Multi-objective optimization design method and system for hydrostatic bearing

Abstract

This invention discloses a multi-objective optimization design method and system for hydrostatic bearings. Relating to the technical field of hydrostatic bearings, this method employs an iterative solution using analytical expressions for bearing performance without the need for mesh generation. While ensuring the accuracy of core physical laws, it avoids cumbersome mesh generation and large-scale numerical calculations, significantly reducing the time required for a single performance evaluation. This makes large-scale population evolution and global optimization possible within a limited timeframe, significantly improving the efficiency of the hydrostatic bearing design process. Simultaneously, a multi-objective optimization model is established, encompassing maximizing hot stiffness, minimizing total system power consumption, and minimizing bearing temperature rise. This outputs a Pareto optimal solution set, providing designers with a set of non-dominated optimal design schemes. This overcomes the shortcomings of traditional single-objective optimization, which focuses on only a single indicator while neglecting other key performance aspects, enabling the final design parameters to achieve an optimal balance among various performance indicators.

Description

Technical Field

[0001] This invention relates to the technical field of hydrostatic bearings, and more particularly to a multi-objective optimization design method and system for hydrostatic bearings. Background Technology

[0002] As a core component of high-end CNC machine tools, the design accuracy of hydrostatic bearings directly affects the machining performance of the machine tool. Most existing design theories are based on the isothermal assumption, treating the viscosity of the lubricating oil as a constant in calculations. However, in high-speed electric spindle applications, such as at speeds exceeding 40,000 rpm, the high-speed rotation of the spindle causes severe shear heat generation in the oil film, leading to a significant decrease in viscosity and subsequent softening of the oil film stiffness. This strong coupling effect of "heat-viscosity-pressure" makes the calculation results of traditional methods tend to be idealized, deviating significantly from actual operating conditions. This can easily cause design parameters to deviate from actual working conditions, and even lead to failures. Therefore, these methods are difficult to directly apply to optimization design processes requiring thousands of iterations, and cannot balance computational accuracy and efficiency. Summary of the Invention

[0003] The purpose of this invention is to overcome the shortcomings of the prior art. This invention provides a multi-objective optimization design method and system for hydrostatic bearings.

[0004] This invention provides a multi-objective optimization design method for hydrostatic bearings, including: The design boundary and initialization variables of the hydrostatic bearing are obtained. The spindle design speed, external working load, system oil supply pressure and oil inlet temperature are input as boundary conditions, and multiple design variables are determined. At the same time, based on the basic equations of fluid mechanics and the law of conservation of energy, the analytical expression of bearing performance without meshing is derived. For any given set of design variables, iterative calculations are performed using the bearing performance analytical formula. At this time, the current temperature is initialized and the corresponding viscosity is solved. The oil chamber pressure is then analyzed by substituting it into the bearing performance analytical formula. Based on the pressure field, shear and pumping power consumption are calculated to correct the temperature until the temperature converges, thereby outputting the analytical results corresponding to the set of design variables. The geometric parameters of the hydrostatic bearing are defined as optimization variables. A multi-objective optimization is established with the goals of maximizing hot oil film stiffness, minimizing total system power consumption, and minimizing bearing temperature rise. Constraints including maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness are set. Initialize a population containing multiple sets of design variables, call the corresponding analysis result for each individual in the population to determine the corresponding fitness value, until the corresponding termination condition is met, thereby outputting the corresponding optimal solution set, and determining the structural design parameters of the hydrostatic bearing based on the optimal solution set.

[0005] This invention provides a multi-objective optimization design system for hydrostatic bearings, which is applied to the aforementioned multi-objective optimization design method for hydrostatic bearings; the multi-objective optimization design system for hydrostatic bearings includes: The design variable module is used to obtain the design boundary and initialization variables of the hydrostatic bearing. The spindle design speed, external working load, system oil supply pressure and oil inlet temperature are input as boundary conditions, and multiple design variables are determined. At the same time, based on the basic equations of fluid mechanics and the law of conservation of energy, the analytical expression of bearing performance without meshing is derived. The analytical results module is used to perform iterative calculations using the bearing performance analytical formula for any given set of design variables. At this time, the current temperature is initialized and the corresponding viscosity is solved. The oil pressure is then substituted into the bearing performance analytical formula to analyze the oil chamber pressure. Based on the pressure field, the shear and pumping power consumption are calculated to correct the temperature until the temperature converges, thereby outputting the analytical results corresponding to the set of design variables. The multi-objective optimization module is used to define the geometric parameters of the hydrostatic bearing as optimization variables, establish multi-objective optimization content with the objectives of maximizing hot oil film stiffness, minimizing total system power consumption and minimizing bearing temperature rise, and set constraints including maximum allowable temperature rise, minimum oil film thickness and minimum stiffness. The design parameter module is used to initialize a population containing multiple sets of design variables. For each individual in the population, the corresponding analysis result is called to determine the corresponding fitness value until the corresponding termination condition is met, thereby outputting the corresponding optimal solution set. Based on the optimal solution set, the structural design parameters of the hydrostatic bearing are determined.

[0006] Compared with the prior art, the beneficial effects of the present invention are: (1) By constructing a pressure-temperature-viscosity closed-loop iterative solution mechanism, the temperature field and viscosity field are corrected in real time during the calculation process, which can accurately reproduce the multi-field coupled physical process of high-speed heat generation-viscosity decrease-stiffness change. This makes the design result no longer the theoretical value under ideal working conditions, but a true reflection of the actual performance under high-speed hot conditions. At the same time, the bearing performance analytical formula without meshing is used for iterative solution. Under the premise of ensuring the accuracy of the core physical laws, the cumbersome meshing and large-scale numerical calculation are avoided, which greatly shortens the time of a single performance evaluation. This makes it possible to carry out large-scale population evolution and global optimization within a limited time, which significantly improves the efficiency of the hydrostatic bearing design process.

[0007] (2) A multi-objective optimization model was established, which includes maximizing hot stiffness, minimizing total system power consumption, and minimizing bearing temperature rise. Given the physical constraints between stiffness, power consumption, and temperature rise, this invention outputs a Pareto optimal solution set through a multi-objective optimization algorithm, providing designers with a set of non-dominated optimal design schemes. This breaks the drawback of traditional single-objective optimization that focuses only on a single index while ignoring other key performance indicators, enabling the final design parameters to achieve the best balance among various performance indicators.

[0008] (3) Constraints such as maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness were introduced. These constraints constitute the safety boundary of the design, and from a physical perspective, prevent failure modes such as lubricant failure due to excessive temperature rise, metal-to-metal contact wear due to excessively thin oil film, or machining vibration due to excessively low stiffness. Through the constraint-guided optimization process, it is ensured that the final output structural design parameters not only have superior theoretical performance, but also have the safety and reliability for engineering implementation. Attached Figure Description

[0009] Figure 1 This is a flowchart illustrating the multi-objective optimization design method for hydrostatic bearings in an embodiment of the present invention. Figure 2 This is a simplified flowchart illustrating a specific application scenario of the multi-objective optimization design method for hydrostatic bearings in this invention embodiment; Figure 3 This is a schematic diagram of the structural parameters of the hydrostatic thrust bearing in an embodiment of the present invention; Figure 4 This is a detailed structural schematic diagram of the hydrostatic radial bearing in an embodiment of the present invention; Figure 5 This is a schematic diagram of the structural composition of the multi-objective optimization design system for the hydrostatic bearing in an embodiment of the present invention. Detailed Implementation

[0010] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0011] Please see Figure 1 to Figure 5 A multi-objective optimization design method for hydrostatic bearings is proposed and applied to hydrostatic bearing scenarios. The multi-objective optimization design method for hydrostatic bearings includes: Step S11: Obtain the design boundary and initialization variables of the hydrostatic bearing. Input the spindle design speed, external working load, system oil supply pressure and oil inlet temperature as boundary conditions, and determine multiple design variables. At the same time, based on the basic equations of fluid mechanics and the law of conservation of energy, derive the analytical expression of bearing performance without meshing. Step S12: For any given set of design variables, iterative calculation is performed using the bearing performance analytical formula. At this time, the current temperature is initialized and the corresponding viscosity is solved. The oil chamber pressure is analyzed by substituting it into the bearing performance analytical formula. The shear and pumping power consumption are calculated based on the pressure field to correct the temperature until the temperature converges, thereby outputting the analytical results corresponding to the set of design variables. Step S13: Define the geometric parameters of the hydrostatic bearing as optimization variables, establish a multi-objective optimization framework with the goals of maximizing hot oil film stiffness, minimizing total system power consumption, and minimizing bearing temperature rise, and set constraints including maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness. Step S14: Initialize a population containing multiple sets of design variables, call the corresponding analysis result for each individual in the population to determine the corresponding fitness value, until the corresponding termination condition is met, thereby outputting the corresponding optimal solution set, and determining the structural design parameters of the hydrostatic bearing based on the optimal solution set.

[0012] In step S11, the specific steps are as follows: S111: Input the spindle design speed, external working load, system oil supply pressure and oil inlet temperature as the boundary conditions for the design, and determine multiple design variables including throttle fluid resistance, oil seal width, oil chamber length, bearing inner diameter, oil film thickness and number of oil chambers. S112: Based on the fundamental equations of fluid mechanics and the law of conservation of energy, derive analytical expressions for bearing performance without the need for mesh generation; The specific analytical formula for bearing performance includes the flow balance equation, the power consumption and shear heat generation equation, the heat balance and temperature rise equation, and the viscosity-temperature coupling equation. By combining the flow rate formulas for the flow through the throttle and the flow rate formulas for the flow through each oil sealing surface of the bearing, a flow balance equation with respect to the oil chamber pressure is constructed. A total power consumption analytical expression is established, which includes the power consumption caused by shear friction due to spindle rotation and the power consumption caused by pumping due to oil pump extrusion, to present the corresponding power consumption and shear heat generation equations. An analytical formula for the energy balance between the heat generated by oil film shear and the heat carried away by lubricating oil flow is established to derive the expression for the average temperature rise of the oil film, thus presenting the corresponding heat balance and temperature rise equations. The Reynolds viscosity-temperature index equation is introduced to construct the corresponding viscosity-temperature coupling equation.

[0013] In step S111, the specific steps are as follows: At this point, input the spindle design speed. n External working load W System oil supply pressure P s and oil inlet temperature T 0Among them, the spindle design speed n is the key excitation source that determines the oil film shear heat generation rate; the external working load W defines the minimum load capacity threshold that the bearing must meet; and the system oil supply pressure... P s This determines the upper limit of the oil chamber pressure and the limit of the oil pump's oil supply capacity; the inlet oil temperature T 0 The initial reference temperature, used for thermal balance iterative calculations, directly affects the initial viscosity setting of the lubricating oil. These four parameters constitute the physical constraints of the bearing operating environment and remain constant during the optimization process.

[0014] Determine including the fluid resistance of the throttle. R 0 Multiple design variables, including the oil sealing edge width, oil cavity length, bearing inner diameter, oil film thickness, and number of oil cavities, collectively define the physical configuration of the bearing. Adjusting their values ​​will directly change the flow resistance distribution of the oil film, the effective bearing area, and the heat dissipation characteristics. The selection of design variables must take into account both manufacturing feasibility and performance sensitivity, aiming to find the best balance among conflicting performance indicators through the ergonomic optimization of parameter combinations.

[0015] In step S112, the specific steps are as follows: At this point, by combining the flow rate formulas for the flow through the throttle and the flow rate formulas for the flow through each sealing surface of the bearing, a flow balance equation for the oil chamber pressure is constructed. The essence of this equation is the manifestation of the law of conservation of mass within the closed oil chamber, that is, the volumetric flow rate of the lubricating oil entering the oil chamber through the throttle must be equal to the sum of the volumetric flow rates overflowing from the sealing surfaces around the oil chamber.

[0016] Since the flow rate of the throttle is proportional to the square root of the difference between the supply pressure and the oil chamber pressure, or a linear relationship, depending on the type of throttle, and the oil flow rate at the sealing surface is proportional to the difference between the oil chamber pressure and the external ambient pressure, the balance between the two establishes a relationship regarding the oil chamber pressure. p r The implicit or explicit algebraic equations can be used to directly and analytically calculate the pressure distribution in the oil cavity under specific oil film thickness and viscosity conditions without the need for meshing, providing pressure field input for subsequent load-bearing capacity and stiffness calculations.

[0017] An analytical expression for total power consumption, including shear friction power consumption caused by spindle rotation and pumping power consumption caused by oil pump compression, is established to present the corresponding power consumption and shear heat generation equations. Shear friction power consumption originates from the heat generated by viscous shear force within the oil film during high-speed spindle rotation. Its value is proportional to the square of the rotational speed, the viscosity of the lubricating oil, and the area of ​​the oil film, and is the dominant factor in high-speed bearing heating. Pumping power consumption originates from the pressure work done by the lubricating oil overcoming flow resistance as it enters the oil chamber from the oil supply system and flows out of the sealing edge. Its value is proportional to the product of the flow rate and the oil supply pressure. The sum of these two constitutes the total power consumption n of the system. This analytical expression directly relates to the bearing's operating parameters and heat generation, providing a heat source term for heat balance calculations.

[0018] An analytical formula for the energy balance between the heat generated by oil film shear and the heat carried away by the lubricating oil flow is established to derive the expression for the average temperature rise of the oil film, thus presenting the corresponding heat balance and temperature rise equations. It is assumed that the bearing is in a steady-state heat transfer process, and all the heat generated inside the oil film due to viscous shear is carried away by the flowing lubricating oil through convection heat transfer, neglecting the small amount of heat conducted to the bearing bush and the main shaft. Based on this assumption, the equations are used to... N=ρcQtΔT Derive the expression for the average temperature rise of the oil film. ΔT=N / (ρcQt) ,in r For the density of lubricating oil, c The specific heat capacity of lubricating oil, Qt The equation establishes a direct relationship between temperature rise, power consumption, and flow rate, and reveals the physical law that oil film temperature rise intensifies with increasing rotational speed and decreases with increasing flow rate.

[0019] A corresponding viscosity-temperature coupling equation is constructed by introducing the Reynolds viscosity-temperature index equation. Because the viscosity of lubricating oil is extremely sensitive to temperature changes and exhibits nonlinear decay characteristics, traditional models assuming constant viscosity will lead to serious calculation errors. Therefore, by introducing the equation... μ(T)=μ 0 e -γ(T-T0) ,in m 0 Reference temperature T 0 The dynamic viscosity below, c As the viscosity-temperature characteristic coefficient, a quantitative description of viscosity change with temperature is established. This equation feeds back the aforementioned temperature rise calculation results to the viscosity parameters in the flow and pressure equations, thereby opening up a multi-field coupled closed loop of "pressure-flow-power consumption-temperature rise-viscosity" and ensuring that the analytical model can truly reflect the bearing performance under high-speed hot conditions.

[0020] In step S12, the specific steps are as follows: S121: For any given set of design variables, perform a closed-loop analytical iterative solution of "pressure-temperature-viscosity": initialize the temperature of the current iteration step to the oil inlet temperature T0 or the value of the previous iteration, and calculate the current dynamic viscosity according to the viscosity-temperature coupling equation; substitute the dynamic viscosity into the flow balance equation and analytically solve for the pressure of each oil chamber; S122: Based on the obtained pressure and shear force, calculate the current heat generation using the power consumption and shear heat generation equation, and then analyze the corrected average oil film temperature using the heat balance and temperature rise equation; check the convergence condition, determine whether the temperature difference between two adjacent iterations is less than the set threshold. If it is not satisfied, update the temperature and viscosity and return to continue the loop. If it is satisfied, determine that the calculation has converged, and output the analytical results corresponding to the set of design variables. The analytical results include the final hot state oil film stiffness, total system power consumption and bearing temperature rise under the set of structural parameters.

[0021] In step S121, the specific steps are as follows: At this point, for any given set of design variables, such as the bearing inner diameter D Oil film thickness H r0 Wait, perform initialization operations: set the temperature of the current iteration step. T (k) The value assigned is "oil inlet temperature" T 0 The average oil film temperature calculated during the first cycle or the previous iteration. T (k-1) Substitute this temperature value into the Reynolds viscosity-temperature index equation constructed in step S112. μ(T)=μ 0 e -γ(T-T0) This equation utilizes the viscosity-temperature characteristic coefficient of lubricating oil. c Calculate the current temperature T (k) Corresponding dynamic viscosity m (k) This step is crucial because it transforms the state variable of temperature into the physical properties of fluid motion, providing the necessary material constant inputs for subsequent solutions based on fluid dynamics equations.

[0022] dynamic viscosity m (k) Substituting the flow balance state equation constructed in step S112, which is essentially about the oil chamber pressure... p r The implicit algebraic equation, in its physical meaning, states that the flow rate of lubricating oil entering the oil chamber through the throttle depends on the fluid resistance of the throttle. R 0 With pressure difference Ps -p r It must equal the flow rate overflowing from the sealing surface around the oil cavity, which depends on the viscosity. m (k) Seal edge geometry and pressure difference p r -P ambient By analytically solving this equation, the pressure values ​​of each oil cavity at the current hot viscosity can be directly obtained without complex mesh generation and numerical integration. pr (k) This pressure value reflects the actual load-bearing capacity and stiffness of the bearing oil film after taking into account the viscosity changes caused by the current temperature. It is a necessary prerequisite for subsequent calculations of shear heat generation and pumping power consumption.

[0023] In step S122, the specific steps are as follows: At this point, based on the pressure of each oil chamber obtained in step S121 pr (k) In addition to the shear force generated by the spindle rotation, the total heat generated in the current iteration step is calculated using the power consumption and shear heat generation equation established in step S112. This heat generation consists of two parts: one is the shear friction power consumption generated by the high-speed rotation of the spindle overcoming the viscous resistance of the oil film. N t Secondly, there is the pumping power consumption Np caused by the oil pump supplying high-pressure oil; substituting the calculated total power consumption into the heat balance and temperature rise equation, i.e. ΔT (k) =（N t +N p ) / rcQ t The current temperature rise is calculated analytically, and the corrected average oil film temperature is derived by combining the oil inlet temperature T0. T new =T 0 +ΔT (k) This step represents a leap from mechanical parameters (pressure and shear force) to thermal parameters (temperature), establishing a direct mapping relationship between energy dissipation and temperature rise.

[0024] Perform convergence condition verification to determine the current corrected average oil film temperature. T new Compared with the initial temperature of the current iteration step T (k) Is the absolute value of the difference less than the set threshold? e 0 That is, to determine | Tnew -T (k) | ≤e 0 Whether this condition is met; if it is not met, it indicates that the temperature field is not yet stable and the coupling relationship between viscosity and pressure field is still fluctuating. In this case, the temperature setpoint needs to be updated, i.e., let T (k +1) =T new Based on this, the dynamic viscosity is corrected, and the pressure is calculated again in step S121 until the temperature difference meets the threshold requirement. If the condition is met, the calculation is considered to have converged, indicating that the multi-field coupling of "pressure-temperature-viscosity" has reached a state of equilibrium, and the iteration loop is exited.

[0025] Once the iteration converges, the analytical results corresponding to this set of design variables are output. These results include the final hot-state oil film stiffness under this set of structural parameters. K Total system power consumption P total and bearing temperature rise ΔT Among them, oil film stiffness K The actual stiffness value takes into account the decrease in viscosity due to temperature rise, and is closer to real-world operating conditions than the traditional isoviscosity assumption; total system power consumption. P total This reflects the energy loss cost during bearing operation; bearing temperature rise. ΔT These are the core indicators for evaluating bearing thermal stability and preventing the risk of lubricating oil vaporization. These output parameters will serve as fitness values ​​in subsequent multi-objective optimization algorithms to evaluate the merits of this set of design variables.

[0026] In step S13, the specific steps are as follows: S131: Define the geometric parameters of the hydrostatic bearing as optimization variables, and then establish a multi-objective optimization mathematical model. In the multi-objective optimization mathematical model, mark the corresponding multi-objective optimization content. The multi-objective optimization content pursues the maximization of hot oil film stiffness, the minimization of total system power consumption, and the minimization of bearing temperature rise. S132: Set constraints to ensure the safety of the bearing during actual high-speed operation. The constraints include the maximum allowable temperature rise constraint, the minimum oil film thickness constraint, and the minimum stiffness constraint.

[0027] In step S131, the specific steps are as follows: At this point, the geometric parameters of the hydrostatic bearing are defined as optimization variables. These variables constitute the individual gene expression in the optimization algorithm and directly determine the physical configuration of the bearing. For each optimization variable, a reasonable range of values ​​and upper and lower bounds need to be set according to the actual engineering situation. For example, the bearing inner diameter needs to conform to the standard shaft system size series, and the oil film thickness needs to be greater than the minimum safety threshold. The selection of optimization variables should follow the sensitivity principle, that is, prioritize the selection of geometric features that have a significant impact on the objective function as variables, such as the throttle parameters that determine the fluid resistance characteristics, the oil sealing edge width that determines the effective bearing area and flow resistance, and the oil film thickness that determines the gap size, so as to reduce the computational dimension while ensuring the optimization effect.

[0028] A multi-objective optimization mathematical model is established, and the corresponding multi-objective optimization content is marked in the model. This multi-objective optimization content aims to solve conflict problems that are difficult to balance in traditional design, specifically including: Pursuing maximum stiffness of hot oil film: Stiffness is the core indicator for evaluating the bearing's ability to resist deformation under external loads, and it directly determines the machining accuracy and dynamic stability of the spindle system; the emphasis here is on "hot state", which specifically refers to the actual stiffness value after convergence through step S12 iterative solution, reflecting the true stiffness characteristics after considering the viscosity decrease caused by temperature rise.

[0029] Pursuing the minimization of total system power consumption: Power consumption reflects the economy and energy efficiency of bearing operation. Total power consumption includes shear power consumption and pumping power consumption, the values ​​of which are directly related to the heat source intensity and cooling cost of the spindle system.

[0030] Minimize bearing temperature rise: Temperature rise is a key indicator for ensuring the safe operation of bearings. Excessive temperature rise can cause the viscosity of lubricating oil to drop sharply or even vaporize, leading to thermal seizure.

[0031] The three objective functions mentioned above often conflict in terms of physical mechanism: increasing stiffness usually requires reducing oil film thickness or increasing viscosity, but this will significantly increase shear power consumption and temperature rise; therefore, it is necessary to establish a vector function that includes the above three objectives and seek the globally optimal Pareto solution set.

[0032] In step S132, the specific steps are as follows: At this point, the maximum allowable temperature rise constraint is set. ΔT≤ΔT max When a hydrostatic bearing operates at high speed, the shear heat generated inside the oil film can cause a sharp rise in temperature. If the temperature rise exceeds the set threshold, it will not only cause the viscosity of the lubricating oil to decrease exponentially, damaging the oil film's load-bearing capacity, but may also cause oxidation and deterioration of the lubricating oil or even vaporization, leading to lubrication failure. In addition, excessively high temperature rise can cause thermal deformation of the bearing and the spindle, compromising the fit accuracy. Therefore, this constraint is designed to prevent the risk of "thermal seizure and burn" and oil film vaporization, ensuring that the bearing operates within the thermal stability boundary.

[0033] Set minimum oil film thickness constraint h min ≥h limit The core advantage of hydrostatic bearings lies in their all-fluid lubrication, meaning the journal and bearing bush are completely separated by a continuous pressure oil film; if "design variables: such as initial oil film thickness..." H r0 "If the oil film thickness is too small, or if the oil film thickness is reduced to less than the safety threshold value of the sum of the roughness of the two surfaces and the geometric error under external load, boundary lubrication or even dry friction will occur, leading to rapid wear or burning of the bearing. This constraint condition ensures the reliability of bearing operation from a geometric dimension by limiting the minimum oil film thickness threshold."

[0034] Setting minimum stiffness constraints K≥Kmin Although maximizing the stiffness of the hot oil film is one of the optimization objectives, an insurmountable minimum threshold must be set during the optimization process. If the optimization algorithm simply pursues low power consumption or low temperature rise, it may tend to give a design scheme with extremely low stiffness, resulting in insufficient ability of the bearing to resist cutting forces under actual working conditions, causing machining chatter and vibration, which seriously affects machining accuracy and surface quality. This constraint ensures that the optimization scheme meets the machine tool spindle's hard stiffness requirements and prevents "insufficient stiffness from causing machining chatter".

[0035] In step S14, the specific steps are as follows: S141: Initialize the structural parameter population using a multi-objective optimization mechanism; take the geometric parameters of each individual in the population as input, call the corresponding analytical results, and calculate their respective fitness values, which include thermal stiffness, power consumption, and temperature rise; S142: Population evolution is carried out through crossover, mutation, and non-dominated sorting selection operations until the maximum number of generations is reached, and then the corresponding optimal solution set is output. Designers select the set of parameters that best fits the specific working conditions from the optimal solution set as the final structural design parameters of the hydrostatic bearing.

[0036] In step S141, the specific steps are as follows: At this point, a multi-objective optimization mechanism (such as NSGA-II or MOEA / D algorithm) is used to initialize the structural parameter population. This process randomly generates a certain number of individuals (i.e., the population size) within the design variable boundary conditions defined in step S131, such as the range of fluid resistance of the throttle and the range of the width of the sealing edge. Each individual represents a set of hydrostatic bearing design schemes to be evaluated, and its gene encoding directly corresponds to the combination of bearing geometric parameters. This process aims to provide a diverse starting point for the optimization search, ensuring that the algorithm can cover the potential design space and avoid getting trapped in local optima.

[0037] Using the geometric parameters of each individual in the population as input, and calling the corresponding analytical results, this process essentially inputs each "virtual design prototype" generated by the algorithm into the "pressure-temperature-viscosity" closed-loop analytical iterative program constructed in step S12. For each individual in the population, the system executes a complete iterative solution process: starting from the initial temperature, it goes through pressure solution, power consumption calculation, temperature rise correction and convergence determination, until the thermal performance value under the specific geometric configuration is output. This step takes advantage of the fast calculation speed of the analytical method to quickly complete the performance evaluation of large-scale design samples.

[0038] The fitness values ​​of each component are calculated. The fitness value is the basis for the optimization algorithm's natural selection, and its value directly reflects the quality of the design scheme. For the multi-objective optimization content of this invention, the output fitness values ​​specifically include: thermal stiffness. K As a positive indicator, a higher value is better; total system power consumption. P total (As a negative indicator, the smaller the value, the better) and bearing temperature rise. ΔT As a negative indicator, the smaller the value, the better. This set of three-dimensional vectors ( K,P total ,ΔT This constitutes the coordinate position of each individual on the Pareto front, providing core data support for subsequent non-dominated sorting and crossover / mutation operations.

[0039] In step S142, the specific steps are as follows: At this point, population evolution is carried out through crossover, mutation, and non-dominated sorting selection operations.

[0040] Crossover operation: Simulating the biological reproduction process, the geometric parameters of superior individuals in the parent population are randomly paired and exchanged to generate offspring individuals with a combination of parental characteristics. The aim is to preserve and combine superior structural features, such as combining the oil sealing edge width of a high-rigidity bearing with the oil film thickness of a low-power bearing.

[0041] Mutation operation: Randomly changes the values ​​of some geometric parameters in an individual with a certain probability, aiming to introduce new genetic characteristics, prevent the algorithm from getting stuck in local optima, and maintain the diversity of the population.

[0042] Selection Operation: Based on "fitness value: thermal stiffness, power consumption, temperature rise: individuals are selected based on their performance to ensure that individuals with superior performance have a greater probability of entering the next generation of reproduction."

[0043] The algorithm uses a non-dominated sorting technique to hierarchically divide individuals in the population. Since the three objective functions of thermal stiffness, power consumption, and temperature rise are mutually conflicting, there is no single global optimal solution. Based on the "Pareto dominance" relationship, the algorithm defines the solution that is not dominated by any other individual, i.e., that cannot improve other indicators without harming one indicator, as the first-level frontier, or Pareto optimal solution. The above operation is repeated until the maximum number of generations is reached. The maximum number of generations is used as the termination condition, indicating that the population has converged to a stable Pareto front, and the calculation process terminates.

[0044] The output is the corresponding optimal solution set, namely the Pareto optimal solution set. This solution set is a set of design schemes with different but non-dominant performance indicators. Designers select the set of parameters that best suits the specific working conditions from the optimal solution set as the final structural design parameters of the hydrostatic bearing. This process integrates the computational power of the algorithm with human engineering experience. Designers can accurately lock the final structural parameters from the solution set according to the actual application scenario of the bearing, such as selecting the stiffness-first scheme if high-precision machining is emphasized, or selecting the power consumption-first scheme if energy saving and consumption reduction are emphasized.

[0045] Please see Figure 2 The specific application scenarios and corresponding embodiments of this application are as follows: In step S11, for scenarios where hydrostatic bearings are used in high-speed electric spindles of high-end CNC machine tools, the R&D personnel input the spindle design speed. n For example, the speed can be set to 40,000 rpm to 50,000 rpm under high-speed conditions to simulate extreme shear heating environments; input the external working load. W This load is typically determined by inverse calculation based on the spindle cutting force, for example, set as a radial cutting load of several kilonewtons; input system oil supply pressure. P s For example, set the high-pressure oil supply condition to 4MPa to 6MPa; input the oil inlet temperature. T 0 The standard temperature is usually set to 20°C or 25°C, controlled by a constant temperature oil supply system. These parameters constitute the boundary constraints of the hydrostatic bearing under high-speed and heavy-load conditions.

[0046] For the unique structure of hydrostatic bearings, a strict parameter system is used to determine the design variables: throttle fluid resistance R 0 Defined as a key variable controlling the pressure drop at the oil chamber inlet, its specific form can be mapped to the orifice diameter of a small-hole throttle. d 0 The ratio of the inner diameter to the length of the capillary throttle directly determines the flow coefficient of the oil chamber; Oil sealing edge width: refined into "oil sealing edge width" on both sides of the oil cavity. L1 The width of the sealing edge between the axial sealing edge and the adjacent oil cavity. L 2 "Zhou Xiang sealed oil edge"; L 1 Changes in these parameters directly affect the axial leakage resistance and flow rate of the lubricating oil. L 2 Changes in this will affect the mutual interference of circumferential flow resistance between oil chambers.

[0047] Oil cavity length L r With bearing inner diameter D : L r The effective load-bearing length of the oil cavity is defined. D This is the fitting diameter of the spindle journal, and together the two determine the projected area and macroscopic dimensions of the bearing.

[0048] Radial oil film thickness H r0 : Defined as the concentric radius clearance between the journal and the bearing bush, it is the most sensitive variable affecting oil film stiffness and flow rate.

[0049] Number of oil chambers k Defined as the number of circumferentially distributed oil chambers, such as the common four-chamber or six-chamber structure, which affects the bearing's load-bearing directionality and stability.

[0050] Through the above steps, the optimization model of the hydrostatic bearing is visualized as follows: at a given rotational speed... n Load W Oil supply pressure P s and oil inlet temperature T 0 Under the boundary constraints, find an optimal set of... {R 0 ,L 1 ,L 2 ,L r ,D,H r0 ,k} The parameters are combined to solve the problems of insufficient hot stiffness and excessive temperature rise in traditional designs under high-speed conditions.

[0051] For hydrostatic bearings, after the lubricating oil enters the oil chamber through the throttle, it needs to overcome the "oil sealing edge width" on both sides of the oil chamber. L 1 : Axial flow resistance and the "sealing edge width" between adjacent oil chambers L 2 "Circumferential flow resistance" outflow; the flow balance equation is specifically constructed as: the inlet flow rate through the throttle.Q in This is equivalent to sealing the edge with oil. L 1 Axial leakage flow Q axial With sealing the edge L 2 Circumferential leakage flow Q circumferential The sum; the analytical expression will explicitly express the oil cavity pressure. p r With oil film thickness H r0 Viscosity m and geometric dimensions L 1 ,L 2 The functional relationship directly reflects the characteristics of the hydraulic resistance network of the radial bearing.

[0052] For the high-speed rotational characteristics of hydrostatic bearings, the analytical formula for shear friction power consumption needs to accurately account for the "linear velocity" of the spindle surface: derived from the rotational speed. n and bearing inner diameter D The determination is based on the integral of the viscous shear stress generated within the oil film thickness Hr0; the pumping power consumption is considered in terms of increasing the lubricating oil pressure from the supply pressure Ps to the oil chamber pressure. p r And the energy consumption to overcome the resistance of the sealing edge; the total power consumption is expressed as follows: P total =N t +N p The main heat source mechanism of hydrostatic bearings in high-speed electric spindle applications was accurately quantified.

[0053] For calculating the oil film temperature rise of hydrostatic bearings, the specific expression of the thermal balance formula is: average oil film temperature rise. ΔT Equal to total power consumption P total Divide by the density of the lubricating oil r Specific heat capacity c With total flow Q t The product of these factors, along with the equation, will serve as the core criterion for iterative solutions. This will be used to correct the viscosity decrease caused by temperature rise, thereby correcting the predicted values ​​of flow rate and pressure, and solving the problem of underestimating temperature rise in traditional design methods.

[0054] For specific grades of lubricating oil used in hydrostatic bearings, the Reynolds viscosity-temperature index equation is introduced. μ(T)= m 0 e -γ(T-T0)During the iterative calculation process, when the temperature rises... ΔT When the average temperature of the oil film increases, the equation rapidly corrects for the dynamic viscosity. m This causes the flow calculation of the hydrostatic bearing to automatically increase, the pressure calculation to automatically adjust due to the decrease in liquid resistance, and the change in flow coefficient to truly reproduce the multi-field coupled physical process of "high-speed rotation leading to temperature rise → viscosity decrease → stiffness and load-bearing capacity change".

[0055] In step S12, for a hydrostatic bearing, the inlet oil temperature is assumed to be... T 0 Set to 25℃; in the initial step of the iterative calculation, such as k =0, the system sets the current oil film temperature. T (0) =25℃; Using the viscosity-temperature coupling equation, substitute the reference viscosity. m 0 and viscosity-temperature coefficient c Calculate the initial dynamic viscosity m (0) If the iteration enters a subsequent loop, such as k If the value is greater than 0, then the average oil film temperature from the previous correction round will be used. T (k) Update viscosity; for example, if the temperature rises to 40°C, the viscosity will decrease significantly according to an exponential law, thus realistically simulating the physical phenomenon of lubricating oil thinning at high temperatures.

[0056] For multi-cavity structures of hydrostatic bearings, such as a typical four-cavity structure, the current viscosity... m (k) Substitute into the flow balance equation; the specific analytical process requires solving the simultaneous equations: Flow rate through the throttle Q in =f(R 0 ,P s ,p r (k) ) .

[0057] Leakage flow through the bearing seal edge Q out =f(μ (k) ,L 1 ,L 2 ,L r ,D,H r0 ,p r(k) ) .

[0058] The analytical equations will directly solve for the pressure values ​​of each oil chamber; for example, at the spindle design speed n, due to the initial assumption of high viscosity, the calculated oil chamber pressure... p r (0) The pressure may be too high; and in subsequent iterations, if the temperature rise causes a decrease in viscosity, the extracted oil chamber pressure will be... p r (k) The pressure field will be dynamically adjusted accordingly to accurately reflect the nonlinear variation law of the load-bearing capacity of the hydrostatic bearing under high-speed hot conditions. This step provides an accurate pressure field input for calculating the subsequent "shear power consumption: dependent on pressure distribution and viscosity".

[0059] For hydrostatic bearings, assuming the oil chamber pressure calculated in the current iteration step is... p r (k) bearing inner diameter D The linear velocity at that point is extremely high, such as a rotational speed of 40,000 rpm; the system calculates the shear power consumption. N t Its value mainly depends on the oil film thickness. H r0 Internal velocity gradient and viscosity m (k) Simultaneously calculate pumping power consumption. N p Its value depends on the oil supply pressure. P s With total flow Q t Using the heat balance equation ΔT=(N t +N p） rcQ t Calculate the temperature rise; for example, if the calculated temperature rise is... N t Extremely large and Q t Smaller, temperature rise ΔT It will increase significantly, leading to a decrease in the corrected average oil film temperature. T new It is much higher than the oil inlet temperature, which truly reflects the heat accumulation effect under high-speed rotation.

[0060] For the iterative process of hydrostatic bearings, a threshold is set. e 0 The value is a minimum, such as 0.01℃. In the early stages of iteration, the assumed low temperature leads to an overestimation of viscosity, which in turn leads to an overestimation of pressure and an underestimation of flow rate, resulting in an inflated temperature rise.T new Often higher than the initial setting value T (k) The convergence condition is not met; the system then updates the temperature. T (k+1) =T new The viscosity setting is reduced, and the pressure and flow rate are recalculated. Through multiple iterations, the temperature difference between the two cycles is reduced to less than 0.01℃. At this point, the oil film is in thermal equilibrium, eliminating the calculation error caused by viscosity-temperature coupling.

[0061] For hydrostatic bearings, the system outputs the final result after the iteration converges; for example, it outputs the set of design variables. {R 0 ,L 1 ,L 2 ,… } Corresponding hot oil film stiffness K This value takes into account the stiffness "softening" phenomenon caused by viscosity decrease at high temperatures, and is more accurate than traditional calculations; it outputs the total power consumption of the system, Ptotal, to evaluate the energy consumption of the high-speed electric spindle; and it outputs the bearing temperature rise. ΔT These three indicators are used to verify whether the operating temperature of the lubricating oil exceeds the upper limit. They will serve as the fitness function of multi-objective optimization algorithms (such as NSGA-II), guiding the algorithm to evolve towards "high stiffness, low power consumption, and low temperature rise".

[0062] In step S13, an optimization variable vector is constructed for the hydrostatic bearing. X=[R 0 ,L 1 ,L 2 ,L r ,H r0 ] Assuming the bearing inner diameter D With the number of oil chambers k Due to limitations in host interface size, it will be treated as a fixed value for now, or included in the variables; specifically defined as: throttle fluid resistance R 0 As an optimization variable, its value is mapped to the orifice diameter or capillary size, which determines the initial flow coefficient.

[0063] Sealing edge width L 1 and L 2As an optimization variable, it directly affects the hydraulic resistance distribution and effective bearing area of ​​the oil cavity in the bearing, and is a key adjustment parameter for balancing flow rate and stiffness.

[0064] Radial oil film thickness H r0 As the most sensitive optimization variable, even small changes in it can non-linearly affect flow rate, stiffness, and power consumption.

[0065] For the multi-objective optimization mathematical model of hydrostatic bearings, the following objective function vector is constructed. F(X) .

[0066] First objective function f 1 (X) Maximize the stiffness of the hot oil film K That is, by using an algorithm to find a set of geometric parameters, the bearing can still maintain sufficient resistance to deformation after the temperature rise and viscosity decrease caused by high-speed rotation, thus ensuring the cutting rigidity of the hydrostatic bearing in the high-speed electric spindle.

[0067] Second objective function f 2 (X) Minimize total system power consumption P total This means reducing the shear friction loss of the hydrostatic bearing and the energy consumption of the oil pump under high-speed conditions such as 40,000 rpm, thereby improving the overall energy efficiency ratio of the machine.

[0068] Third objective function f 3 (X) Minimize bearing temperature rise ΔT By optimizing parameter combinations, such as appropriately increasing the flow rate to remove heat or adjusting the oil seal to reduce shear stress, the average oil film temperature can be controlled to prevent thermal seizure failure of the hydrostatic bearing due to thermal expansion. Therefore, the optimization problem of the hydrostatic bearing is formalized into solving... min[-f 1 (X),f 2 (X),f 3 (X)] The multi-objective programming problem.

[0069] For the application of hydrostatic bearings in high-speed electric spindles, the following settings are made: ΔT max The operating temperature is usually determined based on the characteristics of the lubricating oil; for example, if a specific brand of lubricating oil is used, its maximum permissible operating temperature is 80°C, and the inlet oil temperature is... T 0 If the temperature is 25℃, then the maximum allowable temperature rise is... ΔT maxIt should be set to 55℃; during the optimization iteration process, for any set of design variables, if the temperature rise calculated in step S12 is... ΔT If the temperature exceeds 55℃, the proposed solution is deemed infeasible, and a penalty function is directly applied or the solution is discarded, thereby preventing the vaporization of lubricating oil or thermal seizure of the hydrostatic bearing during actual operation.

[0070] For hydrostatic bearings, considering the surface roughness and form and position tolerances of the spindle journal and bearing bush, a minimum safe oil film thickness needs to be set. h limit For example, according to empirical formulas h limit The overall surface roughness value can be set to 3 to 5 times, such as 10 μm; in the design variables, although the radial oil film thickness is reduced... H r0 It can significantly improve stiffness, but strict limitations must be imposed during the optimization search process. H r0 The lower limit of the value or the calculated minimum working oil film thickness h min Not lower than h limit This ensures that the hydrostatic bearing is always in a state of full film lubrication, avoiding catastrophic failure due to direct metal-to-metal contact under high-speed rotation.

[0071] For high-precision machining scenarios supported by hydrostatic bearings, a minimum stiffness threshold is set. K min For example, based on the machine tool cutting vibration resistance requirements, the oil film stiffness must be set not to be lower than 200 N / μm; during the optimization process, even if a certain set of parameters has extremely low power consumption, if its hot stiffness... K If the calculated value is below 200 N / μm, the scheme will be considered to have violated the constraints and will be eliminated; this ensures that the hydrostatic bearing design scheme must have the basic anti-deformation capability required to support precision machining while pursuing energy saving and consumption reduction.

[0072] In step S14, for the hydrostatic bearing, the population size is set to... n ,For example N =100; the algorithm randomly generates 100 sets of design variable combinations within the set boundaries; for example, individual 1 may correspond to parameter combinations. {R 0 =0.5,L 1 =5mm,H r0 =15m m,… } Individual 2 may correspond to a completely different set of parameters, which include the throttle fluid resistance and the width of the sealing edge. L 1,L 2 Oil cavity length L r and radial oil film thickness H r0 These key geometric features together constitute the initial pool of candidate solutions for hydrostatic bearings.

[0073] For each individual hydrostatic bearing, the system automatically invokes the thermal iterative solver in step S12; for example, for individual bearing 1, input its H r0 With parameters such as 15μm, the iterative program quickly calculates the spindle speed under this structure. n The actual operating state at 40,000 rpm; the calculation result was extracted as the fitness value: If the analysis results show that the individual's viscosity drops sharply due to excessive temperature rise at high speed, the final calculated hot stiffness will be... K It may only be 150 N / μm, and the total power consumption of the system is... P total Up to 2.5kW, bearing temperature rise ΔT It reached 40℃.

[0074] This set of values ​​(150, 2.5, 40) is stored as the fitness vector of individual 1 and used for lateral comparison in the population. Through this step, the optimization algorithm can clearly identify which hydrostatic bearing designs have achieved a better balance between stiffness, power consumption and temperature rise, thereby guiding the subsequent evolutionary direction.

[0075] For hydrostatic bearings, the algorithm performs generational updates on the population; for example, in a certain iteration, parent individual A has a narrower "oil sealing edge width". L 1 "High flow rate and low temperature rise" are related to the thicker "oil film thickness" of parent individual B. H r0 "Low power consumption and low stiffness"—the offspring generated through crossover operations may inherit the narrow sealing edge of A and the thick oil film of B, forming a new design scheme; at the same time, mutation operations may randomly fine-tune the flowmeter's liquid resistance. R 0 To explore undiscovered fluid resistance matching modes, thereby gradually approaching the optimal structural configuration of hydrostatic bearings.

[0076] For the three objective functions of hydrostatic bearings, non-dominated sorting divides the solution space into different levels. For example, one solution has high stiffness but also high temperature rise, while another solution has low temperature rise but slightly higher power consumption. If the two solutions cannot completely dominate each other, they are both retained on the Pareto front. The algorithm iterates continuously. If the maximum number of generations is set to 200, as the number of generations increases, the individuals in the population gradually gather towards a "specific curve: the Pareto front", indicating that the algorithm has found the solution set region that achieves the best balance between stiffness, power consumption, and temperature rise. At this point, the calculation is terminated.

[0077] The algorithm ultimately outputs the Pareto optimal solution set for hydrostatic bearings; designers face multiple alternative solutions: Option A: Extremely high hot stiffness, such as 300 N / μm, but high temperature rise, such as 35℃; Option B: Extremely low temperature rise, such as 20℃, low power consumption, but moderate stiffness, such as 200N / μm.

[0078] If a hydrostatic bearing is used in a high-precision grinding machine, it is extremely sensitive to machining vibration. Designers will select the solution with better stiffness, option A, from the solution set as the final structural design parameter. If it is used in a high-speed electric spindle, heat dissipation is difficult, so option B with lower temperature rise is selected. Through this step, the final implementation from mathematical solution set to engineering product is realized.

[0079] Please see Figure 5 The multi-objective optimization design system for hydrostatic bearings is applied to the aforementioned multi-objective optimization design method for hydrostatic bearings; the multi-objective optimization design system for hydrostatic bearings includes: The design variable module 21 is used to obtain the design boundary and initialization variables of the hydrostatic bearing. The spindle design speed, external working load, system oil supply pressure and oil inlet temperature are input as boundary conditions, and multiple design variables are determined. At the same time, based on the basic equations of fluid mechanics and the law of conservation of energy, the analytical formula of bearing performance without meshing is derived. The analysis result module 22 is used to perform iterative calculations using the bearing performance analytical formula for any given set of design variables. At this time, the current temperature is initialized and the corresponding viscosity is solved. The oil chamber pressure is analyzed by substituting it into the bearing performance analytical formula. The shear and pumping power consumption are calculated based on the pressure field to correct the temperature until the temperature converges, thereby outputting the analysis result corresponding to the set of design variables. The multi-objective optimization module 23 is used to define the geometric parameters of the hydrostatic bearing as optimization variables, establish multi-objective optimization content with the objectives of maximizing hot oil film stiffness, minimizing total system power consumption and minimizing bearing temperature rise, and set constraints including maximum allowable temperature rise, minimum oil film thickness and minimum stiffness. The design parameter module 24 is used to initialize a population containing multiple sets of design variables, call the corresponding analysis results for each individual in the population to determine the corresponding fitness value, until the corresponding termination condition is met, thereby outputting the corresponding optimal solution set, and determining the structural design parameters of the hydrostatic bearing based on the optimal solution set.

[0080] It should be noted that although multiple modules are mentioned in the detailed description above, this division is not mandatory; in fact, according to the embodiments of this disclosure, the features and functions of two or more modules or described above can be embodied in one module; conversely, the features and functions of one module described above can be further divided into multiple modules to be embodied.

[0081] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A multi-objective optimization design method for hydrostatic bearings, characterized in that, include: The design boundary and initialization variables of the hydrostatic bearing are obtained. The spindle design speed, external working load, system oil supply pressure and oil inlet temperature are input as boundary conditions, and multiple design variables are determined. At the same time, based on the basic equations of fluid mechanics and the law of conservation of energy, the analytical expression of bearing performance without meshing is derived. For any given set of design variables, iterative calculations are performed using the bearing performance analytical formula. At this time, the current temperature is initialized and the corresponding viscosity is solved. The oil chamber pressure is then analyzed by substituting it into the bearing performance analytical formula. Based on the pressure field, shear and pumping power consumption are calculated to correct the temperature until the temperature converges, thereby outputting the analytical results corresponding to the set of design variables. The geometric parameters of the hydrostatic bearing are defined as optimization variables. A multi-objective optimization is established with the goals of maximizing hot oil film stiffness, minimizing total system power consumption, and minimizing bearing temperature rise. Constraints including maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness are set. Initialize a population containing multiple sets of design variables, call the corresponding analysis result for each individual in the population to determine the corresponding fitness value, until the corresponding termination condition is met, thereby outputting the corresponding optimal solution set, and determining the structural design parameters of the hydrostatic bearing based on the optimal solution set.

2. The multi-objective optimization design method for hydrostatic bearings according to claim 1, characterized in that, The process involves obtaining the design boundaries and initialization variables of the hydrostatic bearing by inputting the spindle design speed, external working load, system oil supply pressure, and oil inlet temperature as boundary conditions, and determining multiple design variables. Simultaneously, based on the fundamental equations of fluid mechanics and the law of conservation of energy, analytical expressions for bearing performance without mesh generation are derived, including: Input the spindle design speed, external working load, system oil supply pressure and oil inlet temperature as design boundary conditions, and determine multiple design variables including throttle fluid resistance, oil seal width, oil chamber length, bearing inner diameter, oil film thickness and number of oil chambers.

3. The multi-objective optimization design method for hydrostatic bearings according to claim 2, characterized in that, The process involves obtaining the design boundaries and initialization variables of the hydrostatic bearing by inputting the spindle design speed, external working load, system oil supply pressure, and oil inlet temperature as boundary conditions, and determining multiple design variables. Furthermore, based on the fundamental equations of fluid mechanics and the law of conservation of energy, the analytical expression for bearing performance without mesh generation is derived, including: Based on the fundamental equations of fluid mechanics and the law of conservation of energy, we derive analytical expressions for bearing performance that do not require mesh generation. The specific analytical formula for bearing performance includes the flow balance equation, the power consumption and shear heat generation equation, the heat balance and temperature rise equation, and the viscosity-temperature coupling equation. By combining the flow rate formulas for the flow through the throttle and the flow rate formulas for the flow through each oil sealing surface of the bearing, a flow balance equation with respect to the oil chamber pressure is constructed. A total power consumption analytical expression is established, which includes the power consumption caused by shear friction due to spindle rotation and the power consumption caused by pumping due to oil pump extrusion, to present the corresponding power consumption and shear heat generation equations. An analytical formula for the energy balance between the heat generated by oil film shear and the heat carried away by lubricating oil flow is established to derive the expression for the average temperature rise of the oil film, thus presenting the corresponding heat balance and temperature rise equations. The Reynolds viscosity-temperature index equation is introduced to construct the corresponding viscosity-temperature coupling equation.

4. The multi-objective optimization design method for hydrostatic bearings according to claim 1, characterized in that, For any given set of design variables, iterative calculations are performed using the bearing performance analytical formula. At this point, the current temperature is initialized and the corresponding viscosity is solved. The oil chamber pressure is then analyzed by substituting this into the bearing performance analytical formula. Shear and pumping power consumption are calculated based on the pressure field to correct for the temperature until the temperature converges. The analytical results corresponding to this set of design variables are then output, including: For any given set of design variables, perform a closed-loop analytical iterative solution of "pressure-temperature-viscosity": initialize the temperature of the current iteration step to the inlet temperature T0 or the value of the previous iteration, and calculate the current dynamic viscosity according to the viscosity-temperature coupling equation; substitute the dynamic viscosity into the flow balance equation and analytically solve for the pressure of each oil chamber.

5. The multi-objective optimization design method for hydrostatic bearings according to claim 4, characterized in that, For any given set of design variables, iterative calculations are performed using the bearing performance analytical formula. This involves initializing the current temperature and solving for the corresponding viscosity, substituting the viscosity into the bearing performance analytical formula to analyze the oil chamber pressure, calculating shear and pumping power consumption based on the pressure field to correct the temperature, until temperature convergence, and then outputting the analytical results corresponding to that set of design variables. The process also includes: Based on the obtained pressure and shear force, the current heat generation is calculated using the power consumption and shear heat generation equations. Then, the corrected average oil film temperature is calculated analytically using the heat balance and temperature rise equations. The convergence condition is checked by determining whether the temperature difference between two adjacent iterations is less than the set threshold. If it is not met, the temperature and viscosity are updated and the loop continues. If it is met, the calculation is considered to have converged, and the analytical results corresponding to the set of design variables are output. The analytical results include the final hot-state oil film stiffness, total system power consumption, and bearing temperature rise under the set of structural parameters.

6. The multi-objective optimization design method for hydrostatic bearings according to claim 1, characterized in that, The geometric parameters of the hydrostatic bearing are defined as optimization variables. A multi-objective optimization framework is established with the goals of maximizing hot oil film stiffness, minimizing total system power consumption, and minimizing bearing temperature rise. Constraints are set, including maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness. The geometric parameters of the hydrostatic bearing are defined as optimization variables, and a multi-objective optimization mathematical model is established. In the multi-objective optimization mathematical model, the corresponding multi-objective optimization content is marked. The multi-objective optimization content pursues the maximization of hot oil film stiffness, the minimization of total system power consumption, and the minimization of bearing temperature rise.

7. The multi-objective optimization design method for hydrostatic bearings according to claim 6, characterized in that, The process defines the geometric parameters of the hydrostatic bearing as optimization variables, establishes a multi-objective optimization framework with the goals of maximizing hot oil film stiffness, minimizing total system power consumption, and minimizing bearing temperature rise, and sets constraints including maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness. It also includes: Constraints are set to ensure the safety of the bearing during actual high-speed operation. These constraints include maximum allowable temperature rise, minimum oil film thickness, and minimum stiffness.

8. The multi-objective optimization design method for hydrostatic bearings according to claim 1, characterized in that, The initialization process involves a population containing multiple sets of design variables. For each individual in the population, the corresponding analytical result is called to determine its fitness value, until the corresponding termination condition is met. This process outputs the optimal solution set, and the structural design parameters of the hydrostatic bearing are determined based on this optimal solution set, including: A multi-objective optimization mechanism is used to initialize the structural parameter population; the geometric parameters of each individual in the population are used as input, and the corresponding analytical results are called to calculate their respective fitness values, which include thermal stiffness, power consumption, and temperature rise.

9. The multi-objective optimization design method for hydrostatic bearings according to claim 8, characterized in that, The initialization process involves a population containing multiple sets of design variables. For each individual in the population, the corresponding analytical result is called to determine its fitness value, until the corresponding termination condition is met, thereby outputting the optimal solution set. Based on the optimal solution set, the structural design parameters of the hydrostatic bearing are determined. The process also includes: The population evolves through crossover, mutation, and non-dominated sorting selection operations until the maximum number of generations is reached, and then the corresponding optimal solution set is output. Designers select the set of parameters that best suits the specific working conditions from the optimal solution set as the final structural design parameters for the hydrostatic bearing.

10. A multi-objective optimization design system for hydrostatic bearings, characterized in that, The multi-objective optimization design system for the hydrostatic bearing is applied to the multi-objective optimization design method for the hydrostatic bearing as described in any one of claims 1-9; the multi-objective optimization design system for the hydrostatic bearing includes: The design variable module is used to obtain the design boundary and initialization variables of the hydrostatic bearing. The spindle design speed, external working load, system oil supply pressure and oil inlet temperature are input as boundary conditions, and multiple design variables are determined. At the same time, based on the basic equations of fluid mechanics and the law of conservation of energy, the analytical expression of bearing performance without meshing is derived. The analytical results module is used to perform iterative calculations using the bearing performance analytical formula for any given set of design variables. At this time, the current temperature is initialized and the corresponding viscosity is solved. The oil pressure is then substituted into the bearing performance analytical formula to analyze the oil chamber pressure. Based on the pressure field, the shear and pumping power consumption are calculated to correct the temperature until the temperature converges, thereby outputting the analytical results corresponding to the set of design variables. The multi-objective optimization module is used to define the geometric parameters of the hydrostatic bearing as optimization variables, establish multi-objective optimization content with the objectives of maximizing hot oil film stiffness, minimizing total system power consumption and minimizing bearing temperature rise, and set constraints including maximum allowable temperature rise, minimum oil film thickness and minimum stiffness. The design parameter module is used to initialize a population containing multiple sets of design variables. For each individual in the population, the corresponding analysis result is called to determine the corresponding fitness value until the corresponding termination condition is met, thereby outputting the corresponding optimal solution set. Based on the optimal solution set, the structural design parameters of the hydrostatic bearing are determined.

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