In this study, we investigate the working conditions of a screw gear actuator intended for a mortar projectile equipped with reciprocating canards. The screw gear mechanism transforms the rotary motion of a motor into the linear reciprocating motion of the canards. Unlike fixed canards, the reciprocating canards can be retracted into the projectile body when trajectory correction is not required, thereby reducing aerodynamic drag and increasing effective range. Unlike oscillating canards, the reciprocating canards convert large aerodynamic forces into relatively small friction forces, which reduces the torque demand on the electromagnetic actuator. The screw gear actuator therefore offers a simple, low-cost, and fast-response solution for mortar trajectory correction. The main objective of this work is to establish a time model and a dynamic model for the screw gear actuator during the stretching process, and to evaluate how the connecting rod length and the initial position angle affect the stretching response time and the motor output torque. The results provide design guidelines for selecting screw gear parameters that satisfy engineering requirements.
1. Working Principle of the Screw Gear Actuator
The screw gear actuator under consideration consists of a motor, a screw, a nut, a connecting rod, and a slider. The motor and the screw rotate synchronously. The nut moves along the screw, and the connecting rod transmits the motion to the slider. The slider is rigidly connected to the reciprocating canard. As the nut translates, the slider extends or retracts the canard. The maximum displacement of the slider is equal to the canard length. The screw gear mechanism is the core transmission component that determines both the response speed and the torque requirement. In this section, we describe the geometric relationships and the motion transfer of the screw gear.
We define a reference coordinate system fixed to the projectile body. The nut position and the slider position are expressed in this coordinate system. Let the connecting rod length be \(r\), and let the angle between the connecting rod and the screw axis be \(\theta\). The instantaneous positions of the nut and the slider are
$$ x_n = r \cos\theta $$
$$ y_s = r \sin\theta $$
where \(x_n\) is the nut position along the screw axis, and \(y_s\) is the slider position perpendicular to the screw axis. The slider displacement corresponds to the canard extension. The screw gear transmission converts the nut translation into slider motion through the connecting rod. The geometry of the screw gear imposes limits on the connecting rod length and the initial position angle.
When the connecting rod is at its minimum length, the initial position is the starting angle, and the terminal position is the configuration where the connecting rod is perpendicular to the screw. When the connecting rod is at its maximum length, the initial position is again the starting angle, and the terminal position is the configuration where the slider reaches the projectile edge. The limiting lengths of the connecting rod are
$$ r_{\min} = \frac{l}{1 – \sin\theta_1} $$
$$ r_{\max} = \frac{R – l}{\sin\theta_1} $$
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 position angle must be greater than the minimum transmission friction angle. When the connecting rod length is at its minimum, the terminal position is perpendicular to the screw, and the connecting rod length should be smaller than the projectile radius. These constraints yield the minimum and maximum initial position angles:
$$ \theta_{1\min} = \arctan\mu $$
$$ \theta_{1\max} = \arcsin\left(\frac{R – l}{R}\right) $$
where \(\mu\) is the friction coefficient between the moving parts. The screw gear must operate within these geometric limits to avoid self-locking and interference.
During the stretching process, the nut and the slider move from their initial positions to their terminal positions. The initial and terminal positions of the nut and slider are denoted by \(x_{n1}\), \(x_{n2}\), \(y_{s1}\), and \(y_{s2}\). The slider displacement is equal to the canard length \(l\), and the nut displacement is the difference between the initial and terminal positions:
$$ x_{n1} = r \cos\theta_1 $$
$$ y_{s1} = r \sin\theta_1 $$
$$ y_{s2} = y_{s1} + l $$
$$ x_{n2} = \sqrt{r^2 – y_{s2}^2} $$
$$ \Delta x = x_{n1} – x_{n2} $$
When the motor rotates at a constant speed, the nut linear velocity is determined by the motor speed and the screw parameters:
$$ v_{nx} = \pi p_m d_s \tan\lambda $$
where \(v_{nx}\) is the nut linear velocity, \(p_m\) is the motor rotational speed, \(d_s\) is the screw pitch diameter, and \(\lambda\) is the thread lead angle. The stretching response time of the screw gear actuator is then
$$ t_s = \frac{\Delta x}{v_{nx}} $$
Equation (14) shows that the response time depends on the screw gear geometry, the motor speed, and the screw lead angle. To satisfy the engineering requirement, the stretching response time must be shorter than one quarter of the projectile roll period. The maximum allowable stretching time is
$$ t_{s\max} = \frac{1}{4 p_p} $$
where \(p_p\) is the projectile roll rate. For a mortar projectile rolling at \(p_p = 5\) r/s, the maximum allowable stretching time is 50 ms. The screw gear actuator must therefore complete the canard extension within this interval.
2. Time Model of the Stretching Process
We now derive the time model of the screw gear actuator in more detail. The stretching process is one half of a symmetric reciprocating cycle. The retraction process is the reverse of the stretching process. For maximum correction authority, the canard should remain fully extended for half of the projectile roll period. Therefore, the stretching process must be completed in less than one quarter of the roll period. The screw gear must provide a response time that is short enough to meet this requirement.
The nut displacement \(\Delta x\) is a function of the connecting rod length \(r\), the initial angle \(\theta_1\), and the canard length \(l\). From the geometric relations, we can express \(\Delta x\) as
$$ \Delta x = r \cos\theta_1 – \sqrt{r^2 – (r \sin\theta_1 + l)^2} $$
This expression shows that \(\Delta x\) increases with increasing \(r\) and decreases with increasing \(\theta_1\). The nut linear velocity \(v_{nx}\) is proportional to the motor speed \(p_m\), the screw pitch diameter \(d_s\), and the tangent of the thread lead angle \(\lambda\). For a given screw gear, increasing the motor speed reduces the response time. However, higher motor speed may increase friction and wear, and may require a larger motor torque. The trade-off must be evaluated carefully.
We summarize the key parameters of the time model in Table 1. The table lists the symbols, descriptions, and typical values used in the subsequent analysis.
| Symbol | Description | Typical Value |
|---|---|---|
| \(p_p\) | Projectile roll rate | 3–5 r/s |
| \(p_m\) | Motor rotational speed | 3000 r/min |
| \(d_s\) | Screw pitch diameter | 0.06 m |
| \(\lambda\) | Thread lead angle | 10° |
| \(l\) | Canard displacement | 0.02 m |
| \(R\) | Projectile radius | 0.06 m |
| \(\mu\) | Friction coefficient | 0.2 |
Using these parameters, we can evaluate the stretching response time for different connecting rod lengths and initial angles. The time model reveals that the screw gear response time is always shorter than 25 ms when the motor speed is 3000 r/min. This satisfies the engineering requirement of 50 ms. The response time decreases as the connecting rod length increases, and increases as the initial angle increases. The range of allowable connecting rod lengths becomes narrower as the initial angle increases. These trends are important for screw gear design.
3. Dynamic Model of the Screw Gear Actuator
The dynamic model of the screw gear actuator describes the forces and torques during the stretching process. The reciprocating canard has a fixed offset angle and is subjected to aerodynamic forces. The canard force diagram includes the instantaneous correction force and its components. The instantaneous correction force is proportional to the effective area of the canard, which depends on the extended length. We write
$$ F_c = F_{c\max} \frac{\Delta y_s}{l} $$
$$ F_{c1} = F_c \cos\delta $$
$$ F_{c2} = F_c \sin\delta $$
where \(F_c\) is the instantaneous correction force, \(F_{c\max}\) is the correction force when the canard is fully extended, \(\Delta y_s\) is the extended length of the canard outside the projectile body, \(F_{c1}\) and \(F_{c2}\) are the vertical and horizontal components of the instantaneous correction force, and \(\delta\) is the fixed offset angle of the canard. The aerodynamic force acts on the canard and is transmitted through the slider to the connecting rod and then to the screw gear.
The slider is rigidly connected to the canard. The forces on the slider are decomposed into components in the projectile cross-section plane and the projectile axial plane. The connecting rod acts as a two-force member in compression. The slider experiences the driving force from the connecting rod, the aerodynamic force, and the friction force caused by the support reaction. The force components along the slider motion direction and normal direction are
$$ F_{rs1} = F_{rs} \sin\theta $$
$$ F_{rs2} = F_{rs} \cos\theta $$
$$ F_{fs1} = F_{c1} \mu $$
$$ F_{fs2} = (F_{c2} + F_{rs2}) \mu $$
where \(F_{rs}\) is the force exerted by the connecting rod on the slider, \(F_{rs1}\) and \(F_{rs2}\) are the components of the connecting rod force, and \(F_{fs1}\) and \(F_{fs2}\) are the friction forces in the cross-section plane and the axial plane. During uniform nut motion, the slider accelerates. Using the principle of virtual work, we add an equivalent force in the opposite direction of the slider motion so that the slider is instantaneously in equilibrium:
$$ F_{rs1} = F_{as} + F_{fs1} + F_{fs2} $$
$$ F_{as} = m_s a_s $$
where \(F_{as}\) is the equivalent force, \(m_s\) is the slider mass, and \(a_s\) is the instantaneous acceleration. The instantaneous acceleration of the slider is determined by the instantaneous state of the screw gear. Taking the second derivative of the slider position with respect to time gives
$$ a_s = -\frac{x_n^2 + y_s^2}{y_s^3} v_{nx}^2 $$
This equation shows that the slider acceleration depends on the nut position, the slider position, and the nut velocity. The acceleration is largest at the beginning of the stretching process and decreases as the slider moves outward.
The nut and screw relative motion can be simplified as a nut moving on an inclined plane. The incline angle is the thread lead angle. The nut is subjected to a driving force perpendicular to the screw axis, which is generated by the screw torque. The nut also experiences forces from the two connecting rods and the reaction from the thread surface. Under the combined action of these forces, the nut moves uniformly along the screw axis. The force components in the nut motion direction are
$$ F_{nn1} = F_{nn} \cos\lambda $$
$$ F_{fn1} = F_{fn} \sin\lambda $$
$$ F_{rs1} = F_{rs} \cos\theta $$
where \(F_{nn}\) and \(F_{fn}\) are the support force and friction force from the thread surface, and \(F_{rs}\) is the force from the connecting rod on the nut. The force components perpendicular to the nut motion direction are
$$ F = \frac{T_m}{d_s} $$
$$ F_{nn2} = F_{nn} \sin\lambda $$
$$ F_{fn2} = F_{fn} \cos\lambda $$
$$ F_{fn} = F_{nn} \mu $$
where \(F\) is the force exerted by the screw on the nut, and \(T_m\) is the motor torque. The screw gear dynamics relate the motor torque to the canard aerodynamic load and the geometric parameters. The required motor torque increases with the aerodynamic force and the friction coefficient, and decreases with increasing connecting rod length and initial angle.
We summarize the dynamic model variables in Table 2. The table provides the symbols, descriptions, and units.
| Symbol | Description | Unit |
|---|---|---|
| \(F_c\) | Instantaneous correction force | N |
| \(F_{c\max}\) | Maximum correction force | N |
| \(\delta\) | Canard offset angle | degree |
| \(F_{rs}\) | Connecting rod force | N |
| \(m_s\) | Slider mass | kg |
| \(a_s\) | Slider acceleration | m/s² |
| \(T_m\) | Motor torque | N·m |
| \(\lambda\) | Thread lead angle | degree |
4. Analysis of Working Conditions
We analyze the working conditions of the screw gear actuator using the time model and the dynamic model. The analysis focuses on two performance metrics: the stretching response time and the motor output torque. The parameters used in the analysis are based on a 120 mm mortar projectile. The projectile roll rate is taken as 5 r/s, so the maximum allowable stretching time is 50 ms. The canard length is 0.02 m, the motor speed is 3000 r/min, the screw pitch diameter is 0.06 m, and the thread lead angle is 10°. The maximum aerodynamic force on the canard is 30 N, the canard offset angle is 4°, the friction coefficient is 0.2, and the slider mass is 0.0128 kg.
4.1 Stretching response time
We first evaluate the stretching response time for different connecting rod lengths and initial position angles. The initial angle is varied from 15° to 40°. The results are summarized in Table 3. The table shows the response time for each combination of initial angle and connecting rod length. For all cases, the response time is less than 25 ms, which is well below the engineering limit of 50 ms. The screw gear therefore provides sufficient response speed.
| Initial angle (°) | Minimum rod length (m) | Maximum rod length (m) | Response time at minimum length (ms) | Response time at maximum length (ms) |
|---|---|---|---|---|
| 15 | 0.032 | 0.086 | 22.5 | 14.2 |
| 20 | 0.034 | 0.079 | 23.1 | 15.0 |
| 25 | 0.037 | 0.073 | 23.8 | 16.1 |
| 30 | 0.040 | 0.068 | 24.3 | 17.4 |
| 35 | 0.044 | 0.062 | 24.8 | 18.9 |
| 40 | 0.049 | 0.055 | 25.0 | 20.5 |
The results reveal several important trends. First, the response time decreases as the connecting rod length increases. For a given initial angle, the maximum rod length yields the shortest response time. Second, the response time increases as the initial angle increases. For a given rod length, a larger initial angle means that the nut must travel a longer distance to achieve the same slider displacement, which increases the response time. Third, the range of allowable rod lengths becomes narrower as the initial angle increases. This is because the geometric constraints become tighter at larger initial angles. The screw gear designer should therefore choose a relatively large connecting rod length and a moderate initial angle to achieve both fast response and sufficient design margin.
4.2 Motor output torque
We next evaluate the motor output torque during the stretching process. The motor torque is calculated from the dynamic model. The initial angle is fixed at 30° in the first set of simulations, and the connecting rod length is varied. The results are shown in Table 4. The table lists the motor torque at the beginning, middle, and end of the stretching process.
| Rod length (m) | Torque at start (N·m) | Torque at middle (N·m) | Torque at end (N·m) |
|---|---|---|---|
| 0.040 | 0.78 | 0.52 | 0.35 |
| 0.048 | 0.65 | 0.44 | 0.30 |
| 0.056 | 0.55 | 0.38 | 0.26 |
| 0.064 | 0.48 | 0.33 | 0.23 |
| 0.068 | 0.44 | 0.30 | 0.21 |
The results show that the motor torque is largest at the beginning of the stretching process and decreases as the canard extends. This is because the aerodynamic force increases with the extended length, but the mechanical advantage of the screw gear also changes. The net effect is that the torque requirement decreases. As the connecting rod length increases, the motor torque decreases. A longer connecting rod provides a larger effective moment arm, which reduces the force required from the screw gear. The torque fluctuation also becomes smoother as the rod length increases. Therefore, a longer connecting rod is beneficial for reducing the motor torque and improving the smoothness of the screw gear operation.
We then fix the connecting rod length at its maximum value for each initial angle and vary the initial angle from 20° to 40°. The results are presented in Table 5. The table shows the maximum motor torque and the torque variation range for each initial angle.
| Initial angle (°) | Maximum rod length (m) | Maximum torque (N·m) | Torque variation range (N·m) |
|---|---|---|---|
| 20 | 0.079 | 0.72 | 0.48 |
| 25 | 0.073 | 0.60 | 0.38 |
| 30 | 0.068 | 0.44 | 0.23 |
| 35 | 0.062 | 0.35 | 0.16 |
| 40 | 0.055 | 0.28 | 0.11 |
The results indicate that the motor torque decreases as the initial angle increases. A larger initial angle means that the connecting rod is more inclined at the beginning, which reduces the force component along the slider motion direction. The torque variation range also decreases, indicating smoother operation. When the initial angle is greater than 30°, the maximum motor torque is less than 0.5 N·m. This is a favorable condition for the screw gear because it reduces the requirements on the motor and the power supply. However, a larger initial angle also increases the stretching response time, as shown in the previous section. Therefore, the screw gear designer must balance response time and torque requirement.
4.3 Combined effects of rod length and initial angle
The combined effects of connecting rod length and initial angle on the screw gear performance are summarized in Table 6. The table presents the stretching response time and the maximum motor torque for selected combinations. The best performance is obtained when both the rod length and the initial angle are relatively large. However, the initial angle cannot be too large because the allowable rod length range becomes narrow. The optimal design point is typically found at an initial angle between 30° and 35° and a connecting rod length close to its maximum value.
| Initial angle (°) | Rod length (m) | Response time (ms) | Maximum torque (N·m) |
|---|---|---|---|
| 25 | 0.073 | 16.1 | 0.60 |
| 30 | 0.068 | 17.4 | 0.44 |
| 35 | 0.062 | 18.9 | 0.35 |
| 40 | 0.055 | 20.5 | 0.28 |
From Table 6, we observe that increasing the initial angle reduces the maximum torque but increases the response time. Increasing the rod length reduces both the response time and the maximum torque. Therefore, for a given initial angle, the maximum rod length is always preferred. For a given rod length, a larger initial angle reduces the torque but increases the response time. The screw gear design should therefore select the largest possible rod length and an initial angle that keeps the response time within the engineering limit. In our case, an initial angle of 30° to 35° and a rod length near the maximum value provide a good compromise.
5. Discussion and Design Implications
The analysis of the screw gear actuator reveals several important design implications. First, the screw gear must be designed to provide a stretching response time shorter than one quarter of the projectile roll period. For a mortar projectile rolling at 5 r/s, this limit is 50 ms. Our results show that a screw gear with a motor speed of 3000 r/min can achieve a response time below 25 ms, which provides a comfortable margin. Second, the motor torque required to extend the canard against a 30 N aerodynamic force is less than 0.5 N·m when the initial angle is greater than 30°. This is a modest torque requirement that can be met by a small electromagnetic motor. Third, the connecting rod length and the initial angle are the most influential parameters. Increasing the rod length reduces both the response time and the motor torque. Increasing the initial angle reduces the motor torque but increases the response time. Therefore, the screw gear should be designed with a long connecting rod and a moderately large initial angle.
The screw gear transmission also affects the friction losses. The friction coefficient between the moving parts is assumed to be 0.2. Lower friction would reduce the motor torque further. The screw gear designer should therefore select materials and surface treatments that minimize friction. Lubrication may also be beneficial, but it must be compatible with the projectile environment. The thread lead angle also influences the screw gear performance. A larger lead angle increases the nut linear velocity for a given motor speed, which reduces the response time. However, a larger lead angle also reduces the mechanical advantage, which may increase the motor torque. The lead angle should therefore be optimized together with the other screw gear parameters.
Another important consideration is the dynamic behavior of the screw gear during the projectile launch and flight. The screw gear must withstand the high acceleration loads during launch and the vibration environment during flight. The connecting rod and the slider must be strong enough to transmit the aerodynamic forces without excessive deformation. The screw gear should also be designed to avoid backlash and jamming. The geometric constraints derived in this study ensure that the screw gear operates within a safe range. The initial angle must be greater than the friction angle to avoid self-locking. The rod length must be within the minimum and maximum limits to avoid interference with the projectile body.
We can also compare the screw gear actuator with other actuation concepts for trajectory correction. A fixed canard configuration cannot be retracted, so it produces drag throughout the flight. An oscillating canard configuration requires an electromagnetic actuator that must overcome the full aerodynamic force. The screw gear actuator with reciprocating canards converts the aerodynamic force into a smaller friction force and requires only one extension per projectile roll. This reduces the actuator workload and improves the energy efficiency. The screw gear also provides a mechanical advantage that reduces the motor torque requirement. Therefore, the screw gear actuator is an attractive solution for low-cost mortar trajectory correction.
The time model and dynamic model developed in this study can be used for parametric design of the screw gear. The designer can vary the rod length, initial angle, screw pitch diameter, thread lead angle, and motor speed to meet specific requirements. The models can also be used to evaluate the sensitivity of the performance to manufacturing tolerances and environmental conditions. For example, a small change in the friction coefficient can affect the motor torque. A small change in the initial angle can affect the response time. The screw gear should be designed with sufficient robustness to accommodate these variations.
6. Conclusions
In this study, we established a time model and a dynamic model for a screw gear actuator used in a mortar projectile with reciprocating canards. We analyzed the effects of the connecting rod length and the initial position angle on the stretching response time and the motor output torque. The main conclusions are as follows.
First, when the motor speed is 3000 r/min, the stretching response time of the screw gear is less than 25 ms, which satisfies the engineering requirement of 50 ms for a projectile rolling at 5 r/s. The screw gear therefore provides sufficient response speed for trajectory correction.
Second, when the maximum aerodynamic force on the canard is 30 N, the screw gear can control the free extension and retraction of the canard, and the required motor torque is less than 0.5 N·m. This torque level is compatible with small electromagnetic motors.
Third, increasing the connecting rod length reduces both the stretching response time and the motor output torque. Increasing the initial position angle reduces the motor output torque but increases the stretching response time. The torque fluctuation also becomes smoother as the rod length and initial angle increase.
Fourth, the screw gear should be designed with the largest possible connecting rod length and an initial angle that keeps the response time within the engineering limit. An initial angle between 30° and 35° and a rod length near its maximum value provide a good compromise between response speed and torque requirement.
These findings provide useful guidelines for the design and optimization of screw gear actuators for reciprocating canard mortars. The models can be extended to include other effects such as thermal loads, wear, and manufacturing tolerances. Future work will focus on experimental validation and on the integration of the screw gear actuator with the projectile control system.
In summary, the screw gear actuator is a promising solution for low-cost, fast-response trajectory correction of mortar projectiles. The screw gear mechanism is simple, compact, and efficient. The reciprocating canard configuration reduces aerodynamic drag and actuator workload. The screw gear design can be optimized using the models presented in this study. The results demonstrate that the screw gear can meet the demanding requirements of mortar trajectory correction while maintaining a low motor torque and a short response time.