Worm grinding wheel gear grinding machine main shaft thermal error mechanism-data modeling method considering electric-mechanical-thermal coupling effect

Abstract

The invention discloses a worm grinding wheel gear grinding machine main shaft thermal error mechanism-data modeling method considering an electric-mechanical-thermal coupling effect. The method comprises the following steps: 1) constructing an incidence relation between main shaft system thermal expansion deformation and a temperature variable; 2) establishing a main shaft system heat balance equation, and deducing an incidence relation between a main shaft heat error and main shaft structure heat absorption; 3) constructing a mathematical expression of the heat production quantity of the main shaft system about an electric-mechanical-thermal variable; 4) constructing a mathematical expression of the heat dissipating capacity of the main shaft system about an electric-mechanical-thermal variable; 5) establishing a theoretical model of the thermal error of the main shaft system with respect to electric-mechanical-thermal variables; and 6) converting the thermal error model solution of the main shaft system into an optimization problem, and solving by using an HPSO-GA optimization algorithm to obtain the thermal error of the main shaft system. According to the method, the influence of the electro-mechanical-thermal coupling effect of the main shaft system on the thermal error can be revealed while the thermal error of the main shaft is accurately predicted.

Description

Technical Field

[0001] The present invention relates to the technical field of machine tool control, in particular to a method for modeling a thermal error mechanism-data of a main shaft of a worm wheel gear grinding machine taking into account an electrical-mechanical-thermal coupling effect. Background Art

[0002] The main factors affecting machine tool processing accuracy include: geometric error, thermal error, motion error, and processing error, among which thermal error accounts for 40% to 70% of the total error. For gear processing machines, such as worm wheel gear grinding machines, the spindle is the most critical functional component, directly affecting the gear processing accuracy and efficiency. However, due to the integrated design of the spindle system and the combined effects of internal and external heat sources, the internal thermal environment of the spindle is unstable and the temperature distribution is uneven, ultimately causing thermal deformation of the spindle structure. Therefore, revealing the thermal deformation mechanism of the worm wheel gear grinding machine spindle and establishing an accurate thermal error model are crucial for improving processing accuracy and ensuring precision consistency.

[0003] However, due to the unclear electro-mechanical-thermal coupling mechanism of the worm wheel gear grinding machine spindle system and the laws of the spindle thermal characteristics, it is still challenging to establish an accurate spindle thermal error model. Traditional thermal error modeling methods mainly include mechanism-based models (black box models) and data-driven models (white box models). The former is based on the basic theories of thermodynamics and thermoelasticity and can reveal the formation mechanism of spindle thermal errors, but the modeling process is complex, requires a large number of assumptions and simplifications, and has poor robustness. The latter is committed to establishing a mapping relationship between thermal errors and temperature variables through intelligent algorithms. The model coefficients are difficult to obtain, and the model cannot provide intuitive explanations and physical meanings. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle and data considering the electro-mechanical-thermal coupling effect, comprising the following steps:

[0005] 1) Analyze the electrical, mechanical, and thermal multi-energy coupling characteristics of the spindle system, and establish the correlation between the thermal expansion deformation and temperature variables of the spindle system based on thermoelastic mechanics;

[0006] 2) Combining thermodynamic analysis and the law of conservation of energy, the thermal balance equation of the spindle system is established, and the correlation between the spindle thermal error and the heat absorption of the spindle structure is derived;

[0007] 3) Analyze the heat generation characteristics of the spindle system and construct a mathematical expression for the heat generation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0008] 4) Analyze the heat dissipation characteristics of the spindle system and construct a mathematical expression for the heat dissipation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0009] 5) Derive the thermal balance equation of the spindle system and establish a theoretical model of the spindle system thermal error with respect to the motor-mechanical-thermal variables;

[0010] 6) Conduct cutting experiments to obtain real-time data of electro-mechanical-thermal elements, and train the theoretical model of the spindle system thermal error with respect to electro-mechanical-thermal variables to obtain the spindle system thermal error model;

[0011] 7) The thermal error model of the spindle system is converted into an optimization problem, and the HPSO-GA optimization algorithm is used to solve it to obtain the thermal error of the spindle system.

[0012] Furthermore, in step 1), the step of establishing a correlation between the thermal expansion deformation of the spindle system and the temperature variable includes:

[0013] 1.1) Simplify the spindle system into a thin-walled cylinder and construct the thermal expansion deformation expression of the spindle housing at r = b, that is:

[0014] (1)

[0016] Where u r＝b represents the thermal expansion coefficient, r represents the radius, a and b represent the inner radius and outer radius of the spindle respectively, α represents the linear expansion coefficient of the spindle material, c1 represents a constant; T is the spindle temperature; T a 、T b Represent the spindle inner diameter and outer diameter temperatures respectively.

[0017] 1.2) Set all variables except temperature as constants, simplify the expression of thermal expansion deformation of the spindle housing at r = b, and construct the correlation between the thermal expansion deformation of the spindle system and the temperature variable, namely:

[0018] δ spindle =u r＝b =k1(T b -T a )+c1 (2)

[0019] Where, k1 represents the coefficient; δ spindle The thermal error of the spindle.

[0020] Furthermore, in step 2), the heat balance equation of the spindle system is as follows:

[0021]

[0022] Where, Indicates the heat absorbed by the spindle structure; Q accum Q is the heat accumulation of the spindle system; gen and Q dis Respectively represent the heat generation and heat dissipation of the spindle system, and They represent the heat generated by the spindle system bearings and the motor respectively. and They represent the heat dissipation of the spindle system through heat conduction, heat convection, and heat radiation respectively.

[0023] Among them, the spindle system absorbs heat As shown below:

[0024]

[0025] Where m, c, ρ, and l0 represent the spindle structure mass, specific heat capacity, average density, and length, respectively; k2 and k3 represent coefficients; c2 represents a constant; ΔT represents the temperature difference; d represents the bearing diameter; and T0 represents the initial temperature of the external environment.

[0026] Furthermore, in step 2), the correlation between the spindle thermal error and the heat absorption of the spindle structure is as follows:

[0027]

[0028] Where k2 and k3 represent coefficients.

[0029] Furthermore, in step 3), the step of constructing a mathematical expression of the heat generation of the spindle system with respect to the electro-mechanical-thermal variables includes:

[0030] 3.1) Calculate the heat generated by the spindle bearing, that is:

[0031]

[0032]

[0033] Where M and n represent the total friction torque and speed of the bearing, respectively; M0 and M1 represent the friction torque caused by the viscosity of the bearing lubricant and the friction torque caused by the bearing load, respectively; f0 and f1 are constants; γ is the kinematic viscosity of the lubricant; and d m represents the average diameter of the bearing, and F1 is the equivalent load; Indicates the heat generated by the spindle system bearings.

[0034] 3.2) Combining formulas (6)-(7), we get:

[0035]

[0036] Where X0 and Y0 represent the radial and axial static load factors of the bearing respectively, F R and F A Represent the bearing radial load and axial load respectively.

[0037] 3.3) Simplify formula (8) and obtain:

[0038]

[0039] Wherein, k4 and k5 represent coefficients; c3 represents a constant.

[0040] 3.4) Calculate the heat generated by the spindle motor, that is:

[0041]

[0042]

[0043] Where, and Respectively represent the heat generation caused by the rotor, stator, and windage loss, Indicates the heat per unit time, that is, the first-order differential of heat, the same below, P motor and η motor Represent motor power and efficiency respectively, D rotor 、L rotor and f rotor Represent the rotor diameter, length and frequency respectively, μ air represents the dynamic viscosity of air, and h represents the distance between the rotor and the stator. express and The first differential of

[0044] 3.5) Simplifying formula (10), we can obtain the mathematical expression of the heat generation of the spindle system with respect to the motor-mechanical-thermal variables, namely:

[0045]

[0046] Where k6 and k7 are coefficients.

[0047] Furthermore, in step 4), the step of constructing a mathematical expression of the heat dissipation of the spindle system with respect to the electro-mechanical-thermal variables includes:

[0048] 4.1) Calculate the cooling water heat dissipation used for motor heat dissipation Right now:

[0049]

[0050] Where c W and Represent the specific heat capacity and mass flow rate of cooling water, T W,out and T W,in Respectively represent the inlet and outlet temperatures of cooling water.

[0051] 4.2) Simplify formula (11) to obtain:

[0052]

[0053] Where k8 is the coefficient.

[0054] 4.3) Calculate the heat dissipation of compressed air in the motor Right now:

[0055]

[0056]

[0057] Where, T rotor 、T stator and T ca,in Represent the rotor, stator and compressed air temperatures respectively; Respectively represent the rotor and stator convective heat transfer coefficients; A rotor 、A stator Represent the rotor and stator convection heat transfer areas, D cg 、L cg and h W Respectively represent the cooling tank diameter, length and convection heat transfer coefficient; k9, k 10 、k 11 、k 12 represents the coefficient. κ represents the intermediate coefficient;

[0058] 4.4) Simplify formula (14) to obtain:

[0059]

[0060] Where k 13 、k 14 、k 15 Represents the coefficient.

[0061] 4.5) Calculate the natural convection heat dissipation in the motor Right now:

[0062]

[0063] Where k am and are the thermal conductivity and average Nusselt coefficient of natural convection, D sh and A sh Respectively represent the outer diameter and surface area of ​​the spindle motor housing, T sh and T am Respectively represent the spindle housing temperature and external environment temperature; k 16 Represents the coefficient.

[0064] 4.6) Calculate the heat dissipation between the bearing and the compressed air, and the heat dissipation between the bearing and the ambient air by natural convection, that is:

[0065]

[0066] Where, They represent the convection heat dissipation between the compressed air and the front bearing, the convection heat dissipation between the compressed air and the rear bearing, the convection heat dissipation between the air and the front bearing, and the convection heat dissipation between the air and the rear bearing respectively; Respectively represent the convective heat transfer coefficient of compressed air in the front and rear bearings; T front 、T rear Respectively represent the front and rear bearing temperatures; D front 、D rear Respectively represent the front and rear bearing diameters; A front 、A rear Respectively represent the convective heat transfer areas of the front and rear bearings; represent the average Nusselt numbers of compressed air flowing in the front and rear bearings, respectively.

[0067] 4.7) Simplify formula (17) to obtain:

[0068]

[0069] Where k 17 、k 18 、k 19 、k 20 represents the coefficient;

[0070] 4.8) Calculate the heat dissipation of the spindle through heat conduction Right now:

[0071]

[0072] Where λ and A SC denote the thermal conductivity and the equivalent surface area of ​​the spindle system, respectively. l0 denotes the spindle length;

[0073] 4.9) Simplify formula (19) to obtain:

[0074]

[0075] Where k 21 represents the coefficient;

[0076] 4.10) Calculate the heat dissipation of the spindle:

[0077]

[0078] Where ξ represents the thermal emissivity of the material, k B represents a constant, T represents temperature; ξ sh Indicates the emissivity of the spindle motor housing material, A sh represents the outer surface area of ​​the spindle motor housing, ξ rearrepresents the emissivity of the front bearing material, ξ front Indicates the emissivity of the rear bearing material. T front、 T rear Represent the front and rear bearing temperatures respectively; T sh Indicates the spindle motor housing temperature;

[0079] 4.11) Simplifying formula (21), we can obtain the mathematical expression of the heat dissipation of the spindle system with respect to the motor-mechanical-thermal variables, namely:

[0080]

[0081] Where k 22 、k 23 、k 24 Represents the coefficient.

[0082] Furthermore, in step 5), the step of establishing a theoretical model of the spindle system thermal error with respect to the electro-mechanical-thermal variables includes:

[0083] 5.1) According to the law of conservation of energy, the heat balance equation of the spindle system is established, namely:

[0084]

[0085] 5.2) Arrange the heat balance equation of the spindle system and obtain:

[0086]

[0087] In the formula, K1, K2, K3, K4, K5, K6, K7, K8, K9, K 10 , K 11 , K 12 , K 13 , K 14 , K 15 , K 16 represents the electro-mechanical-thermal coefficient of variation, and C represents a constant.

[0088] ΔT ca-fb , ΔT ca-rb , ΔT am-fb , ΔT am-rb , ΔT am-sh , ΔT b-0 They respectively represent the temperature difference between the compressed air and the front bearing, the temperature difference between the compressed air and the rear bearing, the temperature difference between the external environment and the front bearing, the temperature difference between the external environment and the rear bearing, and the temperature difference between the spindle housing and the initial temperature of the external environment.

[0089] 5.3) Considering the time characteristics of the thermal error of the spindle system, a matrix thermal error model is constructed:

[0090]

[0091] Where, K=[K1…K7 K8…K 16 ] represents the coefficient matrix, C represents the constant matrix, Represents the electro-mechanical-thermal element matrix, the elements in this matrix are the electro-mechanical-thermal real-time data measured experimentally. i , t n Represents the time step.

[0092] Furthermore, in step 6), the real-time data of the electrical-mechanical-thermal elements include temperature data of the worm grinding wheel gear grinder, current and voltage signals, and displacement signals of thermal deformation characteristic points of the spindle system.

[0093] Furthermore, the temperature data of the worm wheel gear grinder is obtained through a temperature sensor and a temperature acquisition card;

[0094] The current and voltage signals are obtained through current transformers, voltage transformers, and power meters;

[0095] The displacement signal of the thermal deformation characteristic point of the spindle system is obtained through a laser displacement sensor and a displacement acquisition card.

[0096] Furthermore, in step 7), the spindle system thermal error model solution is converted into an optimization problem, and the steps of solving the problem using the HPSO-GA optimization algorithm include:

[0097] 7.1) Establish the spindle thermal error model coefficient to solve the equation, namely:

[0098]

[0099] Where, represents the spindle thermal error value obtained from experimental measurement, Indicates the spindle thermal error value obtained by the built model. F(K1, K2,…, K 15 ,K 16 , C) is the coefficient function to be solved;

[0100] 7.2) Embed the genetic algorithm into the particle swarm optimization algorithm (Particle Swarm Optimization) to obtain the hybrid optimization algorithm HPSO-GA;

[0101] 7.3) Use the hybrid optimization algorithm HPSO-GA to solve equation (26) and obtain the thermal error.

[0102] The technical effect of the present invention is unquestionable. This discovery first establishes a theoretical model of the spindle thermal error with respect to the electro-mechanical-thermal variables based on mechanism analysis using the basic theories of thermoelasticity and thermodynamics; then the solution of the model coefficients is converted into an optimization problem, and the HPSO-GA algorithm is proposed to solve the model coefficients; finally, a cutting experiment is carried out to obtain the electro-mechanical-thermal real-time data required for the model, and to train the above-mentioned model to obtain an accurate thermal error model of the spindle system. The present invention combines the advantages of the mechanism model and the data-driven model, and proposes a thermal error mechanism-data modeling method. This method has strong interpretability, generalization ability, and high accuracy. While accurately predicting the spindle thermal error, it can reveal the influence of the spindle system's electro-mechanical-thermal coupling effect on the thermal error. Therefore, the present invention will be widely applicable to different worm wheel gear grinders. BRIEF DESCRIPTION OF THE DRAWINGS

[0103] Figure 1 This is a flow chart of the thermal error mechanism-data modeling of the worm wheel gear grinder proposed in the present invention;

[0104] Figure 2 This is a schematic diagram of the electrical-mechanical-thermal coupling of the worm wheel gear grinding machine spindle system;

[0105] Figure 3 This is a schematic diagram of the thermal characteristics of the spindle system of the worm wheel gear grinding machine;

[0106] Figure 4 This is the flow chart of the HPSO-GA algorithm;

[0107] Figure 5 The model fitting curve is shown in Figure 2. DETAILED DESCRIPTION

[0108] The present invention will be further described below with reference to the following examples, but it should not be understood that the scope of the present invention is limited to the following examples. Without departing from the above technical ideas of the present invention, various substitutions and modifications can be made according to common technical knowledge and customary means in the art, and all should be included in the scope of protection of the present invention.

[0109] Example 1:

[0110] See also Figures 1 to 5 A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect includes the following steps:

[0111] 1) Analyze the electrical, mechanical, and thermal multi-energy coupling characteristics of the spindle system, and establish the correlation between the thermal expansion deformation and temperature variables of the spindle system based on thermoelastic mechanics;

[0112] 2) Combining thermodynamic analysis and the law of conservation of energy, the thermal balance equation of the spindle system is established, and the correlation between the spindle thermal error and the heat absorption of the spindle structure is derived;

[0113] 3) Analyze the heat generation characteristics of the spindle system and construct a mathematical expression for the heat generation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0114] 4) Analyze the heat dissipation characteristics of the spindle system and construct a mathematical expression for the heat dissipation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0115] 5) Derive the thermal balance equation of the spindle system and establish a theoretical model of the spindle system thermal error with respect to the motor-mechanical-thermal variables;

[0116] 6) Conduct cutting experiments to obtain real-time data of electro-mechanical-thermal elements, and train the theoretical model of the spindle system thermal error with respect to electro-mechanical-thermal variables to obtain the spindle system thermal error model;

[0117] 7) The thermal error model of the spindle system is converted into an optimization problem, and the HPSO-GA optimization algorithm is used to solve it to obtain the thermal error of the spindle system.

[0118] In step 1), the steps of establishing the correlation between the thermal expansion deformation of the spindle system and the temperature variable include:

[0119] 1.1) Simplify the spindle system into a thin-walled cylinder and construct the thermal expansion deformation expression of the spindle housing at r = b, that is:

[0120]

[0121]

[0122] Where u r＝b represents the thermal expansion coefficient, r represents the radius, a and b represent the inner radius and outer radius of the spindle respectively, α represents the linear expansion coefficient of the spindle material, c1 represents a constant; T is the spindle temperature; T a 、T b Represent the spindle inner diameter and outer diameter temperatures respectively.

[0123] 1.2) Set all variables except temperature as constants, simplify the expression of thermal expansion deformation of the spindle housing at r = b, and construct the correlation between the thermal expansion deformation of the spindle system and the temperature variable, namely:

[0124] δ spindle =u r＝b =k1(T b -T a )+c1 (2)

[0125] Where, k1 represents the coefficient; δ spindle The thermal error of the spindle.

[0126] In step 2), the heat balance equation of the spindle system is as follows:

[0127]

[0128] Where, Indicates the heat absorbed by the spindle structure; Q accum Q is the heat accumulation of the spindle system; gen and Q dis Respectively represent the heat generation and heat dissipation of the spindle system, and They represent the heat generated by the spindle system bearings and the motor respectively. and They represent the heat dissipation of the spindle system through heat conduction, heat convection, and heat radiation respectively.

[0129] Among them, the spindle system absorbs heat As shown below:

[0130]

[0131] Where m, c, ρ, and l0 represent the spindle structure mass, specific heat capacity, average density, and length, respectively; k2 and k3 represent coefficients; c2 represents a constant; ΔT represents the temperature difference; d represents the bearing diameter; and T0 represents the initial temperature of the external environment.

[0132] In step 2), the correlation between the spindle thermal error and the heat absorption of the spindle structure is as follows:

[0133]

[0134] Where k2 and k3 represent coefficients.

[0135] In step 3), the steps of constructing a mathematical expression of the heat generation of the spindle system with respect to the electro-mechanical-thermal variables include:

[0136] 3.1) Calculate the heat generated by the spindle bearing, that is:

[0137]

[0138]

[0139] Where M and n represent the total friction torque and speed of the bearing, respectively; M0 and M1 represent the friction torque caused by the viscosity of the bearing lubricant and the friction torque caused by the bearing load, respectively; f0 and f1 are constants; γ is the kinematic viscosity of the lubricant; and d m represents the average diameter of the bearing, and F1 is the equivalent load; Indicates the heat generated by the spindle system bearings.

[0140] 3.2) Combining formulas (6)-(7), we get:

[0141]

[0142] Where X0 and Y0 represent the radial and axial static load factors of the bearing respectively, F R and F A Represent the bearing radial load and axial load respectively.

[0143] 3.3) Simplify formula (8) and obtain:

[0144]

[0145] Wherein, k4 and k5 represent coefficients; c3 represents a constant.

[0146] 3.4) Calculate the heat generated by the spindle motor, that is:

[0147]

[0148]

[0149] Where, and Respectively represent the heat generation caused by the rotor, stator, and windage loss, Indicates the heat per unit time, that is, the first-order differential of heat, the same below, P motor and η motor Represent motor power and efficiency respectively, D rotor 、L rotor and f rotor Represent the rotor diameter, length and frequency respectively, μ air represents the dynamic viscosity of air, and h represents the distance between the rotor and the stator. express and The first differential of

[0150] 3.5) Simplifying formula (10), we can obtain the mathematical expression of the heat generation of the spindle system with respect to the motor-mechanical-thermal variables, namely:

[0151]

[0152] Where k6 and k7 are coefficients.

[0153] In step 4), the steps of constructing a mathematical expression of the heat dissipation of the spindle system with respect to the electro-mechanical-thermal variables include:

[0154] 4.1) Calculate the cooling water heat dissipation used for motor heat dissipation Right now:

[0155]

[0156] Where c W and Represent the specific heat capacity and mass flow rate of cooling water, T W,out and T W,in Respectively represent the inlet and outlet temperatures of cooling water.

[0157] 4.2) Simplify formula (11) to obtain:

[0158]

[0159] Where k8 is the coefficient.

[0160] 4.3) Calculate the heat dissipation of compressed air in the motor Right now:

[0161]

[0162]

[0163] Where, T rotor 、T stator and T ca,in Represent the rotor, stator and compressed air temperatures respectively; Respectively represent the rotor and stator convective heat transfer coefficients; A rotor 、A stator Represent the rotor and stator convection heat transfer areas, D cg 、L cg and h W Respectively represent the cooling tank diameter, length and convection heat transfer coefficient; k9, k 10 、k 11 、k 12 represents the coefficient. κ represents the intermediate coefficient;

[0164] 4.4) Simplify formula (14) to obtain:

[0165]

[0166] Where k 13 、k 14 、k 15 Represents the coefficient.

[0167] 4.5) Calculate the natural convection heat dissipation in the motor Right now:

[0168]

[0169] Where, They represent the convection heat dissipation between the compressed air and the front bearing, the convection heat dissipation between the compressed air and the rear bearing, the convection heat dissipation between the air and the front bearing, and the convection heat dissipation between the air and the rear bearing respectively; Respectively represent the convective heat transfer coefficient of compressed air in the front and rear bearings; T front 、T rear Respectively represent the front and rear bearing temperatures; D front 、D rear Respectively represent the front and rear bearing diameters; A front 、A rear Respectively represent the convective heat transfer areas of the front and rear bearings; It represents the average Nusselt number of compressed air flowing in the front and rear bearings.

[0170] 4.6) Calculate the heat dissipation between the bearing and the compressed air, and the heat dissipation between the bearing and the ambient air by natural convection, that is:

[0171]

[0172] Where, They represent the convection heat dissipation between the compressed air and the front bearing, the convection heat dissipation between the compressed air and the rear bearing, the convection heat dissipation between the air and the front bearing, and the convection heat dissipation between the air and the rear bearing respectively; Respectively represent the convective heat transfer coefficient of compressed air in the front and rear bearings; T front 、T rear Respectively represent the front and rear bearing temperatures; D front 、D rear Respectively represent the front and rear bearing diameters; A front 、A rear Respectively represent the convective heat transfer areas of the front and rear bearings; represent the average Nusselt numbers of compressed air flowing in the front and rear bearings, respectively.

[0173] 4.7) Simplify formula (17) to obtain:

[0174]

[0175] Where k 17 、k 18 、k 19 、k 20 represents the coefficient;

[0176] 4.8) Calculate the heat dissipation of the spindle through heat conduction Right now:

[0177]

[0178] Where λ and A SC denote the thermal conductivity and the equivalent surface area of ​​the spindle system, respectively. l0 denotes the spindle length;

[0179] 4.9) Simplify formula (19) to obtain:

[0180]

[0181] Where k 21 represents the coefficient;

[0182] 4.10) Calculate the heat dissipation of the spindle:

[0183]

[0184] Where ξ represents the thermal emissivity of the material, k B represents a constant, T represents temperature; ξ sh Indicates the emissivity of the spindle motor housing material, A sh represents the outer surface area of ​​the spindle motor housing, ξ rear represents the emissivity of the front bearing material, ξ front Indicates the emissivity of the rear bearing material. T front 、T rear Represent the front and rear bearing temperatures respectively; T sh Indicates the spindle motor housing temperature;

[0185] 4.11) Simplifying formula (21), we can obtain the mathematical expression of the heat dissipation of the spindle system with respect to the motor-mechanical-thermal variables, namely:

[0186]

[0187] Where k 22 、k 23 、k 24 Represents the coefficient.

[0188] In step 5), the steps of establishing a theoretical model of the spindle system thermal error with respect to the electro-mechanical-thermal variables include:

[0189] 5.1) According to the law of conservation of energy, the heat balance equation of the spindle system is established, namely:

[0190]

[0191] 5.2) Arrange the heat balance equation of the spindle system and obtain:

[0192]

[0193] In the formula, K1, K2, K3, K4, K5, K6, K7, K8, K9, K 10 , K 11 , K 12 , K 13 , K 14 , K 15 , K 16 represents the electro-mechanical-thermal coefficient of variation, C represents a constant; ΔT ca-fb , ΔTca-rb , ΔT am-fb , ΔT am-rb , ΔT am-sh , ΔT b-0 They respectively represent the temperature difference between the compressed air and the front bearing, the temperature difference between the compressed air and the rear bearing, the temperature difference between the external environment and the front bearing, the temperature difference between the external environment and the rear bearing, and the temperature difference between the spindle housing and the initial temperature of the external environment.

[0194] 5.3) Considering the time characteristics of the thermal error of the spindle system, a matrix thermal error model is constructed:

[0195]

[0196] Where, K=[K1…K7 K8…K 16 ] represents the coefficient matrix, C represents the constant matrix, Represents the electro-mechanical-thermal element matrix, the elements in this matrix are the electro-mechanical-thermal real-time data measured experimentally. i , t n Represents the time step.

[0197] In step 6), the real-time data of the electrical-mechanical-thermal elements include the temperature data, current and voltage signals of the worm grinding wheel gear grinder, and the displacement signal of the thermal deformation characteristic point of the spindle system.

[0198] The temperature data of the worm wheel gear grinder is obtained through temperature sensors and temperature acquisition cards;

[0199] The current and voltage signals are obtained through current transformers, voltage transformers, and power meters;

[0200] The displacement signal of the thermal deformation characteristic point of the spindle system is obtained through a laser displacement sensor and a displacement acquisition card.

[0201] In step 7), the spindle system thermal error model solution is converted into an optimization problem, and the steps of solving it using the HPSO-GA optimization algorithm include:

[0202] 7.1) Establish the spindle thermal error model coefficient to solve the equation, namely:

[0203]

[0204] Where, represents the spindle thermal error value obtained from experimental measurement, Indicates the spindle thermal error value obtained by the established model.

[0205] 7.2) Embed the genetic algorithm into the particle swarm optimization algorithm (Particle Swarm Optimization) to obtain the hybrid optimization algorithm HPSO-GA;

[0206] 7.3) Use the hybrid optimization algorithm HPSO-GA to solve equation (26) and obtain the thermal error.

[0207] Example 2:

[0208] A method for modeling the thermal error mechanism and data of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect comprises the following steps:

[0209] 1) Analyze the electrical, mechanical, and thermal multi-energy coupling characteristics of the spindle system, and establish the correlation between the thermal expansion deformation and temperature variables of the spindle system based on thermoelastic mechanics;

[0210] 2) Combining thermodynamic analysis and the law of conservation of energy, the thermal balance equation of the spindle system is established, and the correlation between the spindle thermal error and the heat absorption of the spindle structure is derived;

[0211] 3) Analyze the heat generation characteristics of the spindle system and construct a mathematical expression for the heat generation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0212] 4) Analyze the heat dissipation characteristics of the spindle system and construct a mathematical expression for the heat dissipation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0213] 5) Derive the thermal balance equation of the spindle system and establish a theoretical model of the spindle system thermal error with respect to the motor-mechanical-thermal variables;

[0214] 6) Conduct cutting experiments to obtain real-time data of electro-mechanical-thermal elements, and train the theoretical model of the spindle system thermal error with respect to electro-mechanical-thermal variables to obtain the spindle system thermal error model;

[0215] 7) The thermal error model of the spindle system is converted into an optimization problem, and the HPSO-GA optimization algorithm is used to solve it to obtain the thermal error of the spindle system.

[0216] Example 3:

[0217] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect. The technical content is the same as that of Example 2. Furthermore, in step 1), the step of establishing the correlation between the thermal expansion deformation of the spindle system and the temperature variable includes:

[0218] 1.1) Simplify the spindle system into a thin-walled cylinder and construct the thermal expansion deformation expression of the spindle housing at r = b, that is:

[0219]

[0220] Where u represents the thermal expansion coefficient, r represents the radius, a and b represent the inner and outer radii of the spindle, α represents the linear expansion coefficient of the spindle material, c1 represents a constant, T represents the spindle temperature, Ta 、T b Represent the spindle inner diameter and outer diameter temperatures respectively.

[0221] 1.2) Set all variables except temperature as constants, simplify the expression of thermal expansion deformation of the spindle housing at r = b, and construct the correlation between the thermal expansion deformation of the spindle system and the temperature variable, namely:

[0222] δ spindle =u r＝b =k1(T b -T a )+c1 (2)

[0223] Where, k1 represents the coefficient; δ spindle The thermal error of the spindle.

[0224] Example 4:

[0225] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect. The technical content is the same as any one of Examples 2-3. Further, in step 2), the thermal balance equation of the spindle system is as follows:

[0226]

[0227] Where, Indicates the heat absorbed by the spindle structure; Q accum Q is the heat accumulation of the spindle system; gen and Q dis Respectively represent the heat generation and heat dissipation of the spindle system, and They represent the heat generated by the spindle system bearings and the motor respectively. and They represent the heat dissipation of the spindle system through heat conduction, heat convection, and heat radiation respectively.

[0228] Among them, the spindle system absorbs heat As shown below:

[0229]

[0230] Where m, c, ρ, and l0 represent the spindle structure mass, specific heat capacity, average density, and length, respectively; k2 and k3 represent coefficients; c2 represents a constant; ΔT represents the temperature difference; d represents the bearing diameter; and T0 represents the initial temperature of the external environment.

[0231] Example 5:

[0232] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect. The technical content is the same as any one of Examples 2-4. Furthermore, in step 2), the correlation between the thermal error of the spindle and the heat absorption of the spindle structure is as follows:

[0233]

[0234] Where k2 and k3 represent coefficients.

[0235] Example 6:

[0236] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect, wherein the technical content is the same as any one of embodiments 2-5. Furthermore, in step 3), the step of constructing a mathematical expression of the heat generation of the spindle system with respect to the electro-mechanical-thermal variables includes:

[0237] 3.1) Calculate the heat generated by the spindle bearing, that is:

[0238]

[0239]

[0240] Where M and n represent the total friction torque and speed of the bearing, respectively; M0 and M1 represent the friction torque caused by the viscosity of the bearing lubricant and the friction torque caused by the bearing load, respectively; f0 and f1 are constants; γ is the kinematic viscosity of the lubricant; and d m represents the average diameter of the bearing, and F1 is the equivalent load; Indicates the heat generated by the spindle system bearings.

[0241] 3.2) Combining formulas (6)-(7), we get:

[0242]

[0243] Where X0 and Y0 represent the radial and axial static load factors of the bearing respectively, F R and F A Represent the bearing radial load and axial load respectively.

[0244] 3.3) Simplify formula (8) and obtain:

[0245]

[0246] Wherein, k4 and k5 represent coefficients; c3 represents a constant.

[0247] 3.4) Calculate the heat generated by the spindle motor, that is:

[0248]

[0249]

[0250] Where, and Respectively represent the heat generation caused by the rotor, stator, and windage loss, Indicates the heat per unit time, that is, the first-order differential of heat, the same below, P motor and η motor Represent motor power and efficiency respectively, D rotor 、L rotor and f rotor Represent the rotor diameter, length and frequency respectively, μ air represents the dynamic viscosity of air, and h represents the distance between the rotor and the stator.

[0251] 3.5) Simplifying formula (10), we can obtain the mathematical expression of the heat generation of the spindle system with respect to the motor-mechanical-thermal variables, namely:

[0252]

[0253] Where k6 and k7 are coefficients.

[0254] Example 7:

[0255] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect, wherein the technical content is the same as any one of embodiments 2-6. Furthermore, in step 4), the step of constructing a mathematical expression of the spindle system heat dissipation with respect to the electro-mechanical-thermal variables includes:

[0256] 4.1) Calculate the cooling water heat dissipation used for motor heat dissipation Right now:

[0257]

[0258] Where c W and Represent the specific heat capacity and mass flow rate of cooling water, T W,out and T W,in Respectively represent the inlet and outlet temperatures of cooling water.

[0259] 4.2) Simplify formula (11) to obtain:

[0260]

[0261] Where k8 is the coefficient;

[0262] 4.3) Calculate the heat dissipation of compressed air in the motor Right now:

[0263]

[0264]

[0265] Where, T rotor 、T stator and T ca,in Represent the rotor, stator and compressed air temperatures respectively; Respectively represent the rotor and stator convective heat transfer coefficients; A rotor 、A stator Represent the rotor and stator convection heat transfer areas, D cg 、L cg and h W Respectively represent the cooling tank diameter, length and convection heat transfer coefficient; k9, k 10 、k 11 、k 12 Represents the coefficient.

[0266] 4.4) Simplify formula (14) to obtain:

[0267]

[0268] Where k 13 、k 14 、k 15 Represents the coefficient.

[0269] 4.5) Calculate the natural convection heat dissipation in the motor Right now:

[0270]

[0271] Where k am and are the thermal conductivity and average Nusselt coefficient of natural convection, D sh and A sh Respectively represent the outer diameter and surface area of ​​the spindle motor housing, T sh and T am Respectively represent the spindle housing temperature and external environment temperature; k 16 Represents the coefficient.

[0272] 4.6) Calculate the heat dissipation between the bearing and the compressed air, and the heat dissipation between the bearing and the ambient air by natural convection, that is:

[0273]

[0274] Where, They represent the convection heat dissipation between the compressed air and the front bearing, the convection heat dissipation between the compressed air and the rear bearing, the convection heat dissipation between the air and the front bearing, and the convection heat dissipation between the air and the rear bearing respectively; Respectively represent the convective heat transfer coefficient of compressed air in the front and rear bearings; Tfront 、T rear Respectively represent the front and rear bearing temperatures; D front 、D rear Respectively represent the front and rear bearing diameters; A front 、A rear Respectively represent the convective heat transfer areas of the front and rear bearings; represent the average Nusselt numbers of compressed air flowing in the front and rear bearings, respectively.

[0275] 4.7) Simplify formula (17) to obtain:

[0276]

[0277] Where k 17 、k 18 、k 19 、k 20 represents the coefficient;

[0278] 4.8) Calculate the heat dissipation of the spindle through heat conduction Right now:

[0279]

[0280] Where λ and A SC represent the thermal conductivity and the equivalent surface area of ​​the principal axis system, respectively.

[0281] 4.9) Simplify formula (19) to obtain:

[0282]

[0283] 4.10) Calculate the heat dissipation of the spindle:

[0284]

[0285] Where ξ represents the thermal emissivity of the material, k B represents a constant, T represents temperature; ξ sh Indicates the emissivity of the spindle motor housing material, A sh represents the outer surface area of ​​the spindle motor housing, ξ rear represents the emissivity of the front bearing material, ξ front Represents the emissivity of the rear bearing material.

[0286] 4.11) Simplifying formula (21), we can obtain the mathematical expression of the heat dissipation of the spindle system with respect to the motor-mechanical-thermal variables, namely:

[0287]

[0288] Example 7:

[0289] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect, wherein the technical content is the same as any one of embodiments 2-6. Furthermore, in step 5), the step of establishing a theoretical model of the spindle system thermal error with respect to the electro-mechanical-thermal variables includes:

[0290] 5.1) According to the law of conservation of energy, the heat balance equation of the spindle system is established, namely:

[0291]

[0292] 5.2) Arrange the heat balance equation of the spindle system and obtain:

[0293]

[0294] Where K represents the electro-mechanical-thermal coefficient of variation, C represents a constant; ΔT ca-fb , ΔT ca-rb , ΔT am-fb , ΔT am-rb , ΔT am-sh , ΔT b-0 They respectively represent the temperature difference between the compressed air and the front bearing, the temperature difference between the compressed air and the rear bearing, the temperature difference between the external environment and the front bearing, the temperature difference between the external environment and the rear bearing, and the temperature difference between the spindle housing and the initial temperature of the external environment.

[0295] 5.3) Considering the time characteristics of the thermal error of the spindle system, a matrix thermal error model is constructed:

[0296]

[0297] Where, K=[K1…K7 K8…K 16 ] represents the coefficient matrix, C represents the constant matrix, Represents the electro-mechanical-thermal element matrix, and the elements in this matrix are the electro-mechanical-thermal real-time data measured experimentally.

[0298] Example 9:

[0299] A method for modeling the thermal error mechanism of a worm wheel gear grinder spindle taking into account the electro-mechanical-thermal coupling effect, the technical content of which is the same as any one of Examples 2-8. Furthermore, in step 6), the real-time data of the electro-mechanical-thermal elements include the temperature data, current and voltage signals of the worm wheel gear grinder, and the displacement signal of the thermal deformation characteristic point of the spindle system.

[0300] Example 10:

[0301] A method for modeling the thermal error mechanism of a worm wheel gear grinder spindle considering the electro-mechanical-thermal coupling effect, wherein the technical content is the same as any one of Examples 2-9, and further, the temperature data of the worm wheel gear grinder is obtained through a temperature sensor and a temperature acquisition card;

[0302] The current and voltage signals are obtained through current transformers, voltage transformers, and power meters;

[0303] The displacement signal of the thermal deformation characteristic point of the spindle system is obtained through a laser displacement sensor and a displacement acquisition card.

[0304] Example 11:

[0305] A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect, wherein the technical content is the same as any one of Examples 2-10. Furthermore, in step 7), the method for converting the thermal error model of the spindle system into an optimization problem and solving the problem using the HPSO-GA optimization algorithm comprises:

[0306] 7.1) Establish the spindle thermal error model coefficient to solve the equation, namely:

[0307]

[0308] Where, represents the spindle thermal error value obtained from experimental measurement, Indicates the spindle thermal error value obtained by the built model. F(K1, K2,…, K 15 ,K 16 , C) is the coefficient function to be solved;

[0309] 7.2) Embed the genetic algorithm into the particle swarm optimization algorithm (Particle Swarm Optimization) to obtain the hybrid optimization algorithm HPSO-GA;

[0310] 7.3) Use the hybrid optimization algorithm HPSO-GA to solve equation (26) and obtain the thermal error.

[0311] Example 12:

[0312] A method for modeling the thermal error mechanism and data of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect, comprising the following steps:

[0313] (1) Analyze the electro-mechanical-thermal multi-energy coupling characteristics of the spindle system to reveal the influence of electro-mechanical-thermal variables on the spindle thermal error;

[0314] (2) Based on thermoelasticity, the correlation between the thermal expansion deformation of the spindle system and the temperature variable is established;

[0315] (3) Combining thermodynamic analysis and the law of conservation of energy, the thermal balance equation of the spindle system is established, and the correlation between the spindle thermal error and the heat absorption of the spindle structure is derived;

[0316] (4) Analyze the heat generation characteristics of the spindle system and derive the mathematical expression of the heat generation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0317] (5) Analyze the heat dissipation characteristics of the spindle system and derive the mathematical expression of the heat dissipation of the spindle system with respect to the electrical-mechanical-thermal variables;

[0318] (6) Derive and organize the thermal balance equation of the spindle system and establish a theoretical model of the thermal error of the spindle system with respect to the electrical-mechanical-thermal variables;

[0319] (7) Convert the thermal error theoretical model coefficient solution into an optimization problem and solve it using the HPSO-GA optimization algorithm;

[0320] (8) Carry out cutting experiments to obtain the real-time data of the electrical, mechanical, and thermal elements required for the model and use it to train the thermal error model to obtain an accurate thermal error model of the spindle system.

[0321] Specifically, for a certain type of worm wheel gear grinding machine, the specific process of the spindle system thermal error mechanism-data modeling proposed in this invention is as follows: Figure 1 As shown in the figure; after determining the specific model and structure of the machine tool, the electro-mechanical-thermal coupling mechanism of the machine tool spindle system is analyzed to reveal the correlation between the spindle thermal error and the electro-mechanical-thermal coupling effect. The electro-mechanical-thermal coupling of the spindle system is shown in the figure. Figure 2 As shown; Combining thermoelasticity and thermodynamic analysis, the thermal characteristics of the spindle system are revealed, focusing on the heat generation and heat dissipation of the spindle motor and bearings. The thermal characteristics analysis of the spindle system is shown as follows Figure 3 As shown in the figure, combined with the law of conservation of energy, a mechanism model of the spindle thermal error with respect to the motor-mechanical-thermal variables is established, the solution of the model coefficients is converted into an optimization problem, and the HPSO-GA algorithm is proposed to solve the model coefficients. The flow chart of the HPSO-GA algorithm is shown in the figure. Figure 4 As shown in the figure, a gear grinding experiment was carried out to measure the real-time data of the motor-mechanical-thermal elements and the spindle thermal error data. The above data were brought into the proposed model to obtain an accurate mechanism-data driven combined spindle system thermal error model. The thermal error prediction effect was compared. Figure 5 shown.

[0322] In the above process, based on thermoelastic mechanics, the correlation between the thermal expansion deformation of the spindle system and the temperature variable is constructed as follows:

[0323] The spindle system is simplified as a thin-walled cylinder, and the thermal expansion deformation of the spindle housing at r = b is:

[0324]

[0325] Where u represents the thermal expansion coefficient, r represents the radius, a and b represent the inner and outer radii of the spindle, α represents the linear expansion coefficient of the spindle material, c1 represents a constant, T represents the spindle temperature, T a 、T b Represent the spindle inner diameter and outer diameter temperatures respectively.

[0326] Spindle radial thermal expansion error δ spindle The thermal error (hereinafter referred to as thermal error) can be expressed by the thermal expansion deformation of the spindle housing at r = b. Considering all variables except the temperature variable as constants, the spindle thermal error can be simplified to:

[0327] δ spindle =u r＝b =k1(T b -T a )+c1 (2)

[0328] In the formula, k1 represents the coefficient, the same below, and will not be repeated in the following text.

[0329] Combining thermodynamic analysis and the law of conservation of energy, the heat balance equation of the spindle system is established, and the correlation between the spindle thermal error and the heat absorption of the spindle structure is derived as follows:

[0330] The heat balance equation of the spindle system is:

[0331]

[0332] Where, Indicates the heat absorbed by the spindle structure, and its value is related to the heat accumulation of the spindle system Q accum Equal, Q gen and Q dis Respectively represent the heat generation and heat dissipation of the spindle system, and They represent the heat generated by the spindle system bearings and the motor respectively. and They represent the heat dissipation of the spindle system through heat conduction, heat convection, and heat radiation respectively.

[0333] The heat absorption of the spindle system is:

[0334]

[0335] Where m, c, ρ, and l0 represent the spindle structure mass, specific heat capacity, average density, and length, respectively; k2 and k3 represent coefficients; c2 represents a constant; ΔT represents the temperature difference; d represents the bearing diameter; and T0 represents the initial temperature of the external environment.

[0336] Combined with formula (2), the heat absorption of the spindle system is:

[0337]

[0338] The above content reveals the thermal energy balance mechanism of the spindle system and establishes the correlation between the thermal error of the spindle system and the thermal energy accumulation and temperature variables, providing a theoretical basis for the subsequent establishment of the spindle thermal error mechanism model related to the motor-mechanical-thermal variables.

[0339] Spindle system thermal characteristics analysis and thermal error mechanism modeling

[0340] Analyze the heat generation characteristics of the spindle system as described in step 4, and derive the mathematical expression of the heat generation of the spindle system with respect to the electro-mechanical-thermal variables, as follows:

[0341] Step 4.1: Calculate the heat generated by the spindle bearings:

[0342]

[0343] Where M and n represent the total friction torque and speed of the bearing, respectively; M0 and M1 represent the friction torque caused by the viscosity of the bearing lubricant and the friction torque caused by the bearing load, respectively; f0 and f1 are constants; γ is the kinematic viscosity of the lubricant; and d m It represents the average diameter of the bearing and F1 is the equivalent load.

[0344] Further deduction:

[0345]

[0346] Where X0 and Y0 represent the radial and axial static load factors of the bearing respectively, F R and F A Represent the bearing radial load and axial load respectively.

[0347] Simplified:

[0348]

[0349] Step 4.2: Calculate the heat generated by the spindle motor:

[0350]

[0351] Where, and Respectively represent the heat generation caused by the rotor, stator, and windage loss, Indicates the heat per unit time, that is, the first-order differential of heat, the same below, P motor and η motor Represent motor power and efficiency respectively, D rotor 、L rotor and f rotor Represent the rotor diameter, length and frequency respectively, μ airrepresents the dynamic viscosity of air, and h represents the distance between the rotor and the stator.

[0352] Simplified:

[0353]

[0354] Analyze the heat dissipation characteristics of the spindle system as described in step 5, and derive the mathematical expression of the heat dissipation of the spindle system with respect to the electro-mechanical-thermal variables, as follows:

[0355] Step 5.1: Calculate the heat dissipation of the spindle by convection, where the heat dissipation of the cooling water used for motor heat dissipation is:

[0356]

[0357] Where c W and Represent the specific heat capacity and mass flow rate of cooling water, T W,out and T W,in Respectively represent the inlet and outlet temperatures of cooling water.

[0358] The cooling water heat dissipation is quantified as follows:

[0359]

[0360] The heat dissipated by the compressed air in the motor is:

[0361]

[0362] Where, T rotor 、T stator and T ca,in Denote the rotor, stator and compressed air temperatures respectively, h and A denote the convection heat transfer coefficient and convection heat transfer area respectively, D cg 、L cg and h W represent the cooling tank diameter, length and convection heat transfer coefficient respectively.

[0363] The heat dissipation of compressed air can be quantified and simplified as follows:

[0364]

[0365] The natural convection heat dissipation in the motor is:

[0366]

[0367] Where k am and are the thermal conductivity and average Nusselt coefficient of natural convection, D sh and A sh Respectively represent the outer diameter and surface area of ​​the spindle motor housing, T sh and Tam Represent the spindle housing temperature and external ambient temperature respectively.

[0368] The heat dissipation between the bearing and the compressed air, and the heat dissipation between the bearing and the ambient air by natural convection are:

[0369]

[0370] The meanings of the symbols in the formula are the same as above and will not be repeated here.

[0371] Simplified:

[0372]

[0373] Step 5.2: Calculate the heat dissipation of the spindle through heat conduction:

[0374]

[0375] Where λ and A SC represent the thermal conductivity and the equivalent surface area of ​​the principal axis system, respectively.

[0376] Simplified:

[0377]

[0378] Step 5.3: Calculate the heat dissipation of the spindle:

[0379]

[0380] Where ξ represents the thermal emissivity of the material, k B represents a constant, T represents temperature; ξ sh Indicates the emissivity of the spindle motor housing material, A sh represents the outer surface area of ​​the spindle motor housing, ξ rear represents the emissivity of the front bearing material, ξ front Represents the emissivity of the rear bearing material.

[0381] Simplified:

[0382]

[0383] The spindle system thermal balance equation described in step 6 is derived and sorted out to establish a theoretical model of the spindle system thermal error with respect to the motor-mechanical-thermal variables, as follows:

[0384] Step 6.1: Establish the heat balance equation of the spindle system according to the law of conservation of energy:

[0385]

[0386] Step 6.2: Arrange the heat balance equation of the spindle system:

[0387]

[0388] ΔT ca-fb , ΔT ca-rb , ΔT am-fb , ΔT am-rb , ΔT am-sh , ΔT b-0 They respectively represent the temperature difference between the compressed air and the front bearing, the temperature difference between the compressed air and the rear bearing, the temperature difference between the external environment and the front bearing, the temperature difference between the external environment and the rear bearing, and the temperature difference between the spindle housing and the initial temperature of the external environment.

[0389] Step 6.3: Considering the time characteristics of the thermal error of the spindle system, a matrix thermal error model is constructed:

[0390]

[0391] Where, K=[K1…K7 K8…K 16 ] represents the coefficient matrix, C represents the constant matrix, Represents the electro-mechanical-thermal element matrix, and the elements in this matrix are the electro-mechanical-thermal real-time data measured experimentally.

[0392] The above content analyzes the thermal characteristics of the spindle system, focusing on the heat generated by the spindle motor and bearings and the heat dissipation corresponding to heat conduction, heat convection, and heat radiation; combined with the law of conservation of energy, the thermal energy balance equation of the spindle system is derived, and a thermal error mechanism model of the spindle system with coefficients is obtained.

[0393] Coefficient solution of the thermal error mechanism model of the spindle system and the thermal error model based on the combination of mechanism and data drive

[0394] The thermal error theoretical model coefficient solution described in step 7 is converted into an optimization problem and solved using the HPSO-GA optimization algorithm as follows:

[0395] Step 7.1: Establish the spindle thermal error model coefficient and solve the equation:

[0396]

[0397] Where, represents the spindle thermal error value obtained from experimental measurement, Indicates the spindle thermal error value obtained by the established model.

[0398] Carry out cutting experiments as described in step 8 to obtain real-time data of the electrical, mechanical, and thermal elements required for the model and use it to train the thermal error model to obtain an accurate thermal error model of the spindle system, as follows:

[0399] According to the position of the temperature sensitive point of the worm wheel gear grinder, a temperature sensor is installed, the temperature sensor is connected to the temperature acquisition card, and the temperature acquisition card is used to collect the temperature of the corresponding temperature sensor; a current transformer and a voltage transformer are installed, the transformer is connected to the power meter, and the power meter is used to collect the current and voltage signals of the machine tool; according to the position of the thermal deformation characteristic point of the spindle system of the worm wheel gear grinder, a laser displacement sensor is installed, the laser displacement sensor is connected to the displacement acquisition card, and the displacement acquisition card is used to collect the displacement of the laser displacement sensor.

[0400] Finally, the accurate thermal error model of the spindle system is obtained as follows:

[0401]

[0402] The spindle thermal error measurement value obtained by experiment is compared with the thermal error prediction value obtained by formula (26). Figure 5 shown.

Claims

1. A method for modeling the thermal error mechanism of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect, characterized in that: The following steps are involved: 1) Analyze the electrical-mechanical-thermal multi-energy coupling characteristics of the spindle system, and build the correlation between the thermal expansion deformation and temperature variables of the spindle system based on thermoelastic mechanics; 2) Combining thermodynamic analysis and the law of conservation of energy, the heat balance equation of the spindle system is established, and the correlation between the spindle thermal error and the heat absorption of the spindle structure is derived; 3) Analyze the heat generation characteristics of the spindle system and construct a mathematical expression of the heat generation of the spindle system with respect to the electrical-mechanical-thermal variables; 4) Analyze the heat dissipation characteristics of the spindle system and construct a mathematical expression of the heat dissipation of the spindle system with respect to the electrical-mechanical-thermal variables; 5) Derive the thermal balance equation of the spindle system and establish a theoretical model of the thermal error of the spindle system with respect to the electrical-mechanical-thermal variables; 6) Carry out cutting experiments to obtain real-time data of electro-mechanical-thermal elements, and train the theoretical model of the thermal error of the spindle system with respect to the electro-mechanical-thermal variables to obtain the thermal error model of the spindle system; 7) The thermal error model of the spindle system is converted into an optimization problem, and the HPSO-GA optimization algorithm is used to solve it to obtain the thermal error of the spindle system.

2. According to claim 1, a worm wheel gear grinding machine spindle thermal error mechanism-data modeling method considering the electro-mechanical-thermal coupling effect is characterized in that: In step 1), the step of establishing the correlation between the thermal expansion deformation of the spindle system and the temperature variable includes: 1.1) Simplify the spindle system into a thin-walled cylinder and construct the thermal expansion deformation expression of the spindle housing at r = b, that is: In the formula, u r＝b represents the thermal expansion coefficient, r represents the radius, a and b represent the inner radius and outer radius of the spindle respectively, α represents the linear expansion coefficient of the spindle material, c1 represents a constant; T is the spindle temperature; T a 、T b Represent the spindle inner diameter and outer diameter temperatures respectively. 1.2) Set the variables other than the temperature variable as constants, simplify the expression of thermal expansion deformation of the spindle housing at r = b, and construct the correlation between the thermal expansion deformation of the spindle system and the temperature variable, that is: δ spindle =u r＝b =k1(T b -T a )+c1 (2) In the formula, k1 represents the coefficient; δ spindle The thermal error of the spindle.

3. The method for modeling the thermal error mechanism-data of the main shaft of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 1, characterized in that: In step 2), the heat balance equation of the spindle system is as follows: In the formula, Indicates the heat absorbed by the main shaft structure; Q accum is the heat accumulation of the spindle system; Q gen and Q dis They represent the heat generation and heat dissipation of the spindle system respectively. and They represent the heat generated by the spindle system bearings and the motor respectively. and They respectively represent the heat dissipation of the spindle system by heat conduction, heat convection, and heat radiation. Among them, the spindle system absorbs heat As shown below: Wherein, m, c, ρ and l0 represent the spindle structure mass, specific heat capacity, average density and length respectively; k2 and k3 represent coefficients; c2 represents a constant; ΔT represents the temperature difference; d represents the bearing diameter; and T0 represents the initial temperature of the external environment.

4. The method for modeling the thermal error mechanism-data of the main shaft of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 3 is characterized in that: In step 2), the correlation between the spindle thermal error and the heat absorption of the spindle structure is as follows: In the formula, k2 and k3 represent coefficients; δ spindle The thermal error of the spindle.

5. The method for modeling the thermal error mechanism-data of the main shaft of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 1, characterized in that: In step 3), the step of constructing a mathematical expression of the heat generation of the spindle system with respect to the electro-mechanical-thermal variables includes: 3.1) Calculate the heat generated by the spindle bearing, that is: Where M and n represent the total friction torque and speed of the bearing, respectively; M0 and M1 represent the friction torque caused by the viscosity of the bearing lubricant and the friction torque caused by the bearing load, respectively; f0 and f1 are constants; γ is the kinematic viscosity of the lubricant; and d m represents the average diameter of the bearing, and F1 is the equivalent load; Indicates the heat generated by the spindle system bearings. 3.2) Combining formulas (6)-(7), we get: Where X0 and Y0 represent the radial and axial static load factors of the bearing, respectively, and F R and F A Respectively represent the radial load and axial load of the bearing; 3.3) Simplify formula (8) and get: In the formula, k4 and k5 represent coefficients; c3 represents a constant. 3.4) Calculate the heat generated by the spindle motor, that is: In the formula, and They represent the heat generated by the rotor, stator, and windage loss, respectively. express and The first-order differential of motor and η motor Represent motor power and efficiency respectively, D rotor , L rotor and f rotor Represent the rotor diameter, length and frequency respectively, μ air represents the dynamic viscosity of air, and h represents the distance between the rotor and the stator. 3.5) Simplify formula (10) to obtain the mathematical expression of the heat generation of the spindle system with respect to the electro-mechanical-thermal variables, namely: In the formula, k6 and k7 are coefficients.

6. The method for modeling the thermal error mechanism-data of a main shaft of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 1, characterized in that: In step 4), the step of constructing a mathematical expression of the heat dissipation of the spindle system with respect to the electro-mechanical-thermal variables includes: 4.1) Calculate the cooling water heat dissipation used for motor heat dissipation Right now: In the formula, c W and Respectively represent the specific heat capacity and mass flow rate of cooling water, T W,out and T W,in Respectively represent the inlet and outlet temperatures of cooling water. 4.2) Simplify formula (11) to obtain: In the formula, k8 is the coefficient. 4.3) Calculate the heat dissipation of compressed air in the motor Right now: Where, T rotor 、T stator and T ca,in Represent the rotor, stator and compressed air temperatures respectively; Respectively represent the rotor and stator convective heat transfer coefficients; A rotor , A stator Respectively represent the rotor and stator convection heat transfer areas, D cg , L cg and h W Respectively represent the cooling tank diameter, length and convection heat transfer coefficient; k9, k 10 , k 11 , k 12 represents the coefficient; κ represents the intermediate coefficient; 4.4) Simplify formula (14) to obtain: In the formula, k 13 , k 14 , k 15 Represents the coefficient. 4.5) Calculate the natural convection heat dissipation in the motor Right now: In the formula, k am and denote the thermal conductivity and average Nusselt coefficient of natural convection, D sh and A sh Respectively represent the outer diameter and surface area of ​​the spindle motor housing, T sh and T am Respectively represent the spindle housing temperature and external environment temperature; k 16 Represents the coefficient. 4.6) Calculate the heat dissipation between the bearing and the compressed air, and the heat dissipation between the bearing and the ambient air by natural convection, that is: In the formula, They respectively represent the convection heat dissipation between the compressed air and the front bearing, the convection heat dissipation between the compressed air and the rear bearing, the convection heat dissipation between the air and the front bearing, and the convection heat dissipation between the air and the rear bearing; Respectively represent the convective heat transfer coefficient of compressed air in the front and rear bearings; T front 、T rear Respectively represent the front and rear bearing temperatures; D front , D rear Respectively represent the front and rear bearing diameters; A front , A rear represent the convective heat transfer areas of the front and rear bearings respectively; represent the average Nusselt number of compressed air flowing in the front and rear bearings, respectively. 4.7) Simplify formula (17) to obtain: In the formula, k 17 , k 18 , k 19 , k 20 represents the coefficient; 4.8) Calculate the heat dissipation of the spindle through heat conduction Right now: In the formula, λ and A SC They represent thermal conductivity and equivalent surface area of ​​the spindle system respectively; l0 represents the spindle length; 4.9) Simplify formula (19) and obtain: In the formula, k 21 represents the coefficient; 4.10) Calculate the heat radiation dissipation of the spindle: Where ξ represents the thermal emissivity of the material, k B represents a constant, ξ sh Indicates the emissivity of the spindle motor housing material, A sh represents the outer surface area of ​​the spindle motor housing, ξ rear represents the front bearing material emissivity, ξ front represents the emissivity of the rear bearing material; T front 、T rear Respectively represent the front and rear bearing temperatures; T sh Indicates the spindle motor housing temperature; 4.11) Simplify formula (21) to obtain the mathematical expression of the heat dissipation of the spindle system with respect to the electro-mechanical-thermal variables, namely: In the formula, k 22 , k 23 , k 24 Represents the coefficient.

7. The method for modeling the thermal error mechanism-data of the spindle of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 1, characterized in that: in step 5), the step of establishing a theoretical model of the thermal error of the spindle system with respect to the electro-mechanical-thermal variables comprises: 5.1) According to the law of conservation of energy, the heat balance equation of the spindle system is established, namely: 5.2) Arrange the heat balance equation of the spindle system and obtain: In the formula, K1, K2, K3, K4, K5, K6, K7, K8, K9, K 10 , K 11 , K 12 , K 13 , K 14 , K 15 , K 16 represents the electro-mechanical-thermal coefficient of variation, C represents a constant; ΔT ca-fb , ΔT ca-rb , ΔT am-fb , ΔT am-rb , ΔT am-sh , ΔT b-0 They respectively represent the temperature difference between the compressed air and the front bearing, the temperature difference between the compressed air and the rear bearing, the temperature difference between the external environment and the front bearing, the temperature difference between the external environment and the rear bearing, and the temperature difference between the spindle housing and the initial temperature of the external environment. 5.3) Considering the time characteristics of the thermal error of the spindle system, a matrix thermal error model is constructed: Where, K=[K1…K7 K8…K 16 ] represents the coefficient matrix, C represents the constant matrix, represents the electro-mechanical-thermal element matrix, the elements in this matrix are the electro-mechanical-thermal real-time data measured experimentally; t1, t i ,t n Represents the time step.

8. The method for modeling the thermal error mechanism-data of a worm wheel gear grinding machine spindle considering the electro-mechanical-thermal coupling effect according to claim 1, characterized in that: In step 6), the real-time data of the electrical-mechanical-thermal elements include the temperature data of the worm wheel gear grinder, the current and voltage signals, and the displacement signals of the thermal deformation characteristic points of the spindle system.

9. The method for modeling the thermal error mechanism-data of the main shaft of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 8, characterized in that: The temperature data of the worm wheel gear grinder is obtained through the temperature sensor and temperature acquisition card; The current and voltage signals are obtained through current transformers, voltage transformers, and power meters; The displacement signal of the thermal deformation characteristic point of the spindle system is obtained through a laser displacement sensor and a displacement acquisition card.

10. The method for modeling the thermal error mechanism-data of a main shaft of a worm wheel gear grinding machine considering the electro-mechanical-thermal coupling effect according to claim 1, characterized in that: In step 7), the thermal error model of the spindle system is converted into an optimization problem, and the steps of solving the problem using the HPSO-GA optimization algorithm include: 7.1) Establish the spindle thermal error model coefficient to solve the equation, that is: In the formula, represents the spindle thermal error value obtained by experimental measurement, Indicates the spindle thermal error value obtained by the established model; F(K1, K2,…, K 15 ,K 16 , C) is the coefficient function to be solved; 7.2) Embed the genetic algorithm into the particle swarm algorithm to obtain the hybrid optimization algorithm HPSO-GA; 7.3) Use the hybrid optimization algorithm HPSO-GA to solve formula (26) and obtain the thermal error.

Images