I investigate the working condition of a screw-driving actuator for a mortar projectile equipped with reciprocating canards. The canard pair is mounted with a same-direction offset, and each canard can be extended or retracted. When no correction is required, the canards retract into the projectile body, reducing aerodynamic drag and increasing effective range. When correction is required, the canards extend and generate a control force. The actuator uses screw gears to convert the rotational motion of a motor into the linear motion of a nut, a connecting rod, and a slider. The slider is fixed to the canard. Therefore, the screw gears, the connecting rod, and the slider together define the extension and retraction behavior of the canard.
I establish a time model and a dynamic model for the extension process. I study the influence of the connecting-rod length and the initial position angle on the stretching response time and the motor output torque. I use a 120 mm mortar projectile as the baseline platform. The screw gears in the actuator are the main transmission path, and their geometry, friction, and lead angle determine the torque requirement. I focus on the condition in which the motor rotates at 3000 rpm and the canard is exposed to a maximum aerodynamic force of 30 N. I also examine whether the screw gears can control the free reciprocation of the canard within the engineering limits.
The working principle of the screw gears is as follows. The motor and the screw rotate together. The nut is engaged with the screw through the screw gears, so the rotation of the screw produces axial translation of the nut. The connecting rod links the nut and the slider. The slider is constrained to move radially. As the nut translates, the connecting rod changes its angle relative to the screw axis, and the slider moves outward or inward. The maximum slider displacement equals the canard length. In this way, the screw gears transform a small rotary input into a controlled linear extension of the canard.
For a rolling projectile, one complete roll cycle contains an extension process, a dwell process, and a retraction process. The extension and retraction processes are symmetric and reversible. The canard produces the largest correction effect when its motion time occupies one half of the roll period. For the canard to be fully exposed and to generate the maximum correction capability, the extension time must be shorter than one quarter of the roll period. I define the maximum allowable extension time as
$$ t_{s,\max} = \frac{1}{4 p_p} $$
where \( p_p \) is the projectile roll rate. For a micro-rolling mortar projectile, the roll rate is typically between 3 r/s and 5 r/s. At 5 r/s, the maximum allowable extension time is 50 ms. This value is the engineering limit I use in the following analysis. The screw gears must therefore complete the extension stroke within this limit while also keeping the motor torque below the available value.
I assemble the principal parameters for the baseline case in Table 1. These values are used consistently in the time model, the dynamic model, and the numerical examples. The parameters include the projectile radius, the canard extension length, the roll rate, the motor speed, the screw-gear mean diameter, the lead angle, the friction coefficient, the maximum aerodynamic force, the cant angle, and the moving mass. The screw gears are represented by the mean diameter and the lead angle because these two quantities dominate the linear velocity and the torque transmission ratio.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Projectile radius | \( R \) | 0.06 | m |
| Canard extension length | \( l \) | 0.02 | m |
| Projectile roll rate | \( p_p \) | 3 to 5 | r/s |
| Maximum allowable extension time | \( t_{s,\max} \) | 50 | ms |
| Motor speed | \( p_m \) | 3000 | r/min |
| Screw-gear mean diameter | \( d_s \) | 0.06 | m |
| Screw lead angle | \( \lambda \) | 10 | deg |
| Friction coefficient | \( \mu \) | 0.2 | – |
| Maximum aerodynamic force | \( F_{c,\max} \) | 30 | N |
| Canard cant angle | \( \delta \) | 4 | deg |
| Slider and canard mass | \( m_s \) | 0.0128 | kg |
I define a two-dimensional reference frame \( Oxy \) for the screw-driving actuator. The origin \( O \) lies on the screw axis. The \( x \)-axis is along the screw axis, and the \( y \)-axis is along the radial direction of the canard. The nut position is denoted by \( x_n \), and the slider position is denoted by \( y_s \). The connecting-rod length is \( r \), and the angle between the connecting rod and the screw axis is \( \theta \). The geometric relations are
$$ x_n = r \cos\theta $$
$$ y_s = r \sin\theta $$
The connecting rod must fit within the available space of the projectile, so its length has both a lower bound and an upper bound. When the connecting rod is shortest, its initial position is the initial angular position of the transmission mechanism, and its terminal position is perpendicular to the screw axis. When the connecting rod is longest, its initial position is again the initial angular position, and its terminal position is the position at which the slider reaches the projectile edge. I write the limiting rod lengths as
$$ r_{\min} = \frac{l}{1 – \sin\theta_1} $$
$$ r_{\max} = \frac{R – l}{\sin\theta_1} $$
where \( r_{\min} \) and \( r_{\max} \) are the minimum and maximum connecting-rod lengths, \( R \) is the projectile radius, \( l \) is the canard displacement, and \( \theta_1 \) is the initial position angle of the connecting rod. The initial angle itself is bounded by friction and geometry. The smallest initial angle must exceed the minimum transmission friction angle, and the largest initial angle is set by the requirement that the connecting rod remains within the projectile radius when it reaches the terminal position. These bounds are
$$ \theta_{1,\min} = \arctan\mu $$
$$ \theta_{1,\max} = \arcsin\left(\frac{R – l}{R}\right) $$
With \( \mu = 0.2 \), the minimum initial angle is about 11.3 deg. With \( R = 0.06 \) m and \( l = 0.02 \) m, the maximum initial angle is about 41.8 deg. Therefore, the initial angle range I examine is 15 deg to 40 deg. Within this range, the screw gears and the connecting rod can operate without losing geometric feasibility. The screw gears must also avoid a singular condition in which the connecting rod is nearly perpendicular to the screw axis and the transmission ratio becomes unfavorable.
The extension process begins when the nut is at \( x_{n1} \) and the slider is at \( y_{s1} \). The process ends when the slider reaches \( y_{s2} \). The slider displacement is exactly the canard length \( l \). I write the initial and final positions as
$$ x_{n1} = r \cos\theta_1 $$
$$ y_{s1} = r \sin\theta_1 $$
$$ y_{s2} = y_{s1} + l $$
$$ x_{n2} = \sqrt{r^2 – y_{s2}^2} $$
The nut displacement during the extension stroke is the difference between the initial and final nut positions:
$$ \Delta x = x_{n1} – x_{n2} $$
When the motor rotates at a constant speed, the linear velocity of the nut is determined by the motor speed and the screw-gear parameters. I express this velocity as
$$ v_{nx} = \pi p_m d_s \tan\lambda $$
where \( v_{nx} \) is the nut linear velocity, \( p_m \) is the motor speed, \( d_s \) is the screw-gear mean diameter, and \( \lambda \) is the screw lead angle. The extension response time is then
$$ t_s = \frac{\Delta x}{v_{nx}} $$
This time model shows that the screw gears control the response speed through \( p_m \), \( d_s \), and \( \lambda \). The connecting rod controls the required nut displacement through \( r \) and \( \theta_1 \). I now use these relations to examine the geometric limits and the response time for different initial angles.
In Table 2, I summarize the limiting connecting-rod lengths for initial angles from 20 deg to 40 deg. As the initial angle increases, both \( r_{\min} \) and \( r_{\max} \) increase, but the usable rod-length range becomes narrower. At 20 deg, the rod length can vary over a wide interval. At 40 deg, the interval is very small. This means that a large initial angle restricts the design freedom of the screw gears and the connecting rod. However, a large initial angle also changes the force transmission angle and can reduce the motor torque, as I show later.
| Initial angle \( \theta_1 \) (deg) | \( r_{\min} \) (m) | \( r_{\max} \) (m) | Usable range (m) |
|---|---|---|---|
| 20 | 0.0304 | 0.1170 | 0.0866 |
| 25 | 0.0346 | 0.0947 | 0.0601 |
| 30 | 0.0400 | 0.0800 | 0.0400 |
| 35 | 0.0469 | 0.0697 | 0.0228 |
| 40 | 0.0560 | 0.0622 | 0.0062 |
Table 3 presents the calculated extension response time for the minimum and maximum connecting-rod lengths. The motor speed is 3000 rpm, the screw-gear mean diameter is 0.06 m, and the lead angle is 10 deg. Under these conditions, the nut velocity is sufficiently high that all response times remain below the 50 ms engineering limit. The shortest response time occurs when the rod length is at its maximum and the initial angle is relatively small. The longest response time occurs when the rod length is at its minimum and the initial angle is large. Even in the most conservative case shown here, the screw gears can complete the extension stroke within the allowable time.
| Initial angle \( \theta_1 \) (deg) | Response time at \( r_{\min} \) (ms) | Response time at \( r_{\max} \) (ms) | Difference (ms) |
|---|---|---|---|
| 20 | 17.2 | 5.7 | 11.5 |
| 25 | 18.9 | 7.6 | 11.3 |
| 30 | 20.9 | 9.8 | 11.1 |
| 35 | 23.1 | 13.0 | 10.1 |
| 40 | 25.8 | 18.8 | 7.0 |
The time model therefore leads to a clear trend. For a fixed initial angle, increasing the connecting-rod length reduces the nut displacement required for a given canard extension. The screw gears then need less time to produce the same slider motion. For a fixed rod length, increasing the initial angle increases the nut displacement and therefore lengthens the response time. The screw gears must be selected so that the resulting response time remains below the quarter-roll limit. In practice, I prefer a relatively long connecting rod and an initial angle that is large enough to reduce torque but not so large that the response time or the geometric range becomes unacceptable.
I next build the dynamic model of the reciprocating canard and the screw-driving actuator. The canard has a fixed cant angle, so the aerodynamic force has both a radial component and a tangential component. During the extension process, the effective exposed area increases with the slider displacement. I assume that the flight speed and the aerodynamic coefficients remain unchanged during the short extension stroke. The instantaneous correction force is then proportional to the exposed length of the canard:
$$ F_c = F_{c,\max}\frac{\Delta y_s}{l} $$
where \( F_c \) is the instantaneous correction force, \( F_{c,\max} \) is the correction force when the canard is fully exposed, and \( \Delta y_s \) is the exposed length of the canard outside the projectile body. I resolve this force into two components:
$$ F_{c1} = F_c \cos\delta $$
$$ F_{c2} = F_c \sin\delta $$
where \( \delta \) is the fixed cant angle of the canard, \( F_{c1} \) is the component in the radial plane, and \( F_{c2} \) is the component in the axial plane. These two components produce friction and normal forces on the slider. The slider is also driven by the connecting rod. I treat the connecting rod as a two-force member in compression. Its force on the slider is \( F_{rs} \). The components of this force along the slider motion direction and normal to it are
$$ F_{rs1} = F_{rs}\sin\theta $$
$$ F_{rs2} = F_{rs}\cos\theta $$
The friction forces on the slider are generated by the aerodynamic components and by the normal component of the connecting-rod force. I write them as
$$ F_{fs1} = F_{c1}\mu $$
$$ F_{fs2} = \left(F_{c2} + F_{rs2}\right)\mu $$
During the nut translation, the slider accelerates. I use the principle of virtual work to add an equivalent inertial force in the direction opposite to the slider motion. The force balance along the slider motion direction is then
$$ F_{rs1} = F_{as} + F_{fs1} + F_{fs2} $$
$$ F_{as} = m_s a_s $$
where \( F_{as} \) is the equivalent inertial force, \( m_s \) is the moving mass of the slider and canard, and \( a_s \) is the instantaneous slider acceleration. The slider acceleration is determined by the instantaneous geometry and the nut velocity. Taking the second time derivative of the slider position gives
$$ a_s = -\frac{x_n^2 + y_s^2}{y_s^3} v_{nx}^2 $$
Substituting the force components into the slider equilibrium equation, I obtain the connecting-rod force required to drive the canard:
$$ F_{rs} = \frac{F_{as} + \mu\left(F_{c1} + F_{c2}\right)}{\sin\theta – \mu\cos\theta} $$
This expression shows that the screw gears and the connecting rod must supply a force that increases with the aerodynamic load and the inertial load. The denominator \( \sin\theta – \mu\cos\theta \) is a transmission-efficiency factor. If this factor becomes small, the required rod force becomes large. Therefore, the initial angle and the instantaneous angle must remain sufficiently far from the friction angle. The screw gears must be designed so that this force can be transmitted without excessive torque or jamming.
I now analyze the nut and the screw gears. The nut is engaged with the screw through a helical thread. I simplify the contact as a nut moving on an inclined plane. The inclination angle is the screw lead angle \( \lambda \). The normal force from the thread on the nut is \( F_{nn} \), and the friction force is \( F_{fn} \). The friction force obeys
$$ F_{fn} = F_{nn}\mu $$
The components of the normal force and the friction force along the nut motion direction are
$$ F_{nn1} = F_{nn}\cos\psi $$
$$ F_{fn1} = F_{fn}\sin\psi $$
The component of the connecting-rod force along the nut motion direction is
$$ F_{rs1} = F_{rs}\cos\theta $$
The components perpendicular to the nut motion direction are
$$ F = \frac{T_m}{d_s} $$
$$ F_{nn2} = F_{nn}\sin\psi $$
$$ F_{fn2} = F_{fn}\cos\psi $$
where \( T_m \) is the motor output torque, \( F \) is the effective tangential force produced by the screw gears, and \( \psi \) is the thread lead angle used in the dynamic relation. I use \( \psi = \lambda \) for the screw gears. Eliminating the normal force from the equilibrium equations gives the motor torque required to drive the nut:
$$ T_m = d_s F_{rs}\cos\theta \frac{\sin\psi + \mu\cos\psi}{\cos\psi – \mu\sin\psi} $$
This relation is the core of the dynamic model. It shows that the motor torque depends on the screw-gear mean diameter, the lead angle, the friction coefficient, the connecting-rod angle, and the rod force. The rod force itself depends on the aerodynamic load, the inertial load, and the transmission angle. Therefore, the screw gears, the connecting rod, and the canard form a coupled system. I cannot optimize one element without considering the others.
I use the dynamic model to compute the peak motor torque for different initial angles. In Table 4, I list representative peak torque values for the maximum rod length case. The maximum aerodynamic force is 30 N, the cant angle is 4 deg, the friction coefficient is 0.2, and the moving mass is 0.0128 kg. The results show that the peak torque decreases as the initial angle increases. When the initial angle is greater than about 30 deg, the required motor torque remains below 0.5 N m. The torque fluctuation also decreases, which means that the screw gears operate more smoothly. A larger initial angle improves the force transmission and reduces the peak load on the screw gears.
| Initial angle \( \theta_1 \) (deg) | Peak motor torque (N m) | Mean motor torque (N m) | Torque fluctuation (N m) |
|---|---|---|---|
| 20 | 0.78 | 0.65 | 0.26 |
| 25 | 0.61 | 0.51 | 0.20 |
| 30 | 0.47 | 0.40 | 0.15 |
| 35 | 0.36 | 0.31 | 0.11 |
| 40 | 0.29 | 0.25 | 0.08 |
The influence of the connecting-rod length on the motor torque is equally important. For a fixed initial angle, increasing the rod length reduces the rate at which the connecting-rod angle changes. The slider acceleration and the normal force on the screw gears become smaller. As a result, the peak torque decreases and the torque curve becomes flatter. I summarize this behavior in Table 5. The table compares the qualitative trends for an increase in rod length and an increase in initial angle. Both changes reduce the motor torque, but they have opposite effects on the response time. A longer rod shortens the response time, while a larger initial angle lengthens it. The screw gears must therefore be selected by balancing these two effects.
| Change | Effect on response time | Effect on peak motor torque | Effect on torque smoothness |
|---|---|---|---|
| Increase connecting-rod length | Decreases | Decreases | Improves |
| Increase initial position angle | Increases | Decreases | Improves |
I now combine the time model and the dynamic model. For the baseline case, the screw gears can extend the canard in less than 25 ms when the motor speed is 3000 rpm. This response time satisfies the engineering requirement because the maximum allowable extension time is 50 ms. The screw gears can also control the free reciprocation of the canard when the maximum aerodynamic force is 30 N. The required motor torque is less than 0.5 N m for initial angles above 30 deg. This torque level is practical for a small electromagnetic or brushless motor. The screw gears do not need to generate an excessive force, and the connecting rod does not need to operate near a singular position.
The optimal design region is therefore clear. I should choose a connecting-rod length close to \( r_{\max} \) and an initial angle greater than 30 deg. This choice minimizes the motor torque and makes the torque variation smooth. It also keeps the response time within the allowable limit, although the response time increases with the initial angle. If the response time becomes too long, I can compensate by increasing the motor speed or by adjusting the screw-gear lead angle. The screw gears provide a direct way to trade speed for torque. A larger lead angle increases the linear velocity but also increases the required torque. A smaller lead angle reduces the torque but slows the extension. Therefore, the screw gears must be matched to the available motor and the required response time.
I also examine the friction condition of the screw gears. The denominator in the rod-force equation is \( \sin\theta – \mu\cos\theta \). For the mechanism to transmit force without self-locking, this denominator must remain positive. With \( \mu = 0.2 \), the minimum angle for positive transmission is about 11.3 deg. All initial angles considered here are above this value. However, during the extension stroke, the instantaneous angle \( \theta \) changes. I must ensure that the terminal angle also remains above the friction angle. The terminal angle is determined by the final slider position and the rod length. A longer rod and a larger initial angle generally keep the terminal angle in a favorable range. The screw gears therefore benefit from a geometry that avoids both the initial and terminal friction limits.
The screw-gear lead angle also affects self-locking. If the lead angle is too small, the screw gears may become self-locking and require additional torque to overcome friction. If the lead angle is too large, the screw gears become more efficient but may reduce the mechanical advantage. In the baseline case, the lead angle is 10 deg. The friction coefficient is 0.2. This combination is close to the self-locking boundary. I can adjust the lead angle to improve efficiency, but I must also consider the motor torque and the response time. A moderately larger lead angle can reduce friction losses but increases the linear velocity, which helps the response time. However, it also reduces the torque multiplication, so the motor must provide more torque. The screw gears must therefore be selected as part of a system-level optimization.
I summarize the baseline working condition in Table 6. The table lists the required response time, the available response time, the required motor torque, the available motor torque, and the design recommendation. The results show that the screw-driving actuator is feasible for the reciprocating canard mortar projectile. The screw gears can extend the canard within the quarter-roll limit, and the motor torque remains below the practical limit. The design should use a long connecting rod and an initial angle greater than 30 deg. The screw gears should have a lead angle that avoids self-locking while maintaining a sufficient mechanical advantage.
| Quantity | Required or computed value | Engineering limit | Status |
|---|---|---|---|
| Extension response time | Less than 25 ms | 50 ms | Satisfied |
| Motor torque at 30 N aerodynamic load | Less than 0.5 N m | 0.5 N m | Satisfied for \( \theta_1 > 30^\circ \) |
| Connecting-rod length | Close to \( r_{\max} \) | Between \( r_{\min} \) and \( r_{\max} \) | Recommended |
| Initial position angle | 30 deg to 40 deg | 11.3 deg to 41.8 deg | Recommended |
| Screw-gear lead angle | Around 10 deg | Avoid self-locking | Acceptable with adjustment |
I also study the sensitivity of the response time to the screw-gear parameters. The nut velocity is proportional to the motor speed, the screw-gear mean diameter, and the tangent of the lead angle. If I increase the motor speed from 3000 rpm to 4000 rpm, the response time decreases by about 25 percent. If I increase the screw-gear mean diameter, the response time also decreases, but the required motor torque increases because the torque relation contains \( d_s \). If I increase the lead angle, the response time decreases, but the torque relation becomes less favorable. Therefore, the screw gears cannot be adjusted freely without considering the motor torque. The best approach is to set the motor speed and the screw-gear diameter first, then tune the lead angle to meet the response time while keeping the torque within the motor limit.
The aerodynamic load also affects the working condition. I use 30 N as the maximum force. If the actual force is lower, the motor torque decreases almost linearly. If the force is higher, the motor torque increases. The screw gears must therefore be designed with a torque margin. A margin of 20 to 30 percent is reasonable because the aerodynamic force may vary with flight speed, angle of attack, and canard exposure. The connecting rod and the slider must also have enough stiffness to avoid deflection under the aerodynamic load. If the connecting rod bends, the effective rod length changes, and the transmission angle becomes less favorable. This effect increases the required motor torque and may cause the screw gears to operate less smoothly.
The inertial load is another important factor. The slider and canard accelerate during the extension process. The acceleration term is proportional to the square of the nut velocity. Therefore, a faster motor speed increases the inertial load. If the motor speed is too high, the screw gears may experience a torque spike at the beginning of the extension stroke. I can reduce this spike by using a longer connecting rod or a larger initial angle. A longer rod reduces the rate of change of the connecting-rod angle, and a larger initial angle improves the force transmission. Both changes make the acceleration term less severe and reduce the peak torque on the screw gears.
I note that the extension and retraction processes are symmetric. The same screw gears, connecting rod, and slider can drive the canard in both directions. The retraction process is the reverse of the extension process, so the time and torque characteristics are similar. In a complete roll cycle, the screw gears must first extend the canard, then hold it during the dwell period, and finally retract it. The dwell period does not require motion, but the screw gears may need to hold the canard against the aerodynamic load. If the screw gears are self-locking, the holding torque can be small. If they are not self-locking, the motor must provide a holding torque. This requirement should be considered in the motor selection and the screw-gear design.
The working condition also depends on the projectile roll rate. I use 5 r/s as the maximum roll rate. At this rate, the roll period is 200 ms, and the quarter-roll limit is 50 ms. If the roll rate is lower, the allowable extension time is longer. If the roll rate is higher, the allowable extension time is shorter. For a roll rate of 10 r/s, the quarter-roll limit would be 25 ms. The screw gears can still meet this limit if the motor speed and lead angle are increased, but the motor torque may also increase. Therefore, the screw gears must be designed for the highest expected roll rate. A design that works at 5 r/s may not work at 10 r/s without adjustment. I recommend evaluating the screw gears over the entire roll-rate range.
I now discuss the design implications of the results. The screw gears are the main transmission element, so their efficiency and wear characteristics are important. A high-efficiency screw gear reduces the motor torque and the heat generation. The lead angle, the thread profile, and the surface finish all affect efficiency. A small lead angle may provide a high mechanical advantage but can also cause self-locking and high friction. A large lead angle improves efficiency but reduces the mechanical advantage. The baseline lead angle of 10 deg is a compromise. If the motor has enough torque, a slightly larger lead angle can improve the response time and reduce friction losses. If the motor torque is limited, a smaller lead angle can reduce the torque requirement but may slow the response. The screw gears should be selected with these trade-offs in mind.
The connecting rod should be as long as the geometry allows. A long connecting rod reduces the nut displacement and the response time. It also reduces the rate of change of the rod angle, which lowers the slider acceleration and the peak motor torque. However, a long rod requires more space along the screw axis. The available space inside the projectile is limited. I must therefore choose the rod length so that it fits within the projectile body while still providing the required extension. The maximum rod length \( r_{\max} \) is a useful upper bound. A rod length close to \( r_{\max} \) gives the best dynamic performance. The minimum rod length \( r_{\min} \) is only a lower bound and should be avoided because it leads to a longer response time and a higher torque.
The initial angle should be large enough to reduce the motor torque but not so large that the response time or the geometric range becomes unacceptable. The results show that an initial angle above 30 deg keeps the motor torque below 0.5 N m. The usable rod-length range at 35 deg and 40 deg is narrower, so the design tolerance becomes tighter. If the manufacturing tolerance is loose, a smaller initial angle such as 30 deg may be more practical. At 30 deg, the peak torque is about 0.47 N m, which is still within the 0.5 N m limit. The response time is about 9.8 ms for the maximum rod length, which is well below the 50 ms limit. Therefore, 30 deg is a robust choice for the initial angle.
I also consider the effect of friction. The friction coefficient is assumed to be 0.2. If the friction coefficient increases, the required motor torque increases, and the transmission efficiency decreases. The minimum initial angle also increases because \( \theta_{1,\min} = \arctan\mu \). For example, if \( \mu = 0.3 \), the minimum initial angle becomes about 16.7 deg. This reduces the available design range. The screw gears should therefore be lubricated or coated to keep the friction coefficient low. A lower friction coefficient reduces the motor torque and improves the smoothness of the extension. It also reduces wear and extends the service life of the screw gears.
The aerodynamic force is another source of uncertainty. I use 30 N as the maximum force, but the actual force may be lower or higher depending on the flight condition. If the force is higher, the motor torque increases, and the screw gears may need a larger motor. If the force is lower, the motor torque decreases, and the design margin increases. I recommend using a conservative aerodynamic force in the design. The screw gears should be sized for the maximum expected force plus a margin. The connecting rod and the slider should also be sized for the maximum force to avoid bending or jamming. The screw gears should be inspected for wear and fatigue because they operate repeatedly during the flight.
The time response is also affected by the motor dynamics. I assume that the motor reaches constant speed immediately. In reality, the motor has a startup transient. The screw gears may experience a torque spike during startup. If the startup time is a significant fraction of the extension time, the response time may be longer than predicted. I recommend using a motor with a fast startup characteristic or adding a soft-start control. The screw gears should also be designed to withstand the startup torque. The control system can adjust the motor speed profile to reduce the peak torque and improve the response time. A trapezoidal speed profile may be better than a step input because it reduces the inertial load and the torque spike.
The control strategy for the reciprocating canard is also important. The screw gears must extend the canard at the correct phase of the roll cycle. If the extension is late, the canard may not produce the desired correction. If the extension is early, the canard may produce an unwanted force. The control system must synchronize the motor command with the projectile roll angle. The screw gears should have a repeatable response time so that the control system can predict the canard position. A repeatable response time also makes the correction more accurate. The screw gears should therefore be manufactured with tight tolerances and tested over the expected temperature range.
I summarize the main findings of this study. First, the screw-driving actuator can extend the reciprocating canard within the quarter-roll limit. At a motor speed of 3000 rpm, the response time is less than 25 ms, which is well below the 50 ms engineering limit. Second, the actuator can control the free reciprocation of the canard when the maximum aerodynamic force is 30 N. The required motor torque is less than 0.5 N m when the initial angle is above 30 deg. Third, increasing the connecting-rod length reduces both the response time and the motor torque. Fourth, increasing the initial angle reduces the motor torque but increases the response time. Fifth, the screw gears should be selected with a lead angle that avoids self-locking and maintains a sufficient mechanical advantage. Sixth, the design should use a long connecting rod and an initial angle around 30 deg to 40 deg. Seventh, the screw gears should be designed with a torque margin to account for aerodynamic and friction uncertainties. Eighth, the control system should synchronize the motor command with the projectile roll angle to ensure correct canard extension.
These findings provide a practical basis for the working-condition analysis of screw-driving actuators in reciprocating-canard mortar projectiles. The screw gears are not merely a transmission component; they determine the response time, the motor torque, and the smoothness of the canard motion. By modeling the time response and the dynamic forces, I can identify the design region that satisfies the engineering requirements. The results show that the screw-driving actuator is feasible and that the screw gears can be optimized by choosing the connecting-rod length and the initial angle appropriately. A long connecting rod and a moderately large initial angle give the best combination of fast response, low torque, and smooth operation. The screw gears should be designed with low friction, a suitable lead angle, and sufficient strength to withstand the aerodynamic and inertial loads.
In future work, I would extend the model to include the full six-degree-of-freedom motion of the projectile and the transient dynamics of the motor. I would also study the effect of temperature on the friction coefficient and the screw-gear efficiency. I would test the actuator under repeated extension and retraction cycles to evaluate wear and reliability. I would optimize the screw-gear profile and the connecting-rod geometry using multi-objective optimization. I would also investigate the interaction between the canard extension and the projectile roll motion. These extensions would provide a more complete understanding of the working condition and would support the design of more efficient screw-driving actuators for reciprocating canards.
In conclusion, I have established a time model and a dynamic model for the extension process of a screw-driving actuator with reciprocating canards. I have examined the influence of the connecting-rod length and the initial position angle on the stretching response time and the motor output torque. The results show that the screw gears can extend the canard in less than 25 ms at 3000 rpm, that the motor torque is less than 0.5 N m when the initial angle is above 30 deg, and that a longer connecting rod and a larger initial angle reduce the motor torque and improve the smoothness of the screw gears. These conclusions provide a useful guide for the design and working-condition assessment of screw-driving actuators in mortar projectiles with reciprocating canards.