# Study on Error Separation of Three-Probe Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory of Three-Probe Measurement

#### 2.1. The Error-Separation Equations

_{1}, v

_{2}and v

_{3}, are placed around the artifact. The indicators v

_{2}and v

_{3}are separated from v

_{1}by angles α and β in the x-y plane. The reading of each sensor is a combination of the roundness error of the artifact and the error-motion of the spindle. s characterizes the roundness error of the artifact; x and y characterize the components of the error-motion of the spindle.

_{0}is the radius of a basic circle superimposed on the radial error motion to avoid the negative error motion values and for visual representation.

#### 2.2. Time-Domain Solution

_{i}= 360(i − 1)/N, where θ

_{i}is the i-th rotational angle, and N is the total number of points in a revolution. The discretized form of Equation (5) is:

_{1}(k); s(k + m

_{1}) and s(k + m

_{2}) are the roundness error sequences contained in probe 2 and probe 3; and m

_{1}and m

_{2}correspond to angle α and β, which can be calculated as:

_{i}is the roundness error at i-th index number.

_{1}) and s(k + m

_{2}) are the same except for time-delay. The time-delay is related to the angles between the sensors; s(k + m

_{1}) and s(k + m

_{2}) can be written as:

_{1}and m

_{2}are integers, the sampling points from the three probes are the same group. Otherwise, the sampling points of the three probes are not the same group, indicating inconsistency. It is further noted that the consistency of sampling points contradicts the full-rank matrix H, i.e., integer m

_{1}and m

_{2}lead to a deficient-rank matrix H. However, rounding non-integer m

_{1}and m

_{2}into integers derives a full-rank matrix H with the introduction of unavoidable rounding errors.

_{2}has sampling points inconsistent with the other two probes.

_{1}to integer, matrix H becomes full and leads to a unique time-domain solution. The rounding error is further quantified in Section 3.

#### 2.3. Frequency-Domain Solution

#### 2.3.1. Symmetry of Transfer Function W(n)

_{v}(n). Otherwise, the resultant F

_{s}(n) is an asymmetrical spectrum and results in a complex sequence of roundness errors in time-domain, which is of no meaning.

_{v}(n) obtained from FFT is symmetric about the X-axis. But generally, in order to avoid the negative-frequency domain, frequency shifting is carried out for F

_{v}(n) and a new axis of symmetry appears.

_{v}(n). By doing so, a symmetrical roundness error F

_{s}(n) results, and leads to a real-number sequence in time-domain through inverse FFT.

#### 2.3.2. Harmonic Suppression

_{s}(n) and thus disappear in the resulted roundness error, which is called harmonic suppression.

_{h}denotes the suppressed harmonic numbers of FFT, n

_{α}= 1, 2, …, and n

_{β}= 1, 2, ….

_{r}is the rotation-harmonic number; Δf is the frequency interval of FFT; f

_{r}is the rotation frequency; f

_{s}is sampling rates; L is the number of points for FFT; N is the number of points sampled per revolution.

#### 2.3.3. The Average Schemes

_{c}N, where N

_{c}is the number of revolutions. This is called the latter averaging scheme; the integer harmonic numbers of FFT turn to rotation interhamonic numbers due to the linear coefficient 1/N

_{c}.

## 3. Experiments

#### 3.1. Test Rig

#### 3.2. Analysis of Time-Domain Solution

#### 3.3. Analysis of Frequency-Domain Solution

## 4. Analysis of Influencing Factors

#### 4.1. Rotational Speed

#### 4.2. Consistency of Sampling Points

#### 4.3. Probe Arrangement

#### 4.4. Number of Revolutions

_{c}defines the rotation interhamonic numbers, and thus leads to influence of frequency-domain solution.

## 5. Conclusions

- (1)
- The separating accuracy between rounding error and spindle error was demonstrated through the rotational speed experiments.
- (2)
- When the three-probe method aim was to obtain the roundness error, the previous average scheme of frequency-domain solution was recommended. When the aim was to measure spindle error, the latter average scheme of frequency-domain solution and time-domain solution was preferred.
- (3)
- The probe arrangement leading to less suppressed harmonic was preferred, and the number of revolutions was suggested to be at least 40 for roundness error to keep constant.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Example of consistency of the sampling points of three probes in probe arrangement of 0–127°–225°: (

**a**) N = 360, (

**b**) N = 200.

**Figure 4.**Polar plot of time-domain solutions in Experiment 1: (

**a**) roundness error; (

**b**) spindle error. Units: μm.

**Figure 5.**Intermediate and final results of the frequency-domain solutions under the two average schemes in Experiment 1: (

**a**) the frequency-domain roundness error in the previous average scheme; (

**b**) the frequency roundness error in the latter average scheme; (

**c**) the time-domain roundness error in the previous average scheme; (

**d**) the time-domain roundness error in the latter average scheme; (

**e**) the spindle error in previous average scheme; (

**f**) the spindle error in the latter average scheme. “upr”, short for undulation per revolution, is used to stress n as a rotation-harmonic number.

**Figure 6.**The effect of rotational speed on the polar plot of the frequency-domain solutions: (

**a**) the roundness error under previous scheme; (

**b**) the roundness error under the latter scheme; (

**c**) the spindle error under the previous scheme; (

**d**) the spindle error under the latter scheme. Units: μm.

**Figure 7.**The effect of consistency of sampling points on the polar plot of frequency-domain solutions: (

**a**) roundness error under the previous scheme; (

**b**) roundness error under the latter scheme; (

**c**) spindle error under the previous scheme; (

**d**) spindle error under the latter scheme. Units: μm.

**Figure 8.**The effect of probe arrangement on the polar plot of the frequency-domain solutions under the two average schemes: (

**a**) roundness error under the previous scheme; (

**b**) roundness error under the latter scheme; (

**c**) spindle error under the previous scheme; (

**d**) spindle error under the latter scheme. Units: μm.

**Figure 9.**The effect of the number of revolutions on the polar plot of frequency-domain solutions: (

**a**) roundness error under the previous scheme; (

**b**) roundness error under the latter scheme; (

**c**) spindle error under previous scheme; (

**d**) spindle error under the latter scheme. Units: μm.

**Figure 10.**The effect of the number of revolutions on values of frequency-domain solutions: (

**a**) roundness error value; (

**b**) spindle error value.

α/° | β/° | N | m_{1} | m_{2} | Consistency | Rank (H) |
---|---|---|---|---|---|---|

127 | 225 | 360 | 128 | 226 | consistency | 358 |

127 | 225 | 200 | [71.6] | 126 | inconsistency | 200 |

Experiments No. | Probe Arrangement | Rotational Speed | N | f_{s} |
---|---|---|---|---|

1 | 0–127–225° | 3000 rpm | 200 dots | 10 kHz |

2 | 0–100–225° | 3000 rpm | 200 dots | 10 kHz |

3 | 0–127–225° | 3000 rpm | 360 dots | 18 kHz |

4 | 0–127–225° | 9000 rpm | 360 dots | 54 kHz |

Experiments No. | Roundness Error Value/μm | Spindle Error Value/μm |
---|---|---|

1 | 51.4791 | 2.6337 |

Experiment No. | α/° | β/° | K_{r} in Previous Average | K_{r} in Latter Average |
---|---|---|---|---|

1, 3, 4 | 127 | 225 | 1, 360 k ± 1 | 0.02, 1.8 k ± 1 |

2 | 100 | 225 | 1, 72 k ± 1 | 0.02, 0.36 k ± 1 |

Rotational Speed/rpm | Roundness Error Values/μm | Spindle Error Value/μm | ||
---|---|---|---|---|

Previous Aver | Latter Aver | Previous Aver | Latter Aver | |

3000 | 1.7973 | 2.0599 | 2.0185 | 3.0874 |

9000 | 1.7939 | 2.0040 | 1.7201 | 2.7147 |

Error % | 0.2 | 2.7 | 14.8 | 12.1 |

N | Roundness Error Values/μm | Spindle Error Value/μm | ||
---|---|---|---|---|

Previous Aver | Latter Aver | Previous Aver | Latter Aver | |

200 | 1.8060 | 1.6426 | 1.8615 | 3.0997 |

360 | 1.7973 | 2.0599 | 2.0185 | 3.0874 |

Error % | 0.5 | 25.4 | 8.4 | 0.4 |

Probe Arrangement | Roundness Error Values/μm | Spindle Error Values/μm | ||
---|---|---|---|---|

Previous Aver | Latter Aver | Previous Aver | Latter Aver | |

0–127–225° | 1.8060 | 1.6426 | 1.8615 | 3.0997 |

0–100–225° | 1.8131 | 1.9820 | 1.9087 | 2.2726 |

Error % | 0.4 | 20.7 | 2.5 | 26.7 |

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**MDPI and ACS Style**

Zhong, C.; Zhuo, M.; Cui, Z.; Geng, J.
Study on Error Separation of Three-Probe Method. *Symmetry* **2022**, *14*, 866.
https://doi.org/10.3390/sym14050866

**AMA Style**

Zhong C, Zhuo M, Cui Z, Geng J.
Study on Error Separation of Three-Probe Method. *Symmetry*. 2022; 14(5):866.
https://doi.org/10.3390/sym14050866

**Chicago/Turabian Style**

Zhong, Chengbao, Ming Zhuo, Zhong Cui, and Jiqing Geng.
2022. "Study on Error Separation of Three-Probe Method" *Symmetry* 14, no. 5: 866.
https://doi.org/10.3390/sym14050866