# Fast and Precise Non-Contact Measurement of Cylindrical Surfaces with Air Gauges

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{D}= 10 µm in the middle and 15 μm in the upper and lower cross-sections. The gauge head was hidden inside the table and moved up inside the cylinder. It was not necessary to precisely adjust the cylinder’s position, because the gauging head is flexible and can self-center when entering the cylinder.

- Stage 0:
- 1.
- Initial position of the measuring head; the top edge is ca. 5 mm over the table surface.
- 2.
- Manual positioning of the measured part over the gauge head edge.
- 3.
- Automatic fixture of the part with the top cover.

- Stage 1:
- 1.
- The measuring head moves up and stops at the first (lower) measurement position.
- 2.
- The gauge head performs a full revolution while collecting measurement data.
- 3.
- Data transmission to a PC.

- Stage 2:
- 1.
- The gauge head moves to the position of the middle intersection.
- 2.
- Revolution and measurement.
- 3.
- Data transmission to a PC.

- Stage 3:
- 1.
- The gauge head move to the upper position.
- 2.
- Revolution and measurement.
- 3.
- Data transmission to a PC.

- Stage 4:
- 1.
- Retraction of the gauging head to the initial position.
- 2.
- Release of the measured part.
- 3.
- Data processing.

- The gauge head is mounted on the flexible rod (a floating head).
- The gauge head contains three independent small-chamber air gauges, G1, G2, and G3, as shown in Figure 3. This is the principal difference compared to conventional solutions with three nozzles connected into one measuring chamber. Initial research proved that this solution offers certain advantages [27].
- In order to assess roundness, a novel algorithm designed for analysis of three independent signals is used.

## 3. Data Processing

#### 3.1. Data Collection

_{w}= 1.020 mm, while the respective inner and outer diameters of the measuring nozzle were d

_{p}= 1.610 mm and d

_{c}= 4.8 mm, in order to ensure that the multiplication factor |K| = 0.5 kPa/μm with non-linearity below δ = 0.5%, in the measuring range z

_{p}= 100 μm. The chamber volume was set as V

_{k}≈ 1.2 cm

^{3}, which provided good dynamic characteristics; the time constant T = 0.008 s; and the dynamic error below 5% at the input frequency of f

_{0.95}= 7 Hz. Importantly, because the inlet nozzles’ diameters were set, no regulation was needed prior to measurement, and the metrological characteristics remained unchanged. When the measurement task is different, the inlet nozzles can be replaced, providing more suitable characteristics for the application.

_{z}= 150 kPa. The back-pressure p

_{k}was measured with piezoresistive transducers (class 0.05) with an integrated AD converter and an independent memory buffer. Each gauge had its calibration data saved in the system’s memory. During the measurement, the gauge head performed a 370° rotation and N = 1000 measurement points were collected. The registered back-pressure p

_{k}(kPa) was then converted to displacement s (μm) values, in accordance with the air gauging principle. Figure 4 presents the displacement plots derived from the three air gauges at each point i during the rotation of the gauging head.

#### 3.2. Data Smoothing, Profile Closure, and Interpolation

_{i}and y′

_{i}are the rough and smoothed coordinates, respectively.

_{i}) and the initial y

_{i}values was minimal. It should be noted, however, that the higher the polynomial degree is, the larger the number of points that must be taken into consideration, and the respective curve is less smooth. The same procedure can be repeated for a particular dataset and the number of repetitions chosen empirically.

_{4}to y

_{N−3}were subjected to this procedure.

_{365°}and y

_{5°}was divided by the number of points in between. The coordinates of each of the points were then corrected with the calculated value, so that the first and last corrected points coincided.

#### 3.3. Calculation of the Profile and Roundness

_{1}, ΔR

_{2}and ΔR

_{3}indications.

_{1}, ΔR

_{2}, and ΔR

_{3}denote changes of indications from G3, G1, and G2, respectively, and 2α indicates the angle shown in Figure 5b between the measurement radiuses R

_{1}and R

_{2}.

_{k}, as described by the following equation [34]:

## 4. Simulations and Measurement Results

_{1}= 15 µm close to edges and T

_{2}= 10 µm in the middle cross-section, all cylinders were measured at these three levels, denoted S1, S2, and S3 in the Table 1. The actual measured values of the roundness deviation appeared to be much smaller than the roundness tolerance T

_{D}: they were no larger than 9.5 µm for the top and bottom positions (levels 1 and 3) and did not exceed 7.0 µm in the middle (level 2). The distribution was asymmetric, with skewness varying between 1.4 and 2.6. In Table 1, the statistical parameters of the analyzed population are presented and examples of histograms for different levels are shown in Figure 7.

_{s}were smaller by 0.12 ÷ 1.11 µm than the results from the reference measurements CYLt

_{r}. The mean value of difference CYLt

_{r}–CYLt

_{s}was 0.69 µm and the standard deviation was 0.39 µm.

_{ag}. Its expected value was E[e

_{ag}] = 0 with standard deviation s = 0.5 µm. A total of 1000 measurements were simulated in order to assess the impact of this error type on the obtained roundness deviation values RONt and the amplitudes of the particular harmonics.

_{1}, R

_{2}, and R

_{3}represent the measurement signals received from the air gauges G1, G2, and G3, respectively (random errors were taken into account), and, lastly, “output” represents the processed data obtained from the simulated measurement.

## 5. Accuracy Analysis

_{RON}can be calculated as follows:

_{RON}covers all the component errors that occur in the measurement system, both random and systematic. Among the main sources of detectable errors, the following should be pointed out in particular:

- (a)
- Measurement signals generated by air gauges (non-linearity of characteristics, air flow instability, etc. [37]), pressure transducers, AD converters, and the geometrical inaccuracy of the gauge head and other components.
- (b)
- The air feed and, above all, pressure instability.
- (c)
- Environmental sources, temperature in particular.
- (d)
- The gauge head angle indication.
- (e)
- The data processing procedures, such as simplified formulas, rounding the values, etc.

_{p}is the quantile of the standardized normal deviation; for P = 0.95, it is u

_{p}= 1.96.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Displacement s registered by the air gauges G1, G2, and G3 during the 370° rotation of the gauging head.

**Figure 5.**Three-point roundness assessment: (

**a**) with the V-block method; (

**b**) with the three-point floating gauge head.

**Figure 6.**Roundness profile derived from variations of indications from the gauges G3, G1, and G2 respectively.

**Figure 8.**Histograms of roundness deviation RONt measured on 100 cylinders: (

**a**) second harmonic; (

**b**) fifth harmonic.

**Table 1.**Results of Talyrond measurement of the cylinders [34].

Out-of-Roundness RON | S1 (μm) | S2 (μm) | S3 (μm) |
---|---|---|---|

Mean $\overline{RONt}$ | 2.93 | 1.88 | 2.08 |

Standard deviation s | 1.40 | 1.17 | 1.33 |

Skewness Skew[RONt] | 1.31 | 2.43 | 2.56 |

Minimum RONt_{min} | 0.79 | 0.54 | 0.65 |

Maximum RONt_{max} | 8.51 | 6.94 | 9.45 |

Out-of-Cylindricity CYLt | (μm) |
---|---|

Mean $\overline{CYLt}$ | 5.58 |

Median m | 5.35 |

Standard deviation s | 1.90 |

Skewness | 0.81 |

Minimum | 2.14 |

Maximum | 10.40 |

Population | 100 | Standard Deviation | 0.00008 |
---|---|---|---|

Maximum and minimum values of the relative error w_{RON} | 0.092/0.085 | Confidence interval of an error of the method | 0.088 ± 0.022 |

Mean value ${\overline{w}}_{RON}$ | 0.088 | Variance for the sample | 0.0000064 |

Confidence interval of the mean value | 0.0880 ± 0.0022 | Overall accuracy MA | 9.29% |

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**MDPI and ACS Style**

Jermak, C.J.; Jakubowicz, M.; Wieczorowski, M.; Rucki, M.
Fast and Precise Non-Contact Measurement of Cylindrical Surfaces with Air Gauges. *Materials* **2021**, *14*, 3728.
https://doi.org/10.3390/ma14133728

**AMA Style**

Jermak CJ, Jakubowicz M, Wieczorowski M, Rucki M.
Fast and Precise Non-Contact Measurement of Cylindrical Surfaces with Air Gauges. *Materials*. 2021; 14(13):3728.
https://doi.org/10.3390/ma14133728

**Chicago/Turabian Style**

Jermak, Czeslaw Janusz, Michal Jakubowicz, Michal Wieczorowski, and Miroslaw Rucki.
2021. "Fast and Precise Non-Contact Measurement of Cylindrical Surfaces with Air Gauges" *Materials* 14, no. 13: 3728.
https://doi.org/10.3390/ma14133728