Horizontal Backward Launch Dynamics Modeling and Analysis

Fighters’ air strikes play a vital role in modern warfare. But, with the development of fighters, the amount of bombs will inevitably become a major disadvantage due to the limitations of the invisible design and other functions, so a combination of the old model bombers and the advanced fifth-generation fighters has been proposed. This paper takes the aircraft missile horizontal backward launch system as the research object; firstly, the finite element model of the missile’s off-track is established, and the simulation calculation under the fixed platform is tested to verify the correctness of the finite element model. Meanwhile, the simulation parameters of the initial trajectory are obtained. Then, by establishing the separation model of the machine under the open state of the aircraft’s rear launcher, the variation of the flow field during the separation process of the missile is analyzed, and the variation of the force and the attitude of the missile is studied. It is found that the pitching motion of the missile is greatly affected by the initial pitch angle, it is always in the heading state during the whole separation process, and the yaw motion is not obvious.


Introduction
Fighters' air strikes play a vital role in modern warfare.Through the preemptive air strike, the enemy's living forces will be destroyed in the first round, and then, the next round of strikes will be carried out.But, with the development of fighters, the amount of bombs will inevitably become a major disadvantage due to the limitations of the invisible design and other functions, which makes the use of fighters full of limitations and cannot fully play its role in the war so a combination of old model bomber and advanced fifthgeneration fighters has been proposed.The establishment of a mature and safe missile weapon launching system in the interior of the large strategic bomber cabin is the key to solving this problem.
Many scholars have done a lot of meaningful research on the separation of the missile aircraft.Schindel [1] systematically described the separation process of unpoweredunguided external objects and proposed the safety criteria for the separation of the external objects from the carrier.Covert [2] studied the flight path of the external objects during the separation process and proposed a safe separation criterion based on the relative initial velocity and relative initial acceleration of the external object.Wang et al. [3] used the aerodynamic modeling method of "free flow model plus interference model" to establish aerodynamic model of missile in the separation process.Zheng and Liao [4] studied the research on the criterion of safe separation of machine bombs and unified the basic concept of machine separation.Anandhanarayanan et al. [5] conducted research on the separation process of powered rail-type aircraft missiles.Zhang and Ni [6] proposed that the rudder bias should be suppressed during the launching process to increase the tendency of the missile to roll, so the missile is as stable as possible during the offtrack process, and the safety of the separation of the missile is guaranteed.Chen et al. [7] based on the contact dynamics between the suspension and the guide rail, established a finite element simulation analysis model of the airborne missile rail launch process and conducted in-depth research on the safety of the launch process.With the rapid development of computational fluid technology, many scholars [8][9][10][11][12][13] use CFD calculation methods to study the separation safety of aircraft missiles.However, the issue of the separation of the machine that is launched horizontally backwards from the tail of a large transport aircraft has not yet been published.In this paper, the multibody dynamics, launch dynamics, and computational fluid dynamics theory are used to carry out modeling and theoretical analysis on the initial disturbance of missile's off-track and safety analysis of missile separation.It provides a theoretical basis and reference for the design and optimization of airborne missile horizontal backward transmission system.

Six Degrees of Freedom Model for Missile-Aircraft Separation
After the missile derails, it will enter the complex flow field below the tail of the aircraft.This paper focuses on the movement of the missile in the flow field around the tail of the aircraft after derailment.Therefore, in order to obtain the motion parameters of the missile after the derail, it is necessary to know the force and the moment acting on the missile and establish and solve the six degrees of freedom model for missile-aircraft separation.And this provides input for the coupled motion of the missile in the flow field after it is offtrack.

Coordinate System Definition and Conversion.
The two coordinate systems that need to be established in the separation are shown in Figure 1.The carrier coordinate system (inertial coordinate system) is fixed to the carrier, the ox-axis points in the horizontal direction toward the tail, the oy-axis is positive in the vertical symmetry plane, and the oz-axis is determined by the right-hand rule.The origin of the missile coordinate system is located at the center of the mass of the missile.O 1 x 1 is the axial direction of the missile and points to the tail of the missile; o 1 y 1 is in the longitudinal symmetry plane of the missile, and the initial moment is upward; and o 1 z 1 can be determined by the right-hand rule.
The transformational relationship between the projectile coordinate system and the carrier coordinate system is required to solve the missile motion.The relative relationship between the projectile coordinate system and the carrier coordinate system is determined by three angles [14], which has been shown in Figure 2.
(1) Pitch Angle ϑ.The angle between the axis of the missile's longitudinal axis o 1 x 1 and the horizontal plane oxz.If the longitudinal axis of the missile is above the horizontal plane, the pitch angle is positive.
(2) Yaw Angle ψ.The angle between the projection of the longitudinal axis of the missile in the horizontal plane and the ox -axis of the carrier coordinate system.If the ox-axis is rotated to the axis ox ′ is counterclockwise, the yaw angle is positive.
(3) Roll Angle γ.The angle between the o 1 y 1 -axis of the missile coordinate system and the vertical plane ox′y′ containing the longitudinal axis of the missile.Looking at the tail of the missile toward the tail, if o 1 y 1 is to the right of the plane, the roll angle is positive.
The conversion from the carrier coordinate system to the missile coordinate system undergoes 3 rotations: first, rotating ψ around the oy-axis, then, rotating ϑ around the oz ′ -axis and finally, rotating the γ angle around the ox 1 -axis.The transformation matrix is The coordinate transformation matrix from the ground coordinate system to the missile coordinate system is   International Journal of Aerospace Engineering aerodynamic force of the entire missile can be expressed as this equation: where î, ĵ, k is the unit basis vector group of the carrier coordinate system.F pressure is the aerodynamic force caused by pressure, and F shear is the aerodynamic force formed by shear stress.

Aerodynamic Moment.
The calculation of the aerodynamic moment is carried out in the missile coordinate system, so the aerodynamic force is first converted into the aerodynamic force in the missile coordinate system, and then, the centroid is taken.
where î1 , ĵ1 , k1 is the unit basis vector group of the missile coordinate system, L is the coordinate transformation matrix of the carrier coordinate system to the missile coordinate system, and r i is the radial diameter of the centroid of the face element i in the missile coordinate system.

Dynamic Equations
(1) Centroid Translation.Establishing the centroid translational motion equations in the ground coordinate system, then equation ( 5) is available according to Newton's second law.
where G x , G y , and G z are the components of gravity, and V x , V y , and V z are the components of the centroid acceleration, respectively.
(2) Missile's Rotation.The missile coordinate system is dynamic.Given ω is the angular velocity relative to the ground coordinate system.The dynamic equation of the missile's rotation around the centroid in this coordinate system is established as The components of ω are set to ω x 1 , ω y 1 , and ω z 1 .The moment of momentum's constituents are set to H x 1 , H y 1 , and H z 1 .
where J is the inertia tensor, and since the three axes of the missile coordinate system are the principal axes of inertia, the inertia of the missile to each axis is zero, so there is Bring equations ( 7) and ( 8) into equation ( 6), and the dynamic scalar equations of the missile's rotation around the centroid is obtained.
2.2.4.Kinematic Equations.Solving the dynamic equations can obtain the velocity and angular velocity of the missile at each time step, and then, the trajectory and attitude of the missile can be gained by solving the kinematic equations.
(1) Translational Equation.Deriving the displacement of the missile in three directions can get the speed in three directions.
(2) Rotational Equation.To obtain the body's pitch angle ϑ, yaw angle ψ, and roll angle γ, it is necessary to establish the relationship between the attitude angle and the rotational coordinate velocity ω of the missile coordinate system relative to the carrier coordinate system.
where ψ and oy have the same direction, ϑ and oz 1 share one direction, and γ and ox 1 have the same direction.So the equation below can be obtained: And then, 3 International Journal of Aerospace Engineering 2.3.Solving the Motion Equations.Equations ( 10) and ( 13) form the kinematic equations of the missile, and with equation ( 9) and ( 5), it constitutes the six degrees of freedom rigid body equation of motion.The equations consist of 12 differential equations and are calculated using the fourth-order Runge-Kutta method.

Modeling and Analysis of Missile Horizontal Backward
Launching System.The aircraft missile horizontal rearward launching system is placed inside the cabin of a large transport aircraft.When the aircraft is operated, the rear cabin door is opened, and the missile is ejected backwards but pointing forward.

System
Composition.The entire launching system is composed of a storage device, a locking device, an ejection device, a sliding device, and a control system.The topology of this system is as shown in Figure 3.
When the missile is launched, the missile acquires a certain ejection speed through the ejection device; then, it performs unpowered sliding on the sliding device (horizontal rail), until it is completely separated from the sliding device.

Finite Element Modeling and Verification.
According to the topological relationship displayed above, the finite element model of the launcher is established, and a single-shot missile-launching device is built to perform the ground ejection test to simulate the offtrack process of the missile under fixed conditions, which can be seen in Figures 4 and 5.By conducting the ground ejection test and comparing with the offtrack results under the fixed conditions of the platform, the finite element analysis model of the horizontal backward-launched missile's off-track is verified.
The Table 1 below gives a comparison of the simulation results of the offtrack speed of the horizontal rear-launched missiles and the experimental results.It can be seen that the simulations agree well with the experimental results.
2.4.Initial Disturbance of Missile's Off-Track.According to the finite element model established above, the motion process of the missile on the launching frame is simulated, and the initial disturbance of the missile's off-track is obtained as shown in Table 2.

Modeling and Analysis of Missile-Platform Separation
In the initial study of the horizontally-launched outer ballistics of aircraft missiles, due to the different launching environments and the relative motion of the launches, the horizontal rearward launch of aircraft missiles and the external launch of aircraft missiles are very different on the initial segment of the outer ballistics.The experience of external aircraft launch research can only provide little reference, and it is necessary to do a more comprehensive calculation and analysis of horizontal backward launch environment and motion laws.

Flow Field Modeling of Initial Ballistics
3.1.1.Geometric Boundary.The computational geometry consists of aircraft, missiles and launchers, as shown in Figure 6.
Because the launching frame is far from the door, its internal flow is very weak, and it has complex structures such as truss structure, guide rails, and ejection devices.The internal space of the launcher is narrow, so the space occupied by the launching rack is simplified to the outer contour of the launching frame as a fully enclosed entity.
3.1.2.Computational Domain.The computing domain used for flow field calculation is shown in Figure 7.The darkershaded part of the figure is the outline of the aircraft.The size of the computational domain is 400 m × 700 m × 150 m, and this size is used to meet the requirements of far-field boundary conditions.
3.1.3.Computational Grid.The role of the missile in the flow field is influenced by its aerodynamic shape, so it is based on the aerodynamic shape of the missile in this study.The surface boundary of the missile is encrypted to account for the viscous effect of the airflow.Figures 8 and 9 show the aircraft surface mesh.Grid encryption is made at the wing edge, belly, and tail to better simulate the carrier flow field.The total number of mesh is 3.78 million, of which the minimum mesh size is 0.03 m.The time step is 0.001 s according to the CFL condition, and 23 processors and 192 hours are used to complete the calculation.

Separation Analysis after Derailment.
According to the safety judgment criterion of the separation of the machine, the separation safety mainly considers: after the ejection, the missile does not collide with the outer envelope of the carrier; the missile have suitable pitch, yaw, and roll movements; and the distance between the missile and the carrier in all directions increases.
When the pitch, yaw, and roll movements of the missile reduce the distance between it to the aircraft, or even cause collision, it can be considered as an unsafe separation.After the launch, the distance from the carrier in all directions is increased, and if the missile does not exhibit obvious pitch, yaw, and roll motion, it can be regarded as the optimal safety separation.The distance between the missile and the carrier in all directions increases slowly with time, and if the missile exhibits a small attitude motion, it can be regarded as a safe separation.If the missile exhibits pitch, yaw, and roll motion after separation, although it still stays in the carrier interference flow field for a while and the distance from the carrier is increased slowly, it can be considered as a basic safety separation.But in engineering practice, it should be avoided.

Flow Field Analysis.
After the missile is derailed, the flow field distribution at the tail of the aircraft is closely related to the relative motion of the missile, and the flow field state in return affects the motion of the missile, so the motion in the interference flow field is very complicated.Therefore, in the study of missile separation safety, it is necessary to clearly understand the interference of the flow field on missile movement.Firstly, the flow field of the carrier in the case of no 4 International Journal of Aerospace Engineering bomb is calculated to reach the steady state, to simulate the preparation process after the aircraft opens the launcher until the missile is launched, and the internal flow field of the cabin is basically stable, providing a stable flow field environment for the missile's launch.Secondly, the missile is added to the flow field model, and the constant calculation is performed again until convergence.This is because the speed of the missile's movement is low relative to the airflow.The missile's launch can be considered as an approximate quasistatic process.In order to eliminate the large disturbance caused by the addition of the missile model to the flow field parameters, the model is calculated to constant convergence after joining the missile, and then, the six degrees of freedom         The offtrack parameters of the missile are shown in Table 2.The atmospheric parameters are selected according to the literature, the flight speed is 80 m/s and the flight tude is 8000 m.The aircraft flies stable (has controlled aircraft movement) and missile control surfaces are locked, until the missile has safely separated from the aircraft.
Figures 10-19 show the velocity vector contour and pressure contour around the missile at 1 second period.It can be seen from the flow field contours that during the whole separation process, the missile gradually enters into the high-speed high pressure field outside the cabin from the stable low-speed low pressure field in the cabin, and the force and motion posture of the missile are affected by the flow field, and changed significantly.It can be seen from Figure 20 that during the 0-0.45 s period, the pitching moment is positive, which causes the missile to have a tendency to bow.During this time period, only the second half of the missile enters the shear mixing zone and is subjected to a large lift force, while the first half is subjected to less lift, so that the overall torque is a head moment.After 0.45 s, the first half of the missile began to gradually enter the shear mixing zone, and under the combined effect of the heading angle and the velocity distribution of the flow field, the angle of attack of the first half of the missile is greater than the angle of attack of the second half; thus, making the dynamic forces at the first half quickly exceeds that at the second half, and the torque changes to the head-up torque.Under the action of the pitching moment, the pitching angular velocity and the pitching angle curve are shown in Figures 21 and 22.Because of the initial

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International Journal of Aerospace Engineering disturbance of the missile caused by a large initial heading angle and angular velocity, the missile always looks up, and the head is increasing during the entire offtrack movement.Figures 23 and 24 show the yaw angular velocity and lateral velocity of the missile versus time.Since the initial yaw angle and lateral velocity of the missile's off-track are very small, and the maximum yaw rate value is 0.68 degrees per second, the lateral maximum speed is 0.425 meters per second during the separation process.It can be seen that the yaw motion of the missile is not significant compared to the pitch motion.Therefore, the pitch attitude of the missile as it leaves the aircraft contribute to safe separation, but when the missile falls off the aircraft, it may not be ideal due to it enters the flow stream around the aircraft.

Analysis of Missile's Separation Safety.
The motion posture of the missile during the separation process will directly determine whether the missile will collide with the carrier or not, especially for the pitching motion after the missile      International Journal of Aerospace Engineering derails, which brings the missile's "head-up" phenomenon, and that will become an important hidden danger of the missile separation accident.In this section, the motion attitude of the missile in 0-1 s is given from the two directions of side view and back view, and the safety of missile separation is analyzed and discussed.
It can be seen from Figures 25-30 that during the whole separation movement, the missile does not directly collide with the carrier, the safety separation is realized.In order to more accurately describe the minimum distance change between the missile and the tail tank, the minimum distance variation curve between the missile and the tailgate of the carrier during the separation process can be seen from Figure 31, and the missile separation process is further clearly described.It can be seen that with this condition, the minimum distance between the missile and the aircraft's upper launcher appears at the initial moment of separation and increases with the progress of the separation process.
Therefore, the missile is considered to be completely safely separated under this condition.International Journal of Aerospace Engineering the guide rail.Both the force and the attitude of the missile are affected by the flow field and have produced significant changes.The variation of the force and motion attitude of the missile is studied.It is found that the pitch motion of the missile is greatly affected by the initial pitch angle.It is always in the heading state during the whole separation process, and the yaw motion is not obvious.The minimum distance between the missile and the carrier is continuously reduced to achieve safe separation.

Figure 3 :
Figure 3: Main structure topology diagram of aircraft missile launching system.

Figure 4 :
Figure 4: Finite element model for launching of airborne missile weapon system.

Figure 7 :
Figure 7: Side view of computational domain.

Figure 9 :
Figure 9: Side view of aircraft's surface grid.

3. 2 . 1 .
Analysis of Force and Motion.After knowing the change of flow velocity and pressure around the missile, the force situation and motion attitude during the missile separation process are analyzed and discussed.

Figure 15 :
Figure 15: Flow field pressure contour at 0 s.

Figure 17 :
Figure 17: Flow field pressure contour at 0.6 s.

Figure 18 :
Figure 18: Flow field pressure contour at 0.8 s.

Table 1 :
Missile's offtrack speed simulation and test