Cooling element of precision worm wheel gear grinding machine and its multi-objective thermal-fluid topology design optimization method

Abstract

The present invention discloses a multi-objective thermal-fluid topological design optimization method for the cooling element of a precision gear grinding machine tool, comprising the following steps: Step 1: Define the design domain: Expand the cooling element into a plane, define a two-dimensional design domain representing the cooling element's geometric structure, and discretize it using the finite element method; Step 2: Numerical modeling: Model the cooling element as a porous medium; Under the conditions of laminar incompressible flow, the fluid dynamics are governed by the dimensionless forms of the continuity equation and the momentum equation; The fluid-solid coupled heat transfer model is governed by the dimensionless energy conservation equation; Step 3: Construct an objective function: The optimization objective is defined as a weighted function of heat transfer and fluid dissipated power, and a fluid-solid heat transfer topological optimization model is constructed; Step 4: Solve the objective function; Step 5: Restore the shape and structure of the cooling element. The present invention also discloses a precision gear grinding machine tool and a cooling element.

Description

Technical Field

[0001] The present invention belongs to the technical field of optimization design, and specifically relates to a cooling element of a precision worm grinding wheel gear grinding machine tool and a multi-objective thermal-fluid topology design optimization method thereof. Background Art

[0002] The significance of precision worm gear grinding machines (GGMTs) lies in their ability to achieve tight tolerances, improving the overall quality and performance of high-performance gears, making them indispensable in industries requiring high precision, such as aerospace, automotive, and engineering. Thermal errors in GGMTs, primarily caused by uneven temperature rises across machine components, are a key factor affecting machining accuracy. In particular, the moving heat source in a GGMT, represented by the moving nut, leads to uneven temperature distribution throughout the GGMT, resulting in significant thermal errors and reduced grinding accuracy. Therefore, the temperature rise and thermal errors caused by these moving heat sources should be reduced.

[0003] The prior art believes that temperature rise is the root cause of thermal errors in GGMT. Therefore, a closed-loop circulating cooling system is used to effectively control the thermal errors caused by the heat generated by the heat source. The coolant is supplied to the hollow screw shaft (SS) in the ball screw feed drive system (BSFDS) through the circulating cooling system, thereby achieving temperature control of the SS. Although the temperature rise of the BSFDS can be reduced with the existing cooling system, the closed-loop cooling system currently used often leads to problems such as insufficient rigidity of stainless steel, uneven axial temperature distribution, and coolant leakage in rotating parts. Then the positioning accuracy and repeatability of the BSFDS, as well as the gear grinding accuracy of the entire machine tool, will not be significantly improved.

[0004] Prior art cooling systems (CEs) with serpentine and spiral channels are used to improve heat transfer efficiency and capacity. These designs are widely used in various industrial applications due to their ability to provide excellent thermal control. Serpentine cooling channels are characterized by their winding paths, which promote turbulence, enhance convective heat transfer, and ensure more uniform cooling. The geometry of these channels, including the curvature and spacing of the bends, is optimized to maximize heat transfer capacity while minimizing pressure drop. Helical cooling channels and their spiral paths offer a different approach to improving heat transfer capacity. Helical channels achieve continuous fluid motion and increased turbulence, significantly improving heat transfer capacity. The pitch, diameter, and helix angle of the spiral channels are optimized to enhance thermal and fluid dynamics. Both serpentine and spiral cooling channels have demonstrated excellent performance in thermal control. However, despite their significant advantages in improving heat transfer efficiency and capacity, serpentine and spiral cooling channels also have significant disadvantages. Serpentine channels have a high pressure drop, while spiral channels face the complexity of design optimization and potential uneven heat transfer capacity.

[0005] In summary, the existing technology has the following deficiencies:

[0006] (1) The hollow stainless steel circulating cooling system effectively controls the temperature and thermal error of the GGMT. However, the non-uniform temperature gradient along the axial direction of the hollow SS in the GGMT leads to dynamic changes in the positioning error (PE) and repeatability error (RPE) of the BSFDS.

[0007] (2) Although CE effectively reduces thermal errors, in the existing technology, CE is mostly designed based on experience and still has limitations in terms of heat transfer and thermal error control efficiency and ability. Summary of the Invention

[0008] To reduce the temperature rise of the moving heat source in precision worm wheel gear grinding machines and the thermal error of the BSFDS, thereby improving the machining accuracy of the entire GGMT, hollow stainless steel can no longer be used as a cooling element (CE). Instead, solid stainless steel should be used to improve the rigidity of the stainless steel. In addition, to ensure uniform axial temperature distribution, avoid coolant leakage in rotating parts, and reduce the temperature of the moving nut (MN), an additional CE should be designed, which contacts the MN. Due to the changes in CE, a more systematic and scientific cooling strategy and CE design method are urgently needed.

[0009] In view of this, the object of the present invention is to provide a cooling element for a precision gear grinding machine tool and a multi-objective thermal-fluid topology design optimization method thereof. By applying the designed cooling element to the moving nuts (MN) of the GGMT's X, Y, and Z axes, the heat transfer capacity and efficiency can be significantly improved, thereby improving the machining accuracy of the GGMT.

[0010] In order to achieve the above object, the present invention provides the following technical solutions:

[0011] The present invention first proposes a multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine tool, comprising the following steps:

[0012] Step 1: Define the design domain

[0013] The cooling element is unfolded into a plane, a two-dimensional design domain representing the geometric structure of the cooling element is defined, and the finite element method is used for discretization;

[0014] Step 2: Numerical modeling

[0015] The cooling element is modeled as a porous medium. Under the condition of laminar incompressible flow, the fluid dynamics is governed by the dimensionless forms of the continuity and momentum equations. The fluid-solid coupled heat transfer model is governed by the dimensionless energy conservation equations.

[0016] Step 3: Construct the objective function

[0017] In order to improve cooling performance and reduce flow resistance, the optimization objective is defined as a weighted function of heat transfer and fluid dissipation power, and a fluid-solid heat transfer topology optimization model is constructed:

[0018]

[0019] Where: J is the optimization target; J th is the heat transfer term; J f is the fluid dissipation work item; w1 and w2 are J th and J f The weight factor of Ω is the design domain; V f is the volume fraction occupied by the fluid flow path; Vol is the total volume contained in the design domain; γ is the design variable; h * is the dimensionless heat transfer coefficient; T * is the dimensionless temperature; is the dimensionless gradient operator; u * is the dimensionless flow velocity; α * is the dimensionless permeability; Vol Ω is the total volume contained in the design domain; Q is the heat load applied to the design domain; is the dimensionless inlet pressure; Γ is the flow channel; Γ in It is the flow channel entrance;

[0020] Step 4: Solve the objective function

[0021] The sensitivity is solved using the adjoint method, and the gradient calculated by the adjoint method is used to form a quadratic subproblem of the sequential quadratic programming method. The design variable γ is updated by solving the quadratic subproblem to ensure that the solution moves towards the optimal configuration until the set iterative convergence conditions are met. The topological design channel is designed in the cooling element.

[0022] Step 5: Restore the shape and structure of the cooling element.

[0023] Furthermore, in step 2, the cooling element is modeled as a porous medium by: within the fluid-solid topology design domain, the design domain is theorized as a porous material, assuming that the fluid resistance F is linearly related to the flow velocity u:

[0024] F=-αu

[0025] Where: α is the permeability;

[0026] The domain of the design domain is divided into a finite number of elements, each of which has a design variable γ, which takes values ​​between 0 and 1; when representing a solid: γ = 0, α → ∞ and F → ∞; when representing a fluid: γ = 1, α → 0 and F → 0.

[0027] Furthermore, in step 3, the fluid dynamics is represented by the continuity equation and the momentum equation by the velocity u*, pressure p*, Reynolds number Re and gradient operator Perform dimensionless representation:

[0028]

[0029] in: is the gradient operator; U and L are characteristic velocity and length respectively; ρ is the density; μ is the dynamic viscosity;

[0030] For incompressible laminar flow, the continuity equation is:

[0031]

[0032] The momentum conservation equation is:

[0033]

[0034] Among them: F * is the volume force;

[0035] According to Darcy's law, the body force F* is proportional to the velocity u*, given by α * The dimensionless permeability represented by is determined by subsequent penalty function interpolation:

[0036] F * =-α * u *

[0037]

[0038]

[0039] The penalty factor q is related to the Darcy number Da and the Reynolds number Re, resulting in the momentum equation:

[0040]

[0041] Where: q is the penalty factor; Da is the Darcy number.

[0042] Furthermore, the dimensionless energy conservation equation is given by the dimensionless temperature T * and the Prandtl number Pr, defined as follows:

[0043]

[0044] Where: T B and T w are the average temperature and the wall temperature respectively; the thermophysical properties of the fluid are characterized by its specific heat capacity C p and thermal conductivity k fis a characteristic and is a key parameter in heat transfer analysis; μ is the fluid viscosity;

[0045] The dimensionless energy conservation equation is expressed as:

[0046]

[0047] Where: u * Indicates speed; is the gradient operator; Q * represents the dimensionless heat generation rate, and the dimensionless temperature T * Proportional to:

[0048]

[0049] By introducing the design variable γ, we get:

[0050]

[0051] Furthermore, in order to avoid the checkerboard phenomenon in the design domain, density filtering in the form of Holmz partial differential equation is used:

[0052]

[0053] Where: R min is the filter radius, which is the size of the cooling element; θ is the design variable before filtering; is the filtered design variable; represents the gradient operator;

[0054] In order to solve the problem of gray elements, the hyperbolic tangent projection technique is used:

[0055]

[0056] in: is the output design variable after projection, η and β are the projection point and slope, respectively.

[0057] Furthermore, in the step 1, the cooling element is symmetrically divided into two equal halves, and the two-dimensional design domain is defined as the geometric structure representing one half of the cooling element; in the step 5, the two cooling elements are symmetrically combined into one.

[0058] The present invention also proposes a cooling element for a precision worm wheel gear grinding machine, which is designed using the multi-objective thermal-fluid topology design optimization method described above.

[0059] The present invention also proposes a precision worm wheel gear grinding machine, comprising an X-axis, a Y-axis and a Z-axis, wherein the X-axis, the Y-axis and the Z-axis are each provided with a ball screw feed drive system, wherein the ball screw feed drive system comprises a screw shaft and a movable nut cooperating with the screw shaft, wherein the movable nut comprises a nut and a nut housing, and a cooling element as described above is provided between the nut and the nut housing.

[0060] The beneficial effects of the present invention are:

[0061] Thermal errors significantly reduce the machining accuracy of precision worm-gear grinding machines, making effective thermal error control imperative. To control thermal errors, an innovative approach to directly cool the moving heat source of a gear grinding machine is proposed, replacing the existing hollow screw cooling method. A cooling element for a precision worm-gear grinding machine and its multi-objective thermal-fluid topology design optimization method are also proposed. The designed cooling element is applied to the moving nuts (MN) of the GGMT's X, Y, and Z axes. The axial thermal expansion of the screw shaft is reduced through the cooling strategy and multi-objective topology design optimization method. Results show that the topology-optimized channel exhibits superior heat transfer capability and a 2-3 times lower pressure drop than conventional serpentine cooling channels. Embedded in the ball screw feed drive system of a precision gear grinding machine, the cooling element reduces the temperature rise of the moving nut by over 3K and the thermal expansion of the screw shaft by 10%. By using the designed cooling element with a topologically optimized channel shape, the improvement rate of repeat positioning accuracy is in the range of [29.03%, 92.59%]. The results of machining cylindrical gears using a gear grinding machine with the designed cooling element show that the grinding accuracy of the precision worm grinding wheel gear grinding machine is improved by about 65%. In summary, the cooling element of the precision worm grinding wheel gear grinding machine and its multi-objective thermal-fluid topological design optimization method of the present invention can significantly improve the heat transfer capacity and efficiency by applying the designed cooling element to the moving nuts (MN) of the X, Y and Z axes of the GGMT, thereby improving the machining accuracy of the GGMT. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] In order to make the purpose, technical solutions and beneficial effects of the present invention more clear, the present invention provides the following drawings for illustration:

[0063] Figure 1 Cooling strategy for GGMT;

[0064] Figure 2 A schematic diagram of the design method;

[0065] Figure 3 Solid domain and fluid domain in topology optimization;

[0066] Figure 4 It is TO model;

[0067] Figure 5 is the mesh division result;

[0068] Figure 6 Simulate data for the cooling plate;

[0069] Figure 7 is the structural dimensions and boundary conditions;

[0070] Figure 8 for CP with TOC;

[0071] Figure 9 Set up and process the experimental platform;

[0072] Figure 10 is the surface temperature of the cooling plate;

[0073] Figure 11 is the average surface temperature;

[0074] Figure 12 It is a curve diagram of pressure drop data;

[0075] Figure 13 is the CE embedding process and boundary conditions;

[0076] Figure 14 is the temperature field;

[0077] Figure 15 For the experimental device;

[0078] Figure 16 To measure the position;

[0079] Figure 17 It is the temperature curve of the moving nut;

[0080] Figure 18 Thermal expansion of the screw shaft support end face;

[0081] Figure 19 is the repeat positioning error;

[0082] Figure 20 A gear grinding device;

[0083] Figure 21 For the measurement results. DETAILED DESCRIPTION

[0084] The present invention will be further described below with reference to the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it. However, the embodiments are not intended to limit the present invention.

[0085] 1. Precision gear grinding machine tools

[0086] GGMT consists of several key components, including three BSFDS axes (X, Y, and Z) and three rotation axes (A, B, and C), such as Figure 1 As shown in the figure, the X-axis adjusts the lateral position between the grinding wheel and the workpiece. The Y-axis adjusts the height between the grinding wheel and the workpiece. The Z-axis adjusts the fore-aft position between the grinding wheel and the workpiece. The A-axis is a rotary axis used to tilt the grinding wheel to accommodate different gear tooth profiles. The B-axis is a rotary axis used to adjust the grinding wheel angle. The C-axis rotates around the workpiece, enabling continuous grinding as the workpiece rotates. The grinding principle is as follows. Gear grinding is achieved by controlling the spatial position of linear and rotary axes. The workpiece rotates along the C-axis, and the A- and B-axes are used to adjust the grinding wheel angle to match the gear tooth profile. The X, Y, and Z axes control the relative position between the grinding wheel and the workpiece, ensuring grinding accuracy. The cooling system plays a vital role in maintaining proper temperatures in the X, Y, and Z axes. The cooling system consists of an industrial chiller, a liquid flow control system, a cooling element (CE) located on the moving nut (MN), a data acquisition system, and a data processing system. The cooling system controls the temperature of the BSFDS in the GGMT and reduces thermal errors in the X, Y, and Z axes. The data acquisition system and data processing system are used to continuously monitor and collect temperature rise, pressure drop, etc.

[0087] GGMT's cooling strategy is as follows Figure 1 As shown. This embodiment focuses on the cooling element topology design of the X, Y and Z axis MN. Here, the Y axis is taken as an example to demonstrate the idea of ​​directly cooling the mobile heat source of the GGMT. The goal is to improve the heat dissipation efficiency by embedding the TO-based CE into the GGMT. The CE of the GGMT is designed using the TO method. Specifically, a CE with a TO-shaped channel (TOC) is designed to fit tightly and be embedded in the nut housing and used to wrap the MN. Based on the dimensionless control equation, an MTO model of fluid-solid heat transfer is proposed. The cooling system consists of an industrial-grade cooler and a liquid flow meter, which together provide a constant coolant inlet velocity. The heat transfer performance of the designed CE with TOC is evaluated through a comprehensive analysis of the temperature variation, pressure drop and thermal elongation of the screw shaft. In addition, the PE and RPE of the BSFDS and the overall machining accuracy of the entire GGMT are used to demonstrate the effectiveness of the MTO method for precision GGMTCE. These indicators jointly determine the efficacy of the MTO applied to the CE within the GGMT.

[0088] 2. Topology optimization design

[0089] The overall design concept is as follows: The process begins with the design of a half-size free-mount cooling plate (CP). This half-size CP features a TOC designed for optimal fluid flow and heat dissipation. Two half-size CPs are then combined into a full-size CP. The full-size plate effectively doubles the cooling capacity and surface area available for heat transfer. Subsequently, the full-size CP is formed and assembled into a cylindrical CE. The overall design concept is as follows Figure 2 shown.

[0090] 2.1. Fluid-Solid Heat Transfer Topology Optimization Problem

[0091] In the fluid-solid TO domain, the design domain is theorized as a porous material, assuming that the fluid resistance F is linearly related to the flow velocity u:

[0092] F=-αu

[0093] Where: α is the permeability.

[0094] The design domain is divided into a finite number of elements, each of which has a design variable γ, which takes a value between 0 and 1; Figure 3 As shown in the figure, the black area represents a solid and the white area represents a fluid. When representing a solid: γ = 0, α → ∞ and F → ∞; when representing a fluid: γ = 1, α → 0 and F → 0.

[0095] 2.1.1 Incompressible laminar flow

[0096] For incompressible laminar flow, the dimensionless continuity equation and momentum conservation equation are expressed by dimensionless physical quantities. The dimensionless physical quantities include velocity u*, pressure p*, Reynolds number Re, and gradient operator The above dimensionless physical quantity is defined as:

[0097]

[0098] in: is the gradient operator; U and L are the characteristic velocity and length, respectively; ρ is the density; and μ is the dynamic viscosity.

[0099] For incompressible laminar flow, the continuity equation is:

[0100]

[0101] The momentum conservation equation is:

[0102]

[0103] Among them: F * It is the volume force.

[0104] According to Darcy's law, the body force F* is proportional to the velocity u*, given by α *The dimensionless permeability represented by is determined by subsequent penalty function interpolation:

[0105] F * =-α * u *

[0106]

[0107]

[0108] The penalty factor q is related to the Darcy number Da and the Reynolds number Re, resulting in the momentum equation:

[0109]

[0110] Where: q is the penalty factor; Da is the Darcy number.

[0111] 2.1.2 Conjugate Heat Transfer

[0112] The dimensionless energy conservation equation is given by the dimensionless temperature T * and the Prandtl number Pr, defined as follows:

[0113]

[0114]

[0115] Where: T B and T w are the average temperature and the wall temperature respectively; the thermophysical properties of the fluid are characterized by its specific heat capacity C p and thermal conductivity k f is a characteristic and is a key parameter in heat transfer analysis; μ is the fluid viscosity.

[0116] The dimensionless energy conservation equation is expressed as:

[0117]

[0118] Where: u * Indicates speed; Q * represents the dimensionless heat generation rate, and the dimensionless temperature T * Proportional to:

[0119]

[0120] By introducing the design variable γ, we get:

[0121]

[0122] 2.1.3 Density filtering and projection

[0123] In order to avoid the checkerboard phenomenon in the design domain, density filtering in the form of Holmz partial differential equation is used:

[0124]

[0125] Where: R min is the filter radius, which is the size of the cooling element; θ is the design variable before filtering; is the filtered design variable; represents the gradient operator.

[0126] The above density filtering method enhances the stability of the numerical solution. However, it also leads to a surge in gray elements. To solve the problem of gray elements, the hyperbolic tangent projection technique is used:

[0127]

[0128] in: is the output design variable after projection, η and β are the projection point and slope, respectively.

[0129] In this embodiment, η = 0.5, β = 8. The penalty factor q is 10 -2 , Darcy number is 10 -4 , Pr is taken as 6.78. Under the same thermal boundary conditions, the temperature of CP decreases with the increase of Reynolds number Re. The Reynolds number Re is taken as 200 and the heat transfer coefficient h* is taken as 100.

[0130] 2.2 Topology Design Optimization Method

[0131] Existing nut shell such as Figure 4 (a) In order to accommodate CE, the structure of the nut shell was modified, as shown in Figure 4 (b) shows the hollow space used to accommodate the CE. The size of the TO design domain is then determined. Specifically, according to Figure 4 (c), the outer diameter Φ1 is 96 mm, the inner diameter Φ2 is 80 mm, the length and thickness of CE are 81 mm and 8 mm respectively. The cross-sectional distance is 40 mm. In order to simplify the TO model, the three-dimensional design domain is expanded, as shown in Figure 4 (d) As shown. The effect of thickness on CE heat transfer efficiency is not significant, so the thickness degree of freedom is not considered to reduce the computational load. Multi-objective topology optimization (MTO) is performed on a two-dimensional rectangular design domain. The design domain dimensions are as follows: width is 243mm, length is 220πmm. In order to further improve the stability and convergence of the TO model, the two-dimensional rectangular design domain is divided into two halves from the middle, and symmetric boundary conditions are set. Therefore, the final design domain is a rectangle of 243aL×110πaL, as shown in Figure 4(e) is shown. L and a represent the characteristic length and the scale factor, respectively. The scale factor a is assigned a value of 1 / 60 to reduce the computational scale. Considering the inconvenience of arranging CE due to the movement of MN in the axial direction of SS, an embedded CE is designed. In addition, the inflow and outflow of CP are arranged on the same parallel side within the design domain. The two-dimensional design domain and boundary conditions are shown in Figure 4 (f) A uniform heat load Q* is applied to the design domain. The volume flow rate at the inlet is set to fully developed flow conditions, the inlet temperature is set to T* = 0 K, and the outlet pressure is set to P* = 0 Pa.

[0132] Specifically, the cooling element of a precision gear grinding machine tool and the multi-objective thermal-fluid topology design optimization method thereof in this embodiment include the following steps:

[0133] Step 1: Define the design domain

[0134] The cooling element is unfolded into a plane, and a two-dimensional design domain representing the cooling element's geometry is defined. This is then discretized using the finite element method. In this embodiment, the cooling element is symmetrically divided into two halves, and the two-dimensional design domain is defined to represent the geometry of one half of the cooling element. This means that only one half of the cooling element needs to be designed in this embodiment.

[0135] Step 2: Numerical modeling

[0136] The cooling element is modeled as a porous medium. Under the condition of laminar incompressible flow, the fluid dynamics is governed by the dimensionless forms of the continuity and momentum equations. The fluid-solid coupled heat transfer model is governed by the dimensionless energy conservation equations.

[0137] Step 3: Construct the objective function

[0138] In order to improve cooling performance and reduce flow resistance, the optimization objective is defined as a weighted function of heat transfer and fluid dissipation power, and a fluid-solid heat transfer topology optimization model is constructed:

[0139]

[0140] Where: J is the optimization target; J th is the heat transfer term; J f is the fluid dissipation work item; w1 and w2 are J th and J f The weight factor of Ω is the design domain; V f is the volume fraction occupied by the fluid flow path; Vol is the total volume contained in the design domain; γ is the design variable; h * is the dimensionless heat transfer coefficient; T * is the dimensionless temperature; is the dimensionless gradient operator; u * is the dimensionless flow velocity; α* is the dimensionless permeability; Vol Ω is the total volume contained in the design domain; Q is the heat load applied to the design domain; is the dimensionless inlet pressure; Γ is the flow channel; Γ in It is the flow channel entrance;

[0141] Step 4: Solve the objective function

[0142] The sensitivity is solved using the adjoint method, and the gradient calculated by the adjoint method is used to form a quadratic subproblem of the sequential quadratic programming method. The design variable γ is updated by solving the quadratic subproblem to ensure that the solution moves towards the optimal configuration until the set iterative convergence condition is met. The topological design channel (TOC) is designed in the cooling element.

[0143] Step 5: Restoring the shape and structure of the cooling element. In this embodiment, the two cooling elements are symmetrically combined into one, and then restored to an arc shape.

[0144] In this embodiment, the iterative convergence condition is:

[0145]

[0146] Where: k represents the number of iterations; Represents the value of the objective function after the kth iteration.

[0147] In order to meet the mesh density and basic uniformity criteria of TO, a free triangular mesh was used during the construction process. A direct correlation between the mesh size and the filter length used in TO was given. When the number of elements is 18087, the effect of the additional element number increment on the objective function becomes negligible while maintaining a high level of computational efficiency. When the number of elements is 18087, the element size ranges from 1.15×10 -4 m to 0.0576 m. Therefore, the total number of components in the entire TO design domain is 18087. Figure 5 The meshing results are depicted.

[0148] This embodiment also proposes a cooling element for a precision gear grinding machine, which is designed using the multi-objective thermal-fluid topology design optimization method described above in this embodiment.

[0149] This embodiment also proposes a precision gear grinding machine tool, including an X-axis, a Y-axis and a Z-axis, wherein the X-axis, the Y-axis and the Z-axis are each provided with a ball screw feed drive system, the ball screw feed drive system including a screw shaft and a movable nut cooperating with the screw shaft, the movable nut including a nut and a nut housing, and a cooling element as described above in this embodiment is provided between the nut and the nut housing.

[0150] 2.3 Topological Results and Experimental Verification

[0151] 2.3.1 TO results

[0152] Table 1 shows the TO design results when the Reynolds number Re is 200. Volume fraction V f The TO results show that as the volume fraction increases, the TOC branches become thinner and more numerous, while the incidence of disconnected channels increases, indicating a decrease in the convergence of the TO model. Furthermore, at the same volume fraction, increasing the weight factor w1 leads to subtle changes in the channel structure. Differences in TO results directly affect the heat transfer performance.

[0153] Table 1 TO design results

[0154]

[0155]

[0156] To verify the validity of the TO results, thermal characteristics analysis was performed for different TOCs at different Reynolds numbers. The specific boundary settings are as follows: Re is set to 200, 500, 1000, 1500, and 2000. The initial temperature at the inlet is 293.15K, and the outlet pressure is maintained at 0Pa. The remaining boundaries are under adiabatic anti-slip conditions, and a power of 8000W / m is applied throughout the design domain. 3 The material properties are shown in Table 2.

[0157] Table 2 Material properties of solids and liquids

[0158]

[0159] When the Reynolds number Re is 500, the surface temperature distribution in the design domain is shown in Table 3. Volume fraction V f The weight ratio w1 has an impact on the surface temperature distribution and temperature uniformity. When Re is 200, 1000 and 2000, the surface temperature distribution is shown in Table A. f The temperature field distribution of TOC obtained at the weight ratio w1 is significantly different. It is necessary to evaluate the temperature distribution of TOC at different volume fractions V f and the temperature field distribution of TOC obtained at the weight ratio w1.

[0160] Table 3 Surface temperature distribution

[0161]

[0162]

[0163] Table 4 lists the steady-state temperature distribution of the cooling fluid when the Reynolds number Re is 500. f The three instances with the lowest average temperatures appear at w1 values ​​of 0.6, 0.8, and 0.9, respectively. When w1 is 0.9, the pressure drop loss is much more significant than the pressure drop loss at other w1 values, thus hindering the normal flow of the fluid in the cooling channel. Therefore, the TO results when w1 is 0.9 will not be considered in the subsequent discussion. According to the temperature data listed in Table 4, when V f When V is set to 0.55 and w1 is set to 0.8, the optimal conditions of the lowest maximum fluid temperature of only 317K and the most uniform temperature distribution are achieved. Therefore, it can be concluded that when Re is 500, when V f When w1 and w2 are 0.55 and 0.8 respectively, the TO cooling channel achieves the best heat dissipation performance. In addition, according to Table A, the same pattern is observed at different Reynolds numbers.

[0164] Table 4 Average surface temperature (K)

[0165]

[0166] 2.3.2 Simulation and experimental verification

[0167] This example uses surface average temperature, outlet temperature, and pressure drop as evaluation indicators. The heat transfer performance of CP containing TOC and CP containing SC was then compared. These evaluation indicators showed similar trends. The independent variable was Re, which was set to 200, 500, 750, 1000, 1500, and 2000. Figure 6 As shown in Figure (a), the average surface temperature of a full-scale CP and a half-scale CP are compared. The average surface temperature of the full-scale CP decreases from approximately 350K to approximately 300K. For the half-scale CP, the average surface temperature decreases from approximately 320K to approximately 296K. The simulated average surface temperatures of the full-scale and half-scale CPs are identical, so it is reasonable to simplify the full-scale CP into a half-scale CP for the design of the TO. Figure 6 (b) shows the pressure drop of the two CPs. The drop in outlet temperature reflects the trend of the average temperature. In terms of pressure drop, the pressure drop of the full-size CP with TOC is about one-quarter to one-third of the pressure drop of the CP with SC. Similarly, the pressure drop of the half-size CP with TOC is about one-third to one-half of the pressure drop of the CP with SC, showing a consistent proportional relationship. In addition, Figure 6 The simulated outlet temperature in (b) shows the same Figure 6 The simulated average surface temperature (shown in (a)) shows a similar trend. Therefore, it is reasonable to replace the full-scale CP with TOC and the half-scale CP with TOC. In addition, the heat transfer capacity and efficiency of the full-scale and half-scale CPs with TOC are stronger than those of the CP with SC.

[0168] Considering the practical application of the designed CE, a half-size CP was used in the actual application. This choice was made to maintain the consistency of the results trend while reducing the experimental cost as much as possible. Therefore, the two-dimensional TO results were expanded to obtain a three-dimensional CP, such as Figure 7 In order to study the cooling performance of the designed CP with TOC, the designed CP with TOC was manufactured using 3D printing technology, as shown in Figure 8 The experimental conditions are as follows: the inlet fluid flow rate is vin. The outlet pressure is maintained at 0 Pa. The bottom of the CP is uniformly heated by a copper plate with a fixed heating power of 200 W. In addition to these conditions, the remaining boundaries are characterized by natural convection, with a heat transfer coefficient of 9.7 W / (m²·K).

[0169] The purpose of the experimental study is to verify the heat transfer capacity and efficiency of the designed CP with TOC. The heat transfer capacity and efficiency of the designed CP with TOC were compared with those of the CP with SC. The experimental study was conducted to verify the heat dissipation performance of cathodic protection with TOC. The experimental setup is as follows Figure 9 As shown in (a). The bottom of the CP is heated by a uniformly heated copper plate, and the distilled water circulation in the cooling channel is driven by an industrial-grade chiller, and Re is kept constant. The flow rate is adjusted by a two-stage control, namely preliminary adjustment by a ball valve and fine-tuning by a needle valve. In addition, a precision flow meter and accumulator are used to continuously collect the flow rate. At the same time, temperature probes are strategically placed at the inlet and outlet of the CP to collect the temperature difference. In addition, a differential pressure sensor is installed to quantify the pressure change between the inlet and outlet of the CP. The collected data is immediately transmitted to the data logger for subsequent processing. In order to capture the thermal distribution on the upper surface of the CP, an infrared thermal imager is used. Figure 9 (b) shows the experimental flow chart.

[0170] The heating power is maintained at 200 W. The inlet Re is 500. The inlet diameter represented by d is set to 9.7 mm. The inlet Re is defined as:

[0171]

[0172] In this way, the flow velocity v can be obtained, so the flow velocity q is determined using q = Av. The cross-sectional area A of the inlet is expressed as A = πd 2 / 4. Then, the corresponding flow rate q under different Reynolds numbers Re is determined, as shown in Table 5.

[0173] Table 5 Flow rate

[0174]

[0175] The inlet temperature is controlled by an industrial chiller at 20°C, which is the baseline temperature of the entire fluid circulation process. The temperature sensor located at the CP inlet is responsible for monitoring and recording the average temperature of the inflow flow, which is represented by T in When cooling water is introduced into the industrial chiller, the cooling cycle begins, ensuring that the tin temperature is maintained at around 20°C. Once the fluid completely fills the cooling channel and the measurement data displayed on the recorder stabilizes, experimental data collection begins.

[0176] During the experiment, experimental data of CP containing SC and TOC were collected. In order to capture the temperature field of CP surface, an infrared thermal imager was used. The temperature of CP surface is as follows Figure 10 In order to accurately represent the temperature field of the CP with SC and TOC, three experiments were carried out under each working condition, and the surface temperature was taken as the average of the three experimental data.

[0177] by Figure 10 The experimental data shown are used as input to calculate the average surface temperature, and then the simulation value is compared with the experimental data, as shown in Figure 11 As shown in Figure 2, the average surface temperature gradually decreases with increasing Reynolds number (Re). In the simulation, the initial average surface temperature of the CP containing TOC is slightly higher than that of the CP containing SC. However, as Re increases, the temperature difference between the CP containing SC and TOC gradually decreases, eventually reaching equilibrium. The initial temperature difference is recorded as 2.8 K, which decreases to 0.3 K as Re increases. In contrast, the experimental data show an opposite trend to the simulation results. Throughout the experiment, the average surface temperature of the CP containing TOC remains lower than that of the CP containing SC. Initially, in the simulation results, the CP temperature of the TOC is 5 K lower than that of the CP containing SC when the inlet Re is 2000. The temperature difference narrows to 0.1 K. The difference between the measured data and the simulation results is not significant and decreases with increasing inlet Re. This discrepancy is attributed to the actual contact between the heating plate and the CP. Specifically, the contact between the heating plate and the CP is not perfect, resulting in reduced heat transfer efficiency. In addition, heat loss to the surrounding air during the experiment may further degrade the experimental data. Therefore, the measured average surface temperature is lower than the simulation predictions.

[0178] The pressure drop between the input and output measured by the differential pressure sensor does not fully reflect the actual pressure loss of the CP due to friction loss during the circulation process. To more accurately evaluate this parameter, an empty pipe is first used, and the pressure drop Pe between the input and output is measured. The empty pipe is then replaced with a CP, and the pressure drop Pc between the input and output is measured. The pressure drop Pe is then subtracted from the pressure drop Pc to determine the actual pressure loss of the CP with SC and TOC. For both simulation results and experimental data, the CP with SC exhibits a higher pressure loss compared to the CP with TOC. Specifically, as Figure 12 As shown in Figure 3, the experimental data show that at the same Reynolds number Re, the pressure loss of the CP with TOC is only 1 / 3 to 1 / 2 of that of the CP with SC, which indicates that cathodic protection with TOC has significant advantages in saving cycle energy and improving heat transfer efficiency.

[0179] The results show that CPs with TOC offer significant advantages over CPs with SC. Specifically, CPs with TOC excel in two key performance indicators: pressure loss and average surface temperature. Lower pressure loss indicates reduced energy consumption during fluid circulation, leading to improved energy efficiency. Furthermore, lower average surface temperature demonstrates excellent heat dissipation, which is crucial to the overall efficacy of the CP. Therefore, CPs with TOC are not only energy-efficient but also exhibit excellent cooling performance, making them promising for application in cooling systems.

[0180] 3. Experimental verification

[0181] 3.1 Heat source load and thermal boundary conditions

[0182] 3.1.1 Heat source load

[0183] In the GGMT, the main heat sources in the BSFDS are the motor, bearing pair, and nut pair. Considering the distance between the motor and other components and the insulation effect of the silicon steel coupling, the heat generated by the motor is unlikely to enter the BSFDS of the GGMT. Since the temperature rise is not significant, heat radiation is not considered. The motor is running at a speed n of 1000 r / min. The lead L of the SS is f The nut reaches its limit position, decelerating, then accelerating to a constant speed, completing the reverse motion. Since the duration of deceleration and acceleration is significantly shorter than that of constant speed motion, the heat generated during the constant speed phase is primarily considered. The heat generated by the bearing and nut pair is calculated using existing methods. The heat generation rate of the bearing and nut pair is calculated based on Q = 1.047 × 10 -4 nM determined.

[0184] 3.1.2 Convection coefficient

[0185] This embodiment takes into account the natural convection coefficient of the stationary surface / air and the forced convection coefficient of the moving surface / air. In addition, the designed CE is in contact with the MN, and this embodiment also takes into account the forced convection coefficient between the CE and the MN. The specific information of the BSFDS of the GGMT is shown in Table 6. The BSFDS of the GGMT consists of several key components, such as the nut pair with model FSC32-20K3. The rolling guide pair has model HGH-30CA. The movement is driven by the SMMA-312G67BDK stepper motor. The motor is supported by the support block ZS25-130 to ensure stable and safe installation. The fixed support seat BK25 is used to fix one end of the nut pair. The servo driver is SD20-452T3M3F0D51B1.

[0186] Table 6 Experimental device models

[0187]

[0188] When the speed is 1000r / min, the heating power and boundary conditions are: the heating power of the fixed bearing, support bearing and nut pair are 13.1W, 13.1W and 45.65W respectively. The convection coefficient of the rotating surface / air and the stationary surface / air is 26.93W / (m 2 ·K), the convection coefficients of linear motion surface / air and CE / MN are 10.63W / (m 2 ·K)、28.4W / (m 2 K) and 1826.4W / (m 2 ·K).

[0189] 3.2 Thermal-fluid-solid behavior simulation

[0190] Figure 13 A modified MN with an embedded CE is presented. The nut pair is the main heat-generating component, and a heating boundary is placed on its inner wall with a specified heating power of 43.65 W. The inlet and outlet of the designed CE are facilitated by M6-sized threaded holes. During the simulation, the Reynolds number Re at the inlet is precisely adjusted to 1000, and the outlet pressure is maintained at 0 Pa. In addition, the convection coefficient of the outer surface of the MN (excluding the area in contact with the CE) is 9.7 W / (m 2 K). The thermal fluid-solid simulation was carried out for 5 hours.

[0191] When the BSFDS of GGMT runs continuously for 5 hours, the temperature distribution in the MN without the influence of CE is as follows: Figure 14 As shown in (a), the temperature distribution in the MN affected by CE and with TOC and SC is shown in Figure 2. Figure 14(b) and (c). It is noteworthy that the CE containing TOC and SC significantly reduced the temperature of the MN, resulting in a temperature drop of approximately 10°C. Under the action of CE with SC and TOC, the temperature difference between the MNs was not obvious, verifying the effectiveness of CP and TOC, that is, verifying the effectiveness of the MTO-based CE design.

[0192] 3.3 Experimental Verification

[0193] 3.3.1 Experimental setup

[0194] The designed CE was manufactured using 3D printing technology and embedded in the BSFDS of GGMT. Figure 15 As shown in the figure, the ambient temperature was controlled at 293.15K to ensure that environmental conditions did not introduce measurement errors. Temperature rise, thermal expansion of the stainless steel end faces, and positioning accuracy of the BSFDS were used as evaluation indicators. To measure the temperature rise, several PT100 temperature sensors were used. The temperatures of the front and rear bearing housings and the mobile nucleus were collected by PT100 sensors. Eddy current displacement sensors were used to measure the thermal expansion of the stainless steel end faces, with particular attention paid to the support end.

[0195] In order to accurately evaluate the positioning accuracy, a Renishaw XL-80 laser interferometer was used. The MN with the interferometer was traversed on a precisely defined path, ranging from 0 to 1200 mm. The measurement range was divided into seven different points, positioned at intervals of 200 mm, to evaluate the positioning accuracy, such as Figure 16 As shown in Figure 1, the MN with the interferometer pauses at each point for 5 seconds. The servo motor is calibrated to run at 1000 rpm.

[0196] Figure 17 Detailed temperature changes of the mobile nuclei (MNs) within the GGMT BSFDS during approximately 5 hours of continuous operation are shown. The temperature of the MNs without CE was significantly higher than that of the MNs with CE and TOC. Specifically, the temperature of the MN with CE and TOC was 3 K lower than that of the MN without CE, and the temperature of the MN with CE and SC was 2.5 K lower than that of the MN without CE. This indicates that the heat transfer capability of the CE with TOC is stronger than that of the CE with SC. Furthermore, the temperature of the MN with TOC CE is lower than that of the MN with SC CE. The designed CE with TOC is capable of reducing the temperature of the MNs. The CE with TOC has a stronger heat dissipation capability than the CE with TOC.

[0197] The axial thermal expansion of the stainless steel support end surface (SES) is as follows Figure 18As shown in Figure 2 , for a GGMT BSFDS without CE, the SS undergoes rapid thermal expansion during the initial stages of operation. Over time, particularly within the first two hours, the temperature rise causes the SES to gradually stabilize. After two hours, the rate of change in thermal expansion of the SS becomes more gradual. However, the thermal expansion process continues, ultimately resulting in an axial thermal expansion of approximately 42 μm. In contrast, when the GGMT BSFDS is subjected to embedded CE, the axial thermal expansion of the SES is reduced compared to the SES without CE. The SES with CE plus TOC is slightly lower than that with CE plus SC. Compared to the unaffected screw shaft support end face, the axial thermal expansion of the SS SES under CE plus TOC and SC exhibits a similar pattern of change, ultimately reducing the expansion from an initial 42 μm to 32 μm. When the GGMT BSFDS operates for two hours, the average axial thermal expansion of the SS SES under CE and TOC is reduced by approximately 10% over the subsequent two to five hours compared to the SS SES without CE. When the SES of the SS undergoes axial thermal expansion, the positioning accuracy and repeatability will decrease. As the axial thermal expansion of the SS SES increases, the pitch of the SS also increases, resulting in an increase in the axial movement distance of the MN and the worktable driven by the SS with the same number of rotations, thereby introducing more significant PE and RPE.

[0198] RPE was measured as Figure 19As shown in the figure. Under the combined effects of CE and TOC, the PE of the GGMT BSFDS is significantly reduced compared to the GGMT BSFDS not affected by CE. Then, the effectiveness of the CE design of the GGMT BSFDS based on MTO is fully verified. The results show that applying the designed CE with TOC on the MN can significantly improve the repeatability accuracy compared with the case without cooling, with the positioning accuracy increased by 31.00% to 92.59%. When the operation time T = 0.5h, the RPE range of the GGMT BSFDS is [14.4μm, 36.2μm] when it is not affected by CE. Then, when the GGMT BSFDS is affected by CE and SC, the RPE range is reduced to [1.6μm, 22.0μm]. When the GGMT is affected by CE and TOC, the RPE range is reduced to [11.1μm, 18.4μm]. At T = 1 hour, when the BSFDS of GGMT was not affected by CE, the RPE range was [31.0 μm, 60.0 μm], when the BSFDS of GGMT was affected by CE and SC, the RPE range decreased to [4.5 μm, 33.7 μm], and when the BSFDS of GGMT was affected by CE and TOC, the RPE range decreased to [3.7 μm, 30.1 μm]. At T = 1.5 hours, when the BSFDS of GGMT was not affected by CE, the RPE range was [52.2 μm, 86.5 μm], when the BSFDS of GGMT was affected by CE and SC, the RPE range decreased to [10.6 μm, 42.6 μm], and when the BSFDS of GGMT was affected by CE and TOC, the RPE range decreased to [9.8 μm, 39.2 μm]. At T = 2 hours, when the BSFDS of GGMT was not affected by CE, the range of RPE was [71.4 μm, 110.2 μm], when the BSFDS of GGMT was affected by CE and SC, the range of RPE decreased to [23.6 μm, 72.0 μm], and when the BSFDS of GGMT was affected by CE and TOC, the range of RP-E decreased to [23.3 μm, 68.1 μm]. At T = 2.5 hours, when the BSFDS of GGMT was not affected by CE, the range of RPE was [95.9 μm, 139.6 μm], when the BSFDS of GGMT was affected by CE and SC, the range of RPE decreased to [21.1 μm, 82.9 μm], and when the BSFDS of GGMT was affected by CE and TOC. At T = 3 h, the range of RPE was [108.5 μm, 159.8 μm] when the GGMT BSFDS was not affected by CE, which decreased to [48.0 μm, 102.6 μm] when the GGMT BSFDS was affected by CE and SC, and decreased to [48.0 μm, 102.6 μm] when the GGMT BSFDS was affected by CE and TOC.At T = 3.5 hours, when the BSFDS of GGMT was not affected by CE, the range of RPE was [120.6 μm, 174.9 μm], when the BSFDS of GGMT was affected by CE and SC, the range of RPE decreased to [72.7 μm, 127.9 μm], and when the BSFDS of GGMT was affected by CE and TOC. At T = 4 hours, when the BSFDS of GGMT was not affected by CE, the range of RPE was [133.5 μm, 187.0 μm], when the BSFDS of GGMT was affected by CE and SC, the range of RPE decreased to [79.2 μm, 135.9 μm], and when the BSFDS of GGMT was affected by CE and TOC. At T = 4.5 hours, when the BSFDS of GGMT was not affected by CE, the range of RPE was [138.6 μm, 208.0 μm], when the BSFDS of GGMT was affected by CE and SC, the range of RPE decreased to [90.6 μm, 146.6 μm], and when the BSFDS of GGMT was affected by CE and TOC. At T = 5 hours, when the BSFDS of GGMT was not affected by CE, the range of RPE was [154.5 μm, 215.4 μm], when the BSFDS of GGMT was affected by CE and SC, the range of RPE decreased to [83.9 μm, 151.1 μm], and when the BSFDS of GGMT was affected by CE and TOC, the range of RP-E decreased to [83.1 μm, 147.9 μm]. At T=0.5 hour, T=1 hour, T=1.5 hours, T=2 hours, T=2.5 hours, T=3 hours, T:3.5 hours, T=4 hours, T=4.5 hours and T=5 hours, the average reduction rates of RPE were 65.79%, 65.95%, 65.99%, 51.02%, 59.83%, 45.96%, 33.46%, 34.11%, 32.47% and 38.03%, respectively.

[0199] The RPE reduction rate of the BSFDS under the influence of CE and TOC was evaluated compared to the BSFDS without CE, as shown in Table 7. From 0.5 to 3.5 hours, under the influence of CE and TOC, the repeatability accuracy of the BSFDS improved by 29.03% to 92.59% compared to the BSFDS without CE. From 4 to 5 hours, the repeatability accuracy under CE with TOC increased by 29.17% to 46.22% compared to the case without CE, significantly improving repeatability accuracy. Furthermore, the RPE of the BSFDS under CE with TOC was much smaller than that of the BSFDS without CE. The reduction rate within the time range [0.5 h, 3.5 h] was lower than that within the time range [4 h, 5 h] because thermal equilibrium was achieved within this time range. When thermal equilibrium is reached, the temperature and thermal error of the BSFDS in the GGMT do not change significantly, and thus the RPE remains stable.

[0200] Table 7 Repeat positioning error reduction rate

[0201]

[0202] The RPE reduction rate of the BSFDS under CE plus TOC was evaluated compared with that under CE plus SC, as shown in Table 8. From 0.5 h to 3.5 h, the repeated positioning accuracy of the BSFDS under CE plus TOC was improved by 9.82% to 79.44% compared with that of the BSFDS under CE plus SC. From 4 h to 5 h, the repeated positioning accuracy of the BSFDS under CE plus TOC was improved by 7.64% to 18.09% compared with that of the BSFD under CE plus SC.

[0203] Table 8 Repeat positioning accuracy improvement rate

[0204]

[0205] From the above analysis, it can be concluded that the temperature of the MN significantly decreased by more than 3K under the influence of CE and TOC compared to the MN without CE. The axial thermal expansion of the SS SES was reduced by nearly 10%. More importantly, the results showed that the CE with TOC designed on the MN significantly improved the repeatability accuracy compared to the case without CE and the case with CE and SC. When the BSFDS of the GGMT was affected by CE and TOC, the improvement rate of repeatability accuracy was in the range of [29.03%, 92.59%], while the BSSDS of the GGMT was not affected by CE. Compared with the BSFDS of the GGMT under the influence of CE and SC, the improvement rate of repeatability accuracy of the BSSDS of the GGMT under the influence of CE and TOC was in the range of [7.28%, 79.44%].

[0206] 3.3.3 Actual processing verification

[0207] In order to verify the effectiveness of the proposed MTO for CE in BSFDS, CE with TOC was embedded into the X, Y, and Z axes of precision GGMT by modifying the nut housings of X, Y, and Z axes. Then, high-performance gears were machined by GGMT with and without performing CE, as shown in Figure 20 Detailed information regarding the gear being machined is as follows. The number of teeth is 50, the gear module is 5 mm, the pressure angle is 20°, the addendum coefficient and the root coefficient are 1.0 and 1.25, respectively. The profile angle is equal to the pressure angle, the face width is 100 mm, the helix angle is 15°, and the pitch diameter is 250 mm. The process parameters are as follows: The rotational speed of the silicon carbide-based grinding wheel is 2000 rpm. The rotational speed of the machined gear is 100 rpm. The radial feed rate is 2 mm / min, and the axial feed rate is 1 mm / min. The grinding depth is 0.002 mm / min. The abrasive grit size of the grinding wheel is #120. The grinding wheel is calibrated at a frequency of 40 L / min after every 10-20 workpieces to maintain its shape and sharpness. A diamond dressing table is used. Oil-based coolant is used during the grinding process. To ensure effective cooling and lubrication, the coolant flow rate is 40 L / min and the pressure is 6 bar.

[0208] The geometric accuracy of the gears is as follows: Figure 21As shown in the figure. By implementing CE with TOC, the maximum errors of the left and right tooth surfaces were observed to decrease from 18.5 μm to 6.3 μm and from 16.7 μm to 5.4 μm, respectively. The implementation of CE and TOC improved the grinding accuracy of GGMT by more than approximately 65%. The combination of CE and SC improved the grinding accuracy of GGMT by approximately 53%. More importantly, the maximum errors of the left and right tooth surfaces were within the tolerance requirements outlined in ISO1328-1:2013. It is worth noting that CE with TOC proved to be more effective than CE with SC in improving grinding accuracy. The effectiveness of the proposed CE MTO was fully verified in the BSFDS of precise GGMT.

[0209] 4. Conclusion

[0210] To reduce thermal errors, a MTO approach for CE in a precision GGMT BSFDS was proposed. The concept of MN direct temperature control was proposed, and an MTO model was established to maximize heat transfer and minimize fluid power dissipation. The main conclusions of this example are summarized as follows:

[0211] (1) A direct cooling strategy for the MN in the GGMT BSFDS was proposed and implemented. Based on the cooling strategy, the CE was designed using the MTO method and then embedded in the nut shell, which significantly improved the machining accuracy of the entire GGMT.

[0212] (2) A fluid-solid heat transfer TO model was established to design the CE. Using the dimensionless equation, its convergence was effectively enhanced, the pressure drop was greatly reduced, and the cooling channel was optimized to achieve energy saving and improve heat dissipation performance. The designed CE has a volume fraction V f The best cooling performance is achieved when the weighting factor w1 is 0.8.

[0213] (3) The effectiveness of the direct cooling of the MN and the TO model in the GGMT BSFDS was confirmed through empirical tests. The results show that when CE is used, the overall temperature of the MN is significantly reduced by more than 3K. The axial thermal expansion of the SS SES is reduced by about 10%. The results show that the repeated positioning accuracy can be significantly improved by using the designed CE with TOC for repeated positioning of the MN in the GGMT compared with the effect of no CE and the effect of CE with SC. When the BSFDS of the GGMT is affected by the CE with TOC, the improvement rate of repeated positioning accuracy is in the range of [29.03%, 92.59%]. The implementation of CE and TOC improves the grinding accuracy of the GGMT by more than about 65%. The cooling and thermal error reduction effect of CE with TOC is much more significant than that of CE with SC.

[0214] In conclusion, the MTO of CE in precision GGMTs provides a new solution to the thermal error control problem. Future research may further explore more complex geometric designs and the application of new materials to further improve the cooling efficiency and thermal control performance of the system.

[0215] Table A

[0216]

[0217]

[0218]

[0219] The above embodiments are merely preferred embodiments for the purpose of fully illustrating the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are within the scope of protection of the present invention. The scope of protection of the present invention shall be subject to the claims.

Claims

1. A multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine, characterized by: The cooling element is applied to a ball screw feed drive system, wherein the ball screw feed drive system comprises a screw shaft and a movable nut matched with the screw shaft, wherein the movable nut comprises a nut and a nut housing, and the cooling element is arranged between the nut and the nut housing; The method comprises the following steps: Step 1: Define the design domain The cooling element is unfolded into a plane, a two-dimensional design domain representing the geometric structure of the cooling element is defined, and the finite element method is used for discretization; Step 2: Numerical modeling The cooling element is modeled as a porous medium. Under the condition of laminar incompressible flow, the fluid dynamics is governed by the dimensionless forms of the continuity and momentum equations. The fluid-solid coupled heat transfer model is governed by the dimensionless energy conservation equations. Step 3: Construct the objective function In order to improve cooling performance and reduce flow resistance, the optimization objective is defined as a weighted function of heat transfer and fluid dissipation power, and a fluid-solid heat transfer topology optimization model is constructed: in: To optimize the goal; is the heat transfer term; is the fluid dissipation work item; and They are and The weight factor of is the design domain; is the volume fraction occupied by the fluid flow path; is the total volume contained in the design domain; is the design variable; is the dimensionless heat transfer coefficient; is the dimensionless temperature; is the dimensionless gradient operator; is the dimensionless flow velocity; is the dimensionless permeability; is the total volume contained in the design domain; is the thermal load applied by the design domain; is the dimensionless inlet pressure; It is the flow channel; It is the flow channel entrance; Step 4: Solve the objective function The sensitivity is solved by the adjoint method, and the gradient calculated by the adjoint method is used to form a quadratic subproblem of the sequential quadratic programming method. The design variables are updated by solving the quadratic subproblem. , ensuring that the solution moves towards the optimal configuration until the set iterative convergence conditions are met, and a topological design channel is designed in the cooling element; Step 5: Restore the shape and structure of the cooling element.

2. The multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine according to claim 1, characterized in that: In the second step, the cooling element is modeled as a porous medium. In the fluid-solid topology design domain, the design domain is theoretically considered as a porous material, assuming that the fluid resistance and flow rate A linear relationship: in: is the permeability; Divide the domain of the design domain into a finite number of elements, each with a design variable , The value is between 0 and 1; when it represents a solid: 、 and ; When representing a fluid: 、 and .

3. The multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine according to claim 1, characterized in that: In step 3, fluid dynamics is determined by the continuity equation and momentum equation by velocity. ,pressure , Reynolds number Re and gradient operator Perform dimensionless representation: in: is the gradient operator; U and L are characteristic velocity and length respectively; is the density; is the dynamic viscosity; For incompressible laminar flow, the continuity equation is: The momentum conservation equation is: in: It is the volume force; According to Darcy's law, the volume force and speed Proportional to The dimensionless permeability represented by is determined by subsequent penalty function interpolation: Penalty Factor Related to the Darcy number Da and the Reynolds number Re, the momentum equation is obtained: in: is the penalty factor; is the Darcy number.

4. The multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine according to claim 1, characterized in that: The dimensionless energy conservation equation is derived from the dimensionless temperature and the Prandtl number Pr, defined as follows: in: and are the average temperature and the wall temperature respectively; the thermophysical properties of the fluid are characterized by its specific heat capacity and thermal conductivity is a characteristic and is a key parameter in heat transfer analysis; is the fluid viscosity; The dimensionless energy conservation equation is expressed as: in: Indicates speed; is the gradient operator; represents the dimensionless rate of heat generation, and the dimensionless temperature Proportional to: By introducing design variables ,get: 。 5. The multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine according to claim 1, characterized in that: In order to avoid the checkerboard phenomenon in the design domain, density filtering in the form of Holmz partial differential equation is used: in: is the filter radius, taking the cooling element size; is the design variable before filtering; is the filtered design variable; represents the gradient operator; In order to solve the problem of gray elements, the hyperbolic tangent projection technique is used: in: is the output design variable after projection, and are the projection point and slope respectively.

6. The multi-objective thermal-fluid topology design optimization method for the cooling element of a precision worm wheel gear grinding machine according to claim 1, characterized in that: In the step 1, the cooling element is symmetrically divided into two halves, and the two-dimensional design domain is defined as the geometric structure representing one half of the cooling element; in the step 5, the two cooling elements are symmetrically combined into one.

7. A cooling element for a precision worm wheel gear grinding machine, characterized by: The method is designed using the multi-objective thermal-fluid topology design optimization method according to any one of claims 1 to 6.

8. A precision worm wheel gear grinding machine, characterized by: It includes an X-axis, a Y-axis and a Z-axis, each of which is provided with a ball screw feed drive system, the ball screw feed drive system includes a screw shaft and a movable nut cooperating with the screw shaft, the movable nut includes a nut and a nut housing, and a cooling element as claimed in claim 7 is provided between the nut and the nut housing.

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