An analysis of the dynamics of a launcher-missile system on a moveable base

Transcription

1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vo. 58, No. 4, 010 DOI: / VARIA An anaysis of the dynamics of a auncher-missie system on a moeabe base Z. DZIOPA, I. KRZYSZTOFIK, and Z. KORUBA Facuty of Mechatronics and Machine Buiding, Kiece Uniersity of Technoogy, ecia Państwa Poskiego Ae., Kiece, Poand Abstract. The anaysis focuses on the dynamics of a hypothetica anti-aircraft system during the missie aunch. The resuts of the computer simuation of motion of the auncher-missie system are represented graphicay. The diagrams proide information about the range of kinematic excitations acting on the opto-eectro-mechanica coordinator of the target in a sef-guided missie. Key words: missie auncher, system dynamics, ibrations. 1. Introduction The motion of the auncher-missie system equipped with a gyroscope was considered in a three-dimensiona Eucidean space and Earth s graitationa fied. The anaysis focused on ertica ibrations with ow aues of generaized dispacements of the auncher. The discrete mode of the missieauncher system with four degrees of freedom was described using anaytica reations in the form of equations of motion, kinematic reations and three equations of equiibrium [1 3]. In the genera case, the auncher-missie system is not symmetric about the ongitudina ertica pane going through the centre of the system mass. The symmetry refers to seected geometric dimensions and properties of the fexibe eements. In the genera case, the inertia characteristic departs from this symmetry. The auncher turret can rotate with respect to the carrier together with the guide rai and the missie. The turret rotates in accordance with the azimuth ange, ψ p, which is the turret yaw ange. The turret and the guide rai mounted on it constitute a rotary kinematic pair. The guide rai can rotate with respect to the turret in accordance with the eeation ange, ϑ p, which is the guide rai pitch ange. Generay, this eads to an asymmetric distribution of masses. After the turret and the guide rai moe to the position of target interception, the turret configuration does not change. The system is reduced to a structura discrete mode describing the phenomena that are mechanica excitations in character. The basic motion of the turret is reduced to the basic motion of the carrier. The turret is an object whose inertia characteristic is dependent on the position of a target with respect to the anti-aircraft system. The mass of the turret remains constant, but its moments of inertia and moments of deiation change. After the target is intercepted, the turret inertia characteristic remains unchanged. The auncher was modeed as two basic masses and four deformabe eements, as shown in Fig. 1. Fig. 1. Physica mode of the auncher The auncher is a perfecty stiff body with mass m, moments of inertia I x and I z and a moment of deiation I xz. The auncher is mounted on the ehice using four passie eastic-attenuating eements with inear parameters k 11 and c 11, k 1 and c 1, k 13 and c 13 and k 14 and c 14, respectiey. The inertia characteristic of the auncher is dependent on the current position of the components, i.e. the turret and the guide rai. The turret is a perfecty stiff body with mass m w and main centra moments of inertia I wξ, I wη, I wζ. The guide rai is aso a perfecty stiff body with mass m pr and main centra moments of inertia I prξp, I prηp, I prζp. If the basic motion of the auncher is not disturbed, then the 0 x y z, S x y z and S ξ η ζ coordinate systems e-mai: zdziopa@tu.kiece.p 645 Downoad Date 11/17/17 10:13 PM

2 Z. Dziopa, I. Krzysztofik, and Z. Koruba coincide at any moment. The turret mode as an eement of the spatia ibrating system performs a compex motion with respect to the 0 x y z reference system consisting of a straightine motion of the mass centre, S, in accordance with a change in the y coordinate, a rotary motion about the S z axis in accordance with a change in the pitch ange ϑ and a rotary motion about the S x axis in accordance with a change in the tit ange ϕ. Prior to the aunch, the missie is rigidy connected with the guide rai. The mounting preents the missie from moing aong the guide rai. Once the missie motor is actiated, the missie moes aong the rai. The guide rai-missie system used in the anaysis ensures the coinearity of points on the missie and the guide rai. It is assumed that the ongitudina axis of the missie coincides with the ongitudina axis of the guide rai at any moment and that the missie is a stiff body with an unchangeabe characteristic of inertia. The missie is a perfecty stiff body with mass m p and main centra moments of inertia I pxp, I pyp, I pzp. The geometric characteristic of the guide rai-missie system shown in Figs. and 3 can be used to anayze the dynamics of the controed system. the top iew of the turret-missie system mode. It incudes the instantaneous position of the missie whie it moes aong the guide rai. The missie mode performs a straightine motion of the mass centre S with respect to the S ξ p η p ζ p reference system according to a change in the ξ p coordinate. The mode of the auncher-missie system has six degrees of freedom, which resuts from the structure of the formuated mode. The positions of the system at any moment were determined assuming four independent generaized coordinates: 1. y ertica dispacement of the turret mass centre, S ;. ϕ ange of rotation of the turret about thes x axis; 3. ϑ ange of rotation of the turret about thes z axis; 4. ξ p straightine dispacement of the missie mass centre S p aong the S ξ p axis.. The mathematica mode of the auncher-missie system The mathematica mode of the system was deeoped basing on the physica mode. The system was reduced to a structura discrete mode, which required appying differentia equations with ordinary deriaties. Four independent generaized coordinates were seected to determine the kinetic and potentia energies of the mode and the distribution of generaized forces basing on the considerations for the physica mode. By using the second order Lagrange equations, it was possibe to derie the equations of motion of the anayzed system. Equations of motion for the turret: m + m p ÿ + m p ξ ξ p ϑ m p ζ ξ p ϕ + Fig.. Side iew of the turret-and-missie mode +m p ξp η + ξ ϑ ζ ϕ + +m p ξp ξ ϑ ζ ϕ + c 11 λ11 + +c 1 λ1 + k 11 λ 11 + k 1 λ 1 + +c 13 λ13 + c 14 λ14 + k 13 λ k 14 λ 14 + m + m p g = 0, 1 [ Iz + I pxp x1 + I pyp y1 + I pzp z1 + m p ξp ξ + η ] ϑ + +I pxp x1 x + I pyp y1 y + I pzp z1 z I xz m p ξ p ξ ζ ϕ + +m p ξ p ξ ÿ + m p ξ p ξ + η ϑ ξ ζ ϕ ] ξp + Fig. 3. Top iew of the turret-and-missie mode +m p ξ p ξ + η ϑ ξp m p ξ p ξ ζ ϕ ξp + +c 11 1 λ11 + c 1 1 λ1 + k 11 1 λ 11 + k 1 1 λ 1 + The main iew of the turret-missie system mode is presented in Fig.. It incudes the instantaneous position of the missie whie it moes aong the guide rai. Figure 3 shows c 13 λ13 c 14 λ14 k 13 λ 13 k 14 λ 14 + m p g ξ ξ p = 0, 646 Bu. Po. Ac.: Tech Downoad Date 11/17/17 10:13 PM

3 An anaysis of the dynamics of a auncher-missie system on a moeabe base [ Ix + I pxp x + I pyp y + I pzp z + m p ξp η + ζ ] ϕ + +I pxp x1 x + I pyp y1 y + I pzp z1 z I xz m p ξ p ξ ζ ϑ + m p ξ p ζ ÿ + m p ξ p η + ζ ϕ ξ ζ ϑ ] ξp + m p ξ p ξ ζ ϑ ξp + m p ξ p η + ζ ϕ ξp + c 11 d 1 λ11 + c 1 d λ1 k 11 d 1 λ 11 + k 1 d λ 1 + c 13 d 1 λ13 + c 14 d λ14 k 13 d 1 λ k 14 d λ 14 m p g ζ ξ p = 0. Equations of motion for the missie: where: m p [ ξ + η + ζ + ξ + η ϑ + η + ζ ϕ ξ ζ ϑ ϕ ] ξ p + +m p ξ p ξ + η ϑ ξ ζ ϕ ] ϑ + +m p ξ p η + ζ ϕ ξ ζ ϑ ] ϕ + +m p ξ + η ϑ ξ ζ ϕ ] ξp ϑ + +m p η + ζ ϕ ξ ζ ϑ ] ξp ϕ + +m p η + ξ ϑ ζ ϕ ÿ + g = P ss, λ 11 = y + y st + 1 ϑ + ϑ st d 1 ϕ + ϕ st y 01, λ 1 = y + y st + 1 ϑ + ϑ st + d ϕ + ϕ st y 0, λ 13 = y + y st ϑ + ϑ st d 1 ϕ + ϕ st y 03, λ 14 = y + y st ϑ + ϑ st + d ϕ + ϕ st y 04, λ 11 = ẏ + 1 ϑ d 1 ϕ ẏ 01, λ 1 = ẏ + 1 ϑ + d ϕ ẏ 0, λ 13 = ẏ ϑ d 1 ϕ ẏ 03, λ 14 = ẏ ϑ + d ϕ ẏ 04, ξ = cosϑ p cosψ p, η = sin ϑ p, ζ = cosϑ p sin ψ p, x1 = cosϑ p sinψ p, x = cosϑ p cosψ p, y1 = sin ϑ p sin ψ p, y = sinϑ p cosψ p, z1 = cosψ p, z = sin ψ p. 3. Sampe parameters of the auncher-missie system The excitations generated during the missie aunch are transmitted from the auncher to the gyroscopic system, which is a mechanica eement of the coordinator unit. It is required that the eectronic system be abe to conert signas in the first feedback oop under disturbance conditions. The conerted signas shoud permit tracking the ine of sight through the gyroscope axis. The photoeectric resistor conerts eectromagnetic radiation in the infrared range into eectric otage 3 4 puses using a moduating shied. These signas are transmitted to the pre-ampifier and then to the carrier frequency ampifier. The noises produced by the photoeectric resistor and the eectronic eements are eiminated, and the carrier frequency signa is separated and ampified. The signa is then sent to the eneope of the ampifier, which is tuned to ampify the frequency of the gyroscope rotationa eocity and to separate the first harmonic of the eneope of puse packages. The signa in the form of a harmonic function is transmitted to the correction ampifier. The signa is ampified to correct the gyroscope measurements and sent to the autopiot. As a resut, the first feedback oop incudes the opto-eectro-mechanica target coordinator. The foowing are parameters seected for the mathematica mode Parameters describing the inertia eements of the auncher m = m w + m pr I x = I wξ + I prξp cos ϑ p + I prηp sin ϑ p cos ψ p + I wζ + I prζp sin ψ p, I z = I wξ + I prξp cos ϑ p + I prηp sin ϑ p sin ψ p + I wζ + I prζp cos ψ p, I xz = I wξ + I prξp cos ϑ p + I prηp sin ϑ p the turret the guide rai I wζ I prζp cosψp sin ψ p, m w = 50 kg, I wξ = 10 kgm, I wη = 7 kgm, I wζ = 1 kgm, m pr = 30 kg, I prξp = 0.6 kgm, I prηp = 4 kgm, I prζp = 3.5 kgm. Parameters describing the inertia eements of the missie m p = 1 kg, I pxp = 0.01 kgm, I pyp = kgm, I pzp = kgm, 3. Parameters describing the non-inertia eements of the turret suspension: k 11 = N/m, k 1 = N/m, k 13 = N/m, k 14 = N/m, 4. Geometric characteristics 5. Thrust of the missie booster c 11 = 150 Ns/m, c 1 = 150 Ns/m, c 13 = 150 Ns/m, c 14 = 150 Ns/m, 1 = 0.3 m, = 0.3 m, d 1 = 0. m d = 0. m, p = 1.6 m sp = 0.8 m, P ss = 4000 N. Bu. Po. Ac.: Tech Downoad Date 11/17/17 10:13 PM

4 Z. Dziopa, I. Krzysztofik, and Z. Koruba Kinematic reations. The position of the missie mass centre, S p, in the 0 x y z coordinate system: r p r px, r py, r pz, 5 taken by the turret [4, 5]. The ariation in the turret inear acceeration shown in Fig. 4 was used to determine the missie aunch time. r px = ξ p ξ ξ p η ϑ, r py = ξ p ξ ϑ ξ p ζ ϕ + ξ p η + y, r pz = ξ p η ϕ + ξ p ζ. The coordinates of the eocity ector of the missie mass centre in the 0 x y z coordinate system: V p V px, V py, V pz, 6 V px = ξ η ϑ ξ p η ξ p ϑ, V py = η + ξ ϑ ζ ϕ ξ p + ξ ξ p ϑ ζ ξ p ϕ + ẏ, V pz = ζ + η ϕ ξ p + η ξ p ϕ. The modue and the direction anges of the eocity ector of the missie mass centre, S p : V p = Vpx + Vpy + Vpz, 7 Fig. 4. Linear acceeration of the turret tan γ p = V py, Vpx + Vpz tan χ p = V pz V px. 8 The coordinates of the anguar missie eocity ector in the S p x p y p z p coordinate system: ω p ωpxp, ω pyp, ω pzp, 9 ω pxp = ϕ cosϑ p, cosψ p ϑ cosϑ p sinψ p ω pyp = ϑ sin ϑ p sin ψ p ϕ sin ϑ p cosψ p, ω pzp = ϑ cosψ p + ϕ sin ψ p. Equations of equiibrium. k 11 a 11 + k 1 a 1 + k 13 a 13 + k 14 a m + m p g = 0, 10 k 11 1 a 11 +k 1 1 a 1 k 13 a 13 k 14 a 14 = 0, 11 where k 1 d a 1 k 11 d 1 a 11 k 13 d 1 a k 14 d a 14 = 0, a 11 = y st + 1 ϑ st d 1 ϕ st, a 1 = y st + 1 ϑ st + d ϕ st, a 13 = y st ϑ st d 1 ϕ st, a 14 = y st ϑ st + d ϕ st. 4. A numerica simuation of the motion of the auncher-missie system 1 The auncher configuration for t = 0 s is defined by the initia position of the turret and the guide rai: ψ p = 45 deg and ϑ p = 45 deg. During the aunch, the missie moes aong the guide rai. Any undesirabe oad reated to the aunch is Fig. 5. Linear acceeration of the turret during the missie aunch The guide rai and the missie constitute a kinematic pair. The missie motion produced by the booster affects the guide rai motion, and accordingy the turret motion. At t = 1 s the booster is fired and the missie begins its motion aong the guide rai. A change in the missie position aong the guide rai resuts from the presence of thrust. The aunch generates disturbances that are transferred onto the turret. The ariation in the inear acceeration characterizing the turret response to the excitation caused by the aunch is discontinuous Fig. 5. The jump discontinuities are reated to the changes in the missie eocity. This resuts from the physica nature of the processes occurring at those moments. The characteristic points are moments when the booster is fired or burns out and when the missie eaes the auncher. The presence of thrust causes an increase in the turret inear acceeration at t = 1 s. The thrust generates the missie motion aong the guide rai. Thus, the auncher-missie system is a time-dependent system. The moement of the missie with a pre-determined mass aong the guide rai eads to a change in the distribution of the mass in the turret-missie system. Owing to the coupings between the missie motion and the turret performance, the generated disturbances affect the ariation in the inear acceeration. When the booster burns out, the missie is no onger oaded 648 Bu. Po. Ac.: Tech Downoad Date 11/17/17 10:13 PM

5 An anaysis of the dynamics of a auncher-missie system on a moeabe base by the thrust. A reduction in the thrust to zero is responsibe for an increase in the turret inear acceeration at t = 1.07 s. The turret-missie system is not ony a system with different distribution of mass but aso a system with different mass aues. At t = 1.07 s the booster burns out and the missie uses the acquired kinetic energy. Howeer, when the missie eaes the guide rai at t = s, the mass of the turretmissie system changes ony once and in a discrete way. The new system no onger constitutes a kinematic pair as there is a change in the system structure, and accordingy, in the order of its objects. On the one hand, it describes the end of the missie motion aong the guide rai; on the other hand, it determines the initia conditions of the missie motion after the aunch. The missie aunch is responsibe for both ertica and anguar ibrations of the turret. Using the ariation in the turret anguar acceeration, one is abe to determine the moment of missie aunch with ease, Figs. 6 and 8. rai. Before the aunch, the missie is rigidy connected with the guide rai. The missie does not moe aong the guide rai unti t = 1 s, thus the reatie acceeration is equa to zero. At t = 1 s thrust is appied to the missie body, the occurrence of thrust resuts in a sudden increase in the acceeration. The missie starts moing aong the guide rai. The change in the missie position is attributabe to thrust. The aunch generates disturbances that are transferred onto the other eements of the unit. As an integra eement of the anti-aircraft system, the missie is exposed to a feedback of the system in the form of an excitation generated by the guide rai. Fig. 8. Anguar acceeration of the turret in the tit motion Fig. 6. Anguar acceeration of the turret in the pitch motion Fig. 9. Anguar acceeration of the turret in the tit motion during the missie aunch Fig. 7. Anguar acceeration of the turret in the pitch motion during the missie aunch The ariation in the anguar acceeration in the pitch and ro motion characterizing the turret response to an excitation caused by the missie aunch is a discontinuous function, Figs. 7 and 9. The missie and the guide rai constitute a kinematic pair, which is responsibe for the motion of the two eements. Figure 10 presents the time-dependent ariation in the inear acceeration of the missie. This acceeration the acceeration characterizes the missie motion aong the guide Fig. 10. Linear acceeration of the missie in the motion aong the guide rai Bu. Po. Ac.: Tech Downoad Date 11/17/17 10:13 PM

6 Z. Dziopa, I. Krzysztofik, and Z. Koruba Fig. 11. Linear eocity of the missie during a aunch Fig. 1. Direction ange of the missie inear eocity ector γ during a aunch When t = 1.07 s the missie moes using the acquired kinetic energy. The reatie inear acceeration changes in time. Figure 11 shows the time-dependent ariation in the missie inear eocity. Firing the booster causes oading the missie with thrust. The thrust is high enough to push and acceerate the missie to a considerabe eocity oer a ery short distance in a fraction of a second. The time the thrust acts on the missie is 0.07 s. The thrust is constant in time and causes the motion of the missie aong the guide rai. The eocity rises unti t = 1.07 s, when the booster burns out and the missie body is affected by some oad. As there is no thrust, the missie inear eocity fuctuates around a constant aue. The fuctuations are due to the disturbances generated in the missie aunch system. The direction ange γ of the missie inear eocity ector fuctuates around 45 deg Fig. 1. A change in this ange is due to the disturbances generated in the missie aunch system. A change in the direction ange χ of the missie inear eocity ector sighty departs from 45 deg Fig Concusions The resuts show that the missie moing aong the guide rai causes ibrations of the system. The ibrations, in turn, constitute kinematic excitations for the drie of the target coordinator contro, i.e. the mechanica gyroscope. It is necessary to anayze the effects of the excitations on the operation of the gyroscope, because its accuracy is responsibe for the effectie performance of the sef-guided missie, i.e. the destruction of a moeabe aeria target. Acknowedgements. The authors acknowedge support from the Ministry of Science and Higher Education through the project No conducted within the years REFERENCES Fig. 13. Direction ange of the missie inear eocity ectorχ during a aunch As a resut, the reatie acceeration changes unti t = 1.07 s. At t = 1.07 s, the booster burns out and the missie body is no onger oaded with thrust. The reduction in thrust to zero eads to a sudden increase in acceeration. Due to arious interactions, the acceeration is not equa to zero. [1] Z. Dziopa, The dynamics of a rocket auncher paced on a sef-propeed ehice, Mechanica Engineering 81 3, [] Z. Dziopa, The missie coordinator system as one of the objects of an anti-aircraft system, 6 th Int. Conf. on Armament Technoogy: Scientific Aspects of Armament Technoogy, Miitary Uniersity of Technoogy 1, [3] V.A. Sietitskiy, Dynamics Start Fying Objects, Nauka, Moscow, [4] Z. Dziopa, The modeing and inestigation of the dynamic properties of the se-propeed anti-aircraft system, Monographs, Studies, Dissertations o. 3, No. M9, Kiece Uniersity of Technoogy, Kiece, 008. [5] Z. Koruba, Z. Dziopa, and I. Krzysztofik, Dynamics and contro of a gyroscope-stabiized patform in a sef-propeed antiaircraft system, J. Theoretica and Appied Mechanics 48 1, Bu. Po. Ac.: Tech Downoad Date 11/17/17 10:13 PM