1. Introduction
1.1 The Significance of Transmission Systems
In the realm of mechanical engineering, transmission systems serve as the backbone of countless 机械设备. Their performance is directly linked to the operational efficiency and longevity of the entire equipment. A well – designed transmission system can optimize power transfer, reduce energy losses, and enhance the overall functionality of the machinery. Among the various components of a transmission system, the 变速器 plays a crucial role. It is the key device that enables the conversion of different speeds and torques, making it an essential element in a wide range of applications, from automotive vehicles to industrial machinery.
1.2 Challenges in Traditional Transmission Control
However, traditional control methods for transmissions often face significant challenges. In real – world applications, the working environment is complex and variable. For example, in automotive applications, the load on the transmission changes constantly depending on the driving conditions, such as acceleration, deceleration, and driving on different terrains. These variations in load can have a profound impact on the transmission ratio and the stability of the output speed. Conventional control strategies struggle to adapt to such dynamic changes, resulting in poor control accuracy and stability in maintaining a constant output speed.
1.3 Research Objectives and Significance
The primary objective of this research is to develop an advanced control strategy for the new bevel gear – roller flat disk transmission to achieve stable and precise constant – speed output. By addressing the nonlinear characteristics and uncertainties inherent in the transmission system, this study aims to improve the performance of the transmission and, ultimately, the entire mechanical system. The proposed fuzzy PID control strategy offers a promising solution, as it can adapt to changing conditions and optimize the control parameters in real – time. This not only enhances the control accuracy but also improves the system’s robustness and adaptability, which is of great significance for modern industrial applications.
2. Overview of the New Bevel Gear – Roller Flat Disk Transmission
2.1 Structure and Working Principle
The new bevel gear – roller flat disk transmission combines the advantages of bevel gear and roller flat disk transmission mechanisms. As shown in Figure 1, the bevel gear part is responsible for altering the transmission ratio. It achieves this through the meshing of bevel gears, which allows for a change in the direction of rotation and the speed ratio. The roller flat disk part, on the other hand, is in charge of smoothly transmitting torque. The rollers roll on the flat disk, and the friction between them enables the transfer of power. This combination enables the transmission to achieve efficient power transfer and speed variation.
[Insert Figure 1: Structure of the New Bevel Gear – Roller Flat Disk Transmission]
2.2 Advantages over Traditional Transmissions
Compared with traditional transmissions, the new bevel gear – roller flat disk transmission has several notable advantages. Firstly, it can provide a wider range of speed ratios, allowing for more flexibility in different applications. Secondly, its unique structure enables higher transmission efficiency, which can lead to energy savings. In ideal conditions, the transmission efficiency can reach over 95%, making it a more environmentally friendly and cost – effective option. Additionally, the smooth torque transmission of the roller flat disk part reduces vibrations and noise, enhancing the overall performance and comfort of the mechanical system.
3. Fuzzy PID Control Strategy
3.1 Basics of PID Control
PID control is a widely used control algorithm in various engineering fields. It consists of three main components: the proportional (P) term, the integral (I) term, and the differential (D) term. The proportional term responds to the current error between the setpoint and the measured value. By adjusting the proportional coefficient , the system can quickly respond to changes in the error. A larger value results in a faster response but may cause overshoot and oscillation. The integral term accumulates the error over time and is used to eliminate the steady – state error. Increasing the integral coefficient can enhance the system’s ability to eliminate the steady – state error, but an excessively large value may lead to a slower response or even instability. The differential term anticipates the change in the error based on its rate of change. Adjusting the differential coefficient can improve the system’s dynamic performance by reducing overshoot and oscillation, but it also makes the system more sensitive to noise. Table 1 summarizes the effects of adjusting the PID parameters.
| PID Parameter | Effect of Increase | Potential Issues |
|---|---|---|
| Faster system response | Overshoot and oscillation | |
| Enhanced ability to eliminate steady – state error | Slower response or instability | |
| Improved dynamic performance, reduced overshoot and oscillation | Increased sensitivity to noise | |
| Table 1: Effects of Adjusting PID Parameters |
3.2 Fuzzy Logic and Its Application in PID Control
Fuzzy logic provides a powerful tool for dealing with uncertainty and imprecision in control systems. In the context of PID control, fuzzy logic can be used to dynamically adjust the PID parameters based on the current state of the system. Instead of using fixed – gain PID controllers, fuzzy PID control adapts the parameters in real – time according to the system’s input and output. This is achieved by fuzzifying the input variables (such as the error and the error change rate), applying fuzzy rules based on expert knowledge or system characteristics, and then defuzzifying the output to obtain the adjusted PID parameters.
3.3 Design of Fuzzy PID Controller for the Transmission
3.3.1 Input Variable Selection
For the new bevel gear – roller flat disk transmission, the input variables of the fuzzy PID controller are selected as the output shaft speed deviation and its change rate . The speed deviation is calculated as the difference between the actual speed and the set speed (, where is the measured value and is the set value). The error change rate is calculated as , where is the current error, is the previous error, and is the time interval. These two variables provide crucial information about the system’s deviation from the desired state and how quickly this deviation is changing.
3.3.2 Fuzzy Set and Membership Function Definition
Fuzzy sets are defined for the error and the error change rate . Commonly used fuzzy sets include {Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Big}, represented as {NB, NM, NS, Z, PS, PM, PB}. Membership functions are then defined to describe the degree to which a particular value belongs to each fuzzy set. For example, a triangular membership function can be used to represent the fuzzy set. Figure 2 shows an example of a triangular membership function for the error .
[Insert Figure 2: Triangular Membership Function for Error ]
3.3.3 Fuzzy Rule Base Establishment
The fuzzy rule base is a crucial part of the fuzzy PID controller. It is established based on expert experience or the dynamic characteristics of the system. The rules are typically in the form of “If is and is , then is , is , is “, where , , , , are elements of the fuzzy sets. Table 2 shows an example of a fuzzy control rule table.
| NB | NM | NS | Z | PS | PM | PB | |
|---|---|---|---|---|---|---|---|
| NB | NB | NB | NM | NS | NS | NS | Z |
| NM | NM | NS | NS | NS | NS | Z | PS |
| NS | NB | NB | NS | NS | Z | PS | PS |
| Z | NM | NM | NS | NS | NS | PS | PS |
| PS | NM | NM | NS | NS | NS | PS | PM |
| PM | NS | Z | PS | NS | PM | PM | PM |
| PB | PS | PS | PS | NS | PM | PB | PB |
| Table 2: Fuzzy Control Rule Table |
3.3.4 Fuzzy Inference and Defuzzification
Fuzzy inference is carried out using a fuzzy inference machine. Based on the input values of and , the fuzzy inference machine determines the adjustment values of the PID parameters according to the established fuzzy rules. After the fuzzy inference, defuzzification is performed to convert the fuzzy output into a precise value. The weighted – average method is commonly used for defuzzification. For example, if the fuzzy set has a membership function and the corresponding universe of discourse is , the defuzzified output expressions for the PID parameter adjustments , , and are , , and respectively.
4. Dynamic Modeling of the Transmission
4.1 Modeling of the Bevel Gear Part
The bevel gear part of the transmission is modeled by considering the impact of load changes on the transmission ratio. Let the input – shaft speed of the bevel gear part be and the output – shaft speed be . A load coefficient is introduced to account for the effect of load on the transmission ratio. The transmission ratio of the bevel gear is expressed as , where and are the numbers of teeth of the two bevel gears, and is a non – linear function of the output torque . The relationship between the input and output speeds is .
4.2 Modeling of the Roller Flat Disk Part
For the roller flat disk part, when the roller rolls on the flat disk, it is mainly affected by the frictional force . The frictional force is related to the normal pressure between the roller and the flat disk and the friction coefficient . The rolling resistance moment is not only related to the frictional force but also to factors such as the roller’s speed , acceleration , and material properties. By introducing a rolling – resistance coefficient , the rolling resistance moment can be expressed as , where is the material density and is the roller radius.
4.3 Integration of the Dynamics Equations
The dynamics equations of the bevel gear part and the roller flat disk part are integrated to obtain the overall dynamics equation of the transmission. The resulting equation is , where is the moment of inertia of the output shaft, and and are the input – shaft torque and output – shaft torque respectively. This integrated equation comprehensively describes the dynamic behavior of the transmission system, taking into account the interactions between different parts and the effects of external factors such as load and friction.
5. Simulation Experiments
5.1 Experimental Setup
5.1.1 Selection of Comparison Methods
To verify the superiority of the proposed fuzzy PID control method for the new bevel gear – roller flat disk transmission’s constant – speed output, two conventional transmission constant – speed control methods are selected as comparison objects. These are the model predictive control (MPC) – based method and the ant – colony – algorithm – based method.
5.1.2 Simulation Model Configuration
The new bevel gear – roller flat disk transmission is simulated using the MATLAB software. The simulation model parameters are carefully configured, as shown in Table 3. These parameters include the number of teeth of the bevel gears, the radius of the rollers and the flat disk, the rolling – resistance coefficient, the load coefficient, and the initial PID parameters.
| 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 |
| Table 3: Transmission Simulation Model Modeling Parameters |
5.1.3 Load Torque Application
To simulate different operating conditions, a load torque of 150/350 N·m is applied to the system. This allows for the evaluation of the control methods’ performance under different load pressures and provides a more realistic scenario for comparing their speed – control capabilities.
5.2 Experimental Results and Analysis
5.2.1 Speed – Tracking Curves
The simulation results of the transmission’s speed output under different control methods are obtained. Figure 3 shows the speed – tracking curves of the proposed method, the MPC – based method, and the ant – colony – algorithm – based method under 150 N·m and 350 N·m load torques. It can be clearly seen from the curves that the proposed fuzzy PID control method can achieve faster and more stable speed – output control. The system can reach the target speed in a shorter time and has less overshoot and oscillation.
[Insert Figure 3: Transmission Constant – Speed Tracking under Different Load Torques]
5.2.2 Overshoot Comparison
The overshoot values of different control methods are calculated and compared, as shown in Table 4. The overshoot is an important indicator to measure the control accuracy of the system. A smaller overshoot indicates better control performance. The experimental results show that the proposed fuzzy PID control method has a significantly lower overshoot compared to the two conventional methods. This indicates that the fuzzy PID control method can more precisely control the output speed of the transmission, reducing the maximum deviation of the controlled variable.
| Experimental Group Number | Design Method | Conventional Method A (MPC – based) | Conventional Method B (Ant – colony – algorithm – based) |
|---|---|---|---|
| 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% |
| Table 4: Overshoot Comparison Results |
6. Conclusion
6.1 Implications and Future Research Directions
The successful implementation of this fuzzy PID control method has far – reaching implications. In the industrial context, it can significantly enhance the performance of various mechanical systems that rely on transmissions, leading to improved energy efficiency, reduced wear and tear, and enhanced reliability. For example, in the automotive industry, this technology can contribute to better – performing vehicles with smoother gearshifts and more stable power delivery.
Looking ahead, several directions for future research can be identified. Firstly, further optimization of the fuzzy rule base could be explored. By incorporating more advanced machine – learning techniques, such as neural networks, the fuzzy rules could be fine – tuned automatically based on real – time data, potentially leading to even better control performance. Secondly, the application of this control strategy in more complex multi – variable transmission systems could be investigated. In real – world scenarios, transmissions often interact with other components in a system, and understanding how to manage these interactions effectively using the fuzzy PID control is crucial.
Another area of interest is the implementation of the proposed control method in physical prototypes. While the simulation results are promising, real – world implementation may face additional challenges, such as sensor noise, actuator limitations, and unforeseen mechanical vibrations. Testing the control strategy in a physical environment will provide valuable insights into its practical viability and help in the development of more robust control algorithms.
In conclusion, the fuzzy PID control for the new bevel gear – roller flat disk transmission represents a significant step forward in transmission system technology. With continuous research and development, this technology has the potential to revolutionize the way transmissions are controlled in a wide range of applications, contributing to a more efficient and sustainable mechanical engineering landscape.
