1. Introduction
In the realm of mechanical engineering, transmission systems play a pivotal role. They are the heart of various ，determining the operational efficiency and service life of the entire equipment. Among them, transmissions, as crucial components for achieving speed and torque conversion, have always been a focal point of research in the field. The new bevel gear – roller flat disk transmission, with its high – efficiency and stable characteristics, offers novel solutions for modern industrial transmission systems. However, due to the complex and variable working environments in practical applications, traditional control methods often struggle to ensure the stability and accuracy of the transmission’s constant – speed output.
1.1 Significance of Transmission Systems
Transmission systems are widely used in numerous industrial fields, such as automotive, aerospace, and manufacturing. In the automotive industry, for example, a well – functioning transmission system can improve fuel efficiency and driving comfort. A stable and efficient transmission can accurately adjust the engine’s power output to meet the diverse speed and torque requirements during vehicle driving, like starting, accelerating, and climbing slopes. In the aerospace field, high – performance transmission systems are essential for the normal operation of aircraft engines and various mechanical components, ensuring the safety and reliability of flight.
1.2 Challenges in Traditional Transmission Control
Traditional control methods for transmissions face several obstacles. For instance, the Model Predictive Control (MPC) method has limitations. As shown in Table 1, although it can predict the speed output of the transmission under different clutch stages through the system model, in practical applications, the uncertainty of the system model and external interference lead to significant control errors. Additionally, MPC is highly dependent on the accuracy of the system model. Even a slight change in model parameters can severely affect the control effect.
| Control Method | Advantages | Disadvantages |
|---|---|---|
| MPC | Can predict speed output based on system model | Large control error due to model uncertainty and external interference; sensitive to model parameter changes |
| Dual Fuzzy PID Control for HMCVT | Combines advantages of fuzzy and PID control | High – complexity algorithm, difficult to adjust multiple fuzzy controller parameters for ideal control effect |
| CVT Speed Output Control based on Driver Intention | Adjusts speed output according to driver behavior | Difficult to accurately capture driver intention due to its diversity and uncertainty, resulting in large control error |
| IPSO – Optimized Fuzzy PID Control | Improves controller’s adaptability and robustness | IPSO algorithm has slow convergence speed and is prone to local optima, affecting speed tracking accuracy and stability |
The dual – fuzzy PID control strategy for Hybrid – type Continuously Variable Transmissions (HMCVT) also has its drawbacks. Although it combines the advantages of fuzzy control and PID control, its complex algorithm requires precise adjustment of multiple fuzzy controller parameters. This complexity makes it difficult to achieve the desired control effect in real – world applications. The CVT speed output control method based on driver intention encounters problems as well. The diversity and uncertainty of driver intentions make it challenging to accurately capture the driver’s true intentions, thus leading to significant control errors. The method of optimizing fuzzy PID controller parameters through the IPSO algorithm also has limitations. The IPSO algorithm itself has issues such as slow convergence speed and a tendency to get trapped in local optima, which can affect the speed tracking accuracy and stability of the control system.
2. Working Principle of the New Bevel Gear – Roller Flat Disk Transmission
The new bevel gear – roller flat disk transmission integrates the mechanisms of bevel gear and roller flat disk transmissions. It mainly realizes power transfer and speed change through the meshing of bevel gears and the rolling of rollers on the flat disk.
2.1 Bevel Gear Component
The bevel gear part is responsible for altering the transmission ratio. As depicted in Figure 1, when the input shaft rotates, the bevel gears engage with each other, and their different tooth numbers result in a change in the rotational speed of the output shaft. For example, if the number of teeth on the input bevel gear is and on the output bevel gear is , the transmission ratio of the bevel gear part is related to the ratio of the tooth numbers. This design enables the transmission to adapt to different speed requirements in various working conditions.
2.2 Roller Flat Disk Component
The roller flat disk part plays a crucial role in smoothly transmitting torque. Rollers are in contact with the flat disk, and as the rollers roll on the flat disk, torque is transferred. The friction between the rollers and the flat disk is a key factor in this process. A proper friction coefficient ensures efficient torque transmission. If the friction coefficient is too low, slippage may occur, reducing the transmission efficiency; if it is too high, excessive wear and heat generation may happen, affecting the service life of the transmission components.
3. Dynamic Modeling of the New Bevel Gear – Roller Flat Disk Transmission
To achieve accurate speed output control, it is essential to establish a dynamic model of the new bevel gear – roller flat disk transmission. This model takes into account various factors that affect the transmission process.
3.1 Influence of Load Changes on the Transmission Ratio
Load changes have a significant impact on the transmission ratio. By introducing a load coefficient , we can better describe this relationship. The expression for the bevel gear transmission ratio considering the load factor is and , where is a non – linear function of the output torque . This indicates that as the output torque changes due to load variations, the transmission ratio also adjusts accordingly. For example, when the load increases, the transmission ratio may change to ensure that the output shaft can still maintain a relatively stable speed.
3.2 Modeling of the Roller Flat Disk Part
When the roller rolls on the flat disk, it is mainly affected by the frictional force . The rolling resistance torque is related to multiple factors. By introducing a rolling resistance coefficient , the rolling resistance torque can be expressed as , where represents the roller linear velocity, represents the roller angular acceleration, is the friction coefficient between the roller and the flat disk, is the material density, and is the roller radius. This formula comprehensively considers the influence of various physical quantities on the rolling resistance torque, providing a more accurate description of the working state of the roller flat disk part.
3.3 Integration of Dynamic Equations
By integrating the dynamic equations of the bevel gear and roller flat disk parts, we can obtain the dynamic equation of the entire transmission: . Here, represents the moment of inertia of the output shaft, and are the input and output shaft torques, respectively. This integrated equation reflects the overall dynamic characteristics of the transmission, taking into account the power input, torque output, rotational inertia, and resistance torque, which is the basis for subsequent control system design.
4. Fuzzy PID Control Strategy
The fuzzy PID control strategy is an effective approach to address the non – linearity and uncertainty of the bevel gear – roller flat disk transmission system.
4.1 Fuzzy Condition Setting
4.1.1 Selection of Input Variables
The input variables of the fuzzy PID controller are the speed deviation and the rate of change of the speed deviation . The speed deviation is calculated as , where is the measured value and is the set value. It reflects the current control deviation of the system. The rate of change of the speed deviation , where is the error at the current time point, is the error at the previous time point, and is the time interval. This value reflects the changing trend of the system deviation.
4.1.2 Fuzzy Sets and Membership Functions
Fuzzy sets are defined for the error and the error change rate , such as {Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Big}, represented by {NB, NM, NS, Z, PS, PM, PB}. Membership functions are used to describe the degree to which a value belongs to a certain fuzzy set. For example, a triangular membership function can be used. As shown in Figure 2, for the fuzzy set “Positive Small”, the membership function defines how likely a particular error value is to be considered “Positive Small”. This allows for a more flexible and intelligent description of the system state.
4.1.3 Fuzzy Rule Base
The fuzzy rule base is established based on expert experience or the dynamic characteristics of the system. The rules are usually in the form of “If is and is , then is , is , is “. For example, if the error is “Positive Big” and the error change rate is “Positive Small”, according to the established rules, the appropriate adjustments to the PID parameters , , and can be determined. These rules are the core of the fuzzy control system, enabling the system to adaptively adjust the control parameters according to different system states.
4.1.4 Fuzzy Inference and Defuzzification
A fuzzy inference machine is used for fuzzy inference. Based on the input and , the adjustment values of the PID parameters are determined. Defuzzification is then carried out to convert the fuzzy output into a precise value. The weighted average method is commonly used for defuzzification. For example, for the adjustment amount of the proportional coefficient , it can be calculated as , where is the membership function of the fuzzy set corresponding to the adjustment amount of the proportional coefficient , and is the value in the corresponding domain. This process bridges the gap between fuzzy control and traditional PID control, making the control system more practical.
4.2 Design of the Fuzzy PID Controller for Transmission Constant – Speed Output
After establishing the dynamic model of the transmission, a fuzzy PID controller is designed. The input of the fuzzy controller is the speed deviation and its rate of change of the output shaft, and the output is the adjustment amount of the controller parameters , , .
The structure of the designed fuzzy controller is shown in Figure 3. Since the system deviation and the deviation change rate are precise values, they need to be fuzzified and mapped to the fuzzy domain. The quantization factors and are used for this purpose. For example, if the basic domain of the system deviation is and the discrete fuzzy domain after fuzzification is , then the quantization factor . This ensures that the precise values can be effectively processed in the fuzzy control system.
4.3 Output of the Fuzzy PID Control Quantity
After completing the fuzzy PID control design, the fuzzy rules are designed, and the fuzzy output is converted into precise PID parameter adjustment amounts through defuzzification. Then, the PID controller parameters are updated, and the control quantity is calculated using the updated parameters: , where , , are the updated PID parameters, is the speed deviation of the output shaft, and is the control quantity. The control quantity is then converted into an adjustment command for the input shaft speed or torque and sent to the transmission actuator to achieve constant – speed control of the output shaft.
5. Simulation Experiments
To verify the superiority of the proposed fuzzy PID control method for the constant – speed output of the new bevel gear – roller flat disk transmission, simulation experiments were carried out.
5.1 Experiment Description
In this experiment, two conventional transmission constant – speed control methods were selected as comparison objects: the conventional transmission constant – speed control method based on MPC and the conventional transmission constant – speed control method based on the ant colony algorithm. A simulation experiment platform was constructed, and the three control methods were used to perform simulation speed output control on the same transmission model. By comparing the actual control effects, the performance of the proposed method could be evaluated.
5.2 Experiment Object
The new bevel gear – roller flat disk transmission selected for this experiment consists of a bevel gear transmission part and a roller flat disk – type continuously variable transmission part. The bevel gear part can meet the requirements of high – speed transmission, and the roller flat disk part can achieve continuous speed change from low to high. The overall speed range is set from 500r/min to 3000r/min, which can meet the requirements of most automotive driving conditions. In an ideal working condition, the transmission efficiency can reach over 95%. The structure of the transmission is shown in Figure 4.
For simulation modeling, the transmission was modeled using the simulation software MATLAB. The modeling parameters were set as shown in Table 2. These parameters were carefully selected based on the actual characteristics of the transmission to ensure the accuracy of the simulation model.
| Parameter | Configuration |
|---|---|
| Number of teeth of input bevel gear 1 | 50 |
| Number of teeth of bevel gears 3/4 | 55 |
| Number of teeth of bevel gears 5/6 | 65 |
| Number of teeth of input bevel gear 2 | 60 |
| Roller radius/mm | 50 |
| Flat disk radius/mm | 70 |
| Rolling resistance coefficient | 0.15 |
| Load coefficient | 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 |
5.3 Comparison Results of Control Accuracy
The simulation speed output curves of the transmission obtained by different methods are shown in Figure 5. It can be seen from the results that under different load torque conditions, the proposed control method can achieve speed output control within a relatively short adjustment time. The adjustment time of the system under 150/350N·m load torque conditions is within 4s.
To further evaluate the control accuracy, the overshoot of different methods was used as a comparison index. The specific experimental comparison results are shown in Table 3. By comparing the overshoot of different methods, it is clear that the proposed new bevel gear – roller flat disk transmission constant – speed output fuzzy PID control method is significantly superior to the two conventional control methods in terms of actual control accuracy, with a lower instantaneous maximum deviation value of the regulated quantity.
| Experimental Group Number | Design Method | Conventional Method A | Conventional Method B |
|---|---|---|---|
| 01 | 3.15 | 4.24 | 5.57 |
| 02 | 3.36 | 4.52 | 5.54 |
| 03 | 3.25 | 4.36 | 5.61 |
| 04 | 3.19 | 4.21 | 5.50 |
| 05 | 3.11 | 4.26 | 4.65 |
| 06 | 3.26 | 4.34 | 4.60 |
| 07 | 3.16 | 4.22 | 5.61 |
| 08 | 3.05 | 4.45 | 4.56 |
| 09 | 3.31 | 4.57 | 5.10 |
| 10 | 3.10 | 4.53 | 5.54 |
| 11 | 3.36 | 4.48 | 5.50 |
| 12 | 3.22 | 4.31 | 5.12 |
6. Conclusion
This research delved deeply into the fuzzy PID control method for the constant – speed output of the new bevel gear – roller flat disk transmission. It not only provided a new solution for optimizing the performance of the transmission but also enriched the research content in the field of transmission system control.
