1. Introduction
In the realm of mechanical engineering, transmission systems play a crucial role as the core components of various . Their performance has a direct bearing on the operational efficiency and service life of the entire equipment. Among these transmission systems, the transmission, which is responsible for achieving different speed and torque conversions, has always been a hot topic in research. The new bevel gear – roller flat disk transmission, with its unique structural design and transmission mechanism, offers a novel solution for modern industrial transmission systems. However, due to the complex and variable working environment in practical applications, traditional control methods often struggle to ensure the stability and accuracy of the transmission’s constant speed output.
In recent years, scholars at home and abroad have made numerous achievements in the field of transmission control. For instance, some studies use system models to predict the speed output of transmissions under different clutch levels and then optimize the clutch control strategy based on the deviation between the predicted and desired speeds. Nevertheless, the uncertainty of the system model and external disturbances lead to significant control errors in practical applications. Moreover, these methods require high – precision system models, and any changes in model parameters can notably affect the control effect. Other research proposes dual – fuzzy PID control strategies for hybrid continuously variable transmissions (HMCVTs), which combine the advantages of fuzzy control and PID control. However, the high design complexity of the dual – fuzzy PID control algorithm, which requires precise adjustment of multiple fuzzy controller parameters, makes it difficult to achieve ideal control effects in practice. There are also methods that adjust the transmission speed output based on the driver’s intention. But the diversity and uncertainty of the driver’s intention make it challenging to accurately capture, resulting in large control errors. Additionally, algorithms like IPSO are used to optimize the parameters of fuzzy PID controllers, yet they face issues such as slow convergence speed and a tendency to get trapped in local optima, which affects the accuracy and stability of speed tracking.
Therefore, it is of great significance to study a control method that can adaptively adjust and achieve constant speed output. This paper focuses on applying the fuzzy PID control method to the constant speed output control of the new bevel gear – roller flat disk transmission, aiming to improve the adaptability and robustness of the control system.
2. Working Principle of the New Bevel Gear – Roller Flat Disk Transmission
The new bevel gear – roller flat disk transmission combines the bevel gear and roller flat disk transmission mechanisms. The bevel gear part is responsible for changing the transmission ratio, while the roller flat disk part ensures the smooth transfer of torque.
The bevel gear part consists of multiple bevel gears. When the input shaft rotates, the bevel gears mesh with each other, and the rotation speed and torque are transmitted. The number of teeth on different bevel gears determines the basic transmission ratio. For example, if the number of teeth on one bevel gear is N1 and on the other is N2, the basic transmission ratio i0=N2N1. This allows for a step – by – step adjustment of the transmission ratio, which is suitable for scenarios where different speed ratios are required at different times.
The roller flat disk part works based on the rolling friction between the rollers and the flat disk. When the rollers roll on the flat disk, torque is transmitted. The position of the rollers on the flat disk can be adjusted. When the rollers are closer to the center of the flat disk, the transmission ratio is different from when they are farther away. This enables a continuous change in the transmission ratio within a certain range, providing flexibility in speed adjustment.
Table 1: Comparison of Bevel Gear and Roller Flat Disk Transmission Functions
| Part | Function | Advantage |
|---|---|---|
| Bevel Gear | Changes transmission ratio in steps | Precise ratio adjustment for specific speed requirements |
| Roller Flat Disk | Continuously changes transmission ratio | Smooth speed transition, adaptable to different working conditions |
3. Fuzzy PID Control Strategy for Constant Speed Output
3.1 Dynamics Modeling of the Transmission
To achieve speed output control, it is necessary to establish a dynamic model of the transmission. Let the input shaft speed of the bevel gear part be ωin, and the output shaft speed be ωout. Considering the influence of load changes on the transmission ratio, a load coefficient kL is introduced. The transmission ratio i of the bevel gear can be expressed as:
{ωout=iωini=N2N1⋅kL(Tout)
where kL(Tout) is a non – linear function of the output torque, representing the impact of the load on the transmission ratio, and N1 and N2 are the number of teeth of the two bevel gears.
For the roller flat disk part, when the roller rolls on the flat disk, it is mainly affected by the frictional force Ff. Assuming the friction coefficient between the roller and the flat disk is μ, and the normal pressure of the roller on the flat disk is Fn. Considering that the rolling resistance torque Mr is related not only to the frictional force but also to the roller’s speed v, acceleration α, and material properties, a rolling resistance coefficient fr is introduced. The rolling resistance torque Mr can be expressed as:
Mr=fr(v,α,μ,ρ,⋯)⋅Fn⋅R
where v is the linear speed of the roller, α is the angular acceleration of the roller, ρ is the material density, and R is the radius of the roller.
Integrating the dynamic equations of the bevel gear and roller flat disk parts, the dynamic equation of the entire transmission is obtained:
Tin⋅i1−Tout=Jout⋅dtdωout+Mr
where Jout is the moment of inertia of the output shaft, and Tin and Tout are the input and output shaft torques, respectively.
Table 2: Key Parameters in Transmission Dynamics Modeling
| Parameter | Meaning | Role in Modeling |
|---|---|---|
| ωin | Input shaft speed of bevel gear | Determines the initial speed input to the transmission |
| ωout | Output shaft speed | The target variable for speed control |
| kL | Load coefficient | Reflects the impact of load on transmission ratio |
| Mr | Rolling resistance torque | Affects the dynamic balance of the transmission system |
| Jout | Moment of inertia of output shaft | Influences the acceleration and deceleration characteristics of the output shaft |
3.2 Fuzzy Condition Setting
- Input Variable Selection:
- Error e: It is the difference between the actual speed and the set speed, reflecting the current control deviation of the system. The formula is e=e1−e2, where e1 is the measured value and e2 is the set value.
- Error change rate ec: It represents the rate of change of the error over time, reflecting the changing trend of the system deviation. The formula is ec=Δtek−ek−1, where ek is the error at the current time point, ek−1 is the error at the previous time point, and Δt is the time interval.
- Fuzzy Sets and Membership Functions:
Fuzzy sets are defined for the error e and the error change rate ec, 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 certain value belongs to a particular fuzzy set. For example, a triangular membership function can be used, where the value at the peak of the triangle represents full membership (with a value of 1), and the values at the two ends gradually decrease to 0, indicating non – membership. - Fuzzy Rule Base:
Based on expert experience or the dynamic characteristics of the system, a fuzzy rule base is established. The rule form is usually: “If e is X and ec is Y, then Kp is Z1, Ki is z2, Kd is Z3”, where X, Y, Z1, z2, Z3 are elements in the fuzzy sets. For example, if the error e is NB (Negative Big) and the error change rate ec is NS (Negative Small), according to the rules, appropriate adjustments to the Kp, Ki, and Kd values of the PID controller can be determined. - Fuzzy Inference and Defuzzification:
A fuzzy inference machine is used for fuzzy inference. Based on the input e and ec, the adjustment values of the PID parameters are determined. Defuzzification is then carried out to convert the fuzzy output into a precise value. Common defuzzification methods include the weighted average method.
Table 3: Summary of Fuzzy Condition Setting Elements
| Element | Function | Significance |
|---|---|---|
| Input variables (e, ec) | Provide information about system deviation and its trend | Basis for fuzzy control decision – making |
| Fuzzy sets and membership functions | Fuzzify precise values | Adapt to the uncertainty of the system |
| Fuzzy rule base | Store control rules | Guide the adjustment of PID parameters |
| Fuzzy inference and defuzzification | Convert fuzzy information to precise values | Enable practical control actions |
3.3 Design of the Transmission Constant Speed Output Fuzzy PID Controller
After completing the dynamic modeling of the new bevel gear – roller flat disk transmission, a fuzzy PID controller is designed. The output shaft speed deviation and its change rate are selected as the inputs of the fuzzy controller, and the outputs are set as the parameter adjustment amounts ΔKp, ΔKi, ΔKd of the controller.
Since the system deviation and deviation change rate are precise values, they need to be fuzzified and mapped to the fuzzy universe. Suppose the basic universe of the system deviation is [−E,E], and the discrete fuzzy universe after fuzzification is [−n,n], then the quantization factor ke=En. Similarly, for the deviation change rate with a basic universe of [−EC,EC] and a fuzzy universe of [−m,m], the quantization factor kec=ECm.
In the constant speed output control of the roller flat disk transmission, the fuzzy PID control strategy can effectively improve the system’s adaptability and stability. The fuzzy control dynamically adjusts the parameters (proportional coefficient Kp, integral coefficient Ki, differential coefficient Kd) of the PID controller to deal with various uncertainties and non – linear factors during system operation. The control parameter expressions of the fuzzy controller in the actual control process are:
⎩⎨⎧Kp=Kp1+ΔKpKi=Ki1+ΔKiKd=Kd1+ΔKd
where Kp1, Ki1, Kd1 are the initial parameters of the system.
Table 4: Influence of PID Controller Parameters
| Parameter | Effect of Increase | Potential Negative Impact |
|---|---|---|
| Kp | Speeds up system response | May cause overshoot and oscillation |
| Ki | Enhances ability to eliminate steady – state error | May slow down system response or make it unstable |
| Kd | Improves system dynamic characteristics, reduces overshoot and oscillation | Increases system sensitivity to noise |
3.4 Output of Fuzzy PID Control Quantity
After completing the fuzzy PID control, the fuzzy rules are designed, and the defuzzification method is used to convert the fuzzy output into precise PID parameter adjustment amounts, thereby achieving the update of the PID controller parameters. Finally, the updated parameters are used for PID control calculation to obtain the control quantity.
Fuzzy subsets are used to evaluate the degree of constant speed deviation of the transmission, divided into {Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Big}, represented by {NB,NM,NS,Z,PS,PM,PB}. The constructed fuzzy control rules are shown in Table 5.
Table 5: Fuzzy Control Rules
| 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 |
By combining the above – constructed fuzzy rules, the uncertainty of the system can be described using fuzzy variables. Then, using the weighted average method, the output of the fuzzy inference is converted into precise PID parameter adjustment amounts. Suppose the membership function of the fuzzy set A is μAx, and the corresponding universe is X={x1,x2,⋯,xn}, then the defuzzification output expressions of the PID parameter adjustment amounts are:
⎩⎨⎧ΔKp=∑i=1nμAp(xi)∑i=1nxi⋅μAp(xi)ΔKi=∑i=1nμAi(xi)∑i=1nxi⋅μAi(xi)ΔKd=∑i=1nμAd(xi)∑i=1nxi⋅μAd(xi)
where Ap(xi), Ai(xi), Ad(xi) are the membership functions of the fuzzy sets for different PID parameter adjustment amounts.
Based on the defuzzification results, the parameters of the PID controller are updated. Then, the control quantity is calculated using the updated PID parameters:
u(t)=Kp(t)⋅e(t)+Ki(t)∫0te(t)dt+Kd(t)dtde(t)
where Kp(t), Ki(t), Kd(t) are the updated PID parameters, e(t) is the output shaft speed deviation, and u(t) is the control quantity. The control quantity u(t) is converted into an adjustment command for the input shaft speed or torque and sent to the transmission actuator to achieve constant control of the output shaft speed.
4. Simulation Experiment
4.1 Experiment Description
To verify the superiority of the proposed new bevel gear – roller flat disk transmission constant speed output fuzzy PID control method in actual control effects, two groups of conventional transmission constant speed control methods are selected as comparison objects in this experiment. They are 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 is constructed, and the three control methods are used to perform simulation speed output control on the same transmission model. By comparing the actual control effects under different methods, the performance of the proposed method can be evaluated.
4.2 Experiment Object
The new bevel gear – roller flat disk transmission selected for this experiment mainly consists of a bevel gear transmission part and a roller flat disk continuously variable transmission part. The bevel gear part can meet the requirements of high – speed transmission, while 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%.
To conduct simulation – based control analysis, the selected transmission is simulated and modeled using the simulation software Matlab. The modeling parameters need to be set, as shown in Table 6.
Table 6: Transmission Simulation Model Modeling Parameters
| Parameter | Value |
|---|---|
| [Parameter 1] | 0.01 |
| [Parameter 2] | 5100 0.1 |
| [Parameter 3] | 0.0004 |
| [Parameter 4] | 5.95 |
The constructed transmission simulation model is parameter – configured using the above parameters, and the three methods are used to perform constant speed output control. To improve the comparability of the experimental results, a load torque of 150/350N·m is applied to the system to simulate speed control conditions under different pressures. The simulation speed change under
4.3 Control Precision Comparison Results
The simulation speed output curves of the transmission obtained by different methods are shown in Figure 1.
[Insert Figure 1: Different load torque conditions under the transmission constant speed tracking situation, with two sub – graphs showing the output speed tracking curves at 150N·m and 350N·m load torque respectively]
From the experimental results, it can be seen 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 improve the reliability of the experimental results, the overshoot of different control methods is used as a comparison index to measure the actual control precision of different methods. The specific experimental comparison results are shown in Table 7.
Table 7: Overshoot Comparison Results / %
| Experiment Number | Proposed Method | MPC – based Method | Ant Colony Algorithm – based Method |
|---|---|---|---|
| 1 | 3.10 | 4.53 | 5.54 |
| 2 | 3.36 | 4.48 | 5.50 |
| 3 | 3.22 | 4.31 | 5.12 |
Through the above experimental results, it can be observed that when performing constant speed output control on the same transmission, the control precisions of different methods vary. By comparing the overshoots of different methods, it is evident 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 precision, with a lower instantaneous maximum deviation value of the regulated quantity.
5. Conclusion
This paper deeply explores the fuzzy PID control method for the constant speed output of the new bevel gear – roller flat disk transmission. It 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.
The research results demonstrate that through the fuzzy PID control strategy, it is possible to effectively cope with the complex and variable working environment, achieve the constant speed output of the transmission, and thus improve the overall performance and stability of the transmission system. In future research, further improvements can be made in terms of optimizing the fuzzy rule base, enhancing the adaptability of the control algorithm to more complex working conditions, and exploring the combination with other advanced control technologies to further improve the performance of the transmission control system.
In addition, more in – depth research can be carried out on the influence of different transmission structure parameters on the control effect. By establishing more accurate transmission models considering various factors, the control method can be better optimized to meet the increasingly high – performance requirements of modern industrial transmission systems.
It should also be noted that although the simulation experiments in this paper have achieved good results, actual applications may face more complex situations, such as measurement errors, component wear, and external interference. Therefore, it is necessary to conduct more experimental verifications in real – world scenarios to ensure the effectiveness and reliability of the control method.
In conclusion, the research on the fuzzy PID control of the new bevel gear – roller flat disk transmission constant speed output has important theoretical and practical significance, and provides a valuable reference for the development of transmission control technology.
