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

Research on related literature shows that in precision machining, manufacturing errors caused by thermal deformation account for 50% to 70% of the total manufacturing error.

The thermal deformation error of the spindle system is the main source of machine tool thermal error, making it a crucial component to study and analyze to ensure manufacturing accuracy.

During machine operation, the components of the spindle system form their own temperature fields under the influence of internal and external heat sources. The inconsistent thermal expansion performance of each component can cause thermal deformation of the mechanical structure, leading to processing errors in the parts.

The instantaneous rotation axis of the CNC lathe spindle in space position constantly changes due to the dynamics, static, thermal deformation, and processing errors of the bearing and journal.

Experimental results demonstrate that the precision turning roundness error is caused by the spindle rotation error, accounting for about 30% to 70% of the total error. The proportion increases with the precision of the machine.

Spindle rotation accuracy reflects the dynamic performance of the lathe, and it is closely related to the machine’s ability to achieve machining accuracy.

Spindle thermal deformation during the lathe machining process has a significant impact on spindle rotation accuracy. Detecting and controlling for compensation can improve machining accuracy.

With the widespread application of high-speed and high-precision machine tools, detection accuracy and efficiency are gradually improving, and the detection method is evolving 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, with the measurement accuracy continuously improving.

Measuring spindle rotation thermal error is challenging since the actual spindle rotation axis is not visible. The trajectory of the spindle axis can only be indirectly obtained by measuring the external contour of the standard checker bar mounted on the spindle. Thus, measurement results inevitably mix with shape and mounting errors of the standard checker bar.

For high-precision lathe spindle rotation error measurement, the shape and installation errors introduced by the measurement system cannot be ignored. An effective method is needed to separate and remove the signal components affecting the measurement accuracy from the measurement signal to obtain spindle slew accuracy.

This article uses the 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, the article extracts the spindle turning accuracy, evaluates the rotary accuracy of the machine spindle thermal deformation, and analyzes its machining accuracy.

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

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**easurement principle of spindle thermal error**

The spindle system undergoes both axial and radial thermal deformation. To measure axial thermal deformation, an eddy current sensor is placed in the spindle overhang. However, radial thermal deformation is a two-dimensional variable and should be indirectly measured using the bidirectional quadrature method. Measurements may include errors in spindle manufacturing, installation, and thermal distortion.

To accurately assess the impact of spindle thermal deformation on machining accuracy, the thermal deformation error must be separated from the comprehensive error.

The measurement principle for machine tool spindle slew accuracy involves the radial runout generated during spindle rotation, which causes a change in distance between the eddy current sensor and the measured part’s surface. The eddy current sensor and signal converter convert these changes into analog voltage signals for timing collection.

Spindle rotation accuracy significantly influences finishing parts’ shape accuracy and surface roughness, making it an important index for evaluating machine tool machining accuracy. It predicts the minimum form error and roughness that a machine can achieve under ideal machining conditions and can be used for machine tool compensation.

Figure 1 shows the amount of radial runout caused by spindle thermal deformation.

- 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.

After a period of motor operation, thermal deformation can cause the center of rotation (Or) of the spindle system to shift in different temperature fields, leading to real-time changes in the distance between the eddy current displacement transducer and the surface of the measuring cylinder during spindle movement.

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

As shown in Figure 1, the two displacement transducers detect displacement signals dx and dy, 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.

During the measurement process, a high-precision detector bar with a shape error significantly 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 represent the displacement components of the center of the circular section in the X and Y directions. In other words, due to mounting eccentricity, dx and dy determine the path of the geometric center of the circular cross-section rather than the rotation axis.

To minimize the impact of eccentricity on dx and dy and obtain a more realistic measurement result (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 error in radial motion exhibits both periodic and radial characteristics.

Periodicity refers to the property of the circular contour signal, which varies periodically with a period of 2π.

Radiality describes the complex curvilinear profile of the actual circular cross-section, which has varying radial dimensions at different points on the profile.

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

n is the maximum harmonic order of the circular contour harmonic component being measured.

S0 represents the DC component of the circular contour data being measured, in relation to the initial position of the sensor mount.

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

The practical implication of Equation (3) is that the periodic radial error motion can be separated into several octave components that perform circular motion.

To obtain the actual radial motion error, the measured data should have the DC component and the eccentricity (e) of the element under test removed.

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

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**Spindle thermal error measurement**

Figure 2 displays the FANUC CNC lathe used as the test object. Magnetic suction type high-precision temperature sensors are arranged in the spindle motor, front flange, and front wall of the spindle case. The changes in ambient temperature are collected simultaneously.

The machine tool runs empty at various rotational speeds, with specific spindle operations outlined in Table 1.

The temperature rise curve of each part of the lathe spindle is depicted in Figure 3. Each part’s temperature rise varies, resulting in different temperature fields.

Under conditions where the room temperature does not fluctuate significantly, the motor heat causes a faster temperature rise, and the front flange also experiences a substantial temperature increase.

Fig. 2 Temperature measurement of the machine tool spindle

Figure 3 Temperature rise diagram of various parts 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

**Motion Detection of Spindle Radial Error **

As illustrated in Figure 4, a two-way measurement method is employed to test the thermal error of the spindle, where two sensors are positioned orthogonally for testing.

During testing, the spindle is driven by the rotation of the check rod, and two groups of non-contact eddy current displacement transducers (four in total, with two per group) are arranged along the axial direction of the gaging mandrel.

Each group of two displacement transducers is installed orthogonally along the X and Y coordinate axis, labeled S1, S2, S3, and S4 in Figure 4. These four displacement transducers collect the rotation error of the spindle, and MX and MY are the high-speed data acquisition devices used for collecting data in the X and Y directions, respectively.

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

Figure 4 Principle of two-way measurement

Since the test part is a cylindrical hole that cannot be directly measured with the table, a precision gaging mandrel is utilized for dynamic measurement by inserting it into the spindle cone hole, as demonstrated in Figure 5.

Figure 5 Error Measurement of Spindle Rotation

Axial error is a one-dimensional error, which means that it is sufficient to install displacement sensors on the gaging mandrel end face to measure it.

The axial runout of the lathe spindle mainly affects the accuracy of the geometry of the workpiece end face. This, in turn, may cause an end face phase for the perpendicularity error of the outer cylindrical surface, but it does not have any impact on the cylindrical workpiece’s outer contour during processing.

The axial thermal elongation of the spindle increases with the temperature field. The end face runout tends to increase at different speeds and temperatures, and the corresponding signal collected by S5 is 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 can significantly impact the accuracy of spindle rotation measurements. As a result, shape error and mounting error are inevitably intertwined in the measurement data.

To ensure accurate evaluation of spindle rotation accuracy, it is essential to effectively separate the shape error and the mounting error.

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

In non-contact measurement, the measurement data mainly includes 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 these, the spindle roundness error is a macroscopic error and a low-frequency signal. Roughness errors are microscopic and high-frequency signals. The ripple error is an intermediate frequency signal between the roundness error and surface roughness.

The spindle rotation error primarily comprises periodic components, mainly consisting of low harmonic signals in the 1st, 2nd, 3rd, and 4th orders.

The roundness error of the test rod used as a reference axis is negligible due to its high processing accuracy. The heat distortion error separation is primarily processed in the radial direction.

In the error separation process, the acquisition signal S(θ) should have the DC component A0 of the measured component removed 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)

This article presents a spectral analysis of error signals using the FFT method.

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

To accomplish this, the data is processed using the Fourier progression’s mounting eccentricity ‘e’ to separate the gaging mandrel.

This method can also isolate the shape error of the gaging mandrel in the sampled data, enabling extraction of the spindle slew error, as illustrated in Figure 6.

Figure 6 Data processing flow chart

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

In Figure 8, the spectral analysis of both X and Y directions is shown, with the frequency having the largest value close to zero, corresponding to the initial mounting position of the sensor. The first-order component is the mounting eccentricity of the gaging mandrel.

Figure 9 illustrates the error measurements in the X and Y directions with the DC component eliminated, mainly comprising mounting eccentricity and motion error.

Figure 10 exhibits the radial rotation error variation at different times (240r/min, 480r/min, and 960r/min) without separating the mounting eccentricity.

Figure 11 showcases the DC component at the end of 240r/min, 480r/min, and 960r/min, respectively, reflecting 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 displays the rotation accuracy of the spindle system using both the circular image method and the least-squares circular method for radial thermal runout at varying rotational speeds and temperature fields.

As depicted in Table 2, the radial motion error caused by thermal deformation increases in tandem with the rising temperature of the spindle system.

Furthermore, the severity of thermal deformation amplifies proportionally with the degree of temperature rise in the spindle system.

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

(2) The spindle undergoes thermal deformation in both axial and radial directions. Therefore, controlling the temperature rise of the machine tool axis system in a timely manner can reduce the thermal deformation of the spindle and improve its machining accuracy.

(3) Through a comprehensive analysis of the thermal error of the machine tool spindle slewing, it can be observed that the slewing error increases at an accelerating rate as a result of thermal temperature rise. By analyzing experimental measurement data and evaluating slewing error, the influence of machine tool thermal deformation on spindle slewing error can be assessed.

Consequently, the spindle can be stabilized at different temperature fields, which provides a more reliable experimental basis for subsequent machine tool thermal deformation compensation, thereby enhancing its machining accuracy.