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Both multicollinearity and utilization deficiency of temperature sensors affect the robustness and the prediction precision of traditional thermal error prediction models. To address the problem, a thermal error prediction model without temperature sensors is proposed. Firstly, the paper analyzes the temperature field and thermal deformation mechanisms of the spindle of a CNC gear grinding machine in accordance with the parameters, efficiencies, and structures of the machine spindle and bearing. A preliminary theoretical model is established on the basis of the mechanism analysis. Secondly, the theoretical model is corrected according to the actual operation parameters of the machine. Thirdly, verification experiments are carried out on machine tools of the same type. It is found that the corrected model has higher precision in predicting thermal errors at the same rotational velocity. The standard deviation and the maximum residual error are reduced by at least 39% and 48% separately. The prediction precision decreases with the increase in prediction range when predicting thermal errors at different rotational velocities. The model has high prediction precision and strong robustness in the case of reasonable prediction range and classified prediction. In a word, prediction precision and robustness of the model without temperature sensors can be effectively ensured by reasonably determining the prediction range and practicing classified prediction and compensation for thermal errors at different rotational velocities. The model established can be applied to machine tools that have difficulties in arranging sensors or those whose sensors are significantly disturbed.

Thermal errors, which have become one of the major impact factors on component manufacturing precision in CNC precision machining, account for 50 ~ 70% of all manufacturing errors [

The core problem of thermal error compensation is to establish a mathematical model that has high prediction precision and strong robustness. Generally, thermal error compensation follows the steps below.

To address the above problems a thermal error compensation model without temperature sensors is proposed in this paper. The theoretical prediction models for temperature field and thermal deformation are established based on the mechanism analysis of the spindle of a CNC gear grinding machine (YKZ7230). The paper simulates actual operation conditions of the spindle and corrects the theoretical model in accordance with the temperature and thermal deformation data collected, thereby obtaining a mathematical model that is consistent with the actual situation. Considering that acquisition of the corrected data is influenced by working environments, necessary measures are taken to minimize the impact of environmental factors on sensors, so as to ensure the modeling precision. The sensorless prediction of thermal errors has the following advantages:

Contributions of this paper lie in the following two aspects:

Spindle is a major supporting component of the gears of CNC gear grinding machines. Therefore, thermal errors caused by spindle motor heating and bearing heating shall inevitably exert a significant impact on the gear machining quality.

As is shown in Figure

Schematic of heat generation of a spindle.

Therefore, the motor heat is expressed as

According to Document [

The overall heat equation of the spindle is expressed as_{B1} is the heat generated by the upper bearing;_{B2} is the heat generated by the lower bearing.

Equation (_{out} is the heat dissipation of the spindle.

Based on (

The spindle is heated during rotation, leading to forced convection between the spindle and the air. The forced convection coefficient _{up} can be expressed as^{2}/s);

Theoretical model for the spindle temperature field in the warming process can be obtained by substituting _{up} and (

The spindle conducts natural convection heat transfer to air when it stops rotating. The natural convection coefficient _{down}=10W/(m^{2}·K) and the heat input

To establish the theoretical model for thermal deformation, it is necessary to analyze the structure and deformation characteristics of the gear grinding machine first. The structure of the machine tool is shown in Figure

Structure of the machine tools.

Thermal deformation of the spindle.

Figure

An abbreviated drawing of spindle thermal deformation.

As is shown in Figure

A structural diagram of the spindle.

B_{1}B_{2}, which represents the distance between the upper and lower bearings of the spindle, is expressed by _{1}, which represents the distance between the upper bearing and the mandrel, is expressed by _{1} is fixed while B_{2} is mobile, indicating that thermal extension in B_{1}B_{2} has no impact on thermal errors. WB_{1} experiences single point heating, with B_{1} as the heated point. Since thermal deformation that takes place in WB_{1} influences the precision of machine tools, Δ_{1 }(see Figure

Triangular temperature distribution.

Equation (_{1}. _{down}

In the following sections, the temperature model correction process is illustrated by the example of the spindle.

The spindle rotates at a constant velocity during machining, leading to an increase in temperature. It stops rotating when the machining is completed and the temperature drops accordingly in that case. The theoretical and actual temperature curves of the machine tools during operation are shown in Figure

Schematic drawing of the theoretical and actual temperature curves.

_{up1} is the correction slope during heating;_{up2} is the temperature correction value during heating.

The optimal correction coefficients _{up1}, and_{up2} can be obtained by solving the optimal solution of the objective function. The objective function during heating can be established in accordance with Document [

_{dn1} is the correction slope during the _{dn2} is the temperature correction value during the

The thermal error model is corrected in a similar way to the temperature model. The thermal error model during heating is corrected in accordance with (

The optimal correction coefficients

According to the theoretical analysis in previous chapters, it is necessary to correct the theoretical models in accordance with the temperature and thermal deformation data acquired. The experiment takes YKZ7230 (a CNC gear grinding machine) as the experimental platform and uses HIOKI8423 as the data acquisition instrument. The temperature and displacement sensors adopted are E-type MG-24E-GW1-ANP (a temperature sensor manufactured by Anritsu Meter) and DGC-8ZG/D (a noncontact displacement sensor manufactured by Zhongyuan Measuring) separately. The experimental system is shown in Figure

Experimental system.

In accordance with the international standard Test Code for Machine Tools-Part 3: Determination of Thermal Effects (IS0 230-3:2001 IDT) [

To avoid the impact of sensor defects on the model proposed, the following measures were taken:

Considering the changes of temperature fields and thermal errors with rotational velocity, experiments were carried out at different spindle velocities. Since the rotational velocity of YKZ7230 ranges between 0 and 300r/min, the experimental rotational velocities are evenly distributed in this range. The grouping of the experimental rotational velocities is shown in Table

Experimental rotational velocity.

Category | Velocity (r/min) | Category | Velocity (r/min) |
---|---|---|---|

1 | 30 | 6 | 180 |

2 | 60 | 7 | 210 |

3 | 90 | 8 | 240 |

4 | 120 | 9 | 270 |

5 | 150 | 10 | 300 |

In order to accurately arrange the temperature sensors,

Infrared image of the spindle and arrangement of the temperature sensors. (a) Before heating. (b) After heating. (c) Arrangement of the temperature sensors.

Experimental temperature curves.

Thermal deformation of the spindle mainly takes place in the Y direction and the Z direction. Therefore, the displacement sensors S1 and S2 are arranged as in Figure

Arrangement of displacement sensors.

Table _{e} in (

Spindle parameters.

Parameters | Units | Value |
---|---|---|

Spindle diameter | D(mm) | 105 |

Spindle bore | D_{K}(mm) | 75 |

Spindle length | L(mm) | 260 |

Spindle density | ^{2}) | 7.6×10^{3} |

specific heat capacity | c(J/(kg ⋅ k)) | 460 |

Linear expansion coefficient | _{t}(1/k) | 1× |

Thermal conductivity | 31.2 |

Correction coefficient of the temperature field model.

Velocity (r/min) | Coefficient | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

| | | | | | | | | | | | |

30 | 0.72 | 1.89 | 1.53 | 1.23 | 0.92 | 1.58 | 1.93 | -2.1 | 1.34 | 1.15 | 1.26 | 1.14 |

60 | 0.96 | 2.03 | 1.32 | -0.5 | 1.52 | 2.3 | 1.41 | 1.77 | 1.62 | 1.9 | 1.75 | 1.73 |

90 | 1.05 | 1.71 | 1.47 | 1.56 | 1.43 | -0.9 | 1.27 | 1.43 | 1.36 | 1.46 | 1.04 | 1.38 |

120 | 1.16 | 1.83 | 0.97 | -2.7 | 1.29 | 1.8 | 1.53 | -1.8 | 1.56 | -0.3 | 1.97 | 1.47 |

150 | 1.21 | 1.9 | 1.23 | 1.91 | 1.34 | 3.2 | 1.36 | -0.9 | 1.8 | -0.5 | 2.1 | 1.6 |

180 | 1.25 | 1.74 | 1.46 | 1.75 | 1.09 | 0.95 | 1.17 | 1.5 | 0.96 | 1.64 | 1.37 | -2.3 |

210 | 1.34 | -0.3 | 0.98 | 0.83 | 0.96 | 2.1 | 1.04 | 1.27 | 1.23 | 1.3 | 1.42 | -1.1 |

240 | 1.38 | 0.64 | 1.12 | 0.97 | 0.78 | 1.55 | 0.95 | -1.7 | 1.53 | -1.5 | 1.54 | 2.41 |

270 | 1.43 | 0.79 | 1.08 | -2.1 | 0.93 | 1.85 | 0.91 | -1.1 | 1.79 | 1.33 | 1.69 | -3.7 |

300 | 1.49 | 0.62 | 1.05 | -1 | 1.93 | 1.42 | 1.12 | 2.7 | 1.02 | 1.39 | 1.72 | -1.1 |

Correction coefficients of the thermal deformation model.

Velocity (r/min) | Coefficient | ||
---|---|---|---|

_{1} | _{2} | | |

30 | 1.43 | -0.79 | 0.065 |

60 | 1.37 | -0.87 | 0.063 |

90 | 1.33 | -0.54 | 0.059 |

120 | 1.34 | -0.72 | 0.052 |

150 | 1.29 | -0.62 | 0.049 |

180 | 1.26 | -0.31 | 0.046 |

210 | 1.2 | 0.15 | 0.044 |

240 | 1.13 | 0.24 | 0.041 |

260 | 1.09 | -0.42 | 0.04 |

300 | 1.05 | 0.47 | 0.038 |

To illustrate the effect of the modified model, one of these models is used as an example to discuss.

Figure

Comparison of temperature curves before and after correction.

Figure

Comparison of △

Table

Prediction results of △

Model | STD | MR | Error sum of squares |
---|---|---|---|

Theory | 2.8 | 7.1 | 512 |

Correction | 1.1 | 1.5 | 68 |

Figure

Comparison of

Table

Prediction results of

Model | STD | MR | Error sum of squares |
---|---|---|---|

Theory | 1.8 | 3.3 | 193 |

Correction | 1.1 | 1.7 | 79 |

According to the above experiments, the maximum residual and the standard derivation of the corrected model are 1.7

Maximum residual and standard deviation of the corrected model increased from 1.6

To verify the effective prediction range of the corrected model, model (150r/min) was adopted to predict thermal deformation at 30r/min, 60r/min, 90r/min, 120r/min, 180r/min, 210r/min, 240r/min, 270r/min, and 300r/min, respectively. The prediction residuals are shown in Figure

Prediction result by correction model (150r/min).

Velocity (r/min) | STD ( | MR ( |
---|---|---|

30 | 8.8 | -12.9 |

60 | 6.7 | 10.7 |

90 | 4.5 | -7.4 |

120 | 1.3 | 2.3 |

180 | 1.4 | 1.8 |

210 | 4.6 | 6.5 |

240 | 7 | 11.2 |

270 | 8.6 | 12.5 |

300 | 9.1 | -13.1 |

Error prediction result by model (150r/min): (a) residual result of 150r/min and (b) profile of X axis and Y axis.

To verify the above conclusions, model (30r/min) and model (300r/min) were adopted to predict △

Thermal prediction result by model (30r/min).

Thermal prediction result by model (300r/min).

Since the prediction results on

It can be concluded from the above analysis that the prediction model corrected based on temperature and thermal errors at a specific velocity has a specific prediction range. Therefore, the prediction range of the model should be limited to a specific range, so as to ensure the precision of the model proposed.

To further improve the prediction precision of the corrected model, the prediction range at different rotational velocities was limited to modeling velocity ±30r/min. Due to the impacts of machine tool assembly, lubrication, and other factors, thermal errors may be different. Therefore, verification experiments were carried out on another CNC gear grinding machine of the same type, so as to verify the robustness of the model.

Model (90r/min), model (150r/min), model (210r/min), and model (270r/min) were employed to predict thermal errors in corresponding velocity ranges and the prediction results were shown in Tables

Prediction result by model (90r/min).

Velocity (r/min) | STD ( | MR ( |
---|---|---|

60 | 1.7 | 2.4 |

70 | 1.5 | -2.3 |

80 | 1.3 | 1.9 |

100 | 1.4 | 1.9 |

110 | 1.5 | -2.2 |

120 | 1.6 | -2.5 |

Prediction result by model (150r/min).

Velocity (r/min) | STD ( | MR ( |
---|---|---|

120 | 1.8 | 2.3 |

130 | 1.7 | -1.9 |

140 | 1.5 | 2 |

160 | 1.3 | 1.7 |

170 | 1.8 | -2.1 |

180 | 2 | -2.5 |

Prediction result by model (210r/min).

Velocity (r/min) | STD ( | MR ( |
---|---|---|

180 | 1.9 | -2.5 |

190 | 1.8 | 2.3 |

200 | 1.5 | -1.9 |

220 | 1.5 | 2.1 |

230 | 1.5 | -2.1 |

240 | 1.9 | 2.5 |

Prediction result by model (270r/min).

Velocity (r/min) | STD ( | MR ( |
---|---|---|

240 | 1.6 | 2.5 |

250 | 1.6 | 2.2 |

260 | 1.3 | -2.0 |

280 | 1.5 | 2.2 |

290 | 1.7 | -2.3 |

300 | 1.8 | -2.7 |

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This study is supported by the National Natural Science Foundation of China (no. 51375382), the Science and Technology Support Plan Project of Sichuan province, China (no. 2016GZ0205), Sichuan Key Research and Development Projects (no. 18FZ0089), and Key Laboratory Project of Shaanxi Education Department (no. 17JS095).