Optimization design method for foil type dynamic pressure gas radial bearing

Abstract

The invention discloses an optimization design method for a foil type dynamic pressure gas radial bearing. The optimization design method comprises the following steps that 1, an optimization target of the dynamic pressure gas radial bearing is determined; 2, dynamic pressure gas radial bearing optimization parameters are selected, and the parameter value range is set; 3, constructing a dynamic pressure gas radial bearing optimization design sample library; step 4, establishing an approximate model for optimization design of the dynamic pressure gas radial bearing; 5, searching an optimization scheme of the dynamic pressure gas radial bearing by using an optimization algorithm; and 6, a dynamic pressure gas radial bearing design scheme is selected. According to the invention, the problem of nonlinear change of the existing dynamic pressure gas bearing in the aspects of bearing capacity, gas film temperature rise, gas film thickness and the like can be solved, and the optimization design precision is improved.

Description

Technical Field

[0001] This invention relates to the field of aerospace technology, and in particular to an optimized design method for foil-type hydrodynamic gas radial bearings. Background Technology

[0002] Hydrodynamic gas bearings have advantages such as high limiting speed, small mass, low friction, and no oil pollution. They have great technical advantages and application potential in aerospace rotating machinery such as refrigeration turbines, ramjet turbines, cooling fans, and small turbine engines. Corrugated foil hydrodynamic gas bearings use air as the lubricating medium between the rotor and the stator. Their structure includes a stator and a rotor with elastic foil installed. Compared with oil-lubricated hydrodynamic bearings and ball bearings, corrugated foil hydrodynamic gas bearings have the following advantages: (1) High stability. The damping effect generated by the elastic foil can suppress the vibration generated by the rotor during high-speed rotation to a certain extent, making the bearing more stable. (2) Strong adaptability. The flexible surface of the elastic foil can adjust its own gas film thickness and stiffness according to the change of working environment, which has strong adaptability. (3) Suitable for high-speed extreme environments. Using air as the lubricating medium greatly reduces the friction in the bearing gap during high-speed rotation, making the bearing more stable than 10 6 The rotational speed of r / min can still operate normally. (4) Long service life. The wear-resistant coating on the elastic foil is only worn during start-up and shutdown and can withstand 10 r / min. 5 The number of start-stop cycles increases the lifespan of the bearings.

[0003] Hydrodynamic gas bearings are essentially complex fluid-thermal-elastic coupled systems. Their performance exhibits significant nonlinear characteristics as a function of rotation, geometry, and elastic foil structure, posing challenges to their application. In the design of hydrodynamic gas bearings, a trial-and-error method is typically used for parameter design and performance analysis. The design process generally involves first determining the bearing parameters based on experience, then calculating its performance under specific operating conditions, and finally verifying the design's suitability to meet requirements. While this method usually yields feasible designs, it struggles to achieve optimal performance while meeting basic requirements. To improve bearing performance, it is necessary to employ appropriate optimization methods to find the best solution, thereby improving design quality and providing methodological and theoretical support for the design of non-contact rotor support structures for high-speed turbine machinery in aerospace. Summary of the Invention

[0004] Purpose of the invention: This invention provides an optimized design method for foil-type hydrodynamic gas radial bearings, which solves the nonlinear variation problems of existing hydrodynamic gas bearings in terms of load capacity, gas film temperature rise, and gas film thickness, and improves the accuracy of the optimized design.

[0005] Technical solution: The present invention provides an optimized design method for a foil-type hydrodynamic gas radial bearing, comprising the following steps:

[0006] Step 1: Determine the optimization objective for the hydrodynamic gas radial bearing;

[0007] Step 2: Select the optimization parameters for the hydrodynamic gas radial bearing and set the parameter value range;

[0008] Step 3: Construct a sample library for optimized design of dynamic pressure gas radial bearings;

[0009] Step 4: Establish an approximate model for the optimized design of the hydrodynamic gas radial bearing;

[0010] Step 5: Use optimization algorithms to find an optimal solution for the hydrodynamic gas radial bearing;

[0011] Step 6: Select a design scheme for a hydrodynamic gas radial bearing.

[0012] Furthermore, in step 1, the optimization objectives for the hydrodynamic gas radial bearing are determined to include maximizing the load-bearing capacity, minimizing the gas film temperature rise, and increasing the minimum gas film thickness; to ensure that the bearing can operate efficiently and stably under different working conditions, and to prevent wear or instability caused by an excessively thin gas film; by setting these optimization objectives, it is ensured that the hydrodynamic gas radial bearing can maintain long-term stability and high efficiency under harsh conditions such as high load, high speed, or high temperature.

[0013] Furthermore, in step 2, the optimized parameters for the hydrodynamic gas radial bearing include rotational speed n, eccentricity ε, average bearing clearance δ, foil overlap angle θ, and corrugated foil stiffness Ks. The rotational speed n is between 12,000 and 100,000 r / min, the eccentricity ε is between 0.6 and 0.9, the average bearing clearance δ is between 5 and 20 μm, the foil overlap angle θ is between 20° and 120°, and the corrugated foil stiffness Ks is between 10 and 100 GPa. The selection of these parameters must consider their impact on bearing performance, such as the effect of rotational speed on gas film stability, the effect of eccentricity on load-bearing capacity, and the effects of clearance and angle on gas film thickness and temperature rise. Setting reasonable parameter ranges ensures the effectiveness of the optimization process and ultimately achieves the design goals.

[0014] Furthermore, in step 3, a certain number of samples are generated using a greedy strategy-based Latin hypercube method. These samples are then used in a simulation platform to calculate performance indicators such as bearing capacity, maximum air film temperature rise, and minimum air film thickness for each sample group. The design space set from step 2 is divided into equally probable intervals, forming a Latin hypercube grid structure. Following the greedy strategy, starting from the initial point, the optimal sample point is selected each time, ensuring that the samples are uniformly distributed across all dimensions. In each step, based on local optimum strategies (maximum bearing capacity, maximum air film temperature rise, and minimum air film thickness), the problem scale is gradually reduced until the design requirements are met. This method effectively covers the design space, ensuring the diversity and representativeness of the samples. These calculation results construct a sample space between the design parameters and the optimization objective, providing data support and basis for subsequent optimization analysis.

[0015] Furthermore, in step 4, a backpropagation (BP) neural network is used to construct a surrogate model. This avoids continued reliance on numerical iterative calculations on the simulation platform. The BP neural network can learn the mapping relationship between design parameters and optimization objectives by training on existing sample data, thereby quickly predicting performance indicators under different design parameters. This surrogate model greatly improves the efficiency of the optimization process and provides a convenient and efficient tool for subsequent optimization analysis.

[0016] Furthermore, the specific steps for constructing a proxy model using a BP neural network include the following:

[0017] Step 41: Determine network parameters; set input parameters according to requirements. and output parameters Then, the number of layers in the neural network and the number of neurons in each layer are set. Then, set the connection weights between the input layer neurons, hidden layer neurons, and output layer neurons. and Set the threshold for hidden layer neurons and the threshold of the output layer neurons Finally, set the learning rate. ;

[0018] Step 42: Calculate the hidden layer output; the input information is forward-propagated through the connection weights and thresholds of each layer to obtain the hidden layer output.

[0019] The output value of the i-th neuron in the input layer is:

[0020] The output value of the j-th neuron in the hidden layer is:

[0021]

[0022] The output value of the k-th neuron in the output layer is:

[0023]

[0024] Step 43: Backpropagation of error; Compare the hidden layer output value from Step 42 with the expected output value, calculate the prediction error, propagate the error information to the input layer, and update the connection weights of each layer. , and , ;

[0025]

[0026] Output layer weight update:

[0027]

[0028] Hidden weight update:

[0029]

[0030] Output layer threshold update:

[0031]

[0032] Hidden layer threshold update:

[0033]

[0034] Step 44: When the error reaches the set threshold or the number of iterations reaches the upper limit, stop the iteration process; otherwise, continue to execute steps 42 and 43.

[0035] Step 45: Complete training and obtain results; obtain the network connection weights after stopping iterations. , and threshold , .

[0036] Furthermore, in step 5, a multi-objective particle swarm optimization (MOPSO) algorithm, improved based on the UCT algorithm, is used to find an optimal solution for the hydrodynamic gas radial bearing. This algorithm combines the advantages of UCT, enabling it to better balance the conflicts between different objectives in multi-objective optimization, thereby efficiently finding the best design solution that meets the optimization requirements.

[0037] Furthermore, the optimization algorithm for finding an optimal solution for the hydrodynamic gas radial bearing includes the following steps:

[0038] Step 51: Set initial parameters for multi-objective particle swarm optimization: In this optimization algorithm, several key parameters need to be set first, including population size, maximum number of iterations, and inertia weight. The population size is set to 100, and the maximum number of iterations is set to 1000. This allows for a thorough exploration of the solution space and ensures that the algorithm has strong global search capabilities. To avoid being troubled by local optima, the inertia weight w is set to 0.4, which helps maintain the breadth of the search. The individual learning factor c1 and the swarm learning factor c2 are set to 2 to ensure that particles can balance the learning between individuals and the swarm. In addition, in multi-objective optimization, multiple conflicting objectives are usually involved (such as increasing load-bearing capacity while reducing temperature rise). Therefore, a non-dominated sorting strategy is needed to guide the optimization process and gradually approach the Pareto front solution in each iteration.

[0039] Step 52: Initialize the population: To improve the uniformity of the initial population's position distribution and avoid the particle swarm from stagnating at local optima, a patrol strategy is adopted. 50% of the individuals in the initial population are randomly distributed in the solution space, while the other 50% are distributed according to a specific pattern. In this way, the particle swarm can cover a wider range of solution spaces, increase the algorithm's global search capability, and avoid early convergence. Especially in the design of dynamic pressure gas radial bearings, such an initialization strategy can help quickly identify potential high-quality design schemes.

[0040] Step 53: Update the external archive: During the optimization process, fast non-dominated sorting is used to evaluate the position of each particle. Non-dominated solutions refer to solutions that cannot be dominated by other solutions under all objectives. They form the so-called Pareto front. By adding these non-dominated solutions to the external archive, not only can the optimal design scheme be preserved, but also the diversity of solutions can be guaranteed, avoiding the loss of other potentially excellent design directions due to over-concentration on a specific objective. In the design of hydrodynamic gas radial bearings, this means that multiple indicators such as bearing load capacity, temperature rise, and gas film thickness can be considered simultaneously during the optimization process, rather than pursuing only one aspect of performance.

[0041] Step 54: Global Optimal Particle Selection: In a particle swarm, there are usually multiple particles that perform well on different objective functions. In order to find the optimal solution, the particles need to be sorted by crowding and the particle with the best fitness needs to be selected. In order to further improve the global search capability of the algorithm, the UCT algorithm is used to select particles. By combining the quality of known solutions and the exploration value of unknown solutions, the UCT algorithm can select the most promising global optimal solution from many candidate particles. This method can effectively guide the optimization process, avoid getting trapped in local optima, and ensure that the global optimal solution is found.

[0042] Step 55, Particle State Update: In each iteration, the velocity and position of the particle are updated by comparing the current individual optimal solution with the global optimal solution. Each particle calculates a new search direction based on its current velocity and position and updates its velocity. By continuously updating the velocity and position, the particle can perform a detailed search in the solution space and gradually approach the optimal solution. In practical applications, this particle update method can ensure that the dynamic pressure gas radial bearing can find a balance point among multiple performance indicators during the design process.

[0043] Step 56, Iterative Process: The entire optimization process continuously updates the particle's solution through multiple iterative steps. In each iteration, the particles in the population will adjust according to the current global optimal solution and further explore in the solution space. This process will continue until the maximum number of iterations is reached, or the optimization process converges to a stable solution. In the design of dynamic pressure gas radial bearings, multiple objective functions (such as load-bearing capacity, gas film thickness, temperature rise, etc.) will be weighed and optimized in each iteration to ensure that the final design meets various requirements.

[0044] Step 57: Selection: A set of non-dominated solutions generated during the optimization process will be used as candidate solutions. Based on the specific working environment and design requirements, such as bearing load, speed, and ambient temperature, the design scheme that best meets the requirements will be selected. These design schemes have the best balance in terms of load capacity, temperature rise control, and gas film thickness, and can meet the performance requirements under different working conditions. In practical applications, these optimal schemes can help improve the overall efficiency of the system, reduce energy consumption, and extend the service life of the hydrodynamic gas radial bearing.

[0045] Furthermore, in step 54, the UCT algorithm selects particles specifically through the following steps:

[0046] Step 541: Establish the search tree; The UCT algorithm first selects candidate particles as the root node of the current situation based on the "crowding" of the particles (i.e., the density of particles in the design space). This selection process ensures that sparser areas in the design space are given priority, because these areas may contain higher potential. For foil-type hydrodynamic gas radial bearings, different design parameters will affect the bearing's load-bearing capacity. By initially selecting the root node, the foundation can be laid for the subsequent construction of the search tree, and the coverage of the design space can be made as comprehensive as possible.

[0047] Step 542: Selecting Nodes; After the search tree is established, the UCT algorithm obtains the UCB1 value of each node through simulation. The UCB1 value is used to evaluate the potential of the current node and the necessity of exploration. The node with the largest UCB1 value is selected for further search. In the design of dynamic pressure gas radial bearings, the UCB1 value can comprehensively evaluate multiple indicators such as the load-bearing capacity, temperature rise, and gas film thickness of the design scheme. Through this process, the algorithm can prioritize those potentially excellent design schemes and further expand its search path.

[0048] Step 543: Repeat the search; After selecting a node, the algorithm will determine whether the node is a leaf node. If the node is not a leaf node, it means that the node can be further expanded, that is, there is still potential for further searching. Therefore, the algorithm will repeat step 542 starting from the current child node to continue to find the best solution. For foil-type hydrodynamic gas radial bearings, many design parameters may be intertwined, so a single search may not be able to completely cover all design spaces. With the help of this step, the algorithm can continuously refine the search range and find more suitable design points.

[0049] Step 544: Update path values; When the algorithm reaches a leaf node, it has completed an exploration of the current design point. At this time, the system will calculate the UCB1 value of the leaf node and update the UCB1 values ​​of all nodes on the path from the root node to the leaf node using this value. This process is particularly important for the design of dynamic pressure gas radial bearings, because each path update means a deeper exploration of the design space. In this process, designers can gradually discover which design schemes can balance multiple objectives, such as improving load-bearing capacity, reducing maximum temperature rise, and improving operational stability.

[0050] Step 545: Continue to expand; After the algorithm completes the evaluation of a leaf node, it will take that node as the new root node and continue to search for the leaf node with the largest UCB1 value in the next round. If the leaf node has been visited, the algorithm will skip the node to avoid repeated calculations. For the design of foil-type hydrodynamic gas radial bearings, this expansion step helps to prevent the algorithm from staying in the region where a good solution has been found for too long, thereby ensuring that a wider design space is explored and further improving the diversity and possibility of the design.

[0051] Step 546: Iterative Search; The core of this step is to iteratively execute the preceding search process until a predetermined simulation upper limit is reached. In the design of dynamic pressure gas radial bearings, this simulation upper limit often corresponds to certain computational resource and time constraints. The aim is to minimize computation time while ensuring search quality. As the number of simulations increases, the search accuracy of the UCT algorithm gradually improves, enabling it to more accurately identify high-quality solutions that meet the design requirements.

[0052] Step 547: Determine the optimal solution; After all search paths are completed, the UCT algorithm will select the node with the largest UCB1 value from all child nodes of the root node as the optimal node. The particle corresponding to this node is the local optimal solution, representing an optimized scheme for a foil-type hydrodynamic gas radial bearing that meets the design requirements. Through this final step, the designer can obtain an optimal design scheme that comprehensively considers multiple objectives such as load-bearing capacity, temperature rise, and gas film thickness, thereby providing a scientific basis for subsequent engineering applications.

[0053] Furthermore, in step 6, among multiple optimization schemes, a design scheme that meets the requirements of performance, stability, and other key indicators is selected based on specific work needs. Through evaluation and comparison of the optimization results, a dynamic pressure gas bearing parameter scheme with high load capacity, low aerodynamic and thermal characteristics, and a wide safety margin is chosen to ensure optimal performance under actual working conditions. In the particle swarm optimization algorithm, the number of particles is between 100 and 10,000, the inertia weight is 0.75, and the control individual experience and group experience coefficients in the learning factor are 0.5 and 0.5, respectively. The penalty function method is used for local optimization. The upper confidence interval algorithm (UCT) is an algorithm that optimizes decision-making by balancing exploration and exploitation, defining an upper confidence level 'a' of 90%. 95%,UL To find the minimum value of 'a' that satisfies the following conditions, the exploration parameter range is specified as π∈(0.98,1). By establishing a multi-objective particle swarm optimization method based on an improved UCT algorithm, the maximum bearing capacity F, the minimum aerodynamic thermal peak value ΔTmax, and the minimum local gap height δ in the dataset are obtained. min Parameter set: By assigning different weight coefficients to the above three optimization parameters, an optimization parameter set matching the requirements is obtained.

[0054] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: The present invention establishes a performance proxy model of a foil-type hydrodynamic gas radial bearing based on a BP neural network, and then establishes a multi-objective particle swarm optimization method based on an improved UCT algorithm to realize the multi-performance objective optimization design of complex circumferential variable stiffness foil bearings; the results show that the improved multi-objective particle swarm algorithm completely dominates the Pareto front of MOPSO, and the multi-objective particle swarm algorithm has a faster convergence speed and higher optimization accuracy. The optimization solution is obtained by taking the maximum load capacity, minimum gas film temperature rise and maximum gas film thickness as optimization objectives, and the obtained solution is significantly improved compared with the benchmark solution. Attached Figure Description

[0055] Figure 1 This is a schematic diagram of the method flow of the present invention.

[0056] Figure 2 This is a schematic diagram of the UCT algorithm flow of the present invention.

[0057] Figure 3 This is a schematic diagram of the multi-objective particle swarm algorithm based on the improved UCT algorithm of the present invention.

[0058] Figure 4(a) is a simplified diagram of the arch wave structure.

[0059] Figure 4(b) shows the deflection variation of the connecting rod spring structure.

[0060] Figure 5(a) shows the distribution of pressure along the circumferential direction in bearing schemes #1, #2 and #3.

[0061] Figure 5(b) shows the distribution of aerodynamic heat rise along the circumferential direction in bearing schemes #1, #2 and #3.

[0062] Figure 5(c) shows the distribution of air film thickness along the circumferential direction in bearing schemes #1, #2 and #3.

[0063] Figure 5(d) shows the distribution of interfacial shear stress along the circumferential direction in bearing schemes #1, #2 and #3. Detailed Implementation

[0064] like Figure 1 As shown, an optimization design method for a foil-type hydrodynamic gas radial bearing includes the following steps:

[0065] Using the average gap, foil overlap angle, and stiffness of each foil as optimization design variables, and maximizing load-bearing capacity, minimizing maximum air film temperature rise, and maximizing minimum air film thickness as optimization objectives, a multi-objective particle swarm optimization algorithm based on UCT algorithm improvement is adopted. A neural network surrogate model is used to accelerate the search efficiency, and the multi-objective particle swarm optimization method based on UCT algorithm improvement avoids getting trapped in local optima, in order to obtain a dynamic pressure gas radial bearing design scheme that takes into account high load-bearing performance, high safety, and low aerodynamic and thermal characteristics.

[0066] Includes the following steps:

[0067] (1) Determine the optimization objectives of the hydrodynamic gas radial bearing to meet specific application requirements;

[0068] In the design process of hydrodynamic gas radial bearings, the first step is to set optimization objectives based on the specific requirements of the actual application. These requirements include the bearing's load-bearing capacity, stability, and durability under high-speed rotation, high load, and large temperature fluctuation environments. Optimization objectives include maximizing load-bearing capacity, minimizing gas film temperature rise, and increasing the minimum gas film thickness to ensure efficient and stable operation of the bearing under different operating conditions and prevent wear or instability caused by an excessively thin gas film. By setting these optimization objectives, the hydrodynamic gas radial bearing can maintain long-term stability and high efficiency under harsh conditions such as high load, high speed, or high temperature.

[0069] (2) Select the optimization parameters for the hydrodynamic gas radial bearing and set the parameter value range;

[0070] In the design optimization process of hydrodynamic gas radial bearings, key design parameters to be optimized are selected, including rotational speed n, eccentricity ε, bearing clearance δ, foil overlap angle θ, and corrugated foil stiffness Ks, and a reasonable range of values ​​is set for each parameter. The selection of parameters must consider their impact on bearing performance, such as the effect of rotational speed on gas film stability, the effect of eccentricity on load-bearing capacity, and the effects of clearance and angle on gas film thickness and temperature rise. Setting a reasonable parameter range ensures the effectiveness of the optimization process and ultimately achieves the design objectives.

[0071] (3) Construct a sample library for the optimized design of dynamic pressure gas radial bearings;

[0072] In the optimization design of hydrodynamic gas radial bearings, a certain number of samples are first generated using a greedy strategy-based Latin hypercube method. This method effectively covers the design space, ensuring the diversity and representativeness of the samples. Subsequently, a simulation platform is used to calculate these samples, obtaining performance indicators such as load-bearing capacity, maximum gas film temperature rise, and minimum gas film thickness for each sample group. These calculation results are used to construct a sample space between design parameters and optimization objectives, providing data support and basis for subsequent optimization analysis.

[0073] (4) Establish an approximate model for the optimized design of the hydrodynamic gas radial bearing;

[0074] Since computation through simulation platforms is time-consuming and inefficient, this method uses a backpropagation (BP) neural network to construct a surrogate model, thus avoiding continued reliance on numerical iterative calculations from simulation platforms. The BP neural network can learn the mapping relationship between design parameters and optimization objectives by training on existing sample data, thereby quickly predicting performance indicators under different design parameters. This surrogate model significantly improves the efficiency of the optimization process, providing a convenient and efficient tool for subsequent optimization analysis. The training steps for constructing the surrogate model are as follows:

[0075] Step 1: Determine network parameters. Set the input parameters according to your requirements. and output parameters Then, the number of layers in the neural network and the number of neurons in each layer are set. Then, set the connection weights between the input layer neurons, hidden layer neurons, and output layer neurons. and Set the threshold for hidden layer neurons and the threshold of the output layer neurons Finally, set the learning rate. .

[0076] Step 2: Calculate the hidden layer output. The input information is forward-propagated through the connection weights and thresholds between each layer to obtain the hidden layer output.

[0077] The output value of the i-th neuron in the input layer is:

[0078] The output value of the j-th neuron in the hidden layer is:

[0079]

[0080] The output value of the k-th neuron in the output layer is:

[0081]

[0082] Step 3: Error Backpropagation. Compare the hidden layer output value from Step 2 with the expected output value to calculate the prediction error. Backpropagate this error information to the input layer and update the connection weights of each layer. , and , .

[0083]

[0084] Output layer weight update:

[0085]

[0086] Hidden weight update:

[0087]

[0088] Output layer threshold update:

[0089]

[0090] Hidden layer threshold update:

[0091]

[0092] Step 4: Stop the iteration process when the error reaches the set threshold or the number of iterations reaches the upper limit. Otherwise, continue executing steps 2 and 3.

[0093] Step 5: Complete training and obtain results. Obtain the network connection weights after stopping iterations. , and threshold , .

[0094] (5) Use optimization algorithms to find the optimal solution for the hydrodynamic gas radial bearing;

[0095] The load-bearing performance of a hydrodynamic gas radial bearing is closely related to its rotation, geometry, and elastic foil parameters, and changes in these factors can lead to significant nonlinear characteristics in the system performance. To effectively find the optimal solution, this method employs a multi-objective particle swarm optimization (MOPSO) algorithm improved from the UCT algorithm. This algorithm combines the advantages of UCT, better balancing the conflicts between different objectives in multi-objective optimization, and thus efficiently finding the optimal design solution that meets the optimization requirements.

[0096] (6) Selection of design scheme for hydrodynamic gas radial bearing;

[0097] Based on the operational requirements, select the most suitable optimization scheme from among numerous options. Among these, choose the design that meets the performance, stability, and other key performance indicators, based on specific operational needs. Through evaluation and comparison of the optimization results, select the most suitable hydrodynamic gas radial bearing design for the actual application to ensure optimal performance under real-world conditions.

[0098] When adjusting a design parameter to improve a performance index, it is often found that other performance indexes may decrease as a result. This phenomenon indicates that there may be a certain conditional or complete conflict between performance indexes. In the design process of hydrodynamic gas radial bearings, their overall performance is determined by multiple factors, including key parameters such as bearing clearance, foil overlap angle, and foil stiffness. Therefore, to obtain an optimal design solution, it is necessary to comprehensively consider multiple target performance indexes through efficient optimization algorithms. Based on this, an improved multi-objective particle swarm optimization algorithm combined with the UCT algorithm is adopted to efficiently perform optimization calculations in a complex design space and find a set of optimal design parameter combinations. This method can ensure the optimal bearing design is achieved while meeting multiple performance requirements. These performance requirements specifically include: maximizing load capacity, minimizing maximum temperature rise, and maximizing gas film thickness. The specific steps include:

[0099] 1) Setting Initial Parameters for Multi-Objective Particle Swarm Optimization: In this optimization algorithm, several key parameters need to be set first, including population size, maximum number of iterations, and inertia weight. Here, the population size is set to 100, and the maximum number of iterations is set to 1000. This allows for sufficient exploration of the solution space and ensures that the algorithm has strong global search capabilities. To avoid being troubled by local optima, the inertia weight w is set to 0.4, which helps maintain the breadth of the search. The individual learning factor c1 and the swarm learning factor c2 are both set to 2 to ensure that particles can balance the learning between individuals and the swarm. In addition, multi-objective optimization often involves multiple conflicting objectives (such as increasing load-bearing capacity while reducing temperature rise). Therefore, a non-dominated sorting strategy is needed to guide the optimization process, gradually approaching the Pareto front solution in each iteration.

[0100] 2) Population Initialization: To improve the uniformity of the initial population's positional distribution and prevent the particle swarm from stagnating at local optima, a patrolling strategy is employed. 50% of the individuals in the initial population are randomly distributed within the solution space, while the other 50% are distributed according to a specific pattern. This allows the particle swarm to cover a wider range of solutions, increasing the algorithm's global search capability and preventing early convergence. This initialization strategy is particularly helpful in the design of hydrodynamic gas radial bearings, as it can quickly identify potentially superior design solutions.

[0101] 3) Updating External Files: During optimization, fast non-dominated sorting is used to evaluate the position of each particle. Non-dominated solutions are those that cannot be dominated by other solutions under all objectives; these form the so-called Pareto front. By adding these non-dominated solutions to the external file, not only can the optimal design be preserved, but the diversity of solutions can also be guaranteed, avoiding the loss of other potentially excellent design directions due to over-concentration on a specific objective. In the design of hydrodynamic gas radial bearings, this means that multiple indicators such as bearing load capacity, temperature rise, and gas film thickness can be considered simultaneously during the optimization process, rather than pursuing only one aspect of performance.

[0102] 4) Global Optimal Particle Selection: In a particle swarm, multiple particles typically perform well on different objective functions. To find the optimal solution, particles need to be sorted by crowding to select the particle with the best fitness. To further improve the algorithm's global search capability, the UCT algorithm is used for particle selection. By combining the quality of known solutions with the exploration value of unknown solutions, the UCT algorithm can select the most promising global optimal solution from numerous candidate particles. This method can effectively guide the optimization process, avoid getting trapped in local optima, and ensure that the global optimal solution is found.

[0103] 5) Particle State Update: In each iteration, the particle's velocity and position are updated by comparing the current particle's individual optimal solution with the global optimal solution. Each particle calculates a new search direction based on its current velocity and position and updates its velocity accordingly. By continuously updating velocity and position, the particle can perform a detailed search within the solution space, gradually approaching the optimal solution. In practical applications, this particle update method ensures that the hydrodynamic gas radial bearing can find a balance among multiple performance indicators during the design process.

[0104] 6) Iterative Process: The entire optimization process continuously updates the particle's solution through multiple iterative steps. In each iteration, the particles in the population are adjusted based on the current global optimal solution and further explored in the solution space. This process continues until the maximum number of iterations is reached, or the optimization process converges to a stable solution. In the design of dynamic pressure gas radial bearings, multiple objective functions (such as load-bearing capacity, gas film thickness, temperature rise, etc.) are weighed and optimized in each iteration to ensure that the final design meets various requirements.

[0105] 7) Selection: A set of non-dominated solutions generated during the optimization process will be used as candidate solutions. Based on the specific working environment and design requirements, such as bearing load, speed, and ambient temperature, the design scheme that best meets the needs will be selected. These design schemes have the optimal balance in terms of load capacity, temperature rise control, and gas film thickness, and can meet the performance requirements under different working conditions. In practical applications, these optimal schemes can help improve the overall efficiency of the system, reduce energy consumption, and extend the service life of the hydrodynamic gas radial bearing.

[0106] The specific steps of the UCT algorithm to select particles are as follows:

[0107] 1) Building the search tree. The UCT algorithm first selects candidate particles as the root node of the current state based on the "crowding" of the particles (i.e., the density of particles in the design space). This selection process ensures that sparser regions in the design space are given priority, as these regions may contain higher potential. For foil-type hydrodynamic gas radial bearings, different design parameters will affect the bearing's load-carrying capacity. By initially selecting the root node, the foundation can be laid for the subsequent construction of the search tree, and the coverage of the design space can be made as comprehensive as possible.

[0108] 2) Node Selection. After the search tree is built, the UCT algorithm obtains the UCB1 value of each node through simulation. The UCB1 value is used to evaluate the potential of the current node and the necessity of exploration, selecting the node with the largest UCB1 value for further searching. In the design of hydrodynamic gas radial bearings, the UCB1 value can comprehensively evaluate multiple indicators such as the load-bearing capacity, temperature rise, and gas film thickness of the design scheme. Through this process, the algorithm can prioritize potentially excellent design schemes and further expand its search path;

[0109] 3) Repeated Search. After selecting a node, the algorithm determines whether it is a leaf node. If the node is not a leaf node, it means that the node can be further expanded, i.e., there is still potential for further searching. Therefore, the algorithm will repeat step S3-2 starting from the current child node to continue searching for the optimal solution. For foil-type hydrodynamic gas radial bearings, many design parameters may be intertwined, so a single search may not completely cover the entire design space. With the help of this step, the algorithm can continuously refine the search range and find more suitable design points;

[0110] 4) Update path values. When the algorithm reaches a leaf node, it has completed one exploration of the current design point. At this time, the system calculates the UCB1 value of that leaf node and updates the UCB1 values ​​of all nodes on the path from the root node to that leaf node using this value. This process is particularly important for the design of hydrodynamic gas radial bearings, because each path update means a deeper exploration of the design space. In this process, designers can gradually discover which design schemes can balance multiple objectives, such as increasing load-bearing capacity, reducing maximum temperature rise, and improving operational stability.

[0111] 5) Continue expanding. After the algorithm completes the evaluation of a leaf node, it will use that node as the new root node and continue searching for the leaf node with the largest UCB1 value in the next round. If the leaf node has already been visited, the algorithm will skip that node to avoid redundant calculations. For the design of foil-type hydrodynamic gas radial bearings, this expansion step helps to prevent the algorithm from staying in the region where a optimal solution has already been found for too long, thereby ensuring that a wider design space is explored and further improving the diversity and possibilities of the design;

[0112] 6) Iterative Search. The core of this step is to iteratively execute the preceding search process until a predetermined simulation upper limit is reached. In the design of hydrodynamic gas radial bearings, this simulation upper limit often corresponds to certain computational resource and time constraints. The aim is to minimize computation time while ensuring search quality. As the number of simulations increases, the search accuracy of the UCT algorithm gradually improves, enabling it to more accurately identify high-quality solutions that meet the design requirements.

[0113] 7) Determining the Optimal Solution. After all search paths have been completed, the UCT algorithm selects the node with the largest total UCB1 value from all child nodes of the root node as the optimal node. The particle corresponding to this node is the local optimum, representing an optimized scheme for a foil-type hydrodynamic gas radial bearing that meets the design requirements. Through this final step, the designer can obtain an optimal design scheme that comprehensively considers multiple objectives such as load-bearing capacity, temperature rise, and gas film thickness, thus providing a scientific basis for subsequent engineering applications.

[0114] This invention overcomes the problem of nonlinear stiffness variation differences in multi-foil stacked structures caused by the traditional Hesmit linear equivalent foil stiffness model by establishing a nonlinear equivalent stiffness model of the deflection-following link spring. The nonlinear equivalent stiffness model of the deflection-following link spring is used to describe the stiffness of the elastic foil structure. All arched wave structures are simplified to equivalent vertical springs, as shown in Figure 4(a). The stiffness k1 of the horizontal spring is calculated using Castigliano's theorem, as follows:

[0115]

[0116] In the formula, R g Let D be the radius of the arch wave, L be the elastic stiffness of the shell, L be the bearing length, θ0 be the half-angle of the arch wave, S be the cross-sectional area of ​​the foil, and E be the elastic modulus. Figure 4(b) shows the deflection variation of the connecting rod spring structure. When the connecting rod moves along the normal of the arch foil, the relationship between the radial deflection Δh of the arch foil and the deformation ΔL of the horizontal spring is established as follows:

[0117]

[0118] In the formula, Δh is the radial deflection of the arch foil, and ΔL is the deformation of the horizontal spring.

[0119] Based on this, a thermo-mechanical coupling performance analysis method for corrugated foil-type hydrodynamic gas radial bearings is proposed. The accuracy of the method is verified by comparison with experimental data. Furthermore, numerical simulation analysis is conducted, revealing the variations in the bearing capacity, gas film temperature rise, and gas film thickness of the hydrodynamic gas bearing with respect to rotational speed, clearance dimensions, and foil circumferential stiffness parameters. Strong nonlinear and multi-peak characteristics are found among these properties, thus demonstrating the necessity of multi-performance objective optimization design for this type of bearing.

[0120] Next, this invention establishes a performance proxy model for a foil-type hydrodynamic gas radial bearing using a BP neural network. Based on this, an improved multi-objective particle swarm optimization method based on the UCT algorithm is proposed. Bearing schemes #1, #2, and #3 under specific operating conditions are obtained, where bearing scheme #1 is the baseline condition, #2 is the optimized bearing scheme, and #3 is the poorly performing bearing scheme. Figures 5(a), 5(b), 5(c), and 5(d) show the circumferential distribution of pressure, temperature rise, gas film thickness, and shear force under different bearing schemes. As shown in the figures, bearing scheme #2 has higher load capacity, lower maximum temperature rise, and smaller minimum gas film thickness compared to bearing scheme #1 (baseline condition), while bearing scheme #3 exhibits lower load capacity, higher maximum temperature rise, and larger minimum gas film thickness. The average gas film thickness of the optimized bearing scheme #2 decreases, and the gas film thickness to the left of the minimum gas film thickness is greater than that to the right, resulting in a decrease in the pressure peak value and a shift to the left, while the minimum pressure value increases and shifts to the right, as shown in Figure 5(a). The shift in pressure peak value affects the position of temperature peak value, while the decrease in pressure peak value reduces the compression temperature rise. As shown in the shear stress distribution in Figure 5(d), bearing scheme #2 exhibits reduced shear stress at the point of maximum temperature rise, leading to weakened viscous dissipation and a lower maximum temperature rise, as shown in Figure 5(b). In contrast, bearing scheme #3 shows an increase in average gas film thickness, and the gas film thickness on the left side of the minimum gas film thickness is less than that on the right side, resulting in an increase in pressure peak value and a shift to the right, while the minimum pressure value decreases and shifts to the left. The shift in pressure peak value affects the position of temperature peak value, while the increase in pressure peak value increases the compression temperature rise. As shown in the shear stress distribution in Figure 5(d), bearing scheme #3 exhibits increased shear stress at the point of maximum temperature rise, leading to enhanced viscous dissipation and an increased maximum temperature rise, as shown in Figure 5(b). This method significantly improves optimization accuracy and convergence speed. By optimizing the three objectives of maximum load-bearing capacity, minimum air film temperature rise, and maximum air film thickness, the optimization results show that the improved multi-objective particle swarm optimization algorithm achieves higher optimization performance compared to the traditional MOPSO method, with a 7.57% increase in load-bearing capacity, a 2.05% reduction in air film temperature rise, and a 15.12% increase in air film thickness.

[0121] The optimization design method of this invention solves the problem of nonlinear performance optimization of corrugated foil hydrodynamic gas bearings under various working conditions, and can provide more accurate performance prediction and optimization schemes for bearing design, which has high engineering application value.

Claims

1. A method for optimizing the design of a foil-type hydrodynamic gas radial bearing, characterized in that, Includes the following steps: Step 1: Determine the optimization objective for the hydrodynamic gas radial bearing; Step 2: Select the optimization parameters for the hydrodynamic gas radial bearing and set the parameter value range; Step 3: Construct a sample library for optimized design of dynamic pressure gas radial bearings; Step 4: Establish an approximate model for the optimized design of the hydrodynamic gas radial bearing; Step 5: Use optimization algorithms to find an optimal solution for the hydrodynamic gas radial bearing; Step 6: Select a design scheme for a hydrodynamic gas radial bearing.

2. The foil-type hydrodynamic gas radial bearing optimization design method as described in claim 1, characterized in that, In step 1, the optimization objectives for the dynamic pressure gas radial bearing are determined to include maximizing the load capacity, minimizing the gas film temperature rise, and increasing the minimum gas film thickness.

3. The foil-type hydrodynamic gas radial bearing optimization design method as described in claim 1, characterized in that, In step 2, the optimized parameters for the hydrodynamic gas radial bearing include rotational speed n, eccentricity ε, average bearing clearance δ, foil overlap angle θ, and corrugated foil stiffness Ks. The rotational speed n is between 12,000 and 100,000 r / min, the eccentricity ε is between 0.6 and 0.9, the average bearing clearance δ is between 5 and 20 μm, the foil overlap angle θ is between 20° and 120°, and the corrugated foil stiffness Ks is between 10 and 100 Gpa.

4. The foil-type hydrodynamic gas radial bearing optimization design method as described in claim 1, characterized in that, In step 3, a certain number of samples are generated using the Latin hypercube method based on a greedy strategy. These samples are then used to calculate the bearing capacity, maximum air film temperature rise, and minimum air film thickness performance indicators for each sample group. The design space set from step 2 is divided into equally probable intervals to form a Latin hypercube grid structure. Following the greedy strategy, starting from the initial point, the optimal sample point is selected each time to ensure that the samples are uniformly distributed in all dimensions. In each step, based on the local optimum strategy, including the maximum bearing capacity, maximum air film temperature rise, and minimum air film thickness, the problem scale is gradually reduced until the design requirements are met.

5. The optimized design method for foil-type hydrodynamic gas radial bearings as described in claim 1, characterized in that, In step 4, a backpropagation (BP) neural network is used to construct the surrogate model.

6. The foil-type hydrodynamic gas radial bearing optimization design method as described in claim 5, characterized in that, Using a backpropagation (BP) neural network to build a proxy model specifically includes the following steps: Step 41: Determine network parameters; set input parameters according to requirements. and output parameters Then, the number of layers in the neural network and the number of neurons in each layer are set. Then, set the connection weights between the input layer neurons, hidden layer neurons, and output layer neurons. and Set the threshold for hidden layer neurons and the threshold of the output layer neurons Finally, set the learning rate. ; Step 42: Calculate the hidden layer output; the input information is forward-propagated through the connection weights and thresholds of each layer to obtain the hidden layer output. The output value of the i-th neuron in the input layer is: The output value of the j-th neuron in the hidden layer is: The output value of the k-th neuron in the output layer is: Step 43: Error backpropagation; The hidden layer output value from step 42 is compared with the expected output value to calculate the prediction error. The error information is then passed to the input layer, and the connection weights of each layer are updated simultaneously. , and , ; Output layer weight update: Hidden weight update: Output layer threshold update: Hidden layer threshold update: Step 44: When the error reaches the set threshold or the number of iterations reaches the upper limit, stop the iteration process; otherwise, continue to execute steps 42 and 43. Step 45: Complete training and obtain results; obtain the network connection weights after stopping iterations. , and threshold , .

7. The optimized design method for foil-type hydrodynamic gas radial bearings as described in claim 1, characterized in that, In step 5, an optimized scheme for the dynamic pressure gas radial bearing was found using the UCT algorithm, namely the improved multi-objective particle swarm optimization algorithm (MOPSO).

8. The foil-type hydrodynamic gas radial bearing optimization design method as described in claim 7, characterized in that, The specific steps for finding an optimal solution for a hydrodynamic gas radial bearing using optimization algorithms include the following: Step 51: Set initial parameters for multi-objective particle swarm optimization: First, several key parameters need to be set, including population size, maximum number of iterations, and inertia weight. The population size is set to 100, the maximum number of iterations is set to 1000, the inertia weight w is set to 0.4, and the individual learning factor c1 and the swarm learning factor c2 are set to 2. A non-dominated sorting strategy is used to guide the optimization process, gradually approaching the Pareto front solution in each iteration. Step 52: Initialize the population: The patrol strategy is adopted; 50% of the individuals in the initial population are randomly distributed in the solution space, while the other 50% of the individuals are distributed according to a specific pattern. Step 53: Update the external archive: During the optimization process, fast non-dominated sorting is used to evaluate the position of each particle. Non-dominated solutions refer to solutions that cannot be dominated by other solutions under all objectives, forming the Pareto front. These non-dominated solutions are added to the external archive. Step 54, Global Optimal Particle Selection: Sort the particles by crowding and select the particle with the best fitness. The UCT algorithm is used to select the particle. Step 55, Particle State Update: In each iteration, the velocity and position of the particle are updated by comparing the current individual optimal solution with the global optimal solution. Each particle calculates a new search direction based on its current velocity and position and updates its velocity accordingly. Step 56, Iteration process: The solution of the particles is continuously updated through multiple iteration steps. In each iteration, the particles in the population will be adjusted according to the current global optimal solution and further explore in the solution space. This process will continue until the maximum number of iterations is reached, or the optimization process converges to a stable solution. Multiple objective functions such as load-bearing capacity, air film thickness, and temperature rise will be weighed and optimized in each iteration to ensure that the final design meets multiple requirements. Step 57: Selection: A set of non-dominated solutions generated during the optimization process will be used as candidate solutions. Based on the specific working environment and design requirements, such as bearing load, speed, and ambient temperature, the design scheme that best meets the requirements will be selected.

9. The foil-type hydrodynamic gas radial bearing optimization design method as described in claim 8, characterized in that, In step 54, the UCT algorithm selects particles by including the following steps: Step 541: Build the search tree; The UCT algorithm first selects candidate particles as the root node of the current situation based on the "crowding" of the particles. Step 542: Selecting Nodes; After the search tree is established, the UCT algorithm obtains the UCB1 value of each node through simulation. The UCB1 value is used to evaluate the potential of the current node and the necessity of exploration. The node with the largest UCB1 value is selected for further search. In the design of dynamic pressure gas radial bearings, the UCB1 value can comprehensively evaluate multiple indicators such as the load-bearing capacity, temperature rise, and gas film thickness of the design scheme. Through this process, the algorithm can prioritize those potentially excellent design schemes and further expand its search path. Step 543: Repeat the search; After selecting a node, the algorithm will determine whether the node is a leaf node. If the node is not a leaf node, it means that the node can continue to expand, that is, there is still potential for further search. Therefore, starting from the current child node, repeat step 542 to continue to find the best solution. Step 544: Update path values; When the algorithm reaches a leaf node, it has completed an exploration of the current design point. At this time, the system will calculate the UCB1 value of the leaf node and update the UCB1 value of all nodes on the path from the root node to the leaf node using this value. Step 545: Continue to expand; After evaluating a leaf node, the node will be used as the new root node, and the search will continue to find the leaf node with the largest UCB1 value in the next round. If the leaf node has been visited, it will be skipped to avoid duplicate calculations. Step 546: Loop Search; Repeat the previous search process until the predetermined simulation limit is reached; Step 547: Determine the optimal solution; After all search paths are completed, the UCT algorithm will select the node with the largest total UCB1 value from all child nodes of the root node as the optimal node, and the particle corresponding to this node is the local optimal solution.

10. The optimized design method for a foil-type hydrodynamic gas radial bearing as described in claim 1, characterized in that, In step 6, among multiple optimization schemes, a design scheme that meets the performance, stability, and other key indicator requirements is selected based on specific operational needs. Through evaluation and comparison of the optimization results, a dynamic pressure gas bearing parameter scheme with high load capacity, low aerodynamic and thermal characteristics, and a wide safety margin is chosen. In the particle swarm optimization algorithm, the number of particles is between 100 and 10,000, the inertia weight is 0.75, and the control individual experience and group experience coefficients in the learning factor are 0.5 and 0.5, respectively. The penalty function method is used for local optimization. The upper confidence interval algorithm (UCT) is an algorithm that optimizes decisions by balancing exploration and utilization, defining an upper confidence level 'a' of 90%. 95%,UL To find the minimum value of 'a' that satisfies the following conditions, the exploration parameter range is specified as π∈(0.98,1). By establishing a multi-objective particle swarm optimization method based on an improved UCT algorithm, the maximum bearing capacity F, the minimum aerodynamic thermal peak value ΔTmax, and the minimum local gap height δ in the dataset are obtained. min Parameter set: By assigning different weight coefficients to the above three optimization parameters, an optimization parameter set matching the requirements is obtained.

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