阅读说明：本技术 一种可预测滚珠丝杠温升及热误差的建模方法 (Modeling method capable of predicting temperature rise and thermal error of ball screw ) 是由 满兵 郭永环 范希营 于 2021-06-23 设计创作，主要内容包括：本发明公开了一种可预测滚珠丝杠温升及热误差的建模方法,针对传统热误差模型中忽略滚珠丝杠螺母副的相对运动环节与滚珠丝杠热误差之间关联关系的问题,提出一种考虑实际工况的瞬态热-结构耦合模型,利用APDL将滚珠丝杠螺母设定为移动热源载荷,定义滚珠丝杠螺母副与滚珠丝杠的位移-时间关系,模拟滚珠丝杠在轴承和滚珠丝杠螺母副热源作用下的温度和热变形分布,并以温升数据为输入、热误差数据为输出建立了基于粒子群算法PSO优化灰色神经网络GNN热误差预测模型,结果表明,该建模方法能够较好地预测进给系统的热误差,进而可以为实际工况下滚珠丝杠热误差的补偿提供可靠的数据支持。(The invention discloses a modeling method capable of predicting temperature rise and thermal error of a ball screw, which aims at the problem that the incidence relation between a relative motion link of a ball screw nut pair and the thermal error of the ball screw is neglected in a traditional thermal error model, provides a transient thermal-structural coupling model considering the actual working condition, sets the ball screw nut as a mobile heat source load by using APDL, defines the displacement-time relation between the ball screw nut pair and the ball screw, simulates the temperature and thermal deformation distribution of the ball screw under the action of a bearing and a heat source of the ball screw nut pair, and a particle swarm optimization PSO-based gray neural network GNN thermal error prediction model is established by taking temperature rise data as input and thermal error data as output, and the result shows that, the modeling method can better predict the thermal error of the feeding system, and further can provide reliable data support for the compensation of the thermal error of the ball screw under the actual working condition.)

1. A modeling method capable of predicting temperature rise and thermal error of a ball screw is characterized by comprising the following steps:

step one, constructing a transient thermal-structure coupling finite element simulation model: firstly, establishing a ball screw theoretical three-dimensional model, carrying out grid division, then carrying out thermal boundary condition calculation according to calorific value calculation of a bearing and calorific value calculation of a ball screw nut pair, finally carrying out mobile thermal load loading by using APDL (active Power distribution device) to take heat generated by the ball screw nut pair as mobile thermal load on the basis of the theoretical three-dimensional model, calculating a temperature field and a thermal deformation field of a ball screw feeding system by segmented loading and unloading of heat flow density and convection coefficient, and constructing a transient thermal-structural coupling finite element simulation model;

step two, constructing a PSO-GNN thermal error prediction model: on the basis of a transient thermal-structure coupling finite element simulation model, firstly, a grey neural network thermal error prediction model which takes temperature rise data as input and thermal error data as output is constructed, and then, the grey neural network prediction model is optimized by a particle swarm algorithm to generate a PSO-GNN thermal error prediction model.

2. The modeling method capable of predicting the temperature rise and the thermal error of the ball screw according to claim 1, wherein in the first step, when the heat generated by the ball screw nut pair is used as the moving heat load to carry out the moving heat load loading, 7200s is selected as the total simulation duration, and in each step, the heat convection coefficient of the contact surface of the ball screw nut pair and the ball screw is firstly deleted, and then the heat flux density is applied to the ball screw and the bearing.

3. The modeling method capable of predicting the temperature rise and the thermal error of the ball screw according to claim 1, wherein in the second step, when the thermal error prediction model of the gray neural network is constructed, the differential equation expression of the gray neural network model with n parameters is

In the formula, y₁,y₂,…,y_nFor the system input parameter, y₁For system output parameters, a, b₁,b₂…,b_n-1Is the differential equation coefficient;

the response formula of the grey neural network model with n parameters is

Order toThen

And mapping the above formula into a BP neural network to obtain a gray neural network with n input parameters and 1 output parameter.

4. The modeling method for predicting the temperature rise and the thermal error of the ball screw according to claim 3, wherein LA, LB, LC and LD respectively represent four layers of a gray neural network, andthen the initial weight of the network is

w₂₁＝a,w₂₁＝-y₁(0),w₂₂＝u₁,w₂₃＝u₂,…,w_2n＝u_n-1

w₃₁＝w₃₂＝…＝w_3n＝1+e^-at

The threshold value of the output node in the LD is (1+ e)^-at)(d-y₁(0) Network prediction error ofWherein N is the number of samples, y is the expected output value of thermal error, and y is the expected output value of thermal error₁An output value is predicted for the thermal error.

5. The modeling method for predicting ball screw temperature rise and thermal error according to claim 4, wherein the training steps of the grey neural network algorithm are as follows:

step I: initializing a network structure, randomly initializing parameters a and b, and calculating u according to the values of a and b;

step II: calculating w according to the network weight definition₂₁,w₂₂,…,w_2n,w₃₁,w₃₂,…,w_3n；

Step III: for each input sequence (t, y (t)), t ═ 1,2,3, … N, the per-layer outputs are calculated

LA layer: a ═ w₁₁t

LB layer:

LC layer: c. C₁＝bw₂₁,c₂＝y₂(t)bw₂₂,c₃＝y₃(t)bw₂₃,c_n＝y_n(t)bw_2n

An LD layer: d ═ w₃₁c₁+w₃₂c₂+w_3nc_n-θ_y1；

Step IV: correcting the weight value, namely updating the connection weight value and the threshold value according to the prediction error e to enable the predicted value to approach the expected value continuously;

step V: and judging whether the algorithm is finished or not, and if not, returning to the step III.

6. The modeling method capable of predicting the temperature rise and the thermal error of the ball screw according to claim 1, wherein in the second step, the specific algorithm flow for generating the PSO-GNN thermal error prediction model by optimizing the gray neural network prediction model by using the particle swarm optimization is as follows:

step I, determining a network topological structure, and giving an initial weight and a threshold;

step II, after initializing the particle swarm, taking an error obtained by training a gray neural network as a fitness value;

step III, after searching individual extreme values and group extreme values, updating the speed and the position;

step IV, recalculating the particle fitness value, updating the individual extreme value and the group extreme value, if the ending condition is met, obtaining the optimal weight and the threshold value, and if the ending condition is not met, repeatedly entering the step III;

step V, calculating network output and errors after obtaining the optimal weight and the threshold;

and VI, updating the weight and the threshold, if the ending condition is met, simulating and predicting an output result, and if the ending condition is not met, repeatedly entering the step V.

7. The modeling method for predicting ball screw temperature rise and thermal error as claimed in claim 1, wherein in step one, grid division is performed by using curvature and approximate size refinement grid in key area.

8. The modeling method capable of predicting the temperature rise and the thermal error of the ball screw according to claim 1, wherein after a transient thermal-structural coupling finite element simulation model is built, the temperature rise of the center of the ball screw is used as a control target, and orthogonal test and variance analysis are performed on the control target and a plurality of input controllable factors to obtain the controllable factors having significant influence on the temperature rise of the center of the ball screw.

Technical Field

The invention relates to a modeling method, in particular to a modeling method capable of predicting ball screw temperature rise and thermal error based on the transient thermal-structural coupling characteristic of a ball screw, and belongs to the technical field of precision machine tool manufacturing.

Background

The ball screw has the characteristics of high precision, high efficiency and high rigidity, so that the ball screw is widely applied to a feeding transmission system of a machine tool. However, during high-speed feeding, a large amount of frictional heat is generated between the bearing and the nut, thereby causing a temperature rise and thermal deformation. Thermal deformation of the ball screw feed transmission system causes positioning deviation, and eventually, the machining accuracy is lowered. Therefore, reasonably modeling the thermal error of the ball screw of the machine tool and realizing the compensation of the thermal error on the basis of the modeling has important significance for improving the machining precision of the machine tool.

In the prior art, the thermal error of a ball screw feeding system is usually researched by researching the relationship between the temperature of the ball screw and the thermal error, and the following two methods are generally researched:

the first approach is to use an empirical model to determine the relationship between thermal error and temperature rise. If the temperature data of the critical heating value point is used for carrying out theoretical modeling on the thermal error by adopting a regression analysis method, and then tests are carried out under different conditions so as to research the influence of thermal expansion on the ball screw feeding transmission system; and establishing the relationship between the bearing temperature rise and the operation condition based on the wavelet neural network and the NARMA-L2 model, and the like. However, the error modeling method based on the experiment usually needs more sensors, so that the cost is higher, the temperature measurement of the key rotating parts is very difficult, and the real data is often difficult to obtain.

The second approach is to use a theoretical model to predict the thermal error. The finite difference method and the finite element method are generally employed to calculate the thermal characteristics of the ball screw feed system. If a three-dimensional finite element analysis method is adopted to carry out transient thermal-structure coupling analysis on the guide rail and the feeding system, and a finite element model is established, so that a finite element simulation result of a thermal error is obtained; also, the method is based on a numerical simulation method that the pretightening force of the thermal three-dimensional finite element model changes along with the temperature rise; and predicting the heat flow rate, the temperature distribution and the thermal error of the ball screw feeding system by combining a finite element method and a Monte Carlo method, and the like. The thermal characteristics of the ball screw feeding system are researched by using a finite element method, and although the method is convenient and low in cost, the calculated values of the heat source and the boundary conditions of numerical simulation often have certain difference from the actual working condition, so that the simulation precision is influenced.

Disclosure of Invention

Aiming at the problems, the invention provides a modeling method capable of predicting the temperature rise and the thermal error of the ball screw, which is characterized in that a transient thermal-structural coupling model is established according to the incidence relation between the relative motion link of the actual working condition of the ball screw nut and the thermal error of the ball screw, so that the thermal error of a feeding system can be well predicted, and reliable data support can be provided for the compensation of the thermal error of the ball screw under the actual working condition.

In order to achieve the purpose, the modeling method capable of predicting the temperature rise and the thermal error of the ball screw specifically comprises the following steps:

step one, constructing a transient thermal-structure coupling finite element simulation model: firstly, establishing a ball screw theoretical three-dimensional model, carrying out grid division, then carrying out thermal boundary condition calculation according to calorific value calculation of a bearing and calorific value calculation of a ball screw nut pair, finally carrying out mobile thermal load loading by using APDL (active Power distribution device) to take heat generated by the ball screw nut pair as mobile thermal load on the basis of the theoretical three-dimensional model, calculating a temperature field and a thermal deformation field of a ball screw feeding system by segmented loading and unloading of heat flow density and convection coefficient, and constructing a transient thermal-structural coupling finite element simulation model;

step two, constructing a PSO-GNN thermal error prediction model: on the basis of a transient thermal-structure coupling finite element simulation model, firstly, a grey neural network thermal error prediction model which takes temperature rise data as input and thermal error data as output is constructed, and then, the grey neural network prediction model is optimized by a particle swarm algorithm to generate a PSO-GNN thermal error prediction model.

Further, in the first step, when heat generated by the ball screw nut pair is used as the moving heat load to carry out moving heat load loading, 7200s is selected as the total simulation duration, the heat convection coefficient of the contact surface of the ball screw nut pair and the ball screw is firstly deleted every step, and then the heat flux density is applied to the ball screw and the bearing.

Further, in the second step, when the grey neural network thermal error prediction model is constructed, the differential equation expression of the grey neural network model with n parameters is

In the formula, y₁,y₂,…,y_nFor the system input parameter, y₁For system output parameters, a, b₁,b₂…,b_n-1Is the differential equation coefficient;

the response formula of the grey neural network model with n parameters is

Then

And mapping the above formula into a BP neural network to obtain a gray neural network with n input parameters and 1 output parameter. Further, LA, LB, LC, LD represent four layers of the gray neural network, respectively, and orderThen the initial weight of the network is

w₂₁＝a,w₂₁＝-y₁(0),w₂₂＝u₁,w₂₃＝u₂,…,w_2n＝u_n-1

w₃₁＝w₃₂＝…＝w_3n＝1+e^-at

The threshold value of the output node in the LD is (1+ e)^-at)(d-y₁(0) Network prediction error ofWherein N is the number of samples, y is the expected output value of thermal error, and y is the expected output value of thermal error₁An output value is predicted for the thermal error.

Further, the grey neural network algorithm training steps are as follows:

step I: initializing a network structure, randomly initializing parameters a and b, and calculating u according to the values of a and b;

step II: calculating w according to the network weight definition₂₁,w₂₂,…,w_2n,w₃₁,w₃₂,…,w_3n；

Step III: for each input sequence (t, y (t)), t ═ 1,2,3, … N, the per-layer outputs are calculated

LA layer: a ═ w₁₁t

LB layer:

LC layer: c. C₁＝bw₂₁,c₂＝y₂(t)bw₂₂,c₃＝y₃(t)bw₂₃,c_n＝y_n(t)bw_2n

An LD layer: d ═ w₃₁c₁+w₃₂c₂+w_3nc_n-θ_y1；

Step IV: correcting the weight value, namely updating the connection weight value and the threshold value according to the prediction error e to enable the predicted value to approach the expected value continuously;

step V: and judging whether the algorithm is finished or not, and if not, returning to the step III.

Further, in the second step, the particle swarm optimization is used for optimizing the grey neural network prediction model to generate a PSO-GNN thermal error prediction model, and the specific algorithm flow is as follows:

step I, determining a network topological structure, and giving an initial weight and a threshold;

step II, after initializing the particle swarm, taking an error obtained by training a gray neural network as a fitness value;

step III, after searching individual extreme values and group extreme values, updating the speed and the position;

step IV, recalculating the particle fitness value, updating the individual extreme value and the group extreme value, if the ending condition is met, obtaining the optimal weight and the threshold value, and if the ending condition is not met, repeatedly entering the step III;

step V, calculating network output and errors after obtaining the optimal weight and the threshold;

and VI, updating the weight and the threshold, if the ending condition is met, simulating and predicting an output result, and if the ending condition is not met, repeatedly entering the step V.

Further, in the first step, the grid is divided by using curvature and approximate size to refine the grid in the key area.

Further, after a transient thermal-structure coupling finite element simulation model is constructed, the central temperature rise of the ball screw is used as a control target, orthogonal test and variance analysis are carried out on the control target and a plurality of input controllable factors, and the controllable factors which have obvious influence on the central temperature rise of the ball screw are obtained.

Compared with the prior art, the modeling method capable of predicting the temperature rise and the thermal error of the ball screw is directed at the problem that the incidence relation between the relative motion link of the ball screw nut pair and the thermal error of the ball screw is neglected in a traditional thermal error model, provides a transient thermal-structural coupling model considering the actual working conditions, sets the ball screw nut as the load of a mobile heat source by using APDL, defines the displacement-time relation between the ball screw nut pair and the ball screw, simulates the temperature and thermal deformation distribution of the ball screw under the action of the heat source of a bearing and the ball screw nut pair, analyzes the influence of different working conditions (the feeding speed, the cutting load and the pretightening force of the ball screw) on the temperature rise of the center of the ball screw on the basis, establishes a gray neural network thermal error prediction model based on a particle swarm algorithm by taking temperature rise data as input and thermal error data as output, the result shows that the modeling method can better predict the thermal error of the feeding system, and further can provide reliable data support for the compensation of the thermal error of the ball screw under the actual working condition.

Drawings

FIG. 1 is a theoretical three-dimensional model diagram of a ball screw;

FIG. 2 is a grid-divided view of a ball screw;

FIG. 3 is a schematic diagram of mobile heat load loading;

FIG. 4 is a graph showing the simulation result of temperature rise, wherein (a) is a cloud graph showing the temperature field distribution of a ball screw, and (b) is a graph showing the temperature rise curve of the ball screw;

FIG. 5 is a graph of node temperature change;

FIG. 6 is a diagram showing the results of thermal deformation simulation, wherein (a) is a thermal deformation profile of the Y-axis of the ball screw, and (b) is a diagram showing the axial thermal deformation profile of the ball screw;

FIG. 7 is a plot of nodal axial thermal deformation;

FIG. 8 is a diagram of experimental verification, in which (a) is a diagram of temperature rise test and simulation comparison of a ball screw nut, and (b) is a diagram of thermal deformation test and simulation comparison of an axial end of the ball screw;

FIG. 9 is a diagram of a gray neural network topology;

FIG. 10 is a flow chart of particle swarm optimization for a gray neural network;

FIG. 11 is a graph of thermal error prediction curves and residual errors under cutting conditions.

Detailed Description

The modeling method for predicting the temperature rise and the thermal error of the ball screw is discussed below by taking the TBI ball screw SFUR1605 in taiwan as an example.

The basic parameters of the ball screw are as follows: nominal diameter d 16mm, lead P_h5mm, ball diameter d_b3.175 mm; modulus of elasticity E ═ 2.19X 10⁵MPa, Poisson's ratio mu of 0.3, density rho of 7810kg/m³The front end and the rear end of the ball screw are erected and installed through a supporting end bearing and a fixed end bearing respectively, and the ball screw nut pair is matched and sleeved on the ball screw.

Step one, constructing a transient thermal-structure coupling finite element simulation model

The ball screw feed system is complicated in structure, especially in the contact portion between the screw-ball-nut. When finite element analysis is performed, the calculation time is too long, and even the calculation result is divergent. In order to improve the calculation efficiency, on the basis of not influencing the analysis result, the following assumptions are made before modeling the finite element model: tool withdrawal grooves, chamfers and threads on the ball screw are omitted; the lubricant has negligible effect on heat transfer; the heat transfer coefficient and the convective heat transfer coefficient of each contact part are constants; in the process of the circular reciprocating operation of the ball screw nut, the friction heat generated by the bearing and the ball screw nut pair is constant.

Firstly, a theoretical three-dimensional model is established by using three-dimensional drawing software SolidWorks as shown in figure 1, meshing is carried out, meshes of a finite element model are reasonably divided, the solving precision and accuracy can be ensured, the convergence speed is accelerated, meshing is carried out by using a 10-node tetrahedral unit SOLID90, and the meshing result is shown in figure 2. In order to improve the grid dividing quality of the ball screw finite element model, the grid is automatically refined in a key area by using the functions of curvature and approximate size, and 30750 nodes and 18597 grid units are generated in total.

Then, setting thermal boundary conditions of the theoretical three-dimensional model:

calculating the calorific value of the bearing, wherein the calorific value of the bearing is generated by the friction between the rolling body and the support ring of the bearing, and the calculation formula is as follows

Wherein Q is the calorific value (W) of the rolling bearing, N is the rotating speed (r/min) of the bearing, and M is the friction torque (N mm);

the calorific value of the ball screw nut pair is calculated, the calorific value of the ball screw nut pair is similar to that of a bearing in the engineering process, the calorific value can still be calculated by the formula, and the total friction moment M at the moment is calculated by the driving moment M of the screw_DAnd the screw resistance moment M of the ball_PThe composition is calculated by the formula:

M＝M_D+0.94M_P

wherein M is friction torque (N.mm)_DIs the driving torque of a ball screw, M_PIs the screw moment of resistance, F, of the ball_aIs the axial force (N, F) applied to the screw nut_pFor pre-tightening (N, P) of the screw nut_hThe lead of the lead screw (mm) is adopted, and eta is the transmission efficiency of the ball screw nut pair;

the thermal boundary condition is calculated, the rotational motion of the ball screw can accelerate the convection of the surface of the screw and the ambient air, and the convection heat transfer coefficient h (W/(m) is calculated according to the Nouchert correlation theory²K)) can be solved and calculated by the following formula

In the formula N_μλ is the air heat conductivity coefficient (W/(m · K)), and L is the characteristic dimension (L ═ 16mm), in knowlett number;

the screw rod belongs to forced convection heat transfer, so the Nu Shert number N_μIs calculated by the formula

In the formula R_eIs Relo number, P_rIs the Plantt number, w is the screw angular velocity (rad/s), d is the screw diameter (mm), upsilon' is the kinematic viscosity of air (mm)²/s)；

The forced convection heat transfer coefficient h of the surface of the ball screw is calculated as shown in table 1.

TABLE 1 forced convection heat transfer coefficient of ball screw surface

And finally, carrying out moving thermal load loading on the basis of the theoretical three-dimensional model, and constructing a transient thermal-structure coupling finite element simulation model:

when the ball screw feeding system is actually operated, the rolling screw nut pair does not stay at one position fixedly but moves circularly and reciprocally, and in order to obtain more accurate temperature field and thermal deformation data, heat generated by the rolling screw nut pair is applied to the surface of the ball screw in the form of moving heat flux density. The loading process is shown in fig. 3 and mainly includes the following two aspects: firstly, the ball screw nut pair circularly moves, DO circulation in ANSYS is adopted to realize reciprocating circular motion of the ball screw nut pair (heat source load), and IF sentences are adopted to judge whether the ball screw nut pair reaches one end of the effective stroke of the ball screw to perform reverse motion; and secondly, applying heat flux density and convection coefficient, and only reading the boundary condition applied last by default when the convection coefficient and the heat flux density are loaded at the same position at the same time, so that the heat convection coefficient h of the contact surface of the ball screw nut pair and the ball screw is deleted and the heat flux density q is applied to the ball screw and the bearing when each step is taken.

The thermal balance of the ball screw when the ball screw runs for 7200s under no load is basically achieved, therefore, 7200s is selected as the total simulation duration, and the temperature rise simulation result when the ball screw 7200s is obtained under the running condition that the feeding speed is 500r/min, the pretightening force is 1104N and the no load is as shown in FIG. 4. It can be seen from fig. 4(a) that the temperature rise distribution of each part of the ball screw is not uniform, and the temperature field distribution exhibits the characteristics of high middle and low two ends. APDL is used for defining a table of 2400 rows, temperature of the selected position point is collected in real time at intervals of 3s, collected data are drawn into a ball screw temperature rise curve graph shown in figure 4(b) through MATLAB, and the temperature rise rule of the ball screw along with the change of time and position is shown in figure 4 (b).

A node is respectively taken from the middle points of the support end bearing, the fixed end bearing and the ball screw of the ball screw feeding system, and the temperature change data of the positions of 3 temperature sensitive points are respectively extracted to generate a temperature change curve as shown in fig. 5. It can be known from fig. 5 that, along with the increase of the circulation operation time of the ball screw nut pair, the temperature of the ball screw is higher and higher, the temperature rise is fast in the first 3000s, the screw reaches the thermal equilibrium state in about 3500s, the temperature of the ball screw fixed end bearing is slightly higher than that of the ball screw supporting end bearing, and the highest temperature of the ball screw middle position is about 24.8 ℃.

The temperature field analysis result is used as a thermal load, transient thermal-structural coupling simulation is performed on the ball screw, and displacement constraint is performed on the ball screw, the ball screw nut pair and the bearings at the two ends according to actual working conditions, so that a thermal deformation simulation result is obtained as shown in fig. 6. As can be seen from the thermal deformation distribution of the Y-axis of the ball screw in fig. 6(a), the thermal deformation law of the ball screw at 7200s is approximately extended toward both ends, the extension amount of the fixed end is small, and the maximum thermal deformation appears at the support end of the screw as the thermal deformation increases further away from the fixed end, which is about 31.6 μm. Fig. 6(b) is an axial thermal error curved surface diagram of each point in the screw stroke range along with the change of time, and the transient thermal deformation rule of the ball screw can be visually seen.

In order to better analyze the axial thermal deformation of each node along with the time change, starting from the position of a bearing at the supporting end by 0mm, taking one node every 100mm, and reading the axial thermal deformation data of 5 nodes along with the time change to generate an axial thermal error curve of the nodes along with the time change shown in figure 7, wherein as can be seen from figure 7, the axial thermal deformation rate of the ball screw is fast in the first 2000s, the ball screw deforms to a stable state approximately in 3500s, and the axial thermal error curve is matched with the temperature rise change rule of the ball screw.

The ball screw feeding system is tested under the same working condition as simulation, the thermal infrared imager is used for collecting temperature information of the ball screw, the eddy current displacement sensor is used for collecting thermal deformation information of the axial tail end, the obtained thermal deformation result is shown in fig. 8, as can be seen from fig. 8(a) and 8(b), under the working condition, the maximum error between the temperature test value and the simulation value of the central position of the ball screw obtained through the test is 1.6 ℃, and the average absolute percentage error is 2.0%. The maximum error between the axial end thermal deformation test value and the simulation value is 2.11 mu m, and the average absolute percentage error is 3.4%. The constructed transient thermal-structural coupling simulation model has better accuracy and reliability in the aspects of temperature distribution and thermal deformation of the ball screw, and can predict the thermal error of the ball screw on the basis of the transient thermal-structural coupling simulation model.

In order to further study the influence rule of different working conditions of the ball screw feeding system on the temperature change of the central position of the ball screw, an orthogonal test is designed by taking the pretightening force A, the feeding speed B and the cutting load C as controllable factors and taking the temperature rise D of the central position of the ball screw as a control target. The levels of the controlled factors were determined according to the range of the respective operating conditions as shown in table 2.

TABLE 2 factor level table

Selecting a five-level orthogonal table L according to the level number of each working condition₂₅(5⁶). The orthogonal table is populated with the above test factors and horizontal settings and the tests are performed sequentially. Finite element simulation of the ball screw transient temperature field is carried out under different working condition combinations, and the temperature data of the ball screw center are collected as shown in the following table 3.

TABLE 3 test protocol and test results

And secondly, carrying out variance analysis: in an orthogonal test of the temperature rise rule of the central position of the ball screw of the feeding system, the influence of the horizontal change of each factor on the temperature rise can be determined through variance analysis, and the influence significance of each factor on the test result is determined. The results of the test were subjected to data processing using analysis of variance, the analysis results being shown in table 4 below.

TABLE 4 ANOVA TABLE

According to the analysis result of the variance, the feeding speed B and the cutting load C have very obvious influence on the temperature rise of the center of the ball screw, the pretightening force A has small influence on the temperature rise, and the simulation result can help an engineer to optimize the structure or adjust the processing process program.

Step two, constructing a PSO-GNN thermal error prediction model

Firstly, a grey neural network PSO thermal error prediction model is constructed:

the grey neural network integrates the grey system and the BP neural network, so that the prediction precision is improved, and the thermal error deformation can be predicted better.

The differential equation expression of the grey neural network model with n parameters is

In the formula, y₁,y₂,…,y_nFor the system input parameter, y₁For system output parameters, a, b₁,b₂…,b_n-1Is the differential equation coefficient;

the response formula of the grey neural network model with n parameters is

The response formula of the grey neural network model with n parameters is

Order toThen

Mapping the above formula into a BP neural network to obtain a gray neural network with n input parameters and 1 output parameter, wherein the topological structure is shown in FIG. 9, where t is the serial number of the input parameter, y is the serial number of the input parameter, and_2(t),…,y_n(t)for inputting parameters to the network, w₂₁,w₂₂,…,w_2n,w₃₁,w₃₂,…,w_3nIs the network weight, y₁For the network prediction values, LA, LB, LC, LD represent four layers of the gray neural network, respectively.

Order toThe network initial weight may be expressed as

w₂₁＝a,w₂₁＝-y₁(0),w₂₂＝u₁,w₂₃＝u₂,…,w_2n＝u_n-1

w₃₁＝w₃₂＝…＝w_3n＝1+e^-at

The threshold value of the output node in the LD is (1+ e)^-at)(d-y₁(0) Network prediction error ofWherein N is the number of samples, y is the expected output value of thermal error, and y is the expected output value of thermal error₁An output value is predicted for the thermal error.

The grey neural network algorithm training steps are as follows:

step I: initializing a network structure, randomly initializing parameters a and b, and calculating u according to the values of a and b;

step II: calculating w according to the network weight definition₂₁,w₂₂,…,w_2n,w₃₁,w₃₂,…,w_3n；

Step III: for each input sequence (t, y (t)), t ═ 1,2,3, … N, the per-layer outputs are calculated

LA layer: a ═ w₁₁t

LB layer:

LC layer: c. C₁＝bw₂₁,c₂＝y₂(t)bw₂₂,c₃＝y₃(t)bw₂₃,c_n＝y_n(t)bw_2n

An LD layer: d ═ w₃₁c₁+w₃₂c₂+w_3nc_n-θ_y1；

Step IV: correcting the weight value, namely updating the connection weight value and the threshold value according to the prediction error e to enable the predicted value to approach the expected value continuously;

step V: and judging whether the algorithm is finished or not, and if not, returning to the step III.

Secondly, a particle swarm algorithm is used for optimizing a grey neural network prediction model to generate a PSO-GNN thermal error prediction model:

due to the fact that the thermal error mechanism of the ball screw feeding system is complex, the single model is difficult to comprehensively consider the variation trend and the influence factors of the thermal error of the feeding system, and the accuracy in actual prediction is low. Therefore, the model is established by combining and optimizing two or more algorithms, and the defects and limitations of a single algorithm can be effectively reduced. The particle swarm optimization has the characteristics of high convergence speed and more accurate local optimal solution, so that the gray neural network prediction model is optimized by adopting the particle swarm optimization.

The particle swarm optimization algorithm is a biological heuristic method in the field of computational intelligence and mainly simulates the predation behavior of a bird swarm. Suppose that in a D-dimensional target search space, there are m particles that make up a population. Wherein the velocity of the particle i in space is V_i＝(V_i1,V_i2，…,V_iD)^TIn the position X_i＝(X_i1,X_i2,…,X_iD)^TRepresenting a potential solution to the problem. Calculating each particle X according to the objective function_iA corresponding fitness value. Let the value of the individual extremum of the ith particle searched so far be P_b＝(P_b1,P_b2,…,P_bD)^TThe population extremum of the population is P_g＝(P_g1,P_g2，…,P_gD)^T. In each iteration the particle passes through P_bAnd P_gDynamically adjusting the speed and the position of the self body, and the formula is

Where w is the inertia factor, D is (1,2, …, D), i is (1,2, …, m), k is the current iteration number, the velocity V of the particle_id∈[-V_max,V_max]Learning acceleration c₁And c₂Is a non-negative constant, r₁And r₂Is at [0,1 ]]A random number in between.

Because the grey neural network learning process is mainly the weight and threshold adjusting process, the connection weight and the threshold of each layer of the grey neural network can be used as the positions of the particles in the particle swarm optimization algorithm, and the fitness value of each particle is calculated according to the network training error, so that the individual extremum and the global extremum are updated, and the network model has better thermal error prediction output. An algorithm flow of the particle swarm optimization for optimizing the gray neural network is shown in fig. 10, that is, a specific algorithm flow is as follows:

step I, determining a network topological structure, and giving an initial weight and a threshold;

step II, after initializing the particle swarm, taking an error obtained by training a gray neural network as a fitness value;

step III, after searching individual extreme values and group extreme values, updating the speed and the position;

step IV, recalculating the particle fitness value, updating the individual extreme value and the group extreme value, if the ending condition is met, obtaining the optimal weight and the threshold value, and if the ending condition is not met, repeatedly entering the step III;

step V, calculating network output and errors after obtaining the optimal weight and the threshold;

and VI, updating the weight and the threshold, if the ending condition is met, simulating and predicting an output result, and if the ending condition is not met, repeatedly entering the step V.

Ball screw thermal error prediction analysis:

the input layer of the grey neural network thermal error prediction model is the temperature rise of the key point position, the temperature rise of the supporting end bearing, the temperature rise of the fixed end bearing and the temperature rise of the ball screw center are included, the number of nodes is 3, the output layer is the axial thermal deformation of the ball screw, and the number of nodes is 1. The learning rate was set to 0.0015 and the number of iterations was 100. For the particle swarm algorithm, the population quantity is too large, the calculation complexity is increased, and the premature phenomenon can occur if the population quantity is too small. The population size was set to 30 and the maximum number of evolutionary events was 100.

Reading the temperature field of the screw rod and the thermal error analysis result every 120s, and reading 60 groups of data samples in total, wherein 40 groups of samples are used for network training, and 20 groups of samples are used for network prediction. In order to verify the accuracy of the method, the same test sample is adopted, the same network input parameters are set, and the predicted performance of the method is compared and analyzed with the thermal error models of the traditional BP neural network and the gray neural network respectively. The ball screw thermal error prediction model is evaluated by adopting the average absolute percentage error MAPE and the root mean square error RMSE, and the fitting accuracy comparison of different models is shown in the following table 5. As can be seen from Table 5, the highest fitting accuracy was for the PSO-GM model, with a MAPE of 4.1% and a RMSE of 1.3 μm. The fitting accuracy of the BP neural network and the grey neural network thermal error prediction model is relatively poor, the average absolute percentage errors are respectively 6.8% and 5.6%, and the root mean square errors are respectively 4.5 μm and 1.7 μm.

TABLE 5 comparison of fitting accuracy of different models

After the fitting performance of the thermal error prediction model is verified, the generalization performance of the thermal error prediction model is further considered, namely whether the thermal error prediction model has higher prediction precision in other unknown working conditions is verified, and the method is a key index for evaluating whether the model is reliable. And predicting thermal errors according to temperature rise data under the cutting condition that the feeding speed is 3m/min, the pretightening force is 276N and the cutting load is 800N, and recording the mean absolute percentage error root mean square error as MAPE1 and RMSE1 respectively. Comparison of prediction accuracy of different models under cutting conditions is shown in Table 6, and it can be seen that MAPE1 and RMSE1 of the PSO-GNN model are the minimum, which are 1.9% and 0.63 respectively.

TABLE 6 comparison of prediction accuracy of different models under cutting conditions

The thermal error prediction curves of different models under the cutting conditions are shown in the example of FIG. 11. The closer the thermal error prediction curve is to the actual measurement curve, the closer the residual error curve is to 0, and the better the prediction effect is. It can be seen that the thermal error prediction performance of the PSO-GNN model is obviously superior to that of other models, and the generalization performance is better.