#### Abstract

In the background of a container-type vertical launch missile, the simulation method of adapter separation in different wind speeds is researched. Based on force analysis of the adapters during their separation from the missile, the dynamic and kinematics equations of the adapter separation are established. The adapter’s aerodynamic parameters at different attitudes getting from the numerical wind tunnel are chosen to be the input. Through the dynamic simulation of the separation process of the adapters, the simulation results are in good agreement with the experimental data. The trajectory and placement distribution of adapters are obtained during the analysis of force and motion stance at different wind speeds. Then the relative distances between the adapter and missile or launch facility are determined. At the same time, it can be estimated that the combined calculation will save about two-thirds of time compared with dynamic grid method computing, which provides a significant guidance for the simulation method of adapter separation.

#### 1. Introduction

Along with the development of science and technology, the container-type launching technology is widely used in tactical missiles. The main function of a launcher is to store, deliver, and launch the missile. Between some missiles and the launchers, there are elastic liners, named adapters. The adapters play roles of supporting, shock-absorption, guidance, and initial disturbance controlling during the process of missile storage, transportation, and launching. Launching with adapters simplifies the structure and facilitates the loading. This technology is applied to some missiles successfully, such as “Polaris,” “Crotale,” “Exocet MM 40,” and “YJ-62” in China [1–3].

A lot of scholars have had researches on the separation of missile adapters or two-body. Niu et al. [4] made numerical analysis on the initial separation trajectory of missile adapters with the six degree of freedom dynamic mesh technology based on the reconstruction method. Li and Chen [5] researched on the simulation method of missile adapters with a 3-D solid motion interference algorithm and analyzed the adapter separation process in the ejection gas flow field. Gao et al. [6] studied the influences that the action points of separating force and the wind speed have on adapter separation and concluded with the adapter separation distance and the value of collision force at the time when the adapters and the projectile collide. Chen and Zhai [7] took advantage of a dynamic simulation software to make the dynamics simulation for the adapter separation process of a certain missile and analyzed the collision between the adapter and the case. Wang [8] analyzed and calculated the reliability of the separation action of the missile and the adapters; in their studies, the reliability of the adapter separating action was measured under different initial conditions with quantitative data. Cicci et al. [9] investigated the dynamics of the separation of a two-body missile. Djerassi and Viderman [10] analyzed the motion of a missile that was separated by explosive bolts into two bodies in atmospheric free fall. Anandhanarayanan et al. [11] simulated the separation of an air-to-air missile and validated with flight data.

For one certain missile, the adapter launching system was adopted and the target practice with the adapter separation was experimented. Limited by the conditions, the experiment is not comprehensive enough. Coupling calculation methods in hydrodynamics and dynamics are used [12–16] to calculate the movement trails of adapter separation under the random winds from different directions with the wind speeds ranging from 0 m/s to 20 m/s to get the space length between the adapters and the missile or the land-based equipment, which is taken to judge if the adapters will hit the missile or the land-based equipment, and get the reliability of adapter separation. The calculation can be divided into several steps: firstly, the simulated wind tunnel test for the adapters was done through the calculation method of hydrodynamics to get the aerodynamic parameters of the adapter under different postures and speeds. Then dynamic simulation calculation was done for the adapter separation. The wind load during the adapter separation was obtained through looking up the aerodynamic parameters’ list according to its speed and posture. After verification, the above coupling calculation can not only reduce the time for simulation to a large extent but also ensure higher reliability.

#### 2. The Layout and Structure of the Adapters

In one certain missile system, four adapters are placed in a circle in a canister. Each adapter is positioned with the missile through pin holes. After the missile is launched out of the canister, the adapters are separated from the missile using the springs [17]. The layout of adapters on the missile is shown in Figure 1.

A single adapter is composed of a foam body, rubber plate, housing, spring, sleeve, and latch. The adapter structure is shown in Figure 2. After the missile is put in the store and launch canister, the springs will store the energy. When the missile is launched out of the canister, the springs release the energy and the adapters are separated from the missile and get certain separation speed. Under the comprehensive action of the pushing force from the springs and aerodynamic force from the air, the adapters float down to the earth [18–20].

#### 3. Theoretical Model for the Adapter Separation from Missile [21]

When the adapters are launched out with the missile, the movement in the air can be taken as the arbitrary space motion of a free rigid body which has six degrees of freedom. The space motion of a rigid body can be decomposed into translational motion along with its barycenter and fixed-point motion around an instantaneous axis of its barycenter. The body axis coordinate system, the earth axis coordinate system, and the hybrid coordinate system of the adapters are set, respectively, to clearly describe the flight path, posture, and position of the adapters and the projectile.

##### 3.1. Force Analysis on the Separation Process of the Adapters and Missile

When the adapters just get out of the canister and the springs have not returned to their original shapes, there are three forces acted on the adapters: elastic force from the separation spring, joint force of aerodynamic force , and the gravity . After translating the elastic force and the joint force of aerodynamic force to the barycenter of the adapters, according to the translation principle, two moments which can rotate the adapters are generated: the moment of the elastic force to the barycenter and the moment of the joint force of aerodynamic force . The adapter force diagram is shown in Figure 3.

##### 3.2. The Dynamic Equation for the Barycenter Motion of the Adapters

According to the force analysis above, this study chooses the hybrid coordinate system as the speed coordinate system to describe the barycenter translation motion of the adapters. The total aerodynamic force which acts on the adapters can be decomposed into the resistance , the lift , and the lateral force , along the speed coordinate system. For the adapters of short-range tactical missiles, the gravity can be taken as the negative direction along Ay axis of the earth-fixed axis system. The elastic force and the axis of the adapter coincide. Thus, the dynamic equation of the adapter barycenter translation is shown as

In the equation, is the trajectory tilt angle, is the trajectory deflection angle, is the angle of attack, is the angle of the side slip, is the velocity inclination angle, is the projection of adapter barycenter acceleration along the trajectory tangent, is the projection of adapter barycenter acceleration along the trajectory normal within the vertical plane, is the horizontal component of the adapter barycenter acceleration, is the absolute velocity of the adapter barycenter, and is the mass of the adapter.

##### 3.3. The Dynamic Equation of the Adapter Rotation around the Barycenter

Suppose the moment of inertia of the adapter is relative to the three coordinate axes in the body axis coordinate system which are , , and , respectively; the projective components of the relative rotational angular velocity vector are , , and . The projections of the resultant external moment on the adapter barycenter on the three axes in the body axis coordinate system are and . The expression of the differential equation of dynamics for the movement of the adapter around the barycenter in the body axis coordinate system can be concluded.

##### 3.4. The Kinematics Equation of the Adapter Barycenter Motion

In order to decide the movement trails (trajectory), the adapter barycenter is relative to the earth-fixed axis system; the kinematics equation of the motion which missile barycenter is relative to the earth-fixed axis system is required.

Suppose the coordinates of the adapter barycenter relative to the earth-fixed axis system are (, , ), at the instantaneous moment ; the projections of the adapter barycenter velocity at the coordinate axes in the earth-fixed axis system are

The velocity vector of the adapter barycenter and the axis in the trajectory coordinate system coincide. The kinematics equation of the adapter barycenter motion can be concluded using the transformation matrix of the coordinates of the earth-fixed axis system and the hybrid coordinate system. The formula is as follows:

##### 3.5. The Kinematics Equation of the Adapter Barycenter Motion

Because the earth-fixed axis system is fixed with the adapters, the rotational angular velocity that the adapter is relative to the earth-fixed axis system can also be shown as

In order to set the relation between the posture angles , and , the adapter is relative to the earth and the projections are , , and of the rotational angular velocity of the adapter on the three coordinate axes in the body axis coordinate system. The formula is as follows:

After the above formula is expanded, the kinematics equation of the missile rotation around the barycenter is obtained.

##### 3.6. The Dynamic Equation of the Missile Barycenter Motion

In the formula, , , and are the projections of the propulsive forces of the missile engine on the three coordinate axes in the instantaneous earth-fixed axis system; , , and are the coordinates of the missile barycenter at the earth-fixed axis system; is the mass of the missile; and is the gravitational acceleration.

#### 4. Analysis on the Influencing Factors of the Adapter Separation

When the missile is launched out, the adapters and the projectile are moving together in the canister. After sliding out of the launch canister, the adapters are separated from the projectile under the action of the elastic force, the aerodynamic force, and their own gravity of the springs and they fly along the preset paths in the air, while the missile flies along the set trajectory under the action of the propulsive force of the engine. According to the adapter separation characteristics, the factors which influence the separation between the adapters and the projectile are the following [22–25]: (1) Aerodynamic shape and mass: when the outlet velocity is certain, if the aerodynamic lift increases, the resistance coefficient decreases, and the adapters can obtain longer separation distance. If the adapter mass is small, the flying motion of the adapter after separation is influenced easily by the air current. (2) The elastic force of the separation springs and the potential energy stored before the separation: if the elastic force increases, the initial acceleration of the adapter separation increases. If the potential energy stored before separation increases, the moving distance in the separation process increases. (3) The distance between the action point of the spring force and the barycenter of the adapter: the position of the action point of the spring force influences the moving posture of the adapter, so it should be selected properly. (4) Wind speed and direction. (5) Launching angle: generally, in order to expand, the launching angle is favorable for the adapter separation. (6) The initial velocity of the adapter when it is out of the canister: when the initial velocity is increased, the adapter separation velocity is accelerated, which is favorable for the separation. The basic requirements of adapter separation are as follows: the adapters do not touch the projectile, and the extending missile wings or rudders during the launching process and a certain safety distance need to be maintained.

In this study, the action point of the spring force and the adapter barycenter coincide and the launching angle is 90^{o}. The focus of this study is the influences of the wind speed and direction on the adapter separation at the launching site.

#### 5. Numerical Wind Tunnel Test and Dynamic Simulation Calculation

##### 5.1. Numerical Wind Tunnel Test

From the moment that the adapters get out of the canister to the moment they fall to the ground, the adapters are influenced by the air as well as the natural wind from one direction with random force, because of their motion to the three directions in the earth-fixed axis system. Therefore, the composition of velocities of the natural wind and the adapters forms a “joint wind” which actually acts on the adapters in which direction and velocity are not fixed (0~50 m/s). The aerodynamic force (the force and the moment of three directions) of the joint wind acted on the adapters changes along with the motion of the adapters. If the coordinate system is fixed with the adapters, the vector joint wind can be decomposed into this coordinate system, as shown in Figure 4.

The force of joint wind acted on the adapters and the wind moment , in which is the coefficient of the wind force, is the coefficient of the wind force moment, is the air density, is the adapter velocity, and is the reference area. The coefficient of the aerodynamic force of the low speed moving object does not change a lot along with the Mach. The velocity mentioned in this thesis is lower than 0.15 Mach, so the objects discussed are low-speed moving objects. Because the wind speed does not change a lot, approximately, and are taken as no changes along with the time and they are only relative to the angles. Through simulation calculation, it can be seen that is gradually getting smaller when the wind speed is increasing in small amount. The comparative deviation based on the got from the direction when the wind speed is 1 m/s is within 3.47%, and the comparative deviation based on the is within 1.43% when the wind speed is 10 m/s. The details are shown in Table 1. (Note that refers to the comparative deviation based on the at the wind speed of 1 m/s and refers to the comparative deviation based on the at the wind speed of 10 m/s.)

In order to get more accurate and practical calculation result, the aerodynamic parameters at the wind speed of 10 m/s are taken as the benchmark. In this way, all the deviations of aerodynamic parameters can be controlled within 1.5%. That is, the changes of aerodynamic parameters along with the speed are not taken into consideration. The aerodynamic parameters at the wind speed of 10 m/s are taken as the standard, and only the changes of the aerodynamic parameters long with the angles and are taken into consideration.

Since the adapter models are symmetric, a symmetry plane can be used to save half of the calculation time. It can be concluded from the simulations that the coefficients of aerodynamic force have the same symmetric relation. Therefore, half of the calculation is saved and half of the calculating time is saved.

Through simulation calculation, the coefficient changes of the aerodynamic force and can be obtained. The coefficient curves of the aerodynamic force are smooth and basically show sinusoidal variation. This is mainly because the shapes of the adapters are close to rectangular and they are comparatively regular.

##### 5.2. Dynamic Simulation for Adapter Separation

The simulation model is composed of the missile, the adapters, and the launching canister. The mass changes during the launching process are neglected, and they are supposed as a constant value. In addition, all the components are rigid bodies which had little distortions.

Through a numerical simulated wind tunnel, the aerodynamic parameters of the adapters under certain wind speed, different attack angles, drift angles, and spin angles are calculated. During the simulation calculation, the velocity and posture the adapters move in the air are read and the ADAMS (dynamic simulation software) user subprograms are defined. The aerodynamic load under this condition is obtained through interpolation and loaded to the adapters to realize the real-time aerodynamic loading for the adapters. The simulation flowchart is shown in Figure 5.

##### 5.3. Model Verification

In order to verify the simulation model, the experiment was built with wind speed below 3 m/s. The comparison between tests and simulation models are shown in Figures 6–8. The distance between the adapters and the projectile, and the postures of the rotation are shown in Figures 6–8 at certain time.

It can be seen from the comparison that the distance between the adapters and the projectile and the rotation posture in the simulation results fit well with the experiment. Therefore, the calculation method combining the hydrodynamics and dynamics is feasible with reliable simulation.

#### 6. The Result and Analysis of the Simulation Calculation

The adapters are affected the most when a certain adapter faces the upwind fully (as shown in Figure 9). This study calculates the trails and the aerodynamic load of the adapters with the test wind speed (3 m/s) and the maximum wind speed (20 m/s). The space distances between the adapters and the missiles or the ground equipment are calculated at the same time, which can be used to determine whether the adapters will hit the missile and the ground equipment or not.

##### 6.1. Simulation Result with the Wind Speed of 3 m/s

The fall point distribution data are shown in Table 2, and the diagrams are shown in Figures 10 and 11.

Adapter 3 falls below the tail of the missile at = 0.5998 s. At this time, the distance between the adapter and the projectile is 1.997 m. The adapter does not collide with the projectile, and all the adapters do not collide with the projectile during the simulation process.

The aerodynamic loads of the adapters are shown in Figures 12 and 13 (only the aerodynamic load acting on adapter 3 is attached).

##### 6.2. Simulation Result with the Wind Speed of 20 m/s

The fall point distribution data is shown in Table 3, and the diagrams are shown in Figures 14 and 15.

Adapter 3 falls below the tail of the missile at = 0.5819 s. At this time, the distance between the adapter and the projectile is 1.196 m. The adapter does not collide with the projectile, and all the adapters do not collide with the projectile during the simulation process.

The aerodynamic loads of the adapters are shown in Figures 16 and 17 (only the aerodynamic load acting on adapter 3 is attached).

#### 7. Conclusions

After the verification on the feasibility of the calculation method combining hydrodynamics and dynamics, a series of simulation calculations is done in this study. Based on the result analysis, the following conclusions are drawn: (1)With the condition of wind speed at 3 m/s, the flight distance of the adapter is calculated as follows: adapter 3 falls 3.7 m away when it is perpendicular to the wind direction, 10 m away at the upwind direction, and 20 m away at the downwind direction. With the land-based equipment size of 12 m × 3 m, there will be three conditions needed to be considered, which are as follows: the land-based equipment direction is perpendicular to the wind direction, following upwind direction, and downwind direction. The fall point diagram can be drawn as shown in Figure 18 It can be seen from the picture that, when the wind speed is 3 m/s with the land-based equipment following the upwind direction, there is a possibility that the adapter may be dropped on the truck head, while the adapter will not be dropped on the land-based equipment with the other two wind directions. Since the fall point scope of the adapter is small, relative equipment shall be placed out of the fall point scope to ensure that the adapter will not fall on the other corollary equipment.(2)It can be seen that from the simulation results, when the land-based equipment following upwind direction, it is easier for the adapter on the side of the truck head to drop on the launching truck. (it happens with the common wind speed of 3 m/s)(3)The collision between the adapter and the projectile does not happen for these two operating conditions(4)Through different simulation calculations, the conclusion can be drawn that the combining calculation method of hydrodynamics and dynamics saves two-thirds of the time in comparison with the traditional dynamic mesh simulation method [4]

#### Data Availability

The test data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.