The machining accuracy of machine parts and machine tool manufacturing errors caused by thermal errors are closely related.

The research of related literature shows that: in precision machining, the manufacturing error caused by thermal deformation accounts for 50% ~ 70% of the total manufacturing error.

As an important component of the machine tool, the thermal deformation error of the spindle system is the main source of machine tool thermal error.

Therefore, the study and analysis of the thermal characteristics of the spindle system is the key to ensure the manufacturing accuracy of the machine tool.

When the machine tool is working, under the action of internal and external heat sources, the components of the spindle system form their own temperature fields, and the inconsistent thermal expansion performance of each component will lead to thermal deformation of the mechanical structure of space, causing the processing error of the parts.

By the dynamics, static, thermal deformation and processing errors of bearing and journal, etc., the instantaneous rotation axis of CNC lathe spindle in space position is constantly changing.

The experimental results show that: precision turning roundness error is about 30% ~ 70%, which is caused by the spindle rotation error, and the higher the precision of the machine, the greater the proportion.

Spindle rotation accuracy reflects the dynamic performance of the lathe, and the lathe can achieve the machining accuracy is closely related.

The spindle thermal deformation produced in the lathe machining process also has a great influence on the spindle rotation accuracy, and its detection and compensation control can improve the machining accuracy.

With the wide application of high-speed and high-precision machine tools, the detection accuracy and efficiency are gradually improved, and the detection method is rapidly developing from static detection to dynamic and online detection.

The measurement of the shaft system slewing error has shifted from one-way measurement to multi-point measurement, and the measurement accuracy is improving.

When measuring the spindle rotation thermal error, the actual spindle rotation axis is not visible.

The trajectory of the spindle axis can only be obtained indirectly by measuring the external contour of the standard checker bar mounted on the spindle.

As a result, the measurement results are inevitably mixed with the shape error and mounting error of the standard checker bar.

For precision spindles with high rotary accuracy, mixed shape or mounting errors can sometimes overwhelm even minor spindle slew errors.

Therefore, for the measurement of high precision lathe spindle rotation error, the shape error and installation error mixed in cannot be ignored, an effective method must be adopted to separate and remove the signal components introduced by the measurement system that affect the measurement accuracy from the measurement signal to obtain the spindle slew accuracy.

This article uses FFT method to decompose signals based on the theory of spindle slewing accuracy described by complex vectors, by analyzing and eliminating the components that have no influence on the spindle turning accuracy to extract the spindle turning accuracy, then evaluate the rotary accuracy of the machine spindle thermal deformation and analyze its machining accuracy.

**1****. M****easurement principle of spindle thermal error **

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**easurement principle of spindle thermal error**

The thermal deformation of the spindle system includes both axial and radial thermal deformation.

To measure the axial thermal deformation, an eddy current sensor is placed in the spindle overhang.

Radial heat deformation of the spindle is a two-dimensional variable and should be measured indirectly using the bidirectional quadrature method.

The measurements include errors in spindle manufacturing and installation, thermal distortion, etc.

If the impact of spindle thermal deformation on machining accuracy is to be accurately assessed, the thermal deformation error needs to be separated from the comprehensive error.

The measurement principle for machine tool spindle slew accuracy is:

The radial runout generated when the spindle motor drives the spindle in a rotary motion causes the distance between the eddy current sensor and the surface of the measured part to change.

The eddy current sensor and signal converter converts them into analog voltage signals for timing collection.

Spindle rotation accuracy has a great influence on the shape accuracy and surface roughness of finishing parts, which is an important index to evaluate the machining accuracy of machine tools.

It predicts the minimum form error and roughness that a machine can achieve under ideal machining conditions, and can also be used for machine tool compensation.

The amount of radial runout due to thermal deformation of the spindle is shown in Figure 1.

- Oo is the ideal center of rotation, which is the mounting center defined by the spindle bearing components.
- Or is the actual rotation center of the spindle.
- Om is the geometric center of the reference sphere.
- Rm is the diameter of the reference section.
- e is the mounting eccentricity of the gaging mandrel.
- θ is the rotation angle of the gaging mandrel.

Thermal deformation of the spindle bearing after a period of motor operation can cause the center of rotation Or of the spindle system to shift at different temperature fields.

Combined with the movement of the spindle, this results in a real-time change in the distance between the eddy current displacement transducer and the surface of the measuring cylinder.

The voltage value of the displacement variation containing the error information is measured by eddy current sensors and signal converters.

As shown in Figure 1, the displacement signals dx and dy detected by the two displacement transducers are respectively:

ｄｘ ＝ｅｃｏｓθ＋ｒｘ（α）＋Ｓｘ（θ） （１）

ｄｙ＝ｅｓｉｎθ＋ｒｙ（α）＋Ｓｙ（θ） （２）

where

- ecosθ and esinθ are the projections of eccentricity e in the X and Y directions, respectively.
- rx(α) and ry(α) are the projections of the radial motion error r(α) in the X and Y directions, respectively.
- Sx(θ) and Sy(θ) are the shape error of two corresponding points where the detection bars differ by 90°, respectively.

In the measurement process, a high-precision detector bar with a shape error much smaller than the rotation error is used as a reference.

Figure 1: signal analysis of heat distortion error

When the shape error of the high-precision detector is negligible, dx and dy are the components of the displacement of the center of the circular section in the X and Y directions.

In other words, due to the existence of mounting eccentricity, dx and dy determine the path of the geometric center of the circular cross-section instead of the path of the rotation axis.

Therefore, to minimize the effect of eccentricity on dx and dy, and to make the measurement result more realistic (α), the eccentricity e must be minimized or eliminated as much as possible.

**2. Mathematical model of error motion**

**2. Mathematical model of error motion**

The radial motion error has periodic and radial characteristics:

Periodicity means that the circular contour signal has the property of varying with a period of 2π.

Radiality refers to the fact that the actual profile of a circular cross-section is a complex closed curvilinear profile with different radial dimensions at various points on the profile, varying in size.

The Fourier progression of the radial rotary motion of the element under test is described as:

where n is the maximum harmonic order of the measured circular contour harmonic component

S0 is the DC component of the measured circular contour data, with respect to the initial sensor mounting position.

Ai and Bi are the amplitudes of the harmonic components of order i along the x- and y-axes, respectively.

The practical implication of Eq. (3) is that the periodic radial error motion can be decomposed into many octave components that do the circular motion.

To obtain the true radial motion error, the DC component and the eccentricity e of the element under test should be removed from the measured data.

**3****. ****Spindle thermal error measurement**

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**Spindle thermal error measurement**

As shown in Figure 2, the test object is a FANUC CNC lathe. magnetic suction type high-precision temperature sensors are arranged in the spindle motor, front flange and front wall of spindle case respectively, and the changes of ambient temperature are collected at the same time.

The machine tool is running empty at different rotational speeds, the specific operation of the spindle is shown in Table 1.

The temperature rise curve of each part of the lathe spindle is shown in Figure 3.

The temperature rise of each part is different, forming different temperature fields.

In the case where the change of room temperature is not big, the motor fever temperature rise is faster and the front flange also has larger temperature rise.

Fig. 2 Temperature measurement of the machine tool spindle

- Motor Y-axis negative direction
- Motor X-axis positive direction
- Front flange X-axis positive direction
- Front flange Y-axis negative direction
- Spindle front wall Y-axis negative direction
- 6.Spindle front wall Y-axis positive direction
- Room temperature

Figure 3 Temperature rise diagram of various parts of the machine tool spindle

**Motion Detection of Spindle Radial Error **

As shown in Figure 4, the two-way measurement method is taken, i.e., two sensors are installed in the orthogonal distribution for testing.

In the test, the spindle is driven by the rotation of the check rod to test the thermal error of the spindle.

Two groups of non-contact eddy current displacement transducers were arranged along the axial direction of the gaging mandrel (2 per group, 4 in total).

Each group of two displacement transducers is installed orthogonally along the X, Y coordinate axis, i.e. S1, S2, S3, S4 in Figure 4.

The rotation error of the spindle is collected by these four displacement transducers, and the MX and MY are high-speed data acquisition devices in the X and Y directions, respectively.

The eddy current displacement sensor has a resolution of 25 nm and the data acquisition device has a sampling frequency of up to 1 MHz.

Figure 4 Principle of two-way measurement

Because the test part is cylindrical hole, which can’t detect directly with the table, so a precision gaging mandrel is used to inserted into the spindle cone hole for dynamic measurement, as shown in Figure 5.

Figure 5 Error Measurement of Spindle Rotation

Axial error is a one-dimensional error, so it is enough to install displacement sensors in the gaging mandrel end face to measure.

The axial runout of the lathe spindle mainly affects the accuracy of the geometry of the workpiece end face, which will produce an end face phase for the perpendicularity error of the outer cylindrical surface, but the cylindrical workpiece of the outer contour of the processing has no influence.

The axial thermal elongation of the spindle increases with the temperature field, and the end face runout tends to increase at different speeds and temperatures, with the corresponding signal collected by S5 shown in Figure 4.

**4****. ****Thermal error separation and spindle rotation accuracy evaluation**

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**Thermal error separation and spindle rotation accuracy evaluation**

The shape error and mounting eccentricity of the measuring element have a large influence on the spindle rotation accuracy measurement results, so shape error and mounting error are inevitably mixed in the measurement data.

Only by effectively separating the shape error and the mounting error, the spindle rotation accuracy can be accurately evaluated.

Radial thermal deformation errors can be decomposed into signals of different orders.

In non-contact measurement, the measurement data mainly consists of the roundness error signal of the measuring rod, the error signal of the cross-section roughness and the error signal of the ripple degree.

Among them, the spindle roundness error is a macroscopic error, low-frequency signal.

Roughness errors are microscopic and high-frequency signals.

The ripple error is an intermediate frequency signal between roundness error and surface roughness.

The spindle rotation error is mainly composed of periodic components, which mainly consist of low harmonic signals in 1st, 2nd, 3rd and 4th order.

The roundness error of the test rod used as a reference axis is negligible because of its high processing accuracy, and the heat distortion error separation is mainly processed in the radial direction.

In the error separation, the DC component A0 of the measured component should be removed from the acquisition signal S(θ) to obtain the radial motion error Sn(θ).

Sn(θ) is both periodic and radial in nature.

Periodicity means that the change in the contour signal of the circumferential workpiece is repeated many times with a duration of 2π/i.

Radiality means that the radius of the same cross-section of the measured part varies at different positions and there is variability.

Therefore, the error motion in the error-sensitive direction when the spindle is rotated can be seen as a superposition of several error signals of different octave frequencies.

The Fourier progression of the rotary motion Sn(θ) in the sensitive direction of the measuring element is expanded as follows:

When i=1, S1 is the first-order harmonic component included in the measurement result, which is the circular motion information at the same frequency as the spindle, due to the mounting eccentricity of the element under test, with

ts initial phase θ1 is

When i ≥ 2, Si is the inner pendulum with i peaks per weekly circle.

The spindle thermal error consists of two main components:

① The deflection of the rotation center due to thermal deformation of the spindle bearing is reflected in the signal as a change in the DC component.

② Error in radial motion due to thermal deformation can be obtained by removing the eccentricity of the measured element from the measurement result:

S2 (θ) = Σni = 2 (Ai cos (iθ) + Bi sin (iθ)) (7)

The spectral analysis of error signals in this paper with the aid of the FFT method.

The discrete error signal collected in the time domain is turned into a frequency domain signal to analyze its error composition.

Therefore, the data are processed using the mounting eccentricity e of the Fourier progression separating the gaging mandrel.

It can also isolate the shape error of the gaging mandrel in the sampled data, thus extracting the spindle slew error, as shown in Figure 6.

Figure 6 Data processing flow chart

Figure 7 shows the raw data in the X and Y directions, where the small noise data is due to random spindle runout.

Figure 8 shows the spectral analysis of both, where the frequency with the largest value is close to zero, which corresponds to the initial mounting position of the sensor, and the first-order component is the mounting eccentricity of the gaging mandrel.

Figure 9 shows the error measurements in the X and Y directions with the DC component removed, mainly consisting of mounting eccentricity and motion error.

Figure 10 shows the variation of the radial rotation error at different times (when the spindle is running at 240r/min, 480r/min, and 960r/min, respectively) without separating the mounting eccentricity.

Figure 11 shows the DC component at the end of 240r/min, 480r/min, and 960r/min respectively, which reflects the shift of the rotation center with temperature change.

Fig. 7 Raw measurement data in X.Y axis

Figure 8 Spectral analysis results of measured data

Figure 9 Error data after removal of the DC component

Table 2 shows the rotation accuracy of the spindle system based on the circular image method and the least-squares circular method for radial thermal runout at different rotational speeds and different temperature fields.

As shown in Table 2, as the temperature of the spindle system increases, the radial motion error caused by the thermal deformation increases accordingly.

The larger the temperature rise of the spindle system, the more severe the thermal deformation.

Figure 10 Motion error at different moments (unseparated mounting eccentricity)

Figure 11 Variation of DC component with temperature

**Table 2 Spindle rotation error at different speeds**

Spindle speed (r/min) | 240 | 240 | 480 | 480 |

Frequency conversion (Hz) | 3.96 | 3.96 | 7.92 | 7.92 |

Running time (min) | 0 | 65 | 86 | 176 |

Spindle system temperature (^{o}C) |
25.57 | 26.85 | 28.10 | 31.39 |

Spindle rotation accuracy (um) | 2.33 | 2.61 | 3.37 | 4.13 |

Spindle speed (r/min) | 960 | 960 | 1200 | 1200 |

Frequency conversion (Hz) | 15.84 | 15.84 | 19.8 | 19.8 |

Running time (min) | 209 | 306 | 319 | 366 |

Spindle system temperature (^{o}C) |
32.46 | 35.19 | 37.21 | 40.45 |

Spindle rotation accuracy (um) | 5.85 | 7.45 | 9.13 | 12.73 |

**5. Conclusion**

**5. Conclusion**

(1) FFT harmonic analysis of the measured data shows that the spindle slewing eccentricity at different rotational speeds remains basically unchanged, and its first-order frequency is consistent with the spindle slewing frequency.

(2) The spindle has thermal deformation in both axial and radial directions, therefore, timely control of the temperature rise of the machine tool axis system can reduce the thermal deformation of the machine tool spindle and improve its machining accuracy.

(3) A comprehensive analysis of the machine tool spindle slewing thermal error, the results can be seen that the lathe spindle in the influence of thermal temperature rise, the slewing error has an accelerating trend of increasing.

Through the analysis of experimental measurement data and slewing error evaluation study, it can evaluate the influence of machine tool thermal deformation on the spindle slewing error, and obtain the spindle in different temperature stability field.

The change of its machining accuracy provides a more reliable experimental basis for the subsequent machine tool thermal deformation compensation.