Keywords

1 Introduction

CNC machine tools are known as “industrial mother-machines”, which significantly impact manufacturing advancement [1]. Many studies have shown that the influence of thermal error on the machining accuracy of machine tools is as high as 40–70% [2]. Therefore, it is important to reduce the influence of thermal error on the machining accuracy of the machine tool.

Researchers [3,4,5,6] at home and abroad have conducted in-depth research on thermal error compensation technology. To effectively implement thermal error compensation, it is crucial to establish a precise thermal error model. Scholars [7,8,9,10] generally use neural networks, support vector machines, multiple regression and grey theory to establish thermal error models. however, a substantial number of temperature sensors must be strategically positioned during testing, and the subsequent sncreeing of heat sensitive points should be carried out.

However, there are many problems in the selection of conventional temperature measurement points, more sensor wiring and collinearity between measurement points, therefore, to address these issues, this study examines the heat source of the linear feed system using the heat transfer mechanism and identifies the optimal thermal sensitive point to enhance the precision of the thermal error model. Moreover, A new algorithm (ROA) combined with the thermal error prediction model of least square support vector Machine (LSSVM) is proposed to optimize the kernel function, penalty function and relaxation variables of the network. Finally, thermal error measurement tests are carried out on a VDL-600A vertical machining center to verify the reliability of the proposed method.

2 Temperature Field Analysis of Feed System Based on Heat Conduction Mechanism

2.1 Temperature Field Analysis of the Feed System

As depicted in Fig. 1, the ball screw is dispersed into m lead screw units Si (i = 1 2… m), each segment length is S, the length of the non-moving end of the lead screw motor end is S0, and the length of the non-moving end of the other side is SM. the heat transfer mode is shown in Fig. 2.

Fig. 1
A schematic of the feed system structure includes a motor, a coupling, bearing housing 1, a leading screw, a nut, bearing housing 2, and a workbench. The leading screw is divided into M screw units.

Structureof the feed system

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Fig. 2
A schematic diagram of the heat transfer through a lead screw. The center segment is labeled delta U. Q f points inward, Q r points inward, Q c v points outward, and Q c d points rightward and leftward.

Lead screw heat transfer diagram

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Take Δs within the travel range of the lead screw, and the energy conservation relationship within Δt time is:

$$\Delta U = Q_{f} \left( {s,t} \right) + Q_{cd} \left( {s,t} \right) + Q_{cv} \left( {s,t} \right) + Q_{r} \left( {s,t} \right)$$
(1)

where, ΔU represents the change of internal energy in the range of Δs; Qf (s,t) represents the friction heat generated by the nut; Qcd(s,t) represents the heat conduction of the micro end of the lead screw Δs to the adjacent sides; Qcv(s,t) represents the heat pair flow between the lead screw and the surrounding air in the state of motion; Qr(s,t) represents the amount of thermal radiation between Δs and the surrounding air.

Take a Si segment of the lead screw for analysis, and the change in internal energy of the lead screw between time (t−Δt, t) is::

$$\Delta U(s,t) = c\rho \pi R^{2} \frac{\partial T(s,t)}{{4\partial t}}\Delta s\Delta t$$
(2)

where, c is the specific heat capacity and the value is 460 J/kg · ℃; ρ indicates the density of the lead screw, the value is 7850 (kg/m3); R is the screw diameter value of 40 mm, Qf (s, t) calculation formula is as follows:

$$Q_{{\text{f}}} (s,t) = \frac{{\pi R^{2} }}{4}q_{f} (s,t)\Delta s\Delta t$$
(3)

where, qf(s,t) represents the average heat generated within the travel of the lead screw, and its value is 42 W.

The calculation for heat conductivity Qcd (s, t) is as follows:

$$Q_{cd} (s,t) = \pi R^{2} k\frac{{\partial \frac{\partial T(s,t)}{{\partial s}}}}{4\partial s}\Delta s\Delta t$$
(4)

where, k value is 71.568 (W/m · ℃), The formula for Qr is:

$$Q_{r} (s,t) = \varepsilon \pi Rc_{r} (T^{4} (s,t) - T_{a}^{4} (t))\Delta s\Delta t$$
(5)

where, Ta(t) represents the ambient temperature around the lead screw at time t. the value of cr is 5.667 × 10–8 W/(m2 · K4). The calculation formula of Qcv(s,t) is as follows:

$$Q_{cv} (s,t) = \pi Rh_{cv} (T(s,t) - T_{a} (t))\Delta s\Delta t$$
(6)

where, hcv value is 60.47 (W/m · ℃). after calculation and analysis, considering the influence of heat source on bearing seat with fixed end, T(0, t) = TS0−τj is required; considering the influence of free end bearing seat, T(L, t) = TSM−τj, and Ta(t) = Ta−τj should be guaranteed.

3 Thermal Error Measurement Experiment of Feed Shaft

3.1 Measurement of Thermal Error of Linear Feed Shaft

This paper takes the X-axis of machining center of VDL-600A vertical milling machine as the research object to measure the thermal positioning error of linear feed shaft of machine tool. field measurement is shown in Fig. 3, the location of the temperature sensor is shown in Fig. 4.

Fig. 3
A photo of an experimental site has the following labeled components, temperature sensor, displacement acquisition software, mirror assembly, interferometer, and temperature acquisition software.

Experimental site diagram

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Fig. 4
A collage of 3 photos. The left photo highlights the location of temperature sensor T 2 on the motor, the top right photo highlights the location of temperature sensor T 3 on the base of the coupling, and the bottom right photo highlights the location of the T 1 sensor on the bearing house.

Location of the temperature sensor

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In the process of experimental measurement, the NI-5910 temperature acquisition system was used for temperature collection, and the XL-80 laser interferometer was used for displacement collection.

The specific test process is as follows:

  1. (1)

    First, the positioning error of the X-axis in the initial state was measured. The test method is reciprocating measurement, each measurement point stays for 2 s, the reverse overstep is 5 mm, and the reverse gap was eliminated.

  2. (2)

    Make the X-axis reciprocate in the range of 50– 550 mm at a speed of 10000mm/min for 20 min;

  3. (3)

    The X-axis stops moving to test the positioning error of the X-axis;

  4. (4)

    Repeat steps (2) and (3) until the X-axis reaches thermal equilibrium.

In this experiment, three kinds of experiments at different constant speed are designed to investigate the impact of temperature on the thermal error of the feed system at various speeds.

Table 1 displayed the test conditions in detail.

Table 1 Experimental design
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3.2 Results of Thermal Error Measurement

Taking a measurement result as an example, The collected data includes temperature values and corresponding thermal error values as follows.

As can be seen from Fig. 5, With the increase of time, the temperature change of the bearing seat at the fixed end increases the most, reaching a maximum of 33.6 ℃, while the temperature rise of the free end changes little. and finally becomes stable after 220 min. It can be seen from Fig. 6 that the displacement curve and temperature change rule of the machine tool are basically the same during operation. With the passage of time, the thermal positioning error increases gradually. The displacement caused by heat tends to be stable after 200 min and reaches a maximum of 70 μm.

Fig. 5
A line graph of temperature versus time plots 3 increasing trends for T 1, T 2, and T 3. All trends begin at (0, 28) and end at different points, with higher values for T 1 and lower values for T 2.

Temperature change curve

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Fig. 6
A line graph of thermal error versus place plots 12 trends for different time values. The curves from the origin exhibit a blunt incline up to 300 millimeters on the x-axis, after which they incline steeply.

Thermal error curve

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4 Heat Error Modeling with Raccoon-Optimized LSSVM

LSSVM is more sensitive to noise during model training, and the processing of noise is relatively poor. This may cause the training results to be affected by noise and the generalization performance is poor. Therefore, this paper uses raccoon optimization algorithm (ROA) to address the aforementioned challenges, Through the iterative process, the algorithm can search the optimal parameter combination of the least squares support vector machine model, thus improving the accuracy and generalization ability of the model.

5 Model Prediction and Evaluation

With the temperature variable and position of the best thermal sensitive points were determined as the input, and the established ROA-LSSVM model, LSSVM model and SVM neural network model were used to predict the thermal positioning error of the X feed axis. Thermal errors of X axis in 60 and 240 min predicted results and residual errors of each network model are shown in Fig. 7:

Fig. 7
3 line graphs of thermal error versus place. All graphs plot 4 inclining trends with fluctuations for predicted and true values of 60 and 240 minutes and a lateral fluctuating trend for residual values of 60 and 240 minutes for R O A L S S V M, L S S V M, and S V M networks.

Comparison of thermal error prediction results

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As Fig. 7 illustrates, it is evident that the predicted residual value of SVM network model is 10.9 μm at most. The maximum residual predicted by LSSVM model is 8.6 μm. After adjusting the LSSVM model using ROA algorithm, the fluctuation becomes significantly smaller, and the maximum residual is 1.9 μm.

The model’s goodness of fit was assessed according to the root-mean-square error RMSE, fitting coefficient R2, mean absolute error MAE and mean deviation MBE. The goodness of fit of each model is shown in Table 2.

Table 2 Goodness of Fit evaluation of each network
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As indicated in Table 2, Compared with SVM network, the RMSE of ROA-LSSVM model is reduced by 45%. Compared with the LSSVM network, the RMSE of the ROA-LSSVM model is reduced by 42%. It is proved that the thermal error prediction model of CNC machine tool linear axis based on ROA-LSSVM network has higher prediction accuracy.

6 Conclusions

  1. (1)

    In view of the difficulty in determining the thermal sensitive points, heat transfer analysis of the feed system is carried out using the heat conduction theory to simulate the heat transfer process more accurately, so that the modeling results are more close to the actual situation.

  2. (2)

    The thermal error prediction model based on ROA-LSSVM is established, and the least square support vector machine is improved by using ROA algorithm to iteratively refine the LSSVM prediction model parameters, which is difficult to determine parameters, easy to fall into overfitting and low prediction accuracy.

  3. (3)

    After the thermal error measurement experiment of VDL-600A vertical machining center, the heat transfer theory adopted provides a strong support for determining the thermal sensitive points of the linear feed system, and greatly enhances the accuracy of the network model. In addition, the ROA-LSSVM model avoids the random error caused by the empirical setting parameters, and the goodness of fit indexes are better than the traditional LSSVM model and SVM model. It has achieved a great breakthrough in forecasting accuracy.