Quantitative hydrostatic thrust bearing multi-objective optimization design method

Abstract

The invention relates to the field of hydrostatic bearings, in particular to a multi-objective optimization design method for a quantitative hydrostatic thrust bearing, and provides the multi-objective optimization design method for the quantitative hydrostatic thrust bearing aiming at the multi-dimensional and nonlinear characteristics of the structural optimization problem of the hydrostatic bearing. According to the method, multiple targets of the hydrostatic bearing are optimized by applying a multi-target particle swarm algorithm, and the problems of subjectivity and complexity of weight determination of a weighted summation method and the problem that an equivalent optimal solution cannot be found when a Pareto leading edge is non-convex are avoided; improving parameters of the multi-target particle swarm algorithm to improve the convergence speed of the algorithm and prevent particles from falling into local optimum; according to the method, on the basis of a deduced theoretical mathematical model of the hydrostatic bearing, an optimization model is established with the axial oil film rigidity, the overturning bearing capacity and the temperature rise performance as objective functions, MATLAB software is used for designing and applying an improved multi-objective particle swarm optimization algorithm for structural parameter optimization, the theoretical rigidity of the bearing is improved by 78.75%, and the theoretical overturning bearing capacity is improved by 37.17%; model division grids are established according to parameters before and after optimization, fluid simulation is carried out, oil cavity pressure distribution is obtained, the bearing rigidity is calculated, it is verified through simulation comparison that the bearing rigidity is improved by 73.81%, and the method is beneficial to improving the design efficiency and the bearing performance of the hydrostatic bearing.

Description

Technical Field

[0001] The invention belongs to the field of liquid hydrostatic bearings, and relates to an optimization design method for a fan-shaped multi-oil pad liquid hydrostatic bearing, and in particular to a multi-objective optimization design method for a quantitative liquid hydrostatic thrust bearing. Background Art

[0002] Based on the advantages of liquid hydrostatic bearings in the field of mechanical processing, such as strong load-bearing capacity, flexible speed range, high precision and outstanding stability, the demand for various bearing structures in high-end CNC machine tools has been increasing to meet the development needs of large-scale, heavy-loaded and fully automatic control of CNC machine tools. Ultra-precision manufacturing equipment has widely used liquid hydrostatic bearings as key motion support components of spindle systems and feed systems.

[0003] The structural parameters of liquid hydrostatic bearings affect the oil film carrying capacity, power consumption, temperature rise, etc., which in turn determine the performance and life of the bearings. Therefore, the design of hydrostatic bearings is a multi-objective optimization process that must comprehensively consider all aspects. Although the empirical design method can ensure that the bearing has a reasonable structure, it does not necessarily ensure that the design result has good performance. Relevant scholars have conducted a lot of research on the optimization design of hydrostatic bearings, mostly solving a single objective or converting multi-objective problems into single-objective problems through different means and then using single-objective optimization to solve them. However, the optimization problem of liquid hydrostatic bearings has multi-dimensional and nonlinear characteristics. The above methods do not achieve multi-objective optimization in the true sense, and it is difficult to ensure that the best optimization results are found. Summary of the invention

[0004] In view of the multi-dimensional and nonlinear characteristics of the structural optimization problem of liquid hydrostatic bearings, the present invention provides a quantitative multi-objective optimization design method for liquid hydrostatic thrust bearings. The multi-objective particle swarm optimization algorithm is used to improve the parameters and applied to the oil cavity structure optimization design of the hydrostatic bearing. The improved multi-objective particle swarm optimization algorithm is used to find the optimal oil cavity structure parameters of the hydrostatic bearing, with improving the oil film stiffness and the maximum overturning bearing moment and reducing the temperature rise as the optimization goals.

[0005] To achieve the above purpose, the implementation process of a quantitative liquid hydrostatic thrust bearing multi-objective optimization design method provided by the present invention is as follows:

[0006] The first step is to establish a theoretical mathematical model based on the principle of liquid hydrostatic bearing combined with the gap flow formula, and gradually derive the expressions for calculating the axial load characteristics, overturning load characteristics, power loss and temperature rise of the bearing; and establish an optimization mathematical model with the axial oil film stiffness, overturning load capacity and temperature rise performance of the bearing as the target;

[0007] The second step is to use MATLAB software to design and optimize the multi-objective particle swarm algorithm for the hydrostatic bearing mathematical model, analyze and improve the key parameters of the algorithm, and perform multi-objective optimization design of the bearing structure parameters;

[0008] The third step is to establish the thrust bearing oil cavity model in SOLIDWORKS software according to the structural parameters before and after optimization, and use ICEM software to divide the thin-wall oil film grid and import it into FLUENT fluid simulation software for flow field steady-state simulation. The simulation compares the performance improvement effect after parameter optimization.

[0009] Compared with the prior art, the method of the present invention uses a multi-objective particle swarm algorithm to optimize multiple objectives of a liquid hydrostatic bearing, avoiding the subjectivity and complexity of determining weights using the weighted summation method and the problem of being unable to find an equivalent optimal solution when the Pareto front is non-convex; improving the parameters of the multi-objective particle swarm algorithm in the method of the present invention can increase the convergence speed of the algorithm while avoiding particles from falling into a local optimum; the method of the present invention is based on a theoretical mathematical model of a hydrostatic bearing, and establishes an optimization model with axial oil film stiffness, overturning bearing capacity and temperature rise performance as objective functions, and uses MATLAB software to design an improved multi-objective particle swarm algorithm to optimize structural parameters, so that the theoretical stiffness of the bearing is increased by 78.75%, and the theoretical overturning bearing capacity is increased by 37.17%; a fluid simulation comparison based on the parameters before and after optimization shows that the bearing stiffness is increased by 73.81%, verifying the feasibility of the method of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 Optimization design flow chart of the method of the present invention

[0011] Figure 2 Schematic diagram of the structure of a hydrostatic bearing

[0012] Figure 3 Axial oil pad structure diagram

[0013] Figure 4 Relationship between overturning force and oil pad position

[0014] Figure 5 The Pareto optimal solution set obtained by the algorithm DETAILED DESCRIPTION

[0015] Taking a bidirectionally opposed four-oil pad thrust bearing with an oil return groove structure as an example, the present invention is specifically described in conjunction with the accompanying drawings. The structure of the liquid hydrostatic bearing is as follows: Figure 2 As shown, when working, the oil pump delivers the hydraulic oil drawn from the oil tank to the oil cavity through each oil supply hole, so that an oil film is formed between the gaps on each side, thereby maintaining a pure liquid friction state.

[0016] Step 1: Establish a theoretical numerical model

[0017] (1) Calculation of axial load capacity

[0018] The thrust bearing adopts a bidirectionally opposed four-oil pad with oil return groove ring fan-shaped oil chamber structure. The single oil chamber structure is as follows Figure 3 As shown, the equivalent bearing area method is used to calculate its bearing capacity. The dotted area represents the equivalent bearing area of ​​the oil pad. The equivalent radius R of the equivalent bearing area e The area of ​​the equivalent region A e It is expressed as:

[0019]

[0020] Where R 1 is the inner diameter of the oil pad; R 2 is the inner diameter of the oil chamber; R 3 is the outer diameter of the oil chamber; R 4 R is the outer diameter of the oil pad. e1 , R e2 ,θ e They represent the inner diameter, outer diameter and center angle of the equivalent load-bearing area respectively, and the calculation formula is as follows:

[0021]

[0022] where θ 1 is the central angle of a single oil pad; θ 2 is the central angle of a single oil cavity. Taking the ratio of the axial oil film offset to the oil film clearance, i.e., the axial offset rate ε, as the design index, the oil film stiffness calculation formula of the uniformly distributed four-oil pad liquid hydrostatic thrust bearing is derived as follows:

[0023]

[0024] Where μ is the dynamic viscosity of the hydraulic oil; q is the oil supply flow rate of a single oil chamber; and h is the initial oil film clearance of the bearing axial design.

[0025] (2) Calculation of overturning bearing capacity

[0026] The equivalent oil film thickness of each oil seal gap is calculated by differential integration, and then the calculation is further simplified, the oil seal gap is simplified to the plate gap, and the load-bearing characteristics of the eccentrically loaded bearing are analyzed efficiently. Under the overturning force, the bearing tilts in the horizontal direction. The angle formed by the overturning force direction and the center line direction of the oil pad is β. After the tilt, the angle between the oil film and the horizontal direction is α, and the oil film has the largest axial displacement e α , the ratio of the axial offset to the axial initial clearance, i.e. the tilt displacement rate ε α As a calculation parameter, the position relationship between the oil chamber and the direction of overturning moment is as follows: Figure 4 As shown. The equivalent oil film thickness expression of each sealing edge on the i-th oil chamber is derived as follows:

[0027]

[0028] The calculation formula for the oil chamber pressure of the i-th oil pad is:

[0029]

[0030] The maximum load-bearing overturning moment calculation formula of the evenly distributed four-oil pad opposed thrust bearing is:

[0031]

[0032] (3) Calculation of power loss and temperature rise

[0033] The power consumption of the oil system is expressed as follows:

[0034]

[0035] The friction loss expression generated by the friction resistance that needs to be overcome when the bearing rotates and shears the oil film is calculated using the differential integration method:

[0036]

[0037] Where n is the rated operating speed. The simplified temperature rise calculation formula of the hydraulic oil is further obtained as:

[0038]

[0039] Where c is the specific heat capacity of the axial pipeline oil, and ρ is the density of the axial pipeline oil.

[0040] (4) Determine variable boundaries and objective functions

[0041] The main design variables that affect bearing performance include axial clearance, oil cavity width, and oil cavity angle. Considering that the oil pad width is quantitative, the ratio of the oil cavity to the oil pad width k is used instead of the oil cavity width. The axial clearance h, oil cavity width ratio k, and oil cavity angle θ are respectively 2 As a variable. The variable oil chamber width ratio k is set as X 1 ; Variable oil chamber angle θ 2 Set to X 2 ; The variable axial clearance h is set to X 3 , the boundary conditions of the oil chamber structure parameters are restricted as follows:

[0042]

[0043] where k min , k max ,θ min ,θ max 、h min With h maxThe limit parameters are limited based on the processing and manufacturing level and to avoid extreme situations. Taking the oil film stiffness G, bearing temperature rise T and maximum load-bearing overturning moment M as the target performance, the optimal solution algorithm is usually to find the mathematical minimum value, so the objective function is set to convert the target performance into the minimum value. The objective function form is as follows:

[0044]

[0045] Step 2: Multi-objective optimization algorithm design

[0046] (1) Improvement of multi-objective particle swarm optimization algorithm

[0047] In the mathematical description of the particle swarm algorithm, pbt represents the optimal position experienced by the particle during the flight, which is called the individual learning sample, and gbt represents the optimal position in the population, which is called the global learning sample of the particle. At the time t of the iteration, the position of the i-th particle is x i (t) , the speed is v i (t) , the optimal position experienced by the i-th individual is pbt i (t) , the global optimal position in the population is gbt i (t) , then the velocity and position of the particle in the next time unit are expressed as follows:

[0048]

[0049] Where t is the current iteration number; w is the set inertia weight, which affects the moving speed of particles in the search space; c 1 With c 2 Represents the learning factor, which adjusts the flying speed of the particle during the iteration process; rd 1 With rd 2 is a random number in the range of 0 to 1. The randomness of speed update is introduced to avoid local particle loops. In order to provide smoother weight changes during the iteration process, achieve a more natural transition between global search and local search, and avoid jumps or mutations, the cosine decreasing inertia strategy is applied to balance global search and local search. The cosine strategy inertia weight expression is as follows:

[0050]

[0051] Where the maximum inertia weight w max With the minimum inertia weight w min ; T is the maximum number of iterations. Individual learning factor c 1 The component of the particle velocity moving toward its optimal position, the group learning factor c 2The components representing the speed of the particle toward the global optimal position correspond to the individual's memory ability and learning ability. In order to ensure that the individual has a strong learning ability in the early stage and speed up the search; in the later stage, the global optimal state is sought and the fine search ability is improved. The expression of the improved learning factor is as follows:

[0052]

[0053] Where c 1max With c 2max is the maximum individual learning factor and the maximum group learning factor; c 1min With c 2min is the minimum individual learning factor and the minimum group learning factor.

[0054] (2) Run the program to obtain the optimization results

[0055] Use MATLAB software to design and run the multi-objective particle swarm optimization program, combine the algorithm accuracy and the performance of the computing tool to set the multi-objective particle swarm algorithm population number N, select and verify the number of iterations T for the solution set to be stable, and obtain the Pareto optimal solution set as follows Figure 5 As shown in the figure, the parameters corresponding to the particle position can be retrieved from the external archive. The parameters corresponding to the optimal solution position conversion are selected from the optimal solution set and rounded. The numerical results of various performance parameters before and after optimization are shown in Table 1.

[0056] Step 3: Simulation verification of oil film performance optimization effect

[0057] In SOLIDWORKS software, a simplified model of a single oil chamber of a hydrostatic thrust bearing was established and converted into IGS format and imported into the ICEM module for manual meshing. The oil inlet, oil chamber and oil film structure were divided into three layers, and different sizes of layers were set to construct a hexahedral structured grid with good quality. The Fluent module was used to perform steady-state analysis on the flow field grid model. After the calculation was completed, the flow field pressure distribution diagram was exported to obtain the pressure in the oil chamber. The same steps were used to establish the model using the parameters before and after optimization, and the oil chamber pressure was simulated, and the corresponding oil film stiffness simulation result G was calculated. f , as shown in Table 1.

[0058] After optimizing the structural parameters using the designed multi-objective algorithm, the numerical calculation results show that the bearing oil film stiffness is increased by 78.75%, the temperature rise is reduced by 0.99%, and the maximum load-bearing overturning moment is increased by 37.17%. At the same time, the simulation verification shows that the bearing oil film stiffness is increased by 73.81%. The simulation results are similar to the numerical calculation results.

[0059] Through the above example analysis, it is concluded that the method of the present invention effectively improves the axial stiffness and overturning load capacity of the bearing in the multi-objective optimization design of the quantitative liquid hydrostatic thrust bearing without increasing the temperature rise, which verifies that the method of the present invention is feasible and effective.

[0060] Table 1 Comparison of some parameters before and after bearing optimization

[0061]

Claims

1. A multi-objective optimization design method for a quantitative hydrostatic thrust bearing, characterized in that: The method includes the following steps: S1 Based on the principle of liquid hydrostatic bearing combined with the gap flow formula, a theoretical mathematical model is established to gradually derive the expressions for calculating the axial load characteristics, overturning load characteristics, power loss and temperature rise of the bearing; an optimization mathematical model is established with the axial oil film stiffness, overturning load capacity and temperature rise performance of the bearing as the target; S2 designs a multi-objective particle swarm optimization algorithm based on the mathematical model of hydrostatic bearings, analyzes and improves the key parameters of the algorithm, and uses MATLAB software to run the algorithm for multi-objective optimization design of bearing structural parameters; S3 established the thrust bearing oil cavity model in SOLIDWORKS software according to the structural parameters before and after optimization, and used ICEM software to divide the thin-wall oil film grid and imported it into FLUENT fluid simulation software for steady-state flow field simulation. The simulation compared the performance improvement effect after parameter optimization.

2. The multi-objective optimization design method for a quantitative hydrostatic thrust bearing according to claim 1 is characterized in that: The bearing is composed of a bearing shaft, a bearing shell and a thrust plate. The working oil chamber in the thrust direction is the axial thrust oil chamber and the axial opposed oil chamber. It adopts a bidirectional opposed four-oil pad fan-shaped oil chamber structure with an oil return groove. Each oil chamber is supplied with equal flow oil by a constant flow oil supply system.

3. The multi-objective optimization design method for a quantitative hydrostatic thrust bearing according to claim 1 is characterized in that: The specific implementation process of this method is as follows: Step 1: Establish a theoretical numerical model (1) Calculation of axial load capacity The thrust bearing adopts a bidirectionally opposed four-oil pad with an oil return groove ring fan-shaped oil cavity structure. The equivalent bearing area method is used to calculate its load capacity. The equivalent radius R of the equivalent bearing area e The area of ​​the equivalent region A e It is expressed as: Where R1 is the inner diameter of the oil pad, R2 is the inner diameter of the oil chamber, R3 is the outer diameter of the oil chamber, and R4 is the outer diameter of the oil pad. e1 R represents the inner diameter of the equivalent load-bearing area; e2 Represents the outer diameter of the equivalent load-bearing area; θ e It represents the central angle of the equivalent load-bearing area, and the calculation formula is as follows: Where θ1 is the center angle corresponding to a single oil pad; θ2 is the center angle corresponding to a single oil cavity. Taking the ratio of the axial oil film offset to the oil film gap, i.e., the axial offset rate ε, as the design index, the oil film stiffness of the uniformly distributed four-oil pad hydrostatic thrust bearing is: Where μ is the dynamic viscosity of the hydraulic oil; q is the oil supply flow rate of a single oil chamber; and h is the initial oil film clearance of the bearing axial design. (2) Calculation of overturning bearing capacity The equivalent oil film thickness of each oil seal gap is calculated by differential integration, and then the calculation is further simplified, the oil seal gap is simplified to the plate gap, and the load-bearing characteristics of the eccentrically loaded bearing are analyzed efficiently. Under the overturning force, the bearing tilts in the horizontal direction. The angle formed by the overturning force direction and the center line direction of the oil pad is β. After the tilt, the angle between the oil film and the horizontal direction is α, and the oil film has the largest axial displacement e α , the ratio of the axial offset to the axial initial clearance, i.e. the tilt displacement rate ε α As the calculation parameter, the equivalent oil film thickness expression of each oil sealing edge on the i-th oil chamber is derived as follows: Among them, β1, β2, β3, and β4 are the angles between the sealing edge of the oil pad and the direction of the overturning force, and β5 and β6 are the angles between the circumferential edge of the oil pad's equivalent bearing area and the direction of the overturning force. The calculation formula for the oil chamber pressure of the i-th oil pad is: The maximum load-bearing overturning moment calculation formula of the evenly distributed four-oil pad opposed thrust bearing is: (3) Calculation of power loss and temperature rise The power consumption of the oil system consists of two parts. The power expression of the oil pump to overcome the resistance of the oil chamber and output the oil to each axial oil chamber is: The friction loss expression generated by the friction resistance that needs to be overcome when the bearing rotates and shears the oil film is calculated using the differential integration method: Where n is the rated operating speed. The simplified temperature rise calculation formula of the hydraulic oil is: Where c is the specific heat capacity of the axial pipeline oil, and ρ is the density of the axial pipeline oil. (4) Determine variable boundaries and objective functions The main design variables that affect bearing performance include axial clearance, oil chamber width, and oil chamber angle. Considering that the oil pad width is quantitative, the ratio of the oil chamber to the oil pad width k is used instead of the oil chamber width. The axial clearance h, the oil chamber width ratio k, and the oil chamber angle θ2 are used as variables. The variable oil chamber width ratio k is set to X1; the variable oil chamber angle θ2 is set to X2; the variable axial clearance h is set to X3, and the boundary conditions for the oil chamber structural parameters are as follows: where k min is the minimum oil cavity width ratio, k max is the maximum oil cavity width ratio, θ min is the minimum angle of the oil chamber, θ max is the maximum angle of the oil chamber, h min is the minimum axial clearance, h max The maximum axial clearance is defined above based on the consideration of the manufacturing level and the avoidance of extreme situations. The oil film stiffness G, bearing temperature rise T and maximum load-bearing overturning moment M are taken as the target performance. The optimal solution algorithm is usually to find the mathematical minimum value. Therefore, the objective function is set to convert the target performance into the minimum. The objective function form is as follows: f=min[-G,-M,T] T (11) Step 2: Multi-objective optimization of parameters (1) Improvement of multi-objective particle swarm optimization algorithm In the mathematical description of the particle swarm algorithm, pbt represents the optimal position experienced by the particle during the flight, which is called the individual learning sample, and gbt represents the optimal position in the population, which is called the global learning sample of the particle. At the time t of the iteration, the position of the i-th particle is x i (t) , the speed is v i (t) , the optimal position experienced by the i-th individual is pbt i (t) , the global optimal position in the population is gbt i (t) , then the velocity and position of the particle in the next time unit are expressed as follows: In the formula, t is the current iteration number; w is the set inertia weight, which affects the movement speed of the particle in the search space; c1 and c2 represent learning factors, which adjust the flight speed of the particle during the iteration process; rd1 and rd2 are random numbers in the range of 0 to 1, and the randomness of speed update is introduced to avoid local particle loops. In order to provide smooth weight changes during the iteration process, achieve a more natural transition between global search and local search, and avoid jumps or mutations, the cosine decreasing inertia strategy is applied to balance global search and local search. The cosine strategy inertia weight expression is as follows: Where the maximum inertia weight w max With the minimum inertia weight w min ; T is the maximum number of iterations. The individual learning factor c1 represents the component of the particle velocity toward its own optimal position, and the group learning factor c2 represents the component of the particle velocity toward the global optimal position, which correspond to the individual's memory ability and learning ability respectively. In order to ensure that the individual has strong learning ability in the early stage and speed up the search speed; in the later stage, seek the global optimum and improve the fine search ability, the improved learning factor expression is as follows: Where c 1max With c 2max is the maximum individual learning factor and the maximum group learning factor; c 1min With c 2min is the minimum individual learning factor and the minimum group learning factor. (2) Run the program to obtain the optimization results The MATLAB software is used to design and run the multi-objective particle swarm optimization program. The number of multi-objective particle swarm algorithm populations N is set in combination with the algorithm accuracy and the performance of the computing tool. The number of iterations T for the stability of the solution set is selected and verified to obtain the Pareto optimal solution set. The parameters corresponding to the location of the particles can be retrieved from the external archive. The parameters corresponding to the optimal solution position are selected from the optimal solution set and rounded, and the numerical results of various performance parameters before and after optimization are calculated. Step 3: Simulation verification of oil film performance optimization effect A single oil chamber model of a hydrostatic thrust bearing was established in SOLIDWORKS software; After the model is simplified, the converted format is imported into the ICEM meshing software module for manual meshing. The oil inlet, oil cavity and oil film structure are divided into three layers, and different sizes of layers are set to construct a hexahedral structured grid with good quality. Import the divided grid model into Fluent fluid simulation software for parameter setting, simulate and analyze the steady-state flow field pressure distribution, and obtain the bearing oil chamber pressure after the calculation is completed; The same steps are used to establish the model with the parameters before and after optimization, and the meshing and flow field simulation are performed to obtain the oil chamber pressure. The corresponding oil film stiffness simulation results are calculated according to the oil chamber pressure, and the consistency of the simulation results is compared with the numerical results.

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