpre-programmedcontrolmomentsM p ϑ andmp ψ atlow(a) and high(b) gain coefficients Fig a. Time-dependent pre-determined and real angular pitches of the launcheratlow(a)andhigh(b)gaincoefficients,ϑ z pv andϑ pv,respectively
15 Dynamics of a controlled anti-aircraft missile Fig. 19. Time-dependent pre-determined and real angular velocities of the launcher atlow(a)andhigh(b)gaincoefficients, ϑz pv and ϑ pv,respectively Fig. 20. Time-dependent pre-determined and real angular displacements of the launcheratlow(a)andhigh(b)gaincoefficients,ψ z pvandψ pv,respectively Fig. 21. Time-dependent pre-determined and real angular velocities of the launcher atlow(a)andhigh(b)gaincoefficients, ψ z pv and ψ pv,respectively
16 294 Z. Koruba et al. 5. Conclusions The following conclusions were drawn from the theoretical considerations and simulation tests: corrective controls must to be applied to prevent the occurrence of transitional processes resulting from terrain unevenness conditions(bumps); pre-programmed controls of the launcher result in negative vibrations of all its elements; the vibrations can be removed effectively by optimal selection of the coefficients of gain and attenuation in the launcher control system; the launcher control system allows immediate positioning of the launcher in space so that the guide-rail longitudinal axis coincides with the target line of sight. A feasibility study should to be conducted for the proposed control system. Particular attention has to be paid to values of the pre-programmed and corrective control moments. References 1. Dziopa Z., 2004, The dynamics of a rocket launcher placed on a self-propelled vehicle, Mechanical Engineering, 91, 3, ISSN , 23-30, Lviv 2. Dziopa Z., 2005, An analysis of physical phenomena generated during the launch of a missile from an anti-aircraft system, The Prospects and Development of Rescue, Safety and Defense Systems in the 21st Century, Polish Naval Academy, Gdynia, ISBN X, Dziopa Z., 2006a, An anti-aircraft self-propelled system as a system determining the initial parameters of the missile flight, Mechanics in Aviation ML-XII 2006, PTMTS, Warsaw, ISBN , Dziopa Z. 2006b, Modelling an anti-aircraft missile launcher mounted on a road vehicle, Theory of Machines and Mechanisms, Vol. 1, University of Zielona Góra and PKTMiM, ISBN , DziopaZ.,2006c,Themissilecoordinatorsystemasoneoftheobjectsofan anti-aircraft system, 6th International Conference on Armament Technology: Scientific Aspects of Armament Technology, Waplewo, Military University of Technology, ISBN X,
17 Dynamics of a controlled anti-aircraft missile Dziopa Z., 2008, The Modelling and Investigation of the Dynamic Properties of the Sel-Propelled Anti-Aircraft System, Published by Kielce University of Technology, Kielce 7.KorubaZ.,2001,DynamicsandControlofaGyroscopeonBoardofanFlying Vehicle, Monographs, Studies, Dissertations No. 25, Kielce University of Technology, Kielce[in Polish] 8.KorubaZ.,DziopaZ.,KrzysztofikI.,2009,Dynamicsandcontrolofa gyroscope-stabilized platform in a self-propelled anti-aircraft system, Journal of Theoretical and Applied Mechanics, 48, 1, Koruba Z., Osiecki J., 1999, Construction, Dynamics and Navigation of Close-Range Missiles Part 1, University Course Book No. 348, Kielce University of Technology Publishing House, PL ISSN [in Polish] 10. Koruba Z., Tuśnio J., 2009, A gyroscope-based system for locating a point source of low-frequency electromagnetic radiation, Journal of Theoretical and Applied Mechanics, 47, 2, Mishin B.P., 1990, Dinamika raket, Mashinostroyenie, Moskva 12. Mitschke M., 1977, Dynamics of a Motor Vehicle, WKŁ, Warszawa[in Polish] 13. Svietlitskiy V.A., 1963, Dinamika starta letatelnykh apparatov, Nauka, Moskva Dynamika sterowanej przeciwlotniczej wyrzutni rakietowej umieszczonej na ruchomej platformie Streszczenie W pracy dokonano analizy dynamiki sterowanej przeciwlotniczej wyrzutni pocisków rakietowych umieszczonej na ruchomej podstawie(pojeździe samochodowym). Zastosowano sterowanie programowe dla ruchu podstawowego wyrzutni realizowanego w procesie przechwytywania celu oraz sterowanie stabilizujące zaburzenia wynikające z działania zestawu przeciwlotniczego. Tego rodzaju sterowanie pozwala na ustawienie osi podłużnej pocisku rakietowego względem linii obserwacji celu niezależnie od ruchu pojazdu, na którym znajduje się wyrzutnia. Drgania spowodowane są wymuszeniami działającymi bezpośrednio na pojazd samochodowy od strony drogi, startem pocisku rakietowego oraz sterowanego ruchu podstawowego samej wyrzutni. Niektóre wyniki badań symulacji komputerowej przedstawione są w graficznej postaci. Manuscript received May 14, 2009; accepted for print June 19, 2009