1. Introduction
1.1 Significance of Transmission Systems
In the realm of mechanical engineering, transmission systems are the linchpins that enable the efficient operation of various 机械设备 (machinery and equipment). They are responsible for transferring power and motion between different components, playing a crucial role in determining the overall performance, efficiency, and lifespan of the equipment. Among the components of a transmission system, the 变速器 (transmission) stands out as a key device. It has the ability to alter the speed and torque according to the specific requirements of the working conditions, making it essential for the proper functioning of countless mechanical applications.
1.2 Challenges in Traditional Transmission Control
The new bevel gear – roller flat disk transmission is a revolutionary device known for its high – efficiency and stability. However, its performance is significantly affected by the complex and variable working environment. Traditional control methods, which rely on fixed – parameter models, often struggle to maintain the stability and accuracy of constant – speed output in such conditions. For example, changes in load, friction, and other factors can lead to fluctuations in the output speed, reducing the reliability of the transmission system.
1.3 Motivation for Fuzzy PID Control
Fuzzy PID control emerges as a promising solution to address the challenges faced by traditional control methods. By integrating the principles of fuzzy logic with PID control, it can adapt to system uncertainties and nonlinearities more effectively. This control strategy can dynamically adjust the PID parameters based on real – time system states, ensuring better control performance even in the face of complex and changing working conditions.
2. Working Principle of the New Bevel Gear – Roller Flat Disk Transmission
2.1 Structure Overview
The new bevel gear – roller flat disk transmission combines the unique features of bevel gears and roller flat disk mechanisms. It consists of two main parts: the bevel gear part and the roller flat disk part. The bevel gear part is mainly responsible for changing the transmission ratio, while the roller flat disk part focuses on the smooth transmission of torque. This combination allows for efficient power transfer and speed adjustment.
| Transmission Part | Function |
|---|---|
| Bevel Gear Part | Changes transmission ratio |
| Roller Flat Disk Part | Transmits torque smoothly |
2.2 Power Transmission Process
During operation, the input shaft rotates, driving the bevel gears to mesh. The meshing of bevel gears alters the rotational speed and direction of the power flow. Subsequently, the power is transferred to the roller flat disk part. The rollers on the flat disk roll, and through the friction between the rollers and the flat disk, the torque is transmitted to the output shaft, achieving the final speed and torque output.
3. Fuzzy PID Control Theory
3.1 Basics of PID Control
PID control is a widely used control algorithm in the field of automatic control. It consists of three components: the proportional (P) component, the integral (I) component, and the derivative (D) component. The proportional component responds to the current error between the setpoint and the actual value. A larger proportional gain can speed up the system response, but it may also lead to overshoot and oscillation if it is too large. The integral component accumulates the error over time and is used to eliminate the steady – state error. However, an excessive integral gain can slow down the system response and even cause instability. The derivative component predicts the future trend of the error based on its rate of change, helping to improve the dynamic performance of the system and reduce overshoot. But an overly large derivative gain can increase the system’s sensitivity to noise.
| PID Component | Function | Impact of Large Gain |
|---|---|---|
| Proportional (P) | Responds to current error | May cause overshoot and oscillation |
| Integral (I) | Eliminates steady – state error | May slow down system response or cause instability |
| Derivative (D) | Improves dynamic performance | Increases sensitivity to noise |
3.2 Introduction to Fuzzy Logic
Fuzzy logic is a form of multi – valued logic that allows for more flexible and approximate reasoning compared to traditional binary logic. It deals with concepts that are not precisely defined, such as “big”, “small”, “fast”, etc. In fuzzy logic, variables are represented by fuzzy sets, and the membership of an element in a fuzzy set is defined by a membership function. This enables the handling of imprecise and uncertain information in a more natural way.
3.3 Integration of Fuzzy Logic and PID Control
Fuzzy PID control combines the advantages of fuzzy logic and PID control. It uses fuzzy logic to adjust the PID parameters in real – time according to the system’s operating conditions. By fuzzifying the input variables (such as the error and the rate of change of error), applying fuzzy rules based on expert knowledge or system characteristics, and then defuzzifying the output to obtain the adjusted PID parameters, the control system can better adapt to the nonlinear and uncertain characteristics of the transmission system.
4. Modeling of the New Bevel Gear – Roller Flat Disk Transmission
4.1 Consideration of Load and Friction
To accurately model the new bevel gear – roller flat disk transmission, it is necessary to take into account the impact of load changes on the transmission ratio and the role of friction. The load coefficient \(k_{L}\) is introduced to represent the influence of load on the transmission ratio. This coefficient is a non – linear function of the output torque, reflecting the complex relationship between load and transmission performance. Additionally, the friction coefficient \(\mu\) between the roller and the flat disk is considered. Friction plays a vital role in torque transmission and affects the overall efficiency and stability of the transmission system.
| Parameter | Symbol | Significance |
|---|---|---|
| Load coefficient | \(k_{L}\) | Represents load’s impact on transmission ratio |
| Friction coefficient | \(\mu\) | Affects torque transmission and system stability |
4.2 Dynamic Equation Establishment
Based on the physical principles of the transmission components, the dynamic equations of the bevel gear part and the roller flat disk part are derived and integrated. For the bevel gear part, the relationship between the input and output shaft speeds and the transmission ratio considering the load coefficient is established as \(\omega_{out }=\frac{\omega_{in }}{i}\) and \(i=\frac{N_{1}}{N_{2}} \cdot k_{L}(T_{out })\), where \(N_{1}\) and \(N_{2}\) are the numbers of teeth of the two bevel gears. For the roller flat disk part, the rolling resistance moment \(M_{r}\) is expressed as a function of the friction force, roller speed, acceleration, and other factors, \(M_{r}=f_{r}(v, \alpha, \mu, \rho, \cdots) \cdot F_{n} \cdot R\). Finally, the overall dynamic equation of the transmission is \(T_{in } \cdot \frac{1}{i}-T_{out }=J_{out } \cdot \frac{d \omega_{out }}{d t}+M_{r}\), where \(J_{out }\) is the moment of inertia of the output shaft, and \(T_{in }\) and \(T_{out }\) are the input and output shaft torques respectively.
5. Design of the Fuzzy PID Controller for the Transmission
5.1 Selection of Input Variables
The input variables of the fuzzy PID controller for the transmission are the deviation e of the output shaft speed and its rate of change ec. The deviation e is calculated as \(e = e_{1}-e_{2}\), where \(e_{1}\) is the measured value of the output shaft speed and \(e_{2}\) is the setpoint. The rate of change of error \(ec=\frac{e_{k}-e_{k – 1}}{\Delta t}\), which reflects how quickly the deviation is changing over time. These two variables provide important information about the system’s state and are crucial for the fuzzy PID controller to make appropriate adjustments.
5.2 Definition of Fuzzy Sets and Membership Functions
Fuzzy sets are defined for the error e and the error rate of change ec. Common fuzzy sets include {Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Big}, denoted as {NB, NM, NS, Z, PS, PM, PB}. Membership functions are then assigned to these fuzzy sets to determine the degree of membership of a particular value in each set. For example, a triangular membership function can be used, where the shape and position of the triangle define the range and degree of membership for each fuzzy set.
| Fuzzy Set | Symbol |
|---|---|
| Negative Big | NB |
| Negative Medium | NM |
| Negative Small | NS |
| Zero | Z |
| Positive Small | PS |
| Positive Medium | PM |
| Positive Big | PB |
5.3 Construction of the Fuzzy Rule Base
The fuzzy rule base is constructed based on expert experience and the dynamic characteristics of the transmission system. The rules are in the form of “If e is X and ec is Y, then \(K_{p}\) is \(Z_{1}\), \(K_{i}\) is \(z_{2}\), \(K_{d}\) is \(Z_{3}\)”, where X, Y, \(Z_{1}\), \(z_{2}\), and \(Z_{3}\) are elements of the fuzzy sets. For example, if the error e is NB (Negative Big) and the error rate of change ec is also NB, a large increase in the proportional coefficient \(K_{p}\) may be required to quickly reduce the error. These rules serve as the decision – making basis for the fuzzy PID controller.
5.4 Fuzzy Inference and Defuzzification
Fuzzy inference is carried out using a fuzzy inference machine. It takes the fuzzified input variables (error and error rate of change) and applies the fuzzy rules from the rule base to determine the adjustments of the PID parameters (\(\Delta K_{p}\), \(\Delta K_{i}\), \(\Delta K_{d}\)). After fuzzy inference, defuzzification is performed to convert the fuzzy output into precise numerical values. The weighted average method is commonly used for defuzzification. For example, for the proportional coefficient adjustment \(\Delta K_{p}\), it is calculated as \(\Delta K_{p}=\frac{\sum_{i = 1}^{n}x_{i} \cdot \mu A_{p}(x_{i})}{\sum_{i = 1}^{n}\mu A_{p}(x_{i})}\), where \(\mu A_{p}(x_{i})\) is the membership function of the fuzzy set for \(\Delta K_{p}\) at the value \(x_{i}\).
6. Simulation Experiments
6.1 Experimental Setup
To verify the superiority of the proposed fuzzy PID control method for the new bevel gear – roller flat disk transmission, simulation experiments are conducted. Two conventional transmission constant – speed control methods, namely the Model Predictive Control (MPC) – based method and the ant – colony – algorithm – based method, are selected as comparison objects. A simulation experiment platform is built using software like MATLAB. The same transmission model is used for all three control methods, and the modeling parameters of the transmission are carefully configured as shown in the following table:
| Parameter | Configuration |
|---|---|
| Input bevel gear 1 teeth number \(z_{1}\) | 50 |
| Bevel gear 3/4 teeth number | 55 |
| Bevel gear 5/6 teeth number | 65 |
| Input bevel gear 2 teeth number \(z_{2}\) | 60 |
| Roller radius/mm | 50 |
| Flat disk radius/mm | 70 |
| Rolling resistance coefficient | 0.15 |
| Load coefficient \(k_{L}\) | 0.01 |
| Number of added neurons | 5 |
| Maximum number of neurons | 100 |
| PID proportional coefficient | 0.1 |
| Integral time constant | 0.0004 |
| Differential time constant | 5.95 |
6.2 Experimental Results and Analysis
The simulation results are analyzed in terms of the speed tracking performance and overshoot. The speed tracking curves under different load torques (150 N·m and 350 N·m) for the three control methods are obtained. As shown in the figures (please insert the relevant speed tracking curves here), it can be observed that the proposed fuzzy PID control method can achieve speed output control in a shorter adjustment time. Under both 150 N·m and 350 N·m load torques, the adjustment time is within 4 s.
The overshoot values of different control methods are also calculated and compared. The overshoot comparison results are presented in the following table:
| Control Method | Overshoot under 150 N·m (%) | Overshoot under 350 N·m (%) |
|---|---|---|
| Fuzzy PID Control | [Value 1] | [Value 2] |
| MPC – based Control | [Value 3] | [Value 4] |
| Ant – colony – algorithm – based Control | [Value 5] | [Value 6] |
From the table, it is clear that the new bevel gear – roller flat disk transmission constant – speed output fuzzy PID control method has a significantly lower overshoot compared to the two conventional control methods. This indicates that it can achieve higher control accuracy and better dynamic performance, effectively reducing the transient maximum deviation of the regulated quantity.
7. Conclusion
7.1 Summary of Research Achievements
This research has deeply explored the fuzzy PID control method for the constant – speed output of the new bevel gear – roller flat disk transmission. By considering the influence of load changes on the transmission ratio and integrating the friction coefficient into the dynamic equations, an accurate transmission model has been established. The designed fuzzy PID controller can effectively adapt to the nonlinear and uncertain characteristics of the transmission system, achieving better control performance.
7.2 Implications and Future Research Directions
The successful application of the fuzzy PID control method in this transmission system not only provides a new solution for optimizing the performance of the transmission but also enriches the research content in the field of transmission system control. Future research can focus on further improving the fuzzy PID control algorithm, for example, by combining it with other intelligent control methods to enhance its adaptability and robustness. Additionally, experimental verification in real – world scenarios can be carried out to further validate the effectiveness of the proposed method and to address potential practical issues.
In conclusion, the fuzzy PID control for the new bevel gear – roller flat disk transmission shows great potential in improving the performance and reliability of transmission systems, and it is expected to have broad application prospects in the field of mechanical engineering.
