A thermal error modeling method for direct-drive feed axis based on multi-physics field coupling

Abstract

A method for modeling thermal errors in direct-drive feed shafts based on multi-field coupling includes the following steps: 1. Modeling the temperature field of the direct-drive feed shaft to obtain a partial differential equation based on multi-field coupling; 2. Solving the partial differential equation based on multi-field coupling to obtain the temperature field equation; and 3. Establishing a thermal error model for the direct-drive feed shaft based on expansion theory. This method models the error mechanism, comprehensively considering the coupling effects of the electromagnetic field, thermal field, and flow field of the direct-drive feed shaft under actual operating conditions. This method establishes a display analytical function that reflects the entire process of thermal error changes.

Description

Technical Field

[0001] The present invention belongs to the technical field of machining accuracy of numerically controlled machine tools, and relates to a direct-drive feed axis thermal error modeling method based on multi-physical field coupling. Background Art

[0002] The feed axis driven by a linear motor has the advantages of large acceleration, fast speed, short response time, high precision, etc. It does not require an intermediate transmission link and can achieve "direct drive". It has broad application prospects in machine tool feed drive.

[0003] However, at high speeds, direct-drive feed axis systems generate more heat than ball screw transmission systems. Their unique open structure directly causes thermal deformation of external components, affecting the system's positioning accuracy. Thermal error caused by thermal deformation is a significant factor affecting the accuracy of machine tool feed axes. Therefore, it is necessary to study the thermal error model for direct-drive feed systems.

[0004] While extensive research has been conducted on thermal errors in ball screw drive systems, direct-drive feed axis systems differ significantly in structure from those with ball screw drive systems. Furthermore, the heating mechanism of linear motors is more complex than that of rotary motors. Therefore, these research methods are not applicable to direct-drive feed axes. Furthermore, traditional finite element methods and thermal resistance network methods struggle to obtain explicit analytical solutions, hindering their integration into CNC systems for real-time compensation. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this paper provides a method for modeling thermal errors in direct-drive feed axes based on multi-physics field coupling. This method models the error mechanism and comprehensively considers the coupling effects of electromagnetic, thermal, and flow fields in direct-drive feed axes under actual operating conditions. This method then establishes a visual analytical function that reflects the entire process of thermal error variation.

[0006] A direct-drive feed axis thermal error modeling method based on multi-physics field coupling includes the following steps:

[0007] 1. Model the temperature field of the direct-drive feed axis and obtain a partial differential equation based on multi-field coupling;

[0008]

[0009] Where T is the surface temperature of the direct drive feed axis, which is the function to be determined; t is the motion time; ρ c =ρc, c is the specific heat capacity, ρ is the density; h is the convection heat dissipation coefficient; A is the heat dissipation area; T f0 is the ambient temperature; η is the dissipation rate; P J is the electromagnetic loss; y is the position of the direct drive feed axis; a is the thermal diffusion coefficient;

[0010] 2. Solve the partial differential equation based on multi-field coupling to obtain the temperature field equation;

[0011] 3. Based on the expansion theory, a thermal error model of the direct-drive feed axis is established;

[0012]

[0013] Divide the entire feed slide into equal parts, record the length of each section as △L, and the midpoint of each section as P i (i=1,2,3......,m), each section of the feed axis is isotropic, and the thermal expansion coefficient of each section is α l are the same, introduce the correction coefficient ξ, and take It is the function of temperature of each segment to time.

[0014] The beneficial effects of the present invention compared to the prior art are:

[0015] The present invention studies the temperature characteristics of the feed shaft under electromagnetic-thermal-flow field coupling, and establishes a thermal error model of the direct-drive feed shaft based on thermal expansion theory. In view of the difficult-to-solve partial differential equations established in the multi-physics field coupling, a simplified coupling equation solution method is proposed, which can solve the displayed analytical solution and obtain a thermal error function model with time and position as independent variables. The model has a clear and relatively simple expression and can be embedded in the CNC model for real-time compensation. Compared with empirical modeling, the modeling method based on the thermal error mechanism is more theoretical and robust, and high-precision prediction can be achieved through a small amount of measurement data, with a prediction error within ±1um.

[0016] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments: BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 This is a process diagram of the direct-drive feed axis thermal error modeling method based on multi-physics field coupling of the present invention;

[0018] Figure 2 This is a diagram of the electromagnetic field analysis process;

[0019] Figure 3 This is the analysis process diagram of the thermal-flow field;

[0020] Figure 4 This is the flow chart of the coupled equation separation method;

[0021] Figure 5 It is a simplified diagram of the direct drive feed axis structure;

[0022] Figure 6 is a flow chart of the parameter identification procedure;

[0023] Figure 7This is an experimental diagram of measuring the thermal error of the direct-drive feed shaft in the embodiment;

[0024] Figure 8 A comparison diagram of the direct-drive feed axis thermal error model prediction value and the actual value in the embodiment;

[0025] Figure 9 This is a cloud diagram of the thermal error model distribution of the direct-drive feed shaft in the embodiment. DETAILED DESCRIPTION

[0026] See also Figure 1 As shown, a direct-drive feed axis thermal error modeling method based on multi-physics field coupling in this embodiment includes the following steps:

[0027] 1. Model the temperature field of the direct-drive feed axis and obtain the partial differential equation based on multi-field coupling:

[0028]

[0029] Where T is the surface temperature of the direct drive feed axis, which is the function to be determined; t is the motion time; ρ c =ρc, c is the specific heat capacity, ρ is the density; h is the convection heat dissipation coefficient; A is the heat dissipation area; T f0 is the ambient temperature; η is the dissipation rate; P J is the electromagnetic loss; y is the position of the direct drive feed axis; a is the thermal diffusion coefficient;

[0030] When establishing a partial differential equation that can reflect the temperature changes of the direct-drive feed axis, the thermal analysis comprehensively considers factors such as heat generation from the heat source, heat conduction from the feed axis slide, and heat dissipation through air convection. This involves studying the electromagnetic field, thermal field, and flow field. In order to analyze the temperature field, these issues need to be considered comprehensively. Therefore, when conducting thermal analysis, it is necessary to first analyze the electromagnetic field of the linear motor, then consider the heat expansion and heat dissipation caused by the thermal-flow field, and comprehensively consider the mutual coupling of the three physical fields.

[0031] 2. Solve the partial differential equation based on multi-field coupling to obtain the temperature field equation;

[0032] 3. Based on the expansion theory, a thermal error model of the direct-drive feed axis is established;

[0033]

[0034] According to the obtained temperature field and based on the thermal expansion theory, the thermal error model of the direct drive feed axis is established. The entire feed slide is divided into m equal parts, the length of each section is recorded as △L, and the midpoint of each section is recorded as P i (i=1,2,3......,m), such as Figure 5 As shown, each section of the direct drive feed axis is isotropic, and the thermal expansion coefficient of each section is α lare the same. Considering the influence of temperature change on thermal expansion coefficient, a correction factor ξ is introduced, and the is a function of each temperature versus time, the positive thermal deformation of the direct-drive feed axis can be simplified to the above thermal error model;

[0035] In step 2: the establishment process of the partial differential equation based on multi-field coupling is:

[0036] a. Analyze the electromagnetic field of the linear motor

[0037] The steps of electromagnetic field analysis are as follows: Figure 2 As shown in the figure, based on Maxwell's equations and the traveling wave magnetic field generated by the energized winding in the air gap, the magnetic field B and electric field E generated by the primary winding of the linear motor can be calculated; the heat source of the permanent magnet linear motor comes from the electromagnetic loss of the electromagnetic field. In order to calculate the heat generated by the heat source, it is necessary to analyze the energy of the electromagnetic field.

[0038] When H and E are known, the Poynting vector S can be used to represent the energy density of the electromagnetic field. To analyze the energy change of the electromagnetic field, the Poynting vector S can be expanded:

[0039]

[0040] in, represents the integral form of the Poynting vector; It represents the electromagnetic energy reduced per unit time, and W represents the total energy of the electromagnetic field; represents the electromagnetic loss generated in the volume V, J represents the current density, represents resistivity; Indicates the energy provided by the power supply within the volume V, E e Indicates that the outside field is strong;

[0041] Separate electromagnetic losses of the electromagnetic field from electromagnetic energy:

[0042]

[0043] Electromagnetic loss can be divided into copper loss and iron core loss according to the structure of the motor. Copper loss is the main heat source of electromagnetic loss:

[0044]

[0045] Where i represents the current; R cu Indicates copper resistance; is the coefficient of resistance change with temperature; △T is the temperature change;

[0046] b. Analyze the thermal flow field of the linear motor;

[0047] The analysis steps of thermal field-flow field are as follows: Figure 3As shown, according to the Navier-Stokes equations (NS equations) and the energy conservation equation, the basic governing equations of the convective heat transfer problem (NS equations and energy conservation equations) are solved:

[0048] :

[0049] Where ρ is the fluid density; v is the fluid velocity; f is the volume force of the fluid; p is the fluid pressure; μ is the dynamic viscosity; E is the total energy (kinetic energy and internal energy) of the fluid; τ ij is the surface force of the fluid; q r It is an internal heat source;

[0050] The electromagnetic-thermal-fluid field interaction is a complex process. To achieve multi-physics coupling, electromagnetic energy flux can be added to the heat-fluid field control equation to achieve energy balance and coupling. For a one-dimensional feed axis, it can be simplified to a one-dimensional, steady-state, incompressible problem without internal heat sources. Only electromagnetic losses in the electromagnetic term are considered, and the effects of fluid kinetic energy, surface forces, and volume forces are ignored. The simplified equation for a one-dimensional direct-drive feed axis is:

[0051]

[0052] Where T is the surface temperature of the direct drive feed axis, which is the function to be determined; t is the motion time; ρ c =ρc, c is the specific heat capacity, ρ is the density; h is the convection heat dissipation coefficient; A is the heat dissipation area; T f0 is the ambient temperature; η is the dissipation rate; P J is the electromagnetic loss, y is the position of the direct drive feed axis, and a is the thermal diffusion coefficient.

[0053] Furthermore, in order to solve the temperature field equation, a simplified coupled equation solution method is proposed to solve the obtained partial differential equation. Considering that the purpose of thermal error modeling is to embed the model into the CNC system to compensate for thermal error, a clear and relatively simple expression is required to achieve the purpose of real-time compensation. Therefore, a coupled separation method is proposed based on the actual mechanism of heat conduction and convection heat transfer. The analysis steps are as follows: Figure 4 As shown:

[0054] First, the coupling equations are separated to obtain the electromagnetic loss equation, convective heat transfer equation, and heat conduction equation, which correspond to the three physical fields respectively; secondly, the electromagnetic dissipation equation and convective heat transfer equation are solved, and they are substituted into the heat conduction equation as boundary conditions to solve the temperature field equation.

[0055] First, due to the rapid propagation of the electromagnetic field, it can be assumed that the heat generated by electromagnetic loss is evenly distributed in the feed shaft. At the same time, at the initial working stage, heat generation is more obvious than the effects of heat conduction and heat convection. It can be assumed that only the electromagnetic field plays a role in the initial heat generation stage, and the electromagnetic loss equation is separated from Equation (4):

[0056]

[0057] Solving equation (5) yields the temperature variation of the feed shaft over time during the heat generation phase:

[0058]

[0059] When the direct-drive feed axis reaches thermal steady state, the effect of heat conduction is no longer so obvious, resulting in the temperature of each zone in the slide no longer changing with time. At the same time, the heat generation of the linear motor also reaches saturation. At this time, it can be considered that only the flow field is acting. The convective heat transfer equation is separated from Equation (4) to obtain Equation (7):

[0060]

[0061] Solving equation (7), we can obtain the change of feed shaft temperature with position due to air convection heat dissipation in the steady state:

[0062] T br (y) = C2·e χ·y +C1·e -χ·y +T fo (8)

[0063] Among them, C1 and C2 are parameters to be identified;

[0064] Considering the temperature rising stage, since the heat conduction efficiency is the highest at this time, the temperature change is affected by both time and position. The heat conduction equation is separated from formula (4):

[0065]

[0066] Take equations (6) and (8) as boundary conditions: T(0,t)=T br ; T(y,0)=T f Substituting into equation (9) and solving it, we can obtain the temperature field of the direct-drive feed axis under the coupling of electromagnetic and thermal three fields:

[0067] T(t,y)=T br +erf(θ)·(T f -T br ) (10)

[0068] Where erf(θ) is the Gaussian error function,

[0069] Through the above method, a rapid separation and coupling of the control equations is achieved, avoiding the solution of complex partial differential equations. Such a solution can not only obtain an explicit analytical solution, but also has a clear physical meaning.

[0070] Furthermore, a laser interferometer is used to measure the thermal error of the direct drive feed axis. Based on the measured data, the parameters to be identified in the thermal error model are identified based on the least squares algorithm, and a complete thermal error model is obtained. The values of the four parameters to be identified ξ, η, C1, and C2 are obtained. The steps of the identification program are shown in the figure below. Figure 6 shown.

[0071] Example

[0072] This implementation case is operated on a linear motor driven feed axis (model: Wanto-MT140-C2-832-CP2-3) and uses a laser interferometer to measure thermal errors, such as Figure 7 As shown, the PC issues control commands, which drive the linear motor to reciprocate through the controller. A laser interferometer and temperature sensor are then used to measure the positioning error and ambient temperature of the direct-drive feed axis (LMDFS), and the measured data is fed back to the PC in real time. The specific steps are as follows:

[0073] (1) Maintaining a constant indoor temperature, a laser interferometer was used to measure the original error Err0 of the direct-drive feed axis. The total length of the slide was 0.8 m, and the positioning error was measured at 16 points at intervals of 0.05 m.

[0074] (2) After the direct drive feed axis works continuously for 20 minutes, 50 minutes, 80 minutes, 110 minutes, 170 minutes, and 230 minutes, step (1) is repeated to measure the errors of the corresponding time. There are a total of six sets of data (Err1~Err6). The original error Err0 is subtracted from these six sets of data to obtain the thermal error matrix Err d .

[0075] (3) Input the thermal error matrix into the identification program ( Figure 6 As shown in Figure 3, the values of the four parameters ξ, η, C1, and C2 in the thermal error model can be identified. Figure 8 The relationship between the predicted value and the actual value of the thermal error model of the direct-drive feed axis is shown (the upper figure shows the measured value of the thermal error of the direct-drive feed axis, the middle figure shows the comparison between the thermal error modeling and the measured data, and the lower figure shows the comparison between the root mean square error of the thermal error modeling based on the middle figure and the measured data). The root mean square error (RMSE) between the two is within 1μm. Figure 9 The thermal error distribution cloud map established using this method is shown.

[0076] Experiments show that the thermal error model of the direct-drive feed axis based on multi-physics field coupling is consistent with the actual value. This method can efficiently and accurately predict the thermal error of the direct-drive feed axis, provide theoretical support for the precise compensation of the direct-drive feed axis, and improve the overall machining accuracy of the machine tool.

[0077] The present invention has been disclosed above with reference to preferred embodiments, but this is not intended to limit the present invention. Any person skilled in the art who, without departing from the scope of the technical solution of the present invention, can make slight changes or modifications to the above-disclosed structures and technical contents to produce equivalent embodiments with equivalent changes, all of which still fall within the scope of the technical solution of the present invention.

Claims

1. A direct-drive feed axis thermal error modeling method based on multi-physics field coupling, characterized by: The following steps are included:

1. Model the temperature field of the direct-drive feed axis and obtain a partial differential equation based on multi-field coupling; The establishment process of the partial differential equation based on multi-field coupling is: a. Analyze the electromagnetic field of the linear motor Based on Maxwell's equations and the traveling wave magnetic field generated by the energized winding in the air gap, calculate the magnetic field B and electric field E generated by the primary winding of the linear motor; When H and E are known, the Poynting vector S is used to represent the energy density of the electromagnetic field. To analyze the energy change of the electromagnetic field, the Poynting vector S is expanded: in, represents the integral form of the Poynting vector; It represents the electromagnetic energy reduced per unit time, and W represents the total energy of the electromagnetic field; represents the electromagnetic loss generated in the volume V, J represents the current density, represents resistivity; Indicates the energy provided by the power supply within the volume V, E e Indicates that the outside field is strong; Separate electromagnetic losses of the electromagnetic field from electromagnetic energy: Due to electromagnetic loss, copper loss is the main heat source: Where i represents the current; R cu Indicates copper resistance; is the coefficient of resistance change with temperature; ΔT is the temperature change; b. Analyze the thermal flow field of the linear motor; Solve the basic governing equations for convective heat transfer problems based on the Navier-Stokes equations and the energy conservation equation: NS equations and energy conservation equations: Where ρ is the fluid density; v is the fluid velocity; f is the volume force of the fluid; p is the fluid pressure; μ is the dynamic viscosity; E is the total energy of the fluid; τ ij is the surface force of the fluid; q r It is an internal heat source; The electromagnetic energy flux is added to the control equation of the thermal flow field to achieve energy balance and coupling; the simplified equation for the one-dimensional feed axis is: Where T is the surface temperature of the direct drive feed axis, which is the function to be determined; t is the motion time; ρ c =ρc, c is the specific heat capacity, ρ is the density; h is the convection heat dissipation coefficient; A is the heat dissipation area; T f0 is the ambient temperature; η is the dissipation rate; P J is the electromagnetic loss, y is the position of the direct drive feed axis, and a is the thermal diffusivity; Where T is the surface temperature of the direct drive feed axis, which is the function to be determined; t is the motion time; ρ c =ρc, c is the specific heat capacity, ρ is the density; h is the convection heat dissipation coefficient; A is the heat dissipation area; T f0 is the ambient temperature; η is the dissipation rate; P J is the electromagnetic loss; y is the position of the direct drive feed axis; a is the thermal diffusion coefficient; 2. Solve the partial differential equation based on multi-field coupling to obtain the temperature field equation; First, separate the electromagnetic loss equation from equation (4): Solving equation (5) yields the temperature variation of the feed shaft over time during the heat generation phase: When the direct drive feed axis reaches thermal steady state, the linear motor heat also reaches saturation state, and the convection heat transfer equation is separated from equation (4) to obtain equation (7): Solving equation (7), we can obtain the change of feed shaft temperature with position due to air convection heat dissipation in the steady state: T br (y)=C2·e χ·y +C1·e -χ·y +T fo (8) Among them, C1 and C2 are parameters to be identified; Considering the temperature rising stage, the temperature change is affected by both time and position, and the heat conduction equation is separated from formula (4): Take equations (6) and (8) as boundary conditions: T(0,t)=T br ; T(y,0)=T f Substituting into equation (9) and solving it, we can obtain the temperature field of the direct-drive feed axis under the coupling of electromagnetic and thermal three fields: T(t,y)=T br +erf(θ) · (T f -T br ) (10) Where erf(θ) is the Gaussian error function, 3. Based on the expansion theory, a thermal error model of the direct-drive feed axis is established; Divide the entire feed slide into equal parts, record the length of each section as ΔL, record the midpoint of each section as Pi, i = 1, 2, 3..., m, each section of the feed axis is isotropic, and the thermal expansion coefficient of each section is α l are the same, introduce the correction coefficient ξ, take T(t,y pi ) is the function of temperature versus time for each segment.

2. The direct-drive feed axis thermal error modeling method based on multi-physics field coupling according to claim 1, characterized in that: The process of solving the temperature field equation in step 2 is: First, the coupling equations are separated to obtain the electromagnetic loss equation, convective heat transfer equation, and heat conduction equation, which correspond to the three physical fields respectively; secondly, the electromagnetic dissipation equation and convective heat transfer equation are solved, and they are substituted into the heat conduction equation as boundary conditions to solve the temperature field equation.

3. The direct-drive feed axis thermal error modeling method based on multi-physics field coupling according to claim 1 is characterized in that: It also includes the parameter identification process; The thermal error of the direct-drive feed axis is measured using a laser interferometer. Based on the measured data and the least squares algorithm, the parameters to be identified in the thermal error model are ξ, η, C1, and C2.

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