# Control Strategies for Flight Stability of Trajectory Correction Projectile with Air-Ducts Structure

Authors: Silin Cui; Xing Liu; Shoushan Jiang; Jiachong Guo
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Aiming at the controllability of spin-stabilized projectile trajectory correction, a new two-dimensional trajectory correction projectile model with a controllable air-ducts structure suitable for the spin-stabilized projectile is proposed in this paper. Furthermore, control strategies of the projectile in the corrected configuration are studied to ensure a stable flight. The correction scheme of the projectile is innovated on the basis of the ram air control mechanism. The air-ducts structure is designed inside the projectile body, and the internal air valve is used to control the lateral air-jet to achieve trajectory correction of the projectile. This is a new two-dimensional trajectory correction scheme with a relatively simple control method. Firstly, the correction mechanism of the new air-ducts structure projectile is analyzed. Secondly, based on the dynamic equation of the projectile, the key parameters of the impulse air valve control, including the safe range of the working pulse width and frequency, are calculated, and the corresponding control strategies are proposed. Finally, the flow field of the projectile in the corrected configuration of the terminal trajectory is numerically simulated, and aerodynamic parameters of the projectile are obtained. Through stability conditions established in this paper and combined with aerodynamic parameters, the validity of the proposed projectile stable flight control strategies is confirmed. The innovations of this paper include: (1) a proposed new aerodynamic configuration model with the air-ducts structure for the spin-stabilized projectile and (2) proposed innovative stable flight control strategies of the corrected configuration projectile. The numerical simulation results show that, compared with the reference projectile without air-ducts structure, the lateral force of the air-ducts structure projectile can produce a radial correction effect on the moving projectile. The simulation results of control strategies indicate that the control strategies can ensure stable flight of the corrected configuration projectile under different conditions.


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## Body

## 1. Introduction

The trajectory correction projectile is characterized by low cost, high damage rate, and high cost-effectiveness ratio, all of which meet the requirements of modern warfare. Therefore, in recent years, trajectory correction technology has been one of the key research directions in the field of intelligent ammunition. Different from the one-dimensional trajectory correction technology that can only correct the range, a two-dimensional correction technology can correct the radial displacement and longitudinal range of the projectile in both directions. This correction technology mainly corrects the radial displacement. The projectile correction mechanism is controlled to perform the correction action based on the deviation between the projectile and the ideal trajectory in flight. Consequently, the aerodynamic force and moment of the projectile are both altered to achieve the two-dimensional trajectory correction.The correction mechanisms can be divided into aerodynamic control and jet thrust control. Examples of aerodynamic controls include umbrella resistance correction fuses, fixed canards, and ram air deflection controls. Examples of jet thrust controls include pulse detonation engines and explosive thrusters. In aerodynamic control methods, the umbrella resistance correction fuse can only be used to achieve one-dimensional motion direction correction of the projectile. The correction technology of the fixed canard, such as the Precision Guidance Kit (PGK), is relatively difficult. However, the ram air control method takes the oncoming high-speed airflow during the projectile flight as the correction power source. The oncoming airflow is introduced into the projectile interior, while its motion direction is modified by controlling the valve. Then, the airflow is exported from the side of the projectile to correct the flight direction. This correction method is similar to the jet thrust control. Both of them employ the lateral impulse of gas to change the motion direction of the projectile. However, the difference is that the ram air control method does not involve the work of initiating explosive devices. Therefore, it is safer. In addition, as flight velocity increases, aerodynamic control can generate larger control authority.Few types of research and applications can be found regarding the research of trajectory correction projectile using the ram air control method. Publicly available information shows that Chandgadkar et al. [1] applied the ram air control mechanism to a direct-fire penetrator projectile, as shown in Figure 1. A rotary sleeve valve mounted at the front end of the projectile directs the ram air from the central inlet in the nose through the exit port on the side of the nose. The fin-stabilized projectile correction is achieved by controlling the rotary sleeve valve and consequently the air outlet direction. The results indicate that this ram air correction mechanism can provide sufficient control force and reduce the shot-to-shot dispersion of a direct-fire projectile to levels commensurate with the accuracy of the sensor system.Figure 1
Schematic of the ram air control mechanism at the front end of the penetrator projectile [1].Moreover, the lateral jet thrust control method similar to the principle of ram air control is widely employed in the military field [2, 3]. For example, high maneuvering missiles use the lateral jet to achieve lateral force control and terminal control. The extended range projectiles use the characteristics of fast response time and large reaction force of jet thrust to improve the performance. Many researchers have studied the lateral jet via numerical simulation and wind tunnel tests. Kang et al. [4] used numerical simulation to investigate the jet interference effects under various conditions for the missile equipped with a continuous type lateral jet thruster. Wang et al. [5] studied the interaction flow fields of a supersonic lateral jet penetrating the cross-flow of projectiles with different fin configurations by detached eddy simulation. In addition, the authors analyzed the flow field structure under different conditions. Brandeis et al. [6] explored the interaction between the lateral jet and external flow of different missile configurations by employing a wind tunnel test. They also analyzed the influence of different Mach numbers, vector angles, and other factors on jet force amplification.By conducting a literature overview, it can be seen that the method of using a lateral impulse to correct projectile also has an adverse effect. During projectile control, the lateral jet interferes with the free stream and forms a complex flow structure that affects the aerodynamic characteristics of the projectile. It also increases the angle of attack and affects the flight stability of the projectile. Therefore, when using the lateral impulse to correct the motion of the projectile, it is necessary to actively control the flight angle of attack. This has to be done to avoid the excessive angle of attack generated by the lateral force and moment, thereby causing the projectile instability.Cao et al. [7] analyzed the flight stability of the mortar projectile using lateral pulse jet control for trajectory correction. The stable flight of the projectile is controlled by limiting the pulse impulse as well as the distance between the pulse and the mass center. Wernert et al. [8–10] studied the stability of canard-guided dual-spin stabilized projectiles. The authors established theoretical models and the flight stability criteria. Based on the stability study of canard-guided projectile, Wang et al. [11] analyzed the influence of the correction fuse structure parameters on the projectile flight stability, and improved the flight stability of the projectile by optimizing the structure parameters.At present, most of the spin-stabilized projectiles use the fixed canard rudder for two-dimensional trajectory correction [12, 13]. When the projectile enters the correction configuration, the trajectory correction is achieved by reducing the rotation speed of the rudder and controlling the canard roll angle [14–16]. However, due to the high-speed rotation characteristics of the spinning projectile, the rudder correction method has high requirements for internal motor performance. Moreover, the correction control is relatively difficult. At the same time, the aerodynamic control force and moment generated by the fixed canards at the front end of the projectile can change the flight attitude (angle of attack). This, in turn, can affect the flight stability of the projectile.Our contributions in this paper are twofold. (1) To design a two-dimensional trajectory correction method suitable for a spin-stabilized projectile, the method based on [1] is innovated in this paper, and a projectile correction scheme with controllable air-ducts structure is proposed. This scheme can reduce the correction control difficulty of projectile and improve the correction accuracy. The projectile structure of this correction scheme adds internal air-ducts based on the original conventional spinning projectile. The oncoming airflow is introduced from the air inlet to the middle of the projectile, and the air exit ports are arranged near the mass center. In the corrected configuration, by opening and closing the internal air valve on the projectile, as the projectile rotates, the oncoming airflow entering internal air-ducts can be successively derived through multiple air outlets. The derived lateral jet impulse acts near the projectile mass center to achieve a two-dimensional trajectory correction. During this correction process, the correction force is mainly used to achieve the translation of the projectile in a two-dimensional direction without producing a large overturn moment. As a result, the projectile still has good flight stability.(2) Due to the change of the projectile structure, the internal airflow introduced into the projectile destroys its original aerodynamic configuration after derivation. Hence, the abovementioned aerodynamic model and stability theory are not fully applicable. In addition, the active position of the correction force at the air exit ports does not fully coincide with the mass center. Consequently, the correction action affects the original flight stability of the projectile. To ensure the stable flight of the projectile in the corrected configuration, the flight stability of the projectile is systematically studied, and the innovative stable flight control strategies are proposed to modulate the working pulse width and frequency of the impulse air valve.In this paper, a dynamic model and stability of the air-ducts’ structure projectile are firstly studied [17–19]. Then, based on the aforementioned research, the control strategies of the projectile in the corrected configuration are studied to ensure the stable flight, and the projectile flow field at the end of the trajectory is analyzed via computational fluid dynamics [20, 21]. Finally, the flight stability is simulated by employing the calculated aerodynamic parameters to verify the control strategies proposed in this paper.


## 2. Air-Ducts Structure Projectile Dynamic Model

### 2.1. Analysis of Projectile Correction Mechanism

The shape structure of a spin-stabilized trajectory correction projectile with the air-ducts structure in the uncorrected configuration is shown in Figure2. The aerodynamic shape of the projectile is almost the same as the conventional projectile shape. Since no air flows in it, the existence of air-ducts structure has a minor effect on the aerodynamic characteristics of the projectile. The external shape and internal air-ducts structure of the projectile in the corrected configuration are shown in Figure 3. The false cap at the front end of the projectile is unlocked and the air inlet is opened. The air-ducts are composed of a single inlet and three outlets that are evenly distributed along the circumference (In the multi-section view of Figure 3, the air outlets I and II with a circumferential angle difference of 120° are shown). According to the deviation between the current projectile position and the ideal trajectory, the control system determines the correction direction of the air valve at the internal air-ducts intersection.Figure 2
Projectile structure schematic for uncorrected configuration.Figure 3
Projectile structure schematic for corrected configuration and projectile position definition.When the air-ducts structure projectile enters the correction state, the oncoming airflow is introduced from the air inlet to the middle of the projectile. By reasonably controlling the working state of the internal air valve, with the rotation of the projectile, the three outlet ducts successively release the airflow at the correction direction to form a two-dimensional impulse with a fixed direction, and the impulse can generate reaction force. Then, a 2D trajectory correction of the projectile can be achieved by superimposing multiple times correction effects. According to Figure3, the Fp is the component of the lateral correction force in any direction of space under the body coordinate of Ox1y1z1.The internal air-ducts of the projectile are put through the airflow in the corrected configuration. The air-jet impulse at the exit port can be expressed as follows:(1)I=F⋅Δt=mair⋅vair,where mair is the air mass that flows out of the exit port during the correction time ∆t, vair is the airflow rate at the exit port, and F is the lateral correction force formed by the air jet at the air outlet on the projectile under the action of the air valve, which can be expressed as follows:(2)F=mair⋅v˙air=∫0Δtqmdt⋅v˙air,(3)qm=ρ⋅qv=ρ⋅vair⋅A=ρ⋅vair⋅π4d2,where ρ is the air density, qm is mass flow, qv is the flow rate, A is the cross-sectional area of the duct, and d is inlet diameter. According to (2) and (3), the correction force for the fixed correction time increases with the inlet diameter.Here, the air exit ports are arranged near the mass center. Assuming that the axial distance between the center of the air jet at the exit port and the mass center of the projectile isLp (Lp is positive when the mass center is located between the air exit port and the projectile nose), the air-jet impulse can produce lateral correction force and form the correction moment perpendicular to the projectile axis. With an increase in the axial distance Lp and the correction force F, the angular momentum is also increased. The aforementioned harms the flight stability of the projectile.The above analysis suggests that the lateral air-jet brought by the air-ducts structure can produce projectile translation and nutation. Structure parametersd and Lp of the air-ducts affect the projectile flight stability. In actual parameter design, the air-ducts structure parameters are designed according to the expected radial aerodynamic correction force and safe attack flight angle of the projectile.


### 2.2. Lateral Aerodynamic Correction Force and Moment

To describe the lateral aerodynamic correction force and moment produced by the new air-ducts structure projectile in the corrected configuration, the relevant coordinate frames are established [22]. The body coordinate frame Ox1y1z1, velocity coordinate frame Ox2y2z2, and axis coordinate frame Oξηζ are shown in Figure 3. Lateral correction force Fp is expressed in the velocity coordinate frame Ox2y2z2. The correction moment Mp is expressed in the axis coordinate frame Oξηζ. The correction force and moment are given in (4) – (7).(4)Fpx2Fpy2Fpz2=Fpξcosδ2cosδ1−Fpηcosδ1sinδ2sinβ+sinδ1cosβ+Fpζ−cosδ1sinδ2cosβ+sinδ1sinβFpξcosδ2cosδ1+Fpη−sinδ1sinδ2sinβ+cosδ1cosβ−Fpζsinδ1sinδ2cosβ+cosδ1sinβFpξsinδ2+Fpηcosδ2sinβ+Fpζcosδ2cosβ,(5)MpξMpηMpζ=ρvSlDmxz′Δγ˙2−FpLpsinγpcosγ−FpLpcosγpsinγ−FpLpsinγpsinγ+FpLpcosγpcosγ,(6)Fp=F⋅cosε,(7)FpξFpηFpζ=Fp⋅tanε−Fpcosγpcosγ+Fpsinγpsinγ−Fpcosγpsinγ−Fpsinγpcosγ,where F is the lateral correction force acting on the projectile, Fp is the component of the correction force F in the plane y1-O-z1. In the uncorrected configuration, values of Fp and Mp are both zero. Fpx2, Fpy2, Fpz2, Fpξ, Fpη, and Fpζ represent correction force components in the velocity coordinate and axis coordinate, respectively. Mpξ, Mpη, and Mpζ represent correction moment components in the axis coordinate. In addition, ɛ is the angle between the lateral jet direction and plane y1-O-z1. Parameters δ1 and δ2 represent the pitch and yaw attack angle of the projectile, respectively. Parameter β is the angle between the second axis coordinate and the first axis coordinate, while γ is the rolling angle between the body coordinate frame and the axis coordinate frame. Parameter γp is the angle between the air outlet and axis Oy1, and Δγ˙ is the roll angular velocity change caused by the lateral correction force. Lastly, parameters S, l, and D are the reference area, length, and diameter of the projectile, respectively.


### 2.3. Flight Model of the Projectile

To simplify the investigation and keep the main motion characteristics of the projectile, it is assumed that the air-ducts structure projectile is an axisymmetric rigid body whose mass center is located on the longitudinal symmetry axis.In the uncorrected configuration, the projectile is affected by gravity, drag, lift, and Magnus force. Complex-number expression of the force is provided in Table1. According to the centroid movement law, the translational kinematic equation of the projectile can be expressed as follows:(8)dvdt=Fx2m=−bxv2−gsinθa,dθadt=Fy2mvcosψ2=byvδ1+bzγ˙δ2−gcosθav,dψ2dt=Fz2mv=byvδ2+bzγ˙δ1,θa=θ+ψ1,where m is the projectile quality, Fx2, Fy2, and Fz2 are components of the aerodynamic force of the projectile on each axis in the velocity coordinate, θ is the trajectory’s inclination angle, θa, and ψ2 represent elevation angle and velocity azimuth angle, respectively, and γ˙ is the angular velocity. The expressions of bx, by, and bz are shown in Table 1.Table 1
The complex-number expressions of the aerodynamic force and moment.
Aerodynamic force and momentExpressionDragRx=mbxv2, bx=ρSCx/2mLiftRy=mbyv2Δ, by=ρSCy′/2mMagnus forceRz=mbzγ˙vΔ, bz=ρSDCz′/2mPitching momentMz=Ikzv2Δ, kz=ρSlmz′/2IEquatorial damping momentMzz=−IkzzvΦ˙, kzz=ρSlDmzz′/2IDamping momentMxz=−Jkxzγ˙v, kxz=ρSlDmxz′/2JMagnus momentMy=Jkyvγ˙Δ, ky=ρSlDmy′/2JGyroscopic momentMg=Jγ˙Φ˙Projectile motions around the mass center are composed of rotation and swing motions. The damping moment, equatorial damping moment, pitching moment, Magnus moment, and the gyroscopic moment all affect the projectile. The Complex-number expression of the moment is provided in Table1. In the uncorrected configuration, the projectile rotational dynamic equation is given by(9)Mξ=Jγ¨=−Jkxzγ˙v,Mη=−Iφ¨2=−Ikzv2δ2+Ikzzvφ˙2+Jkyvγ˙δ1−Jγ˙φ˙a,Mζ=Iφ¨a=Ikzv2δ1−Ikzzvφ˙a+Jkyvγ˙δ2−Jγ˙φ˙2,φa=θ+φ1,where Mξ, Mη, and Mζ are the moment components on each axis within the coordinate system. J and I represent the axial and equatorial moments of inertia, respectively, while φa and φ2 are elevation and axis azimuth angles, respectively. Expressions of kz, kzz, kxz, and ky are shown in Table 1.In Table1, Cx, Cy, and Cz are the drag coefficient, the lift coefficient, and the Magnus force coefficient of the projectile, respectively. Parameter mz, mzz, mxz, and my are the coefficients of pitching moment, equatorial damping moment, damping moment, and Magnus moment, respectively.In the corrected configuration of the terminal trajectory, the projectile is additionally affected by the aerodynamic correction force and the moment analyzed in Section2.2. Dynamic equations of the projectile need to be added with the control force and moment equations. According to (4) and (8), projectile translational kinematic equation in the corrected configuration can be written as follows:(10)dvdt=Fx2+Fpx2m,dθadt=Fy2+Fpy2mvcosψ2,dψ2dt=Fz2+Fpz2mv,θa=θ+ψ1.According to (5) and (9), the projectile rotational dynamic equation in the corrected configuration is given by(11)dωξdt=Mξ+MpξJ,dωηdt=Mη+MpηI−JωξωζI+ωζ2tanφ2,dωζdt=Mζ+MpζI+JωξωηI−ωηωζtanφ2.where ωξ, ωη, and ωζ are the rotational angular velocities of the projectile on each axis.Variation equation of the complex angle of attack with arc length in the corrected configuration is given by(12)Δ'′+H−iPΔ′−M+iPTΔ=−θ¨v2−kzz−iPθ˙v−i⋅Mpη+iMpζIv2+Fpy2+iFpz2mv2⋅iP−kzz−bx−gsinθv2.The related parameters are shown below:(13)H=kzz+by−bx−gsinθv2,P=Jγ˙Iv,M=kz,T=by−IJky.Since projectile correction time is very short in practical work (milliseconds), parameters in the angle of attack equation show minor variation over a short interval. In this paper, (12) is approximated as a constant-coefficient differential equation. Moreover, a correction process can be considered instantaneous. It is equivalent to giving a strong interference to the moving projectile in a relatively short time. Therefore, the effect of the correction on the angular motion of the projectile can be analyzed by employing the projectile motion under the initial disturbance.The homogeneous solution of the angle of attack equation represents angular motion under the initial condition. The instantaneous state after correction is taken as the initial condition. The homogeneous solution is obtained as follows:(14)Δ=C1eλ1+iω1s+C2eλ2+iω2s.The related parameters are shown in (15)and (16):(15)C1=Δ˙0−λ2+iω2Δ0λ1−λ2+iω1−ω2,C2=Δ˙0−λ1+iω1Δ0λ2−λ1+iω2−ω1,(16)λ1,2=12H±P2T−HP2−4M,ω1,2=12P±P2−4M.According to the homogeneous solution of the angle of attack equation, the motion of the projectile is represented by a complex motion of two vectors. WhereC1 and C2 are complex undetermined constants determined by initial conditions, λ1 and λ2 are the damping indexes, and ω1 and ω2 are modal frequencies.


## 2.1. Analysis of Projectile Correction Mechanism

The shape structure of a spin-stabilized trajectory correction projectile with the air-ducts structure in the uncorrected configuration is shown in Figure2. The aerodynamic shape of the projectile is almost the same as the conventional projectile shape. Since no air flows in it, the existence of air-ducts structure has a minor effect on the aerodynamic characteristics of the projectile. The external shape and internal air-ducts structure of the projectile in the corrected configuration are shown in Figure 3. The false cap at the front end of the projectile is unlocked and the air inlet is opened. The air-ducts are composed of a single inlet and three outlets that are evenly distributed along the circumference (In the multi-section view of Figure 3, the air outlets I and II with a circumferential angle difference of 120° are shown). According to the deviation between the current projectile position and the ideal trajectory, the control system determines the correction direction of the air valve at the internal air-ducts intersection.Figure 2
Projectile structure schematic for uncorrected configuration.Figure 3
Projectile structure schematic for corrected configuration and projectile position definition.When the air-ducts structure projectile enters the correction state, the oncoming airflow is introduced from the air inlet to the middle of the projectile. By reasonably controlling the working state of the internal air valve, with the rotation of the projectile, the three outlet ducts successively release the airflow at the correction direction to form a two-dimensional impulse with a fixed direction, and the impulse can generate reaction force. Then, a 2D trajectory correction of the projectile can be achieved by superimposing multiple times correction effects. According to Figure3, the Fp is the component of the lateral correction force in any direction of space under the body coordinate of Ox1y1z1.The internal air-ducts of the projectile are put through the airflow in the corrected configuration. The air-jet impulse at the exit port can be expressed as follows:(1)I=F⋅Δt=mair⋅vair,where mair is the air mass that flows out of the exit port during the correction time ∆t, vair is the airflow rate at the exit port, and F is the lateral correction force formed by the air jet at the air outlet on the projectile under the action of the air valve, which can be expressed as follows:(2)F=mair⋅v˙air=∫0Δtqmdt⋅v˙air,(3)qm=ρ⋅qv=ρ⋅vair⋅A=ρ⋅vair⋅π4d2,where ρ is the air density, qm is mass flow, qv is the flow rate, A is the cross-sectional area of the duct, and d is inlet diameter. According to (2) and (3), the correction force for the fixed correction time increases with the inlet diameter.Here, the air exit ports are arranged near the mass center. Assuming that the axial distance between the center of the air jet at the exit port and the mass center of the projectile isLp (Lp is positive when the mass center is located between the air exit port and the projectile nose), the air-jet impulse can produce lateral correction force and form the correction moment perpendicular to the projectile axis. With an increase in the axial distance Lp and the correction force F, the angular momentum is also increased. The aforementioned harms the flight stability of the projectile.The above analysis suggests that the lateral air-jet brought by the air-ducts structure can produce projectile translation and nutation. Structure parametersd and Lp of the air-ducts affect the projectile flight stability. In actual parameter design, the air-ducts structure parameters are designed according to the expected radial aerodynamic correction force and safe attack flight angle of the projectile.


## 2.2. Lateral Aerodynamic Correction Force and Moment

To describe the lateral aerodynamic correction force and moment produced by the new air-ducts structure projectile in the corrected configuration, the relevant coordinate frames are established [22]. The body coordinate frame Ox1y1z1, velocity coordinate frame Ox2y2z2, and axis coordinate frame Oξηζ are shown in Figure 3. Lateral correction force Fp is expressed in the velocity coordinate frame Ox2y2z2. The correction moment Mp is expressed in the axis coordinate frame Oξηζ. The correction force and moment are given in (4) – (7).(4)Fpx2Fpy2Fpz2=Fpξcosδ2cosδ1−Fpηcosδ1sinδ2sinβ+sinδ1cosβ+Fpζ−cosδ1sinδ2cosβ+sinδ1sinβFpξcosδ2cosδ1+Fpη−sinδ1sinδ2sinβ+cosδ1cosβ−Fpζsinδ1sinδ2cosβ+cosδ1sinβFpξsinδ2+Fpηcosδ2sinβ+Fpζcosδ2cosβ,(5)MpξMpηMpζ=ρvSlDmxz′Δγ˙2−FpLpsinγpcosγ−FpLpcosγpsinγ−FpLpsinγpsinγ+FpLpcosγpcosγ,(6)Fp=F⋅cosε,(7)FpξFpηFpζ=Fp⋅tanε−Fpcosγpcosγ+Fpsinγpsinγ−Fpcosγpsinγ−Fpsinγpcosγ,where F is the lateral correction force acting on the projectile, Fp is the component of the correction force F in the plane y1-O-z1. In the uncorrected configuration, values of Fp and Mp are both zero. Fpx2, Fpy2, Fpz2, Fpξ, Fpη, and Fpζ represent correction force components in the velocity coordinate and axis coordinate, respectively. Mpξ, Mpη, and Mpζ represent correction moment components in the axis coordinate. In addition, ɛ is the angle between the lateral jet direction and plane y1-O-z1. Parameters δ1 and δ2 represent the pitch and yaw attack angle of the projectile, respectively. Parameter β is the angle between the second axis coordinate and the first axis coordinate, while γ is the rolling angle between the body coordinate frame and the axis coordinate frame. Parameter γp is the angle between the air outlet and axis Oy1, and Δγ˙ is the roll angular velocity change caused by the lateral correction force. Lastly, parameters S, l, and D are the reference area, length, and diameter of the projectile, respectively.


## 2.3. Flight Model of the Projectile

To simplify the investigation and keep the main motion characteristics of the projectile, it is assumed that the air-ducts structure projectile is an axisymmetric rigid body whose mass center is located on the longitudinal symmetry axis.In the uncorrected configuration, the projectile is affected by gravity, drag, lift, and Magnus force. Complex-number expression of the force is provided in Table1. According to the centroid movement law, the translational kinematic equation of the projectile can be expressed as follows:(8)dvdt=Fx2m=−bxv2−gsinθa,dθadt=Fy2mvcosψ2=byvδ1+bzγ˙δ2−gcosθav,dψ2dt=Fz2mv=byvδ2+bzγ˙δ1,θa=θ+ψ1,where m is the projectile quality, Fx2, Fy2, and Fz2 are components of the aerodynamic force of the projectile on each axis in the velocity coordinate, θ is the trajectory’s inclination angle, θa, and ψ2 represent elevation angle and velocity azimuth angle, respectively, and γ˙ is the angular velocity. The expressions of bx, by, and bz are shown in Table 1.Table 1
The complex-number expressions of the aerodynamic force and moment.
Aerodynamic force and momentExpressionDragRx=mbxv2, bx=ρSCx/2mLiftRy=mbyv2Δ, by=ρSCy′/2mMagnus forceRz=mbzγ˙vΔ, bz=ρSDCz′/2mPitching momentMz=Ikzv2Δ, kz=ρSlmz′/2IEquatorial damping momentMzz=−IkzzvΦ˙, kzz=ρSlDmzz′/2IDamping momentMxz=−Jkxzγ˙v, kxz=ρSlDmxz′/2JMagnus momentMy=Jkyvγ˙Δ, ky=ρSlDmy′/2JGyroscopic momentMg=Jγ˙Φ˙Projectile motions around the mass center are composed of rotation and swing motions. The damping moment, equatorial damping moment, pitching moment, Magnus moment, and the gyroscopic moment all affect the projectile. The Complex-number expression of the moment is provided in Table1. In the uncorrected configuration, the projectile rotational dynamic equation is given by(9)Mξ=Jγ¨=−Jkxzγ˙v,Mη=−Iφ¨2=−Ikzv2δ2+Ikzzvφ˙2+Jkyvγ˙δ1−Jγ˙φ˙a,Mζ=Iφ¨a=Ikzv2δ1−Ikzzvφ˙a+Jkyvγ˙δ2−Jγ˙φ˙2,φa=θ+φ1,where Mξ, Mη, and Mζ are the moment components on each axis within the coordinate system. J and I represent the axial and equatorial moments of inertia, respectively, while φa and φ2 are elevation and axis azimuth angles, respectively. Expressions of kz, kzz, kxz, and ky are shown in Table 1.In Table1, Cx, Cy, and Cz are the drag coefficient, the lift coefficient, and the Magnus force coefficient of the projectile, respectively. Parameter mz, mzz, mxz, and my are the coefficients of pitching moment, equatorial damping moment, damping moment, and Magnus moment, respectively.In the corrected configuration of the terminal trajectory, the projectile is additionally affected by the aerodynamic correction force and the moment analyzed in Section2.2. Dynamic equations of the projectile need to be added with the control force and moment equations. According to (4) and (8), projectile translational kinematic equation in the corrected configuration can be written as follows:(10)dvdt=Fx2+Fpx2m,dθadt=Fy2+Fpy2mvcosψ2,dψ2dt=Fz2+Fpz2mv,θa=θ+ψ1.According to (5) and (9), the projectile rotational dynamic equation in the corrected configuration is given by(11)dωξdt=Mξ+MpξJ,dωηdt=Mη+MpηI−JωξωζI+ωζ2tanφ2,dωζdt=Mζ+MpζI+JωξωηI−ωηωζtanφ2.where ωξ, ωη, and ωζ are the rotational angular velocities of the projectile on each axis.Variation equation of the complex angle of attack with arc length in the corrected configuration is given by(12)Δ'′+H−iPΔ′−M+iPTΔ=−θ¨v2−kzz−iPθ˙v−i⋅Mpη+iMpζIv2+Fpy2+iFpz2mv2⋅iP−kzz−bx−gsinθv2.The related parameters are shown below:(13)H=kzz+by−bx−gsinθv2,P=Jγ˙Iv,M=kz,T=by−IJky.Since projectile correction time is very short in practical work (milliseconds), parameters in the angle of attack equation show minor variation over a short interval. In this paper, (12) is approximated as a constant-coefficient differential equation. Moreover, a correction process can be considered instantaneous. It is equivalent to giving a strong interference to the moving projectile in a relatively short time. Therefore, the effect of the correction on the angular motion of the projectile can be analyzed by employing the projectile motion under the initial disturbance.The homogeneous solution of the angle of attack equation represents angular motion under the initial condition. The instantaneous state after correction is taken as the initial condition. The homogeneous solution is obtained as follows:(14)Δ=C1eλ1+iω1s+C2eλ2+iω2s.The related parameters are shown in (15)and (16):(15)C1=Δ˙0−λ2+iω2Δ0λ1−λ2+iω1−ω2,C2=Δ˙0−λ1+iω1Δ0λ2−λ1+iω2−ω1,(16)λ1,2=12H±P2T−HP2−4M,ω1,2=12P±P2−4M.According to the homogeneous solution of the angle of attack equation, the motion of the projectile is represented by a complex motion of two vectors. WhereC1 and C2 are complex undetermined constants determined by initial conditions, λ1 and λ2 are the damping indexes, and ω1 and ω2 are modal frequencies.


## 3. Air-Ducts Structure Projectile Stability Control Strategies

If the correction action is not properly controlled, projectile flight stability will be affected. During the stable flight of the terminal trajectory, the spin-stabilized projectile must meet the dynamic stability, and gyro stability. To control the stable flight of the projectile in the corrected configuration, it is also necessary to ensure that the projectile has a stable flight angle of attack under the action of the correction force and moment.Correction air valve inside the air-ducts structure projectile represents an air-duct switch that determines the magnitude and direction of the correction force. The working pulse width and frequency of the correction air valve are different, which causes variation of the angle of attack increment. If the angle of attack increment is too large and superimposed with the original flight angle of the attack, the projectile may become unstable. By limiting the angle of attack increment caused by the correction and controlling the working parameters of the correction air valve, the projectile can be guaranteed to fly steadily. In this paper, two control strategies are proposed.


### 3.1. Control of the Air Valve Working Pulse Width ∆T

The allowable maximum amplitude of projectile angle of attack is referenced to control the angle of attack increment produced by correction action, and controlling the angle of attack increment does not exceed the limited allowable value∆δmax. According to (10) and (11), projectile motion state changes after correction. A change in each parameter is obtained:(17)Δv=Fpx2Δtm=Ipx2m,Δθa=Fpy2Δtmvcosψ2=Ipy2mvcosψ2≈Δδ1,Δψ2=Fpz2Δtmv=Ipz2mv≈Δδ2,Δωξ=MpξΔtJΔωη=MpηΔtI≈Δδ˙2Δωζ=MpζΔtI≈Δδ˙1,where ∆t is the correction time. The increment of the angle of attack and the angular velocity increment of the angle of attack are given by (18) and (19):(18)Δδ0=Δδ12+Δδ22=Ipmv,(19)Δδ˙0=Δδ˙12+Δδ˙22=IpLpI.Then, angular motion amplitudes generated by the increments can be calculated asδm1 and δm2. In the most severe case, the angle of attack generated by the correction is the sum of two absolute values. The angle of attack ∆δ generated by the correction action is given by(20)Δδ=δm1+δm2=Ipmvσ+IpLpIαvσ,where α=Jγ˙/2Iv and σ=1−kz/α2.To ensure flight stability, the angle of attack∆δ must be lower than the limited allowable value ∆δmax. The flight stability criterion of the correction time is given as follows:(21)Δt<ΔδmaxFp/mvσ+FpLp/Iαvσ.The working pulse width ∆t that satisfies (21) represents the stable working pulse width of the air valve.


### 3.2. Control of the Air Valve Working Frequency ∆T

After determining the stable working pulse width of the air valve, the first correction is made. The gyroscopic moment and equatorial damping moment of the projectile both reduce the angle of attack. When the angle is stabilized, a second correction is carried out to ensure projectile stable flight. The disturbance value ∆δ caused by the correction action tends to stabilize under the action of damping. The damping index λ2 of slow circular motion in the two vector’s complex motion is used for calculation. Assuming that the first correction action is made at time t1, the angle of attack increment is continuously reduced due to damping. In the period ∆T between two adjacent correction actions, the attenuation of the attack angle can be represented as:(22)b=Δδeλ2st1−Δδeλ2st1+ΔT.To ensure flight stability, the time interval between two adjacent correction actions should ensure that the angle of attack increment has enough attenuation time. When the angle of attack increment decreases to less than 1/a of the original increment, i.e., b (a - 1) ∆δ/a, the angle of attack is stabilized. Then, the subsequent correction can be carried out. Therefore, the flight stability criterion of the corrected time interval between two correction actions is provided as follows:(23)ΔT>1λ2lneλ2t1−a−1a−t1.The correction time interval∆T that satisfies (23) is the stable working frequency of the air valve. By controlling the working pulse width and frequency of the correction air valve to meet the inequality constraints in different flight states, the increment of the projectile angle of attack can be stabilized within the allowable range.


## 3.1. Control of the Air Valve Working Pulse Width ∆T

The allowable maximum amplitude of projectile angle of attack is referenced to control the angle of attack increment produced by correction action, and controlling the angle of attack increment does not exceed the limited allowable value∆δmax. According to (10) and (11), projectile motion state changes after correction. A change in each parameter is obtained:(17)Δv=Fpx2Δtm=Ipx2m,Δθa=Fpy2Δtmvcosψ2=Ipy2mvcosψ2≈Δδ1,Δψ2=Fpz2Δtmv=Ipz2mv≈Δδ2,Δωξ=MpξΔtJΔωη=MpηΔtI≈Δδ˙2Δωζ=MpζΔtI≈Δδ˙1,where ∆t is the correction time. The increment of the angle of attack and the angular velocity increment of the angle of attack are given by (18) and (19):(18)Δδ0=Δδ12+Δδ22=Ipmv,(19)Δδ˙0=Δδ˙12+Δδ˙22=IpLpI.Then, angular motion amplitudes generated by the increments can be calculated asδm1 and δm2. In the most severe case, the angle of attack generated by the correction is the sum of two absolute values. The angle of attack ∆δ generated by the correction action is given by(20)Δδ=δm1+δm2=Ipmvσ+IpLpIαvσ,where α=Jγ˙/2Iv and σ=1−kz/α2.To ensure flight stability, the angle of attack∆δ must be lower than the limited allowable value ∆δmax. The flight stability criterion of the correction time is given as follows:(21)Δt<ΔδmaxFp/mvσ+FpLp/Iαvσ.The working pulse width ∆t that satisfies (21) represents the stable working pulse width of the air valve.


## 3.2. Control of the Air Valve Working Frequency ∆T

After determining the stable working pulse width of the air valve, the first correction is made. The gyroscopic moment and equatorial damping moment of the projectile both reduce the angle of attack. When the angle is stabilized, a second correction is carried out to ensure projectile stable flight. The disturbance value ∆δ caused by the correction action tends to stabilize under the action of damping. The damping index λ2 of slow circular motion in the two vector’s complex motion is used for calculation. Assuming that the first correction action is made at time t1, the angle of attack increment is continuously reduced due to damping. In the period ∆T between two adjacent correction actions, the attenuation of the attack angle can be represented as:(22)b=Δδeλ2st1−Δδeλ2st1+ΔT.To ensure flight stability, the time interval between two adjacent correction actions should ensure that the angle of attack increment has enough attenuation time. When the angle of attack increment decreases to less than 1/a of the original increment, i.e., b (a - 1) ∆δ/a, the angle of attack is stabilized. Then, the subsequent correction can be carried out. Therefore, the flight stability criterion of the corrected time interval between two correction actions is provided as follows:(23)ΔT>1λ2lneλ2t1−a−1a−t1.The correction time interval∆T that satisfies (23) is the stable working frequency of the air valve. By controlling the working pulse width and frequency of the correction air valve to meet the inequality constraints in different flight states, the increment of the projectile angle of attack can be stabilized within the allowable range.


## 4. Computational Method and Verification

### 4.1. Computational Model and Method

Numerical simulation is performed using the ANSYS FLUENT for a 62 mm spin-stabilized trajectory correction projectile with air-ducts structure. A section and the main dimensions of the correction projectile in the corrected configuration are shown in Figure4. The diameter of the projectile is D, and the length is 4.93D. The air inlet diameter of the projectile model investigated in this paper is D/3, the diameter of the internal air duct is D/6. Angle α between the outlet duct and the axis of the projectile is denoted as 135°.Figure 4
Projectile section and main dimensions schematic.The computational domains for the air-ducts structure projectile consisted of 2.7 million cell structured hexahedral grids which are generated using ANSYS ICEM. The sliding mesh method, which is widely used in engineering, is employed [23]. This method requires that the computational domain has an external static domain and an internal rotating domain, as shown in Figure 5. The mesh at the projectile wall surface and internal air duct wall is locally densified along the normal direction. Mesh division diagrams of the projectile surface and internal air duct are shown in Figures 6 and 7.Figure 5
Computational domains and boundary condition definition.Figure 6
Grid of projectile surface.Figure 7
Grid of internal air duct.Numerical calculation is based on three-dimensional Navier–Stokes equations. The fully coupled density-based solver and second-order Roe-type upwind scheme are used to simulate the air-ducts structure projectile. The SSTk-ω model is used for turbulence closure. The boundary conditions are set according to Figure 5. The no-slip wall boundary conditions are used on the surface of the projectile and internal air ducts. The boundary of the external static domain is defined as the pressure far field. The connection between the static domain and the rotating domain adopts the interface boundary. The angular velocity of the projectile about the X-axis is set as ω = 188.5 rad/s, and the angle of attack is defined in the X-O-Y plane, as shown in Figure 6. The reference point for the moment is defined at the projectile nose, i.e., point O.


### 4.2. Numerical Verification

To verify the validity and reliability of the numerical simulation method when calculating the projectile aerodynamic parameters established in this paper, a 155 mm diameter projectile model used in the wind tunnel test in Ref. [24] is selected. Dimensions of the model are shown in Figure 8. The circumferential surface pressure distribution coefficient curve of the projectile on the boattail is simulated for the Mach number of 0.94, 5 lbf/in2 and an angle of attack of 4°. Distance between the selected section position x and the projectile nose is x/L0 = 0.969 (L0 is the projectile length), which is consistent with Ref. [24]. Comparison with the simulation results for the wind tunnel test data under the same conditions is shown in Figure 9. It can be observed that the numerical calculation curve in this paper is consistent with the variation law of pressure coefficient obtained from the wind tunnel test. This indicates that the numerical simulation method established in this paper is accurate.Figure 8
Geometric parameters of the wind tunnel model.Figure 9
Comparison of circumferential pressure distribution of the projectile obtained by different methods.It is essential that the numerical results should be grid-independent to ensure the calculation accuracy. In this paper, the same projectile model forms four groups of grids with different numbers. The total number of grids ranges from 1,078,998 to 3,227,252l. The grid models are calculated using the same numerical method, and the flight conditions of the projectile are set to 0.6 Ma, 0° angle of attack. Table2 shows the results of grid independency test. As is shown, the drag coefficient does not change significantly when the number of grids exceeds 2,767,232. Therefore, the grid model with the number of 2,767,232 is selected for subsequent numerical calculations.Table 2
Results of grid independency test.
Number of elementsDrag coefficient1,078,9980.174681,981,1090.178502,767,2320.177723,227,2520.17726


## 4.1. Computational Model and Method

Numerical simulation is performed using the ANSYS FLUENT for a 62 mm spin-stabilized trajectory correction projectile with air-ducts structure. A section and the main dimensions of the correction projectile in the corrected configuration are shown in Figure4. The diameter of the projectile is D, and the length is 4.93D. The air inlet diameter of the projectile model investigated in this paper is D/3, the diameter of the internal air duct is D/6. Angle α between the outlet duct and the axis of the projectile is denoted as 135°.Figure 4
Projectile section and main dimensions schematic.The computational domains for the air-ducts structure projectile consisted of 2.7 million cell structured hexahedral grids which are generated using ANSYS ICEM. The sliding mesh method, which is widely used in engineering, is employed [23]. This method requires that the computational domain has an external static domain and an internal rotating domain, as shown in Figure 5. The mesh at the projectile wall surface and internal air duct wall is locally densified along the normal direction. Mesh division diagrams of the projectile surface and internal air duct are shown in Figures 6 and 7.Figure 5
Computational domains and boundary condition definition.Figure 6
Grid of projectile surface.Figure 7
Grid of internal air duct.Numerical calculation is based on three-dimensional Navier–Stokes equations. The fully coupled density-based solver and second-order Roe-type upwind scheme are used to simulate the air-ducts structure projectile. The SSTk-ω model is used for turbulence closure. The boundary conditions are set according to Figure 5. The no-slip wall boundary conditions are used on the surface of the projectile and internal air ducts. The boundary of the external static domain is defined as the pressure far field. The connection between the static domain and the rotating domain adopts the interface boundary. The angular velocity of the projectile about the X-axis is set as ω = 188.5 rad/s, and the angle of attack is defined in the X-O-Y plane, as shown in Figure 6. The reference point for the moment is defined at the projectile nose, i.e., point O.


## 4.2. Numerical Verification

To verify the validity and reliability of the numerical simulation method when calculating the projectile aerodynamic parameters established in this paper, a 155 mm diameter projectile model used in the wind tunnel test in Ref. [24] is selected. Dimensions of the model are shown in Figure 8. The circumferential surface pressure distribution coefficient curve of the projectile on the boattail is simulated for the Mach number of 0.94, 5 lbf/in2 and an angle of attack of 4°. Distance between the selected section position x and the projectile nose is x/L0 = 0.969 (L0 is the projectile length), which is consistent with Ref. [24]. Comparison with the simulation results for the wind tunnel test data under the same conditions is shown in Figure 9. It can be observed that the numerical calculation curve in this paper is consistent with the variation law of pressure coefficient obtained from the wind tunnel test. This indicates that the numerical simulation method established in this paper is accurate.Figure 8
Geometric parameters of the wind tunnel model.Figure 9
Comparison of circumferential pressure distribution of the projectile obtained by different methods.It is essential that the numerical results should be grid-independent to ensure the calculation accuracy. In this paper, the same projectile model forms four groups of grids with different numbers. The total number of grids ranges from 1,078,998 to 3,227,252l. The grid models are calculated using the same numerical method, and the flight conditions of the projectile are set to 0.6 Ma, 0° angle of attack. Table2 shows the results of grid independency test. As is shown, the drag coefficient does not change significantly when the number of grids exceeds 2,767,232. Therefore, the grid model with the number of 2,767,232 is selected for subsequent numerical calculations.Table 2
Results of grid independency test.
Number of elementsDrag coefficient1,078,9980.174681,981,1090.178502,767,2320.177723,227,2520.17726


## 5. Results and Discussion

### 5.1. Flow Visualization

Corrected configuration of the air-ducts structure projectile at terminal trajectory is simulated. The flight speed of the spinning projectile is set to 0.8 Ma and the flight angle of attack is set to 3°. The pressure distribution of flow fields of the projectile is illustrated in Figure10. Among them, the zoomed view shows the pressure distribution near the air duct outlet on the vertical symmetry plane. And, the velocity distribution of internal and external flow fields of the projectile on a vertical symmetry plane under the same conditions is demonstrated in Figure 11. According to Figure 10, a subsonic oncoming airflow forms a local high-pressure region at the air inlet. Airflow velocity decreases, and a small area of obstruction is formed at the projectile nose. When a lateral air-jet enters the flow field, local pressure at the air outlet increases due to interference between the derived airflow and the oncoming airflow on the projectile surface. Based on Figure 11, the velocity of the oncoming airflow in the interior air duct continuously increases. Affected by the turning and expansion of air-ducts structure as well as wall viscosity, the flow separation is produced at the outer wall zone of the duct steering (zone A in Figure 11). The existence of reverse pressure gradient forms a local recirculation zone which results in flow loss. However, according to Figure 11, the lateral air-jet can still be smoothly derived, and the outlet air velocity can reach up to 200 m/s. At the duct air outlet, the velocity component of the lateral air-jet along the -Z-axis and the oncoming airflow interact to form a certain pressure gradient. Moreover, the velocity component along the X-axis is superimposed with the oncoming airflow. Disturbance of the lateral air-jet to the oncoming airflow extends to the middle and rear parts of the projectile along the flow direction. The aforementioned changes the flow field structure and affects the aerodynamic force on the projectile.Figure 10
Pressure distribution of the air-ducts structure projectile.Figure 11
Velocity distribution of the air-ducts structure projectile.Velocity distribution at the cross section of the air outlet when the air-ducts structure projectile is radially corrected in different directions is shown in Figures12 and 13. Lateral force Fz variation curve of correction projectile and reference projectile under the same conditions are shown in Figure 14. The aerodynamic shape of air-ducts structure projectile is the corrected configuration, and the aerodynamic shape of reference projectile is the same as the corrected configuration, but there is no air-ducts structure inside. Within Figure 14, two kinds of projectiles are rotated from -Z-axis to + Z-axis, the angle corresponding to -Z-axis is 0°, and the angle corresponding to the +Z-axis is 180°. Since the projectile spins and has an angle of attack during flight, a cross-flow occurs through the projectile (along the +Y-axis).Figure 12
Velocity distribution of the cross section of the air outlet (projectile corrected along -(Z)-axis).Figure 13
Velocity distribution of the cross section of the air outlet (projectile corrected along +(Z)-axis).Figure 14
Lateral force variation curve of the correction projectile and the reference projectile.The velocity distribution of the circumferential surface at the air outlet when the air-ducts correction mechanism is corrected along the -Z-axis is shown in Figure 12. Since the airflow rotation direction on the left side of the projectile surface is consistent with the cross-flow, the velocity of airflow along the Y-axis is merged due to air viscosity, while the velocity is increased. The projectile on the right side (along -Z-axis) is affected by the air flow at the outlet duct, and the air flow state is relatively complex. After the air jet is introduced into the external flow field, the viscosity of the air jet, the interactive superposition effect of the jet, and the oncoming airflow accelerate the cross-flow on the right side of the projectile. The lateral force Fz caused by the pressure difference on both sides of the projectile decreases accordingly. Based on Figure 14, the lateral force of the correction projectile is equal to 2.55 N, and the lateral force of the reference projectile is equal to 3.67 N. In the corrected state, lateral air-jet causes a change in the pressure difference on both sides of the projectile. Therefore, the value of Fz is lower than at the uncorrected state of the reference projectile.The velocity distribution of the circumferential surface at the air outlet when the air-ducts correction mechanism is corrected along the +Z-axis is shown in Figure 13. The left side of the projectile is affected by the projectile rotation, cross-flow, and duct outlet air jet, and the velocity gradient changes significantly. Since the airflow direction near the wall is opposite to the cross-flow, the local pressure of the projectile on the right side increases. The reaction force of the lateral air-jet acts on the projectile and its action direction is opposite to the Magnus force. Consequently, the projectile lateral force is the smallest at this time. According to Figure 14, the lateral force of the correction projectile is equal to 1.83 N, while the lateral force of the reference projectile is equal to 3.59 N. Disturbance of lateral air-jet to the surrounding flow field alters the lateral force magnitude.By comparing lateral forces of different structural projectiles in the same flight state (Figure14), it can be found that the lateral force of the reference projectile is stable during rotation. However, during the projectile rotation from the -Z-axis to the +Z-axis, the lateral force Fz of the correction projectile is always lower than the reference projectile. This indicates that the air-ducts structure correction scheme can change the lateral force of the projectile in the working state. Through calculation, it can be observed that the lateral air-jet correction action can make a maximum change of 49% of the projectile lateral force and a minimum change of 30.5% at the flight speed of 0.8 Ma. Therefore, this correction scheme plays a correction role. Furthermore, when the air-ducts structure is corrected along with different circumferential angular positions, the lateral force is altered. The change of the air outlet from -Z-axis to the +Z-axis causes the projectile lateral force to gradually decrease. This can be attributed to the interaction between the lateral air-jet and the flow around the spinning projectile. The pressure distribution on the projectile surface changes, and the correction force is superimposed with the overall aerodynamic force of the projectile. Consequently, the lateral force shows the changing trend demonstrated in Figure 14.Figures15 and 16 represent the variation curve of drag and lift coefficient for two different structural projectiles in the same flight state. The drag and lift forces of the reference projectile are stable during rotation. In addition, the drag force of the correction projectile is always higher than the reference projectile. This indicates that the air-ducts structure increases the drag force of the projectile and that the range direction can be corrected. Compared with the reference projectile, the lift force of the correction projectile is not altered much. Therefore, it can be concluded that the correction effect of the lateral jet has a minor effect on the lift force of the projectile.Figure 15
Drag coefficient variation curve of the correction projectile and the reference projectile.Figure 16
Lift coefficient variation curve of the correction projectile and the reference projectile.


### 5.2. Control Strategies Simulation Verification

In this paper, limited allowable value∆δmax of increment amplitude of angle of attack is used to limit the angle of attack increment caused by the correction action. The operating parameters of the correction air valve are modulated to achieve flight stability control of air-ducts structure projectile. According to (21), the structural parameter Lp of the projectile affects the working parameters of the air valve. To verify the validity of flight stability control strategies, Lp -∆δmax combination schemes are simulated in this paper for different states. The purpose of this is to analyze whether the angle of attack increment generated by the correction meets the requirements, after the working parameters of the air valve are modulated by the control strategies under different conditions. An example of verification schemes is shown in Table 3. Since the air duct outlets are designed near the mass center of the projectile, the structural parameter Lp is set to 10 mm, 20 mm, and 30 mm. And, setting the limited allowable value ∆δmax can be adjusted according to the flight state of the projectile before the correction. Considering that manufacturing errors or other factors may lead to a large angle of attack caused by the correction action, the ∆δmax is set to 6°, 7°, and 8°.Table 3
An example of verification schemes.
Simulation schemesLp (mm)∆δmax1 (°)Simulation schemesLp (mm)∆δmax2 (°)Simulation schemesLp (mm)∆δmax3 (°)Case 1106Case 4107Case 7108Case 2206Case 5207Case 8208Case 3306Case 6307Case 9308Firstly, the control strategy for modulating the working pulse width of the air valve is verified. For Case 1, whenLp = 10 mm and ∆δmax1 = 6°, aerodynamic parameters are introduced into (21). Among them, m = 5 kg, v  = 272 m/s, Fp = 3 N, I = 0.0149 kg m2, P = 0.0569, M = 0.00076, H = 0.00758, T = 0.00345, α = 0.0284, σ = 0.062. Then, the range of stable working pulse width of the air valve is obtained as ∆t < 99.57 ms. Since the air valve working time is relatively short during a single correction (measured in milliseconds), it can be assumed that multiple correction actions within 99.57 ms are completed instantaneously. The simulation sets the air-ducts structure projectile to maintain the stable flight in 10 seconds, and the correction action is carried out in the 10th second. In Figure 17, increment amplitude curves of flight angle of attack are shown for Case 1, the air valve is controlled for correction at the 10th second of projectile flight, and the correction action times are 99 ms and 140 ms. The maximum angle of attack formed during the correction time of 99 ms is 5.45°, and the maximum angle of attack formed during the correction time of 140 ms is 7.6°. The results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax1 when the air valve working pulse width is 99 ms, and it satisfies the stability criterion.Figure 17
Angle of attack increment curves generated by different working pulse widths of air valve (case 1).For Case 5, whenLp = 20 mm and ∆δmax2 = 7°, the range of stable working pulse width of the air valve is obtained as ∆t < 57.85 ms. Figure 18 represents a simulation of the increment amplitude curves of the flight angle of attack when the correction action times are 57 ms and 90 ms. The results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax2, when the air valve working pulse width is 57 ms, and it satisfies the stability criterion. Similarly, for Case 9, when Lp = 30 mm and ∆δmax3 = 8°, the range of the stable working pulse width of the air valve is obtained as ∆t < 44.12 ms. According to Figure 19, the results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax3, when the air valve working pulse width is 44 ms, and it satisfies the stability criterion.Figure 18
Angle of attack increment curves generated by different working pulse widths of air valve (case 5).Figure 19
Angle of attack increment curves generated by different working pulse widths of air valve (case 9).In Figure20, the increment maximum values of the projectile angle of attack corresponding to nine groups of schemes under stable and unstable working pulse widths of the air valve are shown. Simulation results indicate that the increment maximum value of the angle of attack obtained by the working pulse width of the air valve that meets the constraints of (21) can be stable within the limited allowable value ∆δmax. This indicates that the control strategy is effective.Figure 20
Maximum value of the angle of attack increment corresponding to different working pulse width in nine schemes.Next, the control strategy for modulating the air valve working frequency is verified. It is assumed that the scale factor is equal toa = 10 in (23). In other words, when the increment amplitude of the angle of attack is attenuated to less than 1/10 of the original angle of attack increment, the secondary correction is carried out. And, aerodynamic parameters are introduced into (23). Among them, λ1 =− 0.0040, λ2 = −0.0035, ω1 = 0.068, ω2 = −0.0112. The safe range of correction time interval of the air valve in each case can be calculated. The simulation indicates that, after the air valve is corrected with the stable working pulse width, it is stabilized with the calculated safe working interval. Then, a second correction is carried out to verify whether the generated angle of attack increment can be controlled within the ∆δmax range.For Case 1, the projectile is first corrected using stable working pulse width∆t = 99 ms. Through calculation, it is obtained that the safe range of the correction time interval of air valve is ∆T > 2.16 s. The increment amplitude change curve of the angle of attack is shown in Figure 21, when the correction time intervals are 1 s and 2.2 s. If the working frequency of the air valve is too fast and the second correction action is carried out at 11 s, that is, the correction time interval is 1 s, the maximum increment value of the attack angle is 6.7°, which is higher than the limited allowable value ∆δmax1. If the air valve is corrected again after the first correction angle of attack is stabilized, that is, the correction time interval is 2.2 s, the increment amplitude of the angle of attack will not exceed the limited allowable value.Figure 21
Angle of attack increment curves generated by different working frequencies of the air valve (case 1).For Case 5, the projectile is first corrected using the stable working pulse width∆t = 57 ms. The safe range of correction time interval of the air valve is obtained as ∆T > 2.66 s. The increment amplitude change curve of the angle of attack is shown in Figure 22, when the correction time intervals are 1 s and 2.7 s. It can be seen that the increment amplitude of angle of attack will not exceed the limited allowable value ∆δmax2 when the correction time interval is 2.7 s and it satisfies (23). Similarly, for Case 9, the projectile is first corrected using stable working pulse width ∆t = 44 ms. The safe range of correction time interval of the air valve is obtained as ∆T > 3.55 s. According to Figure 23, the results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax3 when the correction time interval is 3.6 s, and it satisfies the stability criterion.Figure 22
Angle of attack increment curves generated by different working frequencies of the air valve (case 5).Figure 23
Angle of attack increment curves generated by different working frequencies of the air valve (case 9).In Table4, the increment maximum values of the projectile angle of attack corresponding to nine groups of schemes under the safe and unsafe working frequencies of the air valve are shown. Simulation results indicate that the increment maximum value of the angle of attack obtained by the working frequency of the air valve that meets the constraints of (23) can be stable within the limited allowable value of ∆δmax. This indicates that the control strategy is effective. If the working time of the air valve is too long, and the interval time between two adjacent corrections is relatively short, the maximum increment value of the angle of attack will be higher than the limited allowable value. Then, the stability of the projectile will be poor, and overturning might occur. Therefore, to control the flight stability of the projectile in the corrected configuration, the working pulse width and frequency of the air valve must be simultaneously controlled.Table 4
Maximum value of angle of attack increment corresponding to different working frequencies in nine schemes.
Simulation schemesStable working pulse width (ms)Correction time interval (ms)The maximum value of the angle of attack incrementCase 19910006.7°22005.8°Case 24910007.8°27005.9°Case 33310007.5°36005.6°Case 411510007.8°22006.7°Case 55710009.1°27006.8°Case 63810008.6°36006.5°Case 713210008.9°22007.8°Case 866100010.5°27007.9°Case 944100010.0°36007.5°


## 5.1. Flow Visualization

Corrected configuration of the air-ducts structure projectile at terminal trajectory is simulated. The flight speed of the spinning projectile is set to 0.8 Ma and the flight angle of attack is set to 3°. The pressure distribution of flow fields of the projectile is illustrated in Figure10. Among them, the zoomed view shows the pressure distribution near the air duct outlet on the vertical symmetry plane. And, the velocity distribution of internal and external flow fields of the projectile on a vertical symmetry plane under the same conditions is demonstrated in Figure 11. According to Figure 10, a subsonic oncoming airflow forms a local high-pressure region at the air inlet. Airflow velocity decreases, and a small area of obstruction is formed at the projectile nose. When a lateral air-jet enters the flow field, local pressure at the air outlet increases due to interference between the derived airflow and the oncoming airflow on the projectile surface. Based on Figure 11, the velocity of the oncoming airflow in the interior air duct continuously increases. Affected by the turning and expansion of air-ducts structure as well as wall viscosity, the flow separation is produced at the outer wall zone of the duct steering (zone A in Figure 11). The existence of reverse pressure gradient forms a local recirculation zone which results in flow loss. However, according to Figure 11, the lateral air-jet can still be smoothly derived, and the outlet air velocity can reach up to 200 m/s. At the duct air outlet, the velocity component of the lateral air-jet along the -Z-axis and the oncoming airflow interact to form a certain pressure gradient. Moreover, the velocity component along the X-axis is superimposed with the oncoming airflow. Disturbance of the lateral air-jet to the oncoming airflow extends to the middle and rear parts of the projectile along the flow direction. The aforementioned changes the flow field structure and affects the aerodynamic force on the projectile.Figure 10
Pressure distribution of the air-ducts structure projectile.Figure 11
Velocity distribution of the air-ducts structure projectile.Velocity distribution at the cross section of the air outlet when the air-ducts structure projectile is radially corrected in different directions is shown in Figures12 and 13. Lateral force Fz variation curve of correction projectile and reference projectile under the same conditions are shown in Figure 14. The aerodynamic shape of air-ducts structure projectile is the corrected configuration, and the aerodynamic shape of reference projectile is the same as the corrected configuration, but there is no air-ducts structure inside. Within Figure 14, two kinds of projectiles are rotated from -Z-axis to + Z-axis, the angle corresponding to -Z-axis is 0°, and the angle corresponding to the +Z-axis is 180°. Since the projectile spins and has an angle of attack during flight, a cross-flow occurs through the projectile (along the +Y-axis).Figure 12
Velocity distribution of the cross section of the air outlet (projectile corrected along -(Z)-axis).Figure 13
Velocity distribution of the cross section of the air outlet (projectile corrected along +(Z)-axis).Figure 14
Lateral force variation curve of the correction projectile and the reference projectile.The velocity distribution of the circumferential surface at the air outlet when the air-ducts correction mechanism is corrected along the -Z-axis is shown in Figure 12. Since the airflow rotation direction on the left side of the projectile surface is consistent with the cross-flow, the velocity of airflow along the Y-axis is merged due to air viscosity, while the velocity is increased. The projectile on the right side (along -Z-axis) is affected by the air flow at the outlet duct, and the air flow state is relatively complex. After the air jet is introduced into the external flow field, the viscosity of the air jet, the interactive superposition effect of the jet, and the oncoming airflow accelerate the cross-flow on the right side of the projectile. The lateral force Fz caused by the pressure difference on both sides of the projectile decreases accordingly. Based on Figure 14, the lateral force of the correction projectile is equal to 2.55 N, and the lateral force of the reference projectile is equal to 3.67 N. In the corrected state, lateral air-jet causes a change in the pressure difference on both sides of the projectile. Therefore, the value of Fz is lower than at the uncorrected state of the reference projectile.The velocity distribution of the circumferential surface at the air outlet when the air-ducts correction mechanism is corrected along the +Z-axis is shown in Figure 13. The left side of the projectile is affected by the projectile rotation, cross-flow, and duct outlet air jet, and the velocity gradient changes significantly. Since the airflow direction near the wall is opposite to the cross-flow, the local pressure of the projectile on the right side increases. The reaction force of the lateral air-jet acts on the projectile and its action direction is opposite to the Magnus force. Consequently, the projectile lateral force is the smallest at this time. According to Figure 14, the lateral force of the correction projectile is equal to 1.83 N, while the lateral force of the reference projectile is equal to 3.59 N. Disturbance of lateral air-jet to the surrounding flow field alters the lateral force magnitude.By comparing lateral forces of different structural projectiles in the same flight state (Figure14), it can be found that the lateral force of the reference projectile is stable during rotation. However, during the projectile rotation from the -Z-axis to the +Z-axis, the lateral force Fz of the correction projectile is always lower than the reference projectile. This indicates that the air-ducts structure correction scheme can change the lateral force of the projectile in the working state. Through calculation, it can be observed that the lateral air-jet correction action can make a maximum change of 49% of the projectile lateral force and a minimum change of 30.5% at the flight speed of 0.8 Ma. Therefore, this correction scheme plays a correction role. Furthermore, when the air-ducts structure is corrected along with different circumferential angular positions, the lateral force is altered. The change of the air outlet from -Z-axis to the +Z-axis causes the projectile lateral force to gradually decrease. This can be attributed to the interaction between the lateral air-jet and the flow around the spinning projectile. The pressure distribution on the projectile surface changes, and the correction force is superimposed with the overall aerodynamic force of the projectile. Consequently, the lateral force shows the changing trend demonstrated in Figure 14.Figures15 and 16 represent the variation curve of drag and lift coefficient for two different structural projectiles in the same flight state. The drag and lift forces of the reference projectile are stable during rotation. In addition, the drag force of the correction projectile is always higher than the reference projectile. This indicates that the air-ducts structure increases the drag force of the projectile and that the range direction can be corrected. Compared with the reference projectile, the lift force of the correction projectile is not altered much. Therefore, it can be concluded that the correction effect of the lateral jet has a minor effect on the lift force of the projectile.Figure 15
Drag coefficient variation curve of the correction projectile and the reference projectile.Figure 16
Lift coefficient variation curve of the correction projectile and the reference projectile.


## 5.2. Control Strategies Simulation Verification

In this paper, limited allowable value∆δmax of increment amplitude of angle of attack is used to limit the angle of attack increment caused by the correction action. The operating parameters of the correction air valve are modulated to achieve flight stability control of air-ducts structure projectile. According to (21), the structural parameter Lp of the projectile affects the working parameters of the air valve. To verify the validity of flight stability control strategies, Lp -∆δmax combination schemes are simulated in this paper for different states. The purpose of this is to analyze whether the angle of attack increment generated by the correction meets the requirements, after the working parameters of the air valve are modulated by the control strategies under different conditions. An example of verification schemes is shown in Table 3. Since the air duct outlets are designed near the mass center of the projectile, the structural parameter Lp is set to 10 mm, 20 mm, and 30 mm. And, setting the limited allowable value ∆δmax can be adjusted according to the flight state of the projectile before the correction. Considering that manufacturing errors or other factors may lead to a large angle of attack caused by the correction action, the ∆δmax is set to 6°, 7°, and 8°.Table 3
An example of verification schemes.
Simulation schemesLp (mm)∆δmax1 (°)Simulation schemesLp (mm)∆δmax2 (°)Simulation schemesLp (mm)∆δmax3 (°)Case 1106Case 4107Case 7108Case 2206Case 5207Case 8208Case 3306Case 6307Case 9308Firstly, the control strategy for modulating the working pulse width of the air valve is verified. For Case 1, whenLp = 10 mm and ∆δmax1 = 6°, aerodynamic parameters are introduced into (21). Among them, m = 5 kg, v  = 272 m/s, Fp = 3 N, I = 0.0149 kg m2, P = 0.0569, M = 0.00076, H = 0.00758, T = 0.00345, α = 0.0284, σ = 0.062. Then, the range of stable working pulse width of the air valve is obtained as ∆t < 99.57 ms. Since the air valve working time is relatively short during a single correction (measured in milliseconds), it can be assumed that multiple correction actions within 99.57 ms are completed instantaneously. The simulation sets the air-ducts structure projectile to maintain the stable flight in 10 seconds, and the correction action is carried out in the 10th second. In Figure 17, increment amplitude curves of flight angle of attack are shown for Case 1, the air valve is controlled for correction at the 10th second of projectile flight, and the correction action times are 99 ms and 140 ms. The maximum angle of attack formed during the correction time of 99 ms is 5.45°, and the maximum angle of attack formed during the correction time of 140 ms is 7.6°. The results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax1 when the air valve working pulse width is 99 ms, and it satisfies the stability criterion.Figure 17
Angle of attack increment curves generated by different working pulse widths of air valve (case 1).For Case 5, whenLp = 20 mm and ∆δmax2 = 7°, the range of stable working pulse width of the air valve is obtained as ∆t < 57.85 ms. Figure 18 represents a simulation of the increment amplitude curves of the flight angle of attack when the correction action times are 57 ms and 90 ms. The results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax2, when the air valve working pulse width is 57 ms, and it satisfies the stability criterion. Similarly, for Case 9, when Lp = 30 mm and ∆δmax3 = 8°, the range of the stable working pulse width of the air valve is obtained as ∆t < 44.12 ms. According to Figure 19, the results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax3, when the air valve working pulse width is 44 ms, and it satisfies the stability criterion.Figure 18
Angle of attack increment curves generated by different working pulse widths of air valve (case 5).Figure 19
Angle of attack increment curves generated by different working pulse widths of air valve (case 9).In Figure20, the increment maximum values of the projectile angle of attack corresponding to nine groups of schemes under stable and unstable working pulse widths of the air valve are shown. Simulation results indicate that the increment maximum value of the angle of attack obtained by the working pulse width of the air valve that meets the constraints of (21) can be stable within the limited allowable value ∆δmax. This indicates that the control strategy is effective.Figure 20
Maximum value of the angle of attack increment corresponding to different working pulse width in nine schemes.Next, the control strategy for modulating the air valve working frequency is verified. It is assumed that the scale factor is equal toa = 10 in (23). In other words, when the increment amplitude of the angle of attack is attenuated to less than 1/10 of the original angle of attack increment, the secondary correction is carried out. And, aerodynamic parameters are introduced into (23). Among them, λ1 =− 0.0040, λ2 = −0.0035, ω1 = 0.068, ω2 = −0.0112. The safe range of correction time interval of the air valve in each case can be calculated. The simulation indicates that, after the air valve is corrected with the stable working pulse width, it is stabilized with the calculated safe working interval. Then, a second correction is carried out to verify whether the generated angle of attack increment can be controlled within the ∆δmax range.For Case 1, the projectile is first corrected using stable working pulse width∆t = 99 ms. Through calculation, it is obtained that the safe range of the correction time interval of air valve is ∆T > 2.16 s. The increment amplitude change curve of the angle of attack is shown in Figure 21, when the correction time intervals are 1 s and 2.2 s. If the working frequency of the air valve is too fast and the second correction action is carried out at 11 s, that is, the correction time interval is 1 s, the maximum increment value of the attack angle is 6.7°, which is higher than the limited allowable value ∆δmax1. If the air valve is corrected again after the first correction angle of attack is stabilized, that is, the correction time interval is 2.2 s, the increment amplitude of the angle of attack will not exceed the limited allowable value.Figure 21
Angle of attack increment curves generated by different working frequencies of the air valve (case 1).For Case 5, the projectile is first corrected using the stable working pulse width∆t = 57 ms. The safe range of correction time interval of the air valve is obtained as ∆T > 2.66 s. The increment amplitude change curve of the angle of attack is shown in Figure 22, when the correction time intervals are 1 s and 2.7 s. It can be seen that the increment amplitude of angle of attack will not exceed the limited allowable value ∆δmax2 when the correction time interval is 2.7 s and it satisfies (23). Similarly, for Case 9, the projectile is first corrected using stable working pulse width ∆t = 44 ms. The safe range of correction time interval of the air valve is obtained as ∆T > 3.55 s. According to Figure 23, the results indicate that the increment amplitude of the angle of attack can be stabilized within the limited allowable value ∆δmax3 when the correction time interval is 3.6 s, and it satisfies the stability criterion.Figure 22
Angle of attack increment curves generated by different working frequencies of the air valve (case 5).Figure 23
Angle of attack increment curves generated by different working frequencies of the air valve (case 9).In Table4, the increment maximum values of the projectile angle of attack corresponding to nine groups of schemes under the safe and unsafe working frequencies of the air valve are shown. Simulation results indicate that the increment maximum value of the angle of attack obtained by the working frequency of the air valve that meets the constraints of (23) can be stable within the limited allowable value of ∆δmax. This indicates that the control strategy is effective. If the working time of the air valve is too long, and the interval time between two adjacent corrections is relatively short, the maximum increment value of the angle of attack will be higher than the limited allowable value. Then, the stability of the projectile will be poor, and overturning might occur. Therefore, to control the flight stability of the projectile in the corrected configuration, the working pulse width and frequency of the air valve must be simultaneously controlled.Table 4
Maximum value of angle of attack increment corresponding to different working frequencies in nine schemes.
Simulation schemesStable working pulse width (ms)Correction time interval (ms)The maximum value of the angle of attack incrementCase 19910006.7°22005.8°Case 24910007.8°27005.9°Case 33310007.5°36005.6°Case 411510007.8°22006.7°Case 55710009.1°27006.8°Case 63810008.6°36006.5°Case 713210008.9°22007.8°Case 866100010.5°27007.9°Case 944100010.0°36007.5°


## 6. Conclusions

In this paper, a new two-dimensional trajectory correction projectile model with a controllable air-ducts structure is proposed to control the trajectory correction of spin-stabilized projectiles. The influence of lateral air-jet on the flight stability of projectile in the corrected configuration is investigated, and stable flight control strategies suitable for the projectile are proposed. The contributions of this paper are summarized as follows:(1)
In this paper, the flow field of a new air-ducts structure projectile model is analyzed via computational fluid dynamics. The numerical results indicate that the correction force produced by the air-ducts structure has an obvious effect on the lateral force of the projectile, and the lateral air jet can play a certain radial correction role. In the flight state of 0.8 Ma and 3° angle of attack, airflow velocity at the exit port can reach up to 200 m/s. Compared with the reference projectile, the maximum lateral force of the air-ducts structure projectile under the same conditions can be changed by up to 49%. Furthermore, the existence of an air-ducts structure increases the overall drag force of the projectile but has a minor effect on lift force.(2)
Flight stability control strategies are proposed by modulating the working pulse width and frequency of the air valve. A safe design range of air valve working parameters is derived and verified. The nine groups ofLp-∆δmax verification schemes are designed to simulate the proposed control strategies. The simulation results indicate that, by reasonably controlling the working pulse width and frequency of the air valve, the increment of the angle of attack caused by the correction action can be stabilized within the preset range. Consequently, the projectile in the corrected configuration can meet the requirements of flight stability.



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*Source: 1383294-2022-03-09.xml*