Research on Angular Motion Characteristics and Stability of Trajectory-Corrected Rocket Projectile with Isolated Rotating Tail Rudder

Wang, Chen; Ding, Libo; Feng, Yang; Lu, Tan

doi:https://doi.org/10.1155/2023/5880706

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5880706 | https://doi.org/10.1155/2023/5880706

Research on Angular Motion Characteristics and Stability of Trajectory-Corrected Rocket Projectile with Isolated Rotating Tail Rudder

Chen Wang,¹Libo Ding,¹Yang Feng,²and Tan Lu¹

Academic Editor: Can Huang

Received10 Nov 2022

Revised03 Jan 2023

Accepted31 Jan 2023

Published15 Feb 2023

Abstract

A trajectory-corrected rocket projectile with an isolated rotating tail rudder provides a new concept for a low-cost trajectory correction and guidance due to its simplified guidance principle. To study the angular motion characteristics of a trajectory-corrected rocket projectile with an isolated rotating tail rudder under the action of a periodic control force, the angular motion differential equation described in the complex plane is derived based on the rigid body trajectory equation. Furthermore, the effects of the initial conditions, gravity, and an asymmetric tail on the flight stability of the projectile in an uncorrected trajectory state are studied. The critical rotational speed of the tail rudder, which makes the projectile unstable due to the Magnus moment, and the rotational speed of the tail rudder, which causes projectile resonance, are derived. The transient and steady-state solutions of the angular motion of the projectile under trajectory correction and the variation law of the velocity direction for the projectile centroid are obtained. Finally, an example is given to verify the conclusions of this study. This research has important guiding significance for trajectory analysis, guidance control, and guidance law designs for isolated rotating rudder trajectory-corrected rocket projectiles.

1. Introduction

Modern warfare has placed increasing demands on the cost-effectiveness of weapon systems, both in terms of increasing their accuracy and reducing costs as much as possible. The effective range of an unguided, man-portable, antiarmor weapon such as rocket-propelled grenades is generally within 500 m. Although these devices are inexpensive to build, they have a small effective range with accuracies that decrease significantly over distance. In comparison, man-portable antitank missiles, such as the “Javelin” missile of the United States, can strike fixed or moving targets within 2000 m with high precision using infrared imaging guides and four movable rudder blades [1]. However, because of its high cost, this type of weapon is only suitable to attack high-value targets.

One effective scheme to increase the range and accuracy of unguided, man-portable, antiarmor weapons is to add a simple guidance module. A fixed canard system [2–6] is a low-cost, miniaturized, two-dimensional, trajectory-corrected steering system. Such a system was first used in the U.S. Army’s Precision Guidance Kit (PKG) [7–11] for 155 and 105 mm grenades. It can also be ported to 120 mm mortar rounds [12], and there have been reports in recent years of its application to long-range rockets. The periodic average control force method [13–18] has been adopted in actuator systems to realize two-dimensional trajectory correction using a single channel. This has the advantages of a simple structure, low cost, and high reliability.

The fixed canard system is usually installed on the projectile head. Under noncorrected trajectories, the steering gear rotates at a uniform speed relative to the inertial system. In a trajectory-corrected state, the direction and magnitude of the periodic average aerodynamic force are changed by controlling the rotation speed of the steering gear to realize trajectory correction. Such trajectory correction projectiles are usually called dual-spin projectiles. To date, many works have modelled the external trajectory of dual-spin projectiles and studied their flight stability and control characteristics. Zhu et al. [16] established a seven degree-of-freedom dynamic motion equation for dual-spin projectiles in a fixed plane coordinate system. The differential equation for the complex angle of attack was derived using projectile linear theory without the assumption of a flat trajectory. A revised stability criterion was established from the Hurwitz stability criterion, and analytic solutions to the stability boundaries for trim angles were developed. To investigate the influence of yawing forces on angular motion, Wang et al. [19] derived a theoretical solution for the total yaw angle function with cyclic yawing forces using the designed seven degree-of-freedom model. Furthermore, a detailed simulation was performed to determine the influence rules of the yawing force on the angular motion. The calculated results illustrate that when the rotational speed of the forward part is close to the initial turning rate, the total yaw angle increases, and the flight range decreases sharply.

Chang et al. [20] studied the swerve response of spin-stabilized projectiles on the canard control problem. A complex deviation angle was used to describe the motion of the projectile velocity vector over the controlled trajectory. The effect of distance between the center of pressure of the canard surface and the center of mass of the projectile on the swerving motion was explained from the perspective of the projectile velocity vector response. The results demonstrate that gravity plays a vital role in predicting the swerve response. Li et al. [21] established the exterior ballistic linearized equations to consider the control force by introducing the concept of the angular compensation matrix. The instability boundaries of the control force magnitude were derived. Numerical simulations demonstrate that if the control force magnitude is in the unstable scope, the projectile loses its stability. Furthermore, the effect of the projectile pitch, velocity, and roll rate on the flight stability during correction is investigated using the proposed instability boundaries. Guan and Yi [15] analyzed the ballistic characteristics of the projectile with a seven-degree-of-freedom projectile trajectory model. Their numerical simulations indicated that the dual-spinning projectile is different from the traditional spinning projectile mostly in that a degree of freedom is added in the direction of the projectile axis. The forebody of the projectile also spins at a low speed or even holds still to improve its control precision, while the afterbody spins at a high speed to maintain gyroscopic stability. Zheng and Zhou [22] established a mathematical model for a dual-spin projectile, and the angular motion equation was obtained using some linearized assumptions. The sufficient and necessary conditions of the coning motion stability for the dual-spin projectile with angular rate loops are analytically derived and further verified via numerical simulations.

The above literature shows that the installation of PGK will affect the stability conditions of the original projectile. Before researching trajectory-corrected control algorithms, it is necessary to perform basic research on the flight dynamics and stability of projectiles. A fixed-wing canard system is applied to a grenade with a caliber of 105–155 mm. As the high-speed rotation of the projectile can produce a strong stability torque, installing the steering gear system on the projectile head has little impact on its stability. However, for individual antiarmor rockets, the tail wing is generally used to provide stability torque to the missile body. Installing the steering gear system at the head reduces the flight stability of the projectile [23–25], which readily causes the projectile to lose stability and overturn. Therefore, arranging the steering gear system at the tail improves the projectile stability. If the PGK system is transplanted to the tail of a subsonic single soldier rocket to give it the ability of two-dimensional trajectory correction, the angular motion and stability law of the projectile body will be different completely from that of a supersonic high rotation grenade. Research on the angular motion law and stability of projectiles is the basic concept for trajectory analysis, guidance control, and guidance law design. However, there have been no relevant reports on this type of trajectory-corrected projectile in recent years.

This study analyzes the angular motion and stability of an isolated rotating rudder trajectory-corrected rocket projectile. The characteristic for this type of trajectory-corrected rocket projectile is that the actuator to generate the correction force is at the tail of the projectile and can rotate along its axis. The principle of the actuator is similar to the PGK, but it can provide corrective and stability torques. First, the guidance and control principle of the isolated rotating tail rudder trajectory correction rocket projectile is clarified. Second, the angular motion of the projectile on the complex plane is defined, and the angular motion differential equation is derived based on the rigid body trajectory equation. The angular motion and stability characteristics of the projectile under different tail speeds are studied for the uncorrected and corrected flight states. Finally, the conclusions are verified through an example.

2. Periodic Control Force Principle of Isolated Rotating Tail Rudder Rocket

Research into the development of isolated rotating tail rudder rockets helps compensate for the shortcomings of unguided, man-portable, antiarmor weapons due to their low precision and short range while keeping costs low. The caliber of the considered rocket is approximately 80 mm and is used primarily to strike fixed or low-speed moving targets within 2000 m. The overall layout of the rocket is shown in Figure 1(a) [26]. The device is composed of a seeker, projectile body, and tail segment. The tail segment is driven by a motor and can rotate relative to the projectile body around its longitudinal axis. Four wings are fixed on the tail segment, as shown in Figure 1(b), and one pair of wings has a zero-degree deflection angle relative to the projectile body. These are the stabilizing rudders and provide a stable moment while the rocket is flying. The other pair of wings has a deflection angle of 5° relative to the projectile body and is the correction rudders. In Figure 1(b), the represents the speed of the tail segment, represents the angular displacement of the tail segment, and represents the direction of the correction force generated by the correction rudders.

(a)

(b)

The correction rudders provide the radial aerodynamic force to the projectile body by rotating around its longitudinal axis with the tail segment. If the tail segment is fixed at a specific phase, the correction rudders provide continuous aerodynamic force in a specific direction to control the projectile body and rotate around its center of mass. This kind of rudder uses period-averaged forces to control the rocket’s position and attitude. When the rudder rotates at a constant speed relative to the ground, the average radial aerodynamic force generated by the entire wing is zero over a rotation period. This simplified assumption is generally only applicable to the case of small angles of attack and rotation speeds. For high angles of attack and rotation speeds, the rudder flow field is more complex, and this law is no longer applicable.

3. Dynamic Modelling of Rocket Projectile with Isolated Rotating Tail Rudder

The coordinate systems defined in this study are shown in Figures 2(a) and 2(b). The coordinate system is for the ground and can be regarded as the inertial system. The position of the origin is at the launch port, the axis points to the firing direction along the horizon, the axis points upward along the vertical direction, and the axis is determined based on the right-hand rule. The ground coordinate system is used to determine the spatial position of the projectile centroid. The coordinate system is the reference coordinate system. The origin is at the projectile mass center and is a dynamic coordinate system that moves with the projectile. Each coordinate axis is obtained by translating the ground coordinate system . The reference coordinate system is used to determine the centroid velocity and azimuth of the projectile axis. The coordinate system is the trajectory coordinate system with its origin located at the projectile mass center. The axis points along the velocity direction of the mass center, the axis is vertical to the axis, and the axis is determined based on the right-hand rule. The trajectory coordinate system is determined by rotating the reference coordinate system twice. First, the angle is rotated in a positive direction around the axis to form the coordinate system . Second, the angle is rotated in a negative direction around the axis to form the coordinate system . The is the height angle, and is the deflection angle of the trajectory. The is the projectile axis coordinate system with an origin located at the projectile mass center. The axis points along the forward direction of the projectile axis, the axis is vertical to the axis, and the axis is determined based on the right-hand rule. The projectile axis coordinate system can be determined by rotating the reference coordinate system twice. First, the angle is rotated in a positive direction around the axis to form the coordinate system . Second, the angle is rotated in a negative direction around the axis to form the coordinate system . The is the height angle, and is the deflection angle of the projectile axis.

(a)

(b)

The translational differential equations for the center of mass of the rocket projectile [14, 27] are as follows:where , , and represent the projectile centroid velocity, velocity elevation angle, and velocity deflection angle, respectively; , , and represent the centroid displacements; represents the total projectile mass with and being the mass of the body and tail, respectively; and , , and represent the force components of the projectile in the three axes of the trajectory coordinate system.

The differential equations for the rocket projectile rotation around the center of mass [28–30] are as follows:where , , , and represent the axial angular displacement of the projectile body, axial angular displacement of the tail, height angle of the projectile axis, and deflection angle of the projectile axis, respectively; , , , and represent the components of the axial angular velocity of the projectile body, axial angular velocity of the tail, angular velocity of the projectile axis height angle, and angular velocity of the deflection angle of the projectile axis in the projectile axis coordinate system, respectively; , , , and represent the components of the aerodynamic moment for the projectile body, aerodynamic moment of the tail, moment of the tail acting on the projectile body, and moment of the projectile body acting on the tail in the direction of the projectile axis coordinate system. The and are the components of the aerodynamic moment acting on the projectile body in the and directions of the projectile axis coordinate system, respectively.

4. Angular Motion Modeling of Projectile Body

4.1. Geometric Description of Angular Motion of Projectile Body

It is necessary to analyze and solve for the angular motion of the projectile to study the attitude motion of the projectile in flight and analyze its flight stability. The flight stability of the projectile is usually represented by variations in the angle of attack δ of the projectile with time t. Under normal conditions, the trajectory of the projectile varies around the ideal trajectory. Taking the velocity line of the ideal trajectory as the benchmark, the pitching and yaw motions of the projectile around this benchmark are the angular motion of the projectile.

The unit sphere description of the projectile angular motion is shown in Figure 3. The spherical surface is a unit sphere with the center of mass c. The unit vector has the same direction as the ideal trajectory velocity, and its intersection with the unit sphere is o. The plane cox is the horizontal plane of the trajectory coordinate system, and coy is the vertical plane. The unit vector has the same velocity direction as the projectile, and its intersection with the unit sphere is T. The unit vector has the same direction as the projectile axis, and the intersection with the unit sphere is B. The small range sphere over which the coordinate system xoy is located can be expanded into the plane as shown in Figure 4. If the x-axis is regarded as the imaginary axis and the y-axis as the real axis, the vectors , , and can be represented by the complex numbers , , and , respectively, which satisfies the relationship . The δ₁ represents the height angle of attack, and δ₂ represents the lateral angle of attack.

4.2. Differential Equation of Projectile Body Angular Motion

The following symbols are introduced to simplify the expressions of the forces and moments:where , , , , , , , and represent the drag coefficient, lift coefficient, Magnus force coefficient, static moment coefficient, equatorial damping moment coefficient, projectile body pole damping moment coefficient, tail pole damping moment coefficient, and tail Magnus moment coefficient, respectively, and , , , and represent the air density, characteristic area, reference length, and projectile diameter, respectively. For small caliber rocket projectiles, the order of magnitude for each parameter at small angles of attack and subsonic velocities is shown in Table 1 [26].

The forces and moments on the projectile are as follows:where , , , , , , and represent the rocket engine thrust, tail motor torque, height angle of attack, lateral angle of attack, relative angle of attack, aerodynamic eccentricity angle of additional force, and aerodynamic eccentricity angle of additional moment, respectively. To facilitate the study of the angle of attack equation of the projectile rocket, the following ideal trajectory equation is introduced:where , , , , and represent the center-of-mass velocity in the ideal trajectory, velocity angle, x-direction displacement, y-direction displacement, and acceleration generated by the rocket thrust, respectively. The complex plane description of the angular motion defined in Figure 4 indicates that the following equation holds: , , , and . According to the second and third equations in equation (1) and the second equation in equation (7), and considering that , , , , , and are small quantities, the equation that describes changes in the velocity direction can be written as follows:

Multiply the second equation in equation (8) by the imaginary unit i, and add the two equations in equation (8) to obtain:where is the velocity complex deflection angle, and is the complex angle of attack.

Changes in the projectile axis direction are described by the third and fourth equations in equation (3). Using the relations , , , and and omitting high-order small quantities allow simplification as follows:

Multiply the first equation in equation (10) by −i and add the two equations to obtain:where is the complex deflection angle of the spring axis. To eliminate the factor before the state variable and of equations (9) and (11), the independent variable is transformed from time t to arc length s, the symbol “” is used to represent the derivative of the arc length, and a small amount of is omitted. Equations (9) and (11) are transformed into the following form:

To derive the differential equation for the angle of attack , the relation and using the first equation of (7) transforms into:

Furthermore, taking the coefficient before in equation (14) as a constant, can be obtained as follows:

Substituting equations (14) and (15) into equation (13) and replacing the state variable with , the following differential equation for the attack angle can be derived:where the symbol “” represents the derivative of the arc length s, and the other symbols are as follows:where E at the right end of the complex angle of attack in equation (16) is the disturbance term, which can be divided into the gravity disturbance E_G and wing aerodynamic disturbance E_F. The E_G mostly contains the θ term, which is caused by changes in gravity, and E_F primarily contains the tail-spin angle term .

5. Angular Motion Characteristics of Projectile under Uncorrected Trajectory

The lateral aerodynamic force generated from the fixed wings in the uncorrected trajectory state should be zero in a rotation cycle. Thus, the rotation speed of the tail wing around the projectile axis relative to the ground coordinate system should be a nonzero fixed value. The angular motion characteristics of the projectile in this state can be analyzed from the solution of equation (16), which contains the angular motion characteristics of the projectile. These are affected by three factors: angular motion caused by initial conditions, angular motion caused by gravity, and angular motion caused by asymmetric factors. Therefore, its solution can be decomposed intowhere is the homogeneous solution of equation (16) and represents the angular motion caused by the initial conditions. The form of the solution is , where and are undetermined coefficients determined by the initial conditions. The is the angular motion due to the nonhomogeneous term E_G, and is the angular motion due to the nonhomogeneous term E_F.

5.1. Angular Motion Characteristics Caused by Initial Conditions

Both the initial angular displacement and initial angular velocity of the projectile can affect its angular motion response. Without considering the disturbance E, the angular motion caused by the initial conditions is . Based on the coefficient freezing method, let and be the two characteristic roots of the following homogeneous equation:

Then, can be written as follows:where and are as follows:

The , , , and are determined by the initial conditions and . Then,

The angle of attack is the angular motion caused by the initial disturbance and is expressed as follows:where and are the angular frequencies for one and two circular motions, and and in and represent changes in the circular motion amplitudes. If the circular motion amplitude is gradually reduced, the derivative of must be negative. That is,where is the real part of the characteristic root, which is obtained from the complex square formula:

Equation (24) is the full trajectory dynamic stability condition that the tail rotation speed should satisfy. For the active phase of the rocket, , equation (24) is more easily satisfied; therefore, it is sufficient to consider only the stability of the passive segment. For the passive phase of the rocket, and monotonically increase with ; therefore, the left-hand side of equation (24) takes the maximum value at the minimum velocity. In summary, the stability of the entire trajectory under the initial disturbance can be guaranteed only by satisfying equation (26) at the minimum velocity . Substituting equation (25) into equation (24), the tail rotation speed should satisfywhere is the critical speed. For the convenience of calculations, omitting the small passive section of allows to satisfy the following condition:

5.2. Angular Motion Characteristics Caused by Gravity

As the normal component of gravity acting on the projectile causes the velocity of the mass center to rotate downward, the angular displacement of the velocity vector for the mass center will cause changes in the angle of attack, which impacts the angular motion characteristics of the projectile. The angular motion equation caused by gravity is as follows:

The form of the solution is the superposition of the general solution for the homogeneous equation and the special solution of the nonhomogeneous equation. Namely,where is the homogeneous solution of equation (28) that considers the case, which has the same form as and , differing only in the coefficients. The constant variation method can be used to solve the special solution of the nonhomogeneous equation as follows:

Then,

5.3. Angular Motion Characteristics Caused by Asymmetric Tail

The tail wing provides the control force and torque to change the projectile motion. The radial periodic force is generated during the tail wing rotation due to the same direction deflection angle of a pair of wings. This can cause the projectile to have the angular motion characteristics of dynamic imbalance. If the angular frequency of the dynamic imbalance is close to the angular frequency of one or two circle motions, the projectile may resonate, which makes it unstable. The angular motion equation caused by the asymmetric tail is as follows:

If the additional force eccentricity angle is equal to the additional moment eccentricity angle , the aerodynamic disturbance term E_F of the wing can be expressed as follows:where , and . According to the coefficient freezing method, the solution of equation (32) can be expressed as follows:where

Then, the angle of attack generated by the asymmetric tail is as follows:

There are three circular motions. Namely, circular motion with the angular frequencies of and generated by the initial disturbance and the forced circular motion with an angular frequency of . If the damping factors and of the passive section satisfy the stability conditions, the amplitudes of the first two circular motions tend to 0 with an increased arc length s, and only the forced circular motion with an angular frequency of remains. The amplitude of the forced circle motion is as follows:

6. Angular Motion Characteristics of Projectile in Trajectory Correction State

When the guidance system generates a deviation signal to correct the projectile attitude, the tail motor outputs a specific rotation speed to make the tail axially stable relative to the ground coordinate system. The direction of the additional torque generated by the aerodynamic eccentricity of the tail is fixed at the required phase, which changes the attitude of the projectile. Then, the tail rotation speed changes from constant to . For convenience, the transition process of the tail rotation speed is omitted, and the response of the angle of attack under the step excitation of the tail rotation speed is studied.

6.1. Steady and Transient Solutions for Angle of Attack under Trajectory Correction

As the tail rotation speed is and the projectile body rotation speed is , the small quantity in equation (16) can be omitted, and the projectile angle of attack satisfies the following equation:where

The rotation speed step excitation response of the system is equivalent to the solution of equation (38) when is equal to the fixed value under the initial conditions and . The initial conditions are determined from the uncontrolled flight terminal state. According to the coefficient freezing method, the form of the homogeneous solution of equation (38) is the same as that of equation (20), and the nonhomogeneous solution is as follows:

Then, the solution of equation (38) is as follows:where , , , and . If the condition is satisfied, the system will be stable, and the first two terms of equation (41) will decrease to 0 with an increased s. Then, is the steady-state solution of the system, and the corresponding steady-state angle of attack is as follows:

6.2. Effect of Trajectory Correction on Velocity Deflection Angle of Projectile

In the trajectory correction state, the tail wing of the projectile generates a control force along a fixed direction, which changes the projectile angle of attack. This generates a lift in the angle of attack plane and changes the direction of the velocity for the projectile mass center. The steady-state angle of attack is substituted into equation (9), and the independent variable is changed from time t to arc length s. Considering the trajectory passive section , the complex deflection angle differential equation for the projectile centroid velocity under the steady-state angle of attack is obtained as follows:

From equation (38), the following can be obtained:

Substituting equation (44) into equation (43) and ignoring a small amount of , the following is obtained:

After linearization, the complex deflection angle of the projectile velocity at the steady angle of attack is as follows:

7. Example and Stability Analysis

Numerical simulations are performed based on the known projectile body and aerodynamic parameters to verify the correctness of the angular motion characteristic analysis of the trajectory correction rocket projectile with an isolated rotating tail rudder as well as the stability analysis of a sample projectile. The relevant projectile body and aerodynamic parameters are shown in Table 2 [26].

7.1. Stability Analysis of Projectile under Magnus Moment

In the trajectory-corrected phase, the Magnus moment acting on the projectile body is generated due to the rotation of the tail wing relative to the inertial system as driven by the motor. The Magnus moment is positively correlated with the projectile velocity, tail rotation speed, and angle of attack. When other conditions are certain, a tail rotation speed that is too high will increase the Magnus moment to a degree that cannot be ignored. As the Magnus moment direction is roughly perpendicular to the projectile axis in the angle of attack plane, the moment will make the projectile move in a spiral around the speed line. If the damping moment received by the projectile cannot dissipate the oscillation energy of the projectile, the projectile axis oscillation will increase and result in projectile instability. The stability of the projectile under different tail wing speeds is shown in Figure 5.

(a)

(b)

(c)

According to the characteristic equation of equation (19), the two characteristic frequencies and of the projectile are and , showing can be obtained. Therefore, the shape of the two characteristic frequencies is oval, as shown in Figure 4. According to equation (27), the critical rotation speed of the tail is . Figure 5 shows the movement of the projectile angle of attack under the initial conditions , , , and . Without considering gravity and the correction force, the variable coefficient nonhomogeneous nonlinear equation (16) describing the complex angle of attack degenerates into the approximate second-order homogeneous linear expression of equation (19), which makes the analytic solution of equation (23) for the angle of attack very close to the numerical solution with a correlation coefficient of . Therefore, the analytic solution of equation (23) is considered accurate. Due to the influence of the Magnus moment, although the initial value of the directional angle of attack is 0, an increased arc length s causes the projectile axis to produce the phenomenon of a lateral reciprocating swing. As shown in Figure 5(a), when , the Magnus moment is small, and the swing amplitude of the projectile body gradually decreases to 0. As shown in Figure 5(b), when , the swing amplitude of the projectile eventually stabilizes at a certain angle. As shown in Figure 5(c), when , the Magnus moment is large, and the swing amplitude of the projectile body first decreases before increasing, which eventually leads to instability of the projectile body.

7.2. Stability of Projectile Caused by Gravity

As the projectile is under the action of gravity, the height deflection angles of the projectile velocity bend downward, which results in a changed attack angle. Equation (29) describes this change, and the results are shown in Figures 6 and 7, and the initial conditions are , , and .

(a)

(b)

(c)

As shown in Figure 6, at the initial stage of the trajectory, the projectile angle of attack swings around the velocity line. With an increased arc length s, if the projectile is stable, the swing amplitude of the angle of attack gradually decreases. The motion form in the swing stage of the angle of attack is derived from the homogeneous solution W₂ in equation (28), and the motion form in the steady stage of the angle of attack is derived from the nonhomogeneous solution W_G in equation (28). The movement trend for the angle of attack obtained by the numerical solution is the same as that obtained from the analytic solution, and only the homogeneous solution has slight errors with the numeric solution because of the approximate linearized results. Therefore, it is acceptable to use the analytic solution to analyze the angular motion of the projectile. As the homogeneous solution W₂ in equation (28) is only different from W₁ in its coefficients, its stability is the same as that without considering the influence of gravity. The effect of gravity is equivalent to applying the initial angular velocity to the height angle of attack, indicating the angle of attack produces a swing motion in the initial trajectory stage. Its motion mechanism is the same as that in Section 6.1. When the tail rotation speed is greater than the critical rotation speed, i.e., , the angle of attack diverges, which results in projectile instability, as shown in Figure 7(c). Figure 7 shows changes in the projectile height angle of attack and lateral angle of attack under different tail rotation speeds when the projectile is stable. Changes in the amplitude of the height angle of attack are less affected by the tail rotation speed , but changes in the lateral angle of attack amplitude increase with the tail rotation speed , which is due to the influence of the increased lateral Magnus moment. It is noted that when the projectile is stable, the angular motion amplitude caused by gravity is small, which can be ignored when considering the effects of the correction force.

7.3. Resonance Stability of Projectile

When the projectile is in the uncorrected trajectory state, the periodic rotation correction force causes the projectile to exhibit forced vibrations. When the rotation frequency of the correction force is close to the natural frequency of the projectile swing, the projectile angle of attack amplitude response produces a large positive gain. If the projectile satisfies the stability condition in Subsection 6.1, i.e., , the projectile angle of attack amplitude can be calculated from equation (37), and the initial conditions are , , and as shown in Figure 8.

When considering the rotation correction force that acts on the projectile, the projectile angular motion can be regarded as a damped forced vibration. According to equation (37), the amplitude of the angle of attack is determined by the damping factors and the tail aerodynamic force B. As shown in Figure 8, when the tail rotation angle frequency approaches the characteristic frequencies and of the projectile body, the projectile body angle of attack amplitude can exceed 0.4 rad. Although the angle of attack amplitude does not diverge, the aerodynamic force generated by its amplitude is in the nonlinear regime, which adversely impacts the projectile body control.

The first method to avoid excessive angular amplitude A_m is to reduce the tail correction rudder deflection angle so that the tail aerodynamic force B is reduced. However, this reduces the correction capability of the projectile. The second method is to avoid the rotation speed of the tail to operate near the characteristic frequencies and of the projectile body. As the projectile needs to change between zero and nonzero in the corrected trajectory stage, if the rotation speed of the tail is too high in the uncorrected trajectory stage, the rotation speed of the tail must pass through the characteristic frequency of the projectile body when switching to the corrected state. Considering the critical rotation speed of the tail obtained in Section 6.1, the optimal rotation speed range of the tail wing in the uncorrected phase of the projectile trajectory is .

7.4. Angular Motion of Projectile in Corrected Trajectory State

When the projectile is in the corrected trajectory state, the tail rotation speed is . At this time, the Magnus force and moment can be ignored. Thus, the damping factor can be achieved only if the static stability moment coefficient satisfies . At this time, the projectile is in the stable state. The analytic solution of the angular motion of the projectile in the corrected trajectory state is given by equation (41). Therein, the steering gear control system can stabilize the phase angle of the correction rudder at the position. Figure 9 shows the angular motion curves for the projectile body in the process of changing the corrected trajectory state after flying for 3 s under the uncorrected trajectory. Different colors represent the angular motion of the projectile when the correction rudder is stable at different phase angles . The initial conditions for the simulations are , , and . The tail rotation speed in the uncorrected trajectory state is .

Figure 9 shows that in the first three seconds, the projectile axis processes around the velocity direction with a precession angular velocity that is at the tail rotation speed of under the uncorrected trajectory state. As the tail rotation speed is within the stable speed range, the attack angle amplitude for the projectile body continues to decrease. When the control action is applied, the attack angle of the projectile body quickly stabilizes to a certain angle in the complex plane (direction of the midpoint line in Figure 9). At this time, the rotation speed of the projectile body is relatively small, and the influence of the Magnus moment can be ignored. Therefore, the stable attack angle direction for the projectile body is approximately equal to the phase angle of the tail wing in the corrected trajectory state. The deflection angle of the projectile centroid velocity in the complex plane in the trajectory correction state is given by equation (46). The deflection angle superimposed with the ideal trajectory velocity height angle is the velocity height angle and velocity lateral angle of the projectile in the reference coordinate system, as shown in Figure 10.

In Figure 10, the different colors represent the projectile velocity height angle and velocity lateral angle when the correction rudder is stabilized at different phase angles . In the uncorrected trajectory stage, the velocity direction of the projectile centroid swings slightly, and the angular velocity of the swing is the tail rotation speed of . Herein, the projectile centroid average deflection angle deviates slightly to the positive direction of due to the influence of the Magnus moment. The superposition of the ideal trajectory velocity height angle causes the coordinate points of the projectile centroid velocity directions to form a circle on the complex plane (the dotted ellipse shown in Figure 10 is caused by different scale intervals of the horizontal and vertical coordinates). The center of the circle is below the origin, which represents the direction range in which the projectile centroid velocity can be controlled.

8. Conclusions

(1)The trajectory correction principle of a rocket projectile with an isolated rotating tail rudder is proposed, and the differential equation for the angular motion of the projectile body under a complex angle of attack plane is derived based on the rigid body trajectory equation. The analytic solution of the angular motion is approximately equal to the numeric solution with a correlation coefficient of , which verifies the accuracy of the analytic solution for the angular motion.(2)Angular motion of the projectile as caused by the initial conditions, gravity, and asymmetric rotating tail is studied. The critical rotation speed and optimal rotation speed range of the tail are derived, which are key factors to ensure the stability of the projectile in the uncorrected trajectory state.(3)The transient and steady-state solutions of the projectile in the corrected trajectory state are derived. The angular motion of the projectile in the process from uncorrected to corrected trajectories is explained through numerical simulations. In the state of the trajectory correction, an approximately equal relationship between the angle of attack and phase angle of the tail wing is visualized, and the variation range of the projectile centroid direction under controlled action is noted.

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This research was funded by the Domain Fund (No. 2021-JCJQ-JJ-0588).

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Copyright © 2023 Chen Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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