# Three-Dimensional Printing of Cylindrical Nozzle Elements of Bernoulli Gripping Devices for Industrial Robots

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{n}followed the gap between its end surface and the surface of the object of manipulation (OM) (label 3). At the same time, at radius r

_{n}, at h

_{c}< r

_{n}/2, the flow experienced its greatest constriction. At the point of greatest constriction, at excessive gripping power pressures of more than 30 kPa, the flow reached a critical speed equal to the speed of sound, for those conditions. As a result of the further increase in area of radial flow, its supersonic velocity increased, and static pressure on the surface of the OM decreased to a lower atmospheric value. At some distance from the center of the nozzle, a sharp braking of the supersonic flow occurred, followed by its transition to the subsonic flow, which was accompanied by a pressure jump. As a result of further expansion, the rate of subsonic flow dropped, and the static pressure in the gap smoothly increased to the value of atmospheric p

_{a}. The effect of the vacuum on the surface of the OM led to its levitation. Lateral displacement of the OM prevented abutments (label 4).

_{n}, the roughness of the nozzle surface and the active grip surface, the external grip radius r

_{g}, and the distance h

_{c}from the nozzle edge to the OM.

## 3. Results and Discussion

#### 3.1. Coefficient Quality of Model

_{n}= 6 mm (Figure 4).

_{max}is the maximum value of the position of the extruder along the Y axis when 3D printing the outer layer of the nozzle opening wall, Y

_{min}is the minimum value of the position of the extruder along the Y axis when 3D printing the outer layer of the nozzle opening wall, and w

_{e}is the extrusion width set during printing (in our case, 0.4 mm).

_{n}is the nominal diameter of an opening of a nozzle of capture that was set in the CAD model; X

_{max}, Y

_{max}represent the maximum value of provision of an extruder on axis X and Y, respectively, at the 3D printing of an external layer of a wall of an opening of a nozzle; and X

_{min}, Y

_{min}represent the minimum value of provision of an extruder on axis X and Y, respectively, at the 3D printing of an external layer of a wall of an opening of a nozzle.

#### 3.2. Coefficient Overexstrusion under Arc Path Motion

_{o}and S

_{i}are filled with plastic:

_{1}is the internal radius of the extruded arc, r is the radius of movement of the center of the extruder, and r

_{2}is the external radius of the extruded arc.

_{e}is the extrusion width.

_{t}and S

_{l}can be equated and a dependence can be obtained:

_{1}= R, then we find Equation (3), and if we specify the radius r

_{2}for the radius r we obtain:

Algorithm 1 The displacement of the inner layer of the hole for different wall thickness | |

1: | INPUT: r1, x, w_{e} |

2: | n ← $\mathbf{floor}\left(\left(x+0.1\right)/{w}_{e}\right)$ |

3: | FindF(n, x) |

4: | $r{2}_{0}\leftarrow r1+x$ |

5: | $\mathbf{for}i\in 1\dots n$ |

6: | ${r}_{i}\leftarrow -{w}_{e}/2+\sqrt{{w}_{e}^{2}/4+r{2}_{i-1}^{2}}$ |

7: | $r{2}_{i}\leftarrow \sqrt{{r}_{i}({r}_{i}-{w}_{e})}$ |

8: | $r{2}_{n}-r1$ |

9: | a ← x, b ← $x+{w}_{e}$, eps ← 10^{−10} |

10: | Findr3 |

11: | ${\mathbf{while}}_{}{\left|F\left(n{,}_{}c\right)\right|}_{}{}_{}eps$ |

12: | $c\leftarrow \left(a+b\right)/2$ |

13: | ${\mathbf{if}}_{}F\left(n{,}_{}a\right)\cdot F\left(n{,}_{}\mathrm{c}\right){0}_{}\mathbf{then}$ |

14: | $b\leftarrow c$ |

15: | $\mathbf{else}$ |

16: | $a\leftarrow c$ |

17: | $c$ |

18: | ${k}_{Arc}\leftarrow 2\cdot \left(c-x\right)$ |

#### 3.3. Coefficient Shrinkage of Material

_{0}is the initial wall volume of the designed nozzle, and V

_{end}is the final wall volume of the printed nozzle.

_{0}is the starting linear size of the designed object, and L

_{end}is the ending linear size of the printed object. The shrinkage size of the printed model was determined by the calculation presented in Appendix A.

_{end}varied within the error range of the measuring instruments by ±0.01 mm, which confirmed the adequacy of the methodology for calculating the arc compensation coefficient, since—depending on the wall thickness and the size of the hole—the error factor of volumetric and linear shrinkage did not exceed 0.5%. From the results given in Table 3, the average volume shrinkage for PLA (Wanhao) plastic was 1.207%, and the average linear shrinkage was 5.058% for PLA (Wanhao) plastic. According to the above method, volumetric and linear shrinkages for PLA filaments of manufacturers were experimentally determined for Wanhao [90], Plexiwire [97], 3D Plast [98], and MonoFilament [99] (Figure 13).

#### 3.4. Effect of 3D Printer Surface Geometry on Power Characteristics of Gripping Devices

#### 3.5. Description of the Finite Element Method for BGD Research

_{1}and the height of the gap between the object of manipulation and the gripper h

_{c}. To ensure adequate operation of the SST model, to model the wall air flows, it was necessary that the minimum number of elements between the walls of the model was three elements. Therefore, the number of nodes was in the range of 0.9–3 million, and the number of elements was 4.5–12 million, and for the range of distances between BGD and OM h

_{c}= 0.1…0.4 mm. Ideal gas from the program library was used during the simulation. Boundary conditions for model of air flow are presented in Figure 17.

#### 3.6. Methods of Experimental Research

_{1}, m

_{2}, m

_{3}), which set the height of the gap h

_{c}between the surfaces of the gripping device—5 and the object of manipulation—6, 7—motion sensor (electronic caliper) on the x-axis, 8—y-axis movement device, 9—stepper motor for movement on the x-coordinate, 10—pressure sensor, which was connected to a measuring nozzle with a diameter of 0.3 mm in the center of the manipulation object–6, 11—a device for moving along the z axis, and 12—a controller built on the board Raspberry Pi 3B, and other peripherals for work with the installation.

_{1}, m

_{2}and m

_{3}. This provided a pre-set with a micrometer (m

_{1}, m

_{2}and m

_{3}) gap between the GD and the OM. After gripping the object of manipulation—6, the pressure distribution along the x-axis was measured using a measuring nozzle—13, which moved the OM relative to the GD by means of a mechanical transmission screw-nut, which was driven by a stepper motor. The system and the measurement process were controlled by a single-board Raspberry Pi 3B microcomputer. The schematic diagram and the process of signal processing are described in Appendix C.

#### 3.7. Results of the Influence of 3D Printing on BGD

_{c}between the manipulation object and the Bernoulli grips, swirls through the surface roughness will possibly form. Therefore, the effect of the gap height h

_{c}between the OM and the BGD, on the lifting force at constant supply pressure p = 300 kPa, was investigated (Figure 25).

_{c}between the Bernoulli grips and the object of manipulation on the power characteristics of the gripper. In particular, in all samples of grippers, the maximum lifting force was achieved at a height of h

_{c}= 0.4 mm. The best values of the lifting force were observed in all areas of the gap height h

_{c}of gripping devices with a printing height of 0.05 mm. In addition, the supply pressure of the Bernoulli grippers on the lifting force at different heights was important in 3D printing the gripper (Figure 26).

## 4. Conclusions

## Author Contributions

## Funding

^{2}ID: 824964 and Better Factory ID: 951813.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{l}represents the coefficients of thermal compression at linear shrinkage; α

_{v}represents the coefficients of thermal compression at volume shrinkage, which are due to the physical properties of the body; t

_{0}is the temperature of the beginning of linear shrinkage; and t

_{end}is the final temperature equal to the ambient temperature.

_{l}is the liquidus temperature above which the alloy is completely in the liquid state, and t

_{s}is the solidus temperature below which the alloy is in the solid phase.

## Appendix B

_{length}is the empirical correlation which controls the length of the transitional area (accept F

_{length}= 100), and F

_{onset}is the function controlling the provision of the beginning of transition.

_{a}

_{2}= 0.06, c

_{e}

_{2}= 50—the empirical constants; $\Omega =\sqrt{2{\Omega}_{i,j}{\Omega}_{i,j}}$—the invariant of the tensor of vorticity; ${F}_{turb}={e}^{-{\left(\frac{{R}_{T}}{2}\right)}^{4}}$; and ${R}_{T}=\frac{\rho k}{\mu \omega}$.

_{onset}was similar to that used in the γ-Reθ model. It was used to start the intermittency production (i.e., activate source term (A11)). It contains the ratio of the Reynolds number to the local vortex ReV (in the current formulation, the strain rate was actually used within ReV, which was equivalent to the boundary layers), to the critical Reynolds number Reθc. However, Reθc was not calculated from the transport equation but algebraically, using k and other local variables. As a result, the beginning of the transition was controlled by the following functions:

_{k}was counted by means of the Kato–Launder formula:

## Appendix C

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**Figure 1.**Example of using 3D printing to create vacuum suction cups. Reprinted with permission from ref. [55]. Copyright 2020 IEEE.

**Figure 3.**Schematic of the proposed methodology determining the diameter of the designed nozzle (d

_{n}—the required nozzle diameter, x—wall thickness, w

_{e}—extrusion width, d—diameter during design to provide post-print diameter d

_{n}).

**Figure 5.**The effect of chord height and angular tolerance on the number of triangles in the STL file describing the 3D model of the nozzle.

**Figure 7.**Extruder motion straight trajectory: (

**a**) view in Cartesian coordinates; (

**b**) top view of the xy plane.

**Figure 11.**Effect of nozzle wall thickness on printing nozzle diameter: (

**a**) x = 0.4 mm, (

**b**) x = 0.8 mm, (

**c**) x = 1.2 mm, (

**d**) x = 1.6 mm.

**Figure 14.**Cross-sectional geometry of the nozzle element of the gripper under the microscope (nozzle diameter of the gripper—6 mm, nozzle diameter of the extruder—0.4 mm, layer height—0.1 mm, extruder temperature of PLA material—210 deg, table temperature—60 deg, printing speed—60 mm/s): (

**a**) classic light, (

**b**) front light.

**Figure 18.**General view of the experimental setup to study the pressure distribution on the surface of the OM: 1—air preparation device, 2—precision reducer for regulating the pressure in the gripper chamber, 3—pressure sensor in the gripper device chamber, 4—three micrometer heads (m

_{1}, m

_{2}, m

_{3}), which set the height of the gap h

_{c}between the surfaces of the gripping device—5 and the object of manipulation—6, 7—motion sensor (electronic caliper) on the x-axis, 8—y-axis movement device, 9—stepper motor for movement on the x-coordinate, 10—pressure sensor, which was connected to a measuring nozzle with a diameter of 0.3 mm in the center of the manipulation object—6, 11—a device for moving along the z axis, and 12—a controller built on the board Raspberry Pi 3B.

**Figure 19.**The work schematic of the experimental installation to research the distribution of pressure on a surface of an OM: 5—gripping device, 6—object manipulation, 13—measuring nozzle.

**Figure 20.**The pressure distribution on the surface of the manipulated object is formed by a BGD with different 3D printing layer heights (p = 200 kPa, h

_{c}= 0.25 mm, r

_{n}= 3 mm, r

_{g}= 30 mm): (

**a**) r from 0 to 30 mm, (

**b**) r from 0 to 10 mm.

**Figure 22.**Effect of layer height in 3D printing on the lifting force which forms the BGD: h

_{c}= 0.25 mm, r

_{n}= 3 mm, r

_{g}= 30 mm (simulation results).

**Figure 23.**Pressure distribution on the surface of the object of manipulation when a BGD gripping with different 3D printing layer heights (p = 200 kPa, h

_{c}= 0.25 mm, r

_{n}= 3 mm, r

_{g}= 30 mm): (

**a**) r from 0 to 30 mm, (

**b**) r from 0 to 12 mm.

**Figure 24.**Effect of 3D printing layer height on the lifting force which forms a BGD: h

_{c}= 0.25 mm, r

_{n}= 3 mm, r

_{g}= 30 mm (experimental results).

**Figure 25.**Influence of gap height h

_{c}between the Bernoulli gripper device and object of manipulation, on lifting force (p = 300 kPa, r

_{n}= 3mm, r

_{g}= 30 mm).

**Figure 26.**Influence of BGD supply pressure on their power characteristics at different printing heights, h (h

_{c}= 0.25 mm, r

_{n}= 3 mm, r

_{g}= 30 mm).

**Figure 27.**Influence of supply pressure BGD (h = 0.05 mm) on power characteristics at different grip heights, h

_{c}(r

_{n}= 3 mm, r

_{g}= 30 mm).

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

Chord height | 0.03 | mm |

Angular tolerance | 0.5…30 | degrees |

Plastic | Wanhao PLA filament | [90] |

Extruder temperature | 210 | °C |

Platform temperature | 50 | °C |

Printing speed | 60 | mm/s |

Fluidity | 100 | % |

Filling | 15 (grid) | % |

Top and bottom wall Thickness | 1.2 | mm |

Wall thickness | 0.8 | mm |

Layer height | 0.1 | mm |

Nozzle diameter | 0.4 | mm |

Rollback speed | 40 | mm/s |

Rollback distance | 3 | mm |

Outer contour speed | 12 | mm/s |

Angular Tolerance | 30 Degrees | 0.5 Degrees |
---|---|---|

Y_{max} | G1 X107.774 Y103.162 E33.61291 | G1 X107.347 Y103.199 E33.59496 |

Y_{min} | G1 X107.384 Y96.802 E33.76570 | G1 X107.429 Y96.802 E33.75383 |

Y_{max} − Y_{min} | 6.36 | 6.399 |

No | d (mm) | x (mm) | w_{e} (mm) | d+k_{Arc} (mm) | d_{end}(mm) | V_{0}(mm ^{2}) | V_{end}(mm ^{2}) | ε_{v}(%) | ε_{l}(%) |
---|---|---|---|---|---|---|---|---|---|

1 | 6 | 0.4 | 0.4 | 6.002 | 5.7 | 48.27 | 47.835 | 0.901 | 5 |

2 | 6 | 0.8 | 0.4 | 6.003 | 5.69 | 102.587 | 101.314 | 1.241 | 5.167 |

3 | 6 | 1.2 | 0.4 | 6.004 | 5.71 | 162.928 | 161.674 | 0.770 | 4.833 |

4 | 6 | 1.6 | 0.4 | 6.005 | 5.7 | 229.361 | 226.903 | 1.071 | 5 |

5 | 6 | 2 | 0.4 | 6.005 | 5.71 | 301.781 | 298.011 | 1.249 | 4.833 |

6 | 4 | 0.4 | 0.4 | 4.003 | 3.79 | 33.198 | 32.459 | 2.226 | 5.25 |

7 | 4 | 0.8 | 0.4 | 4.007 | 3.79 | 72.488 | 71.767 | 0.994 | 5.25 |

8 | 4 | 1.2 | 0.4 | 4.008 | 3.79 | 117.802 | 116.978 | 0.699 | 5.25 |

9 | 4 | 1.6 | 0.4 | 4.009 | 3.8 | 169.163 | 166.836 | 1.376 | 5 |

10 | 4 | 2 | 0.4 | 4.01 | 3.8 | 226.572 | 223.083 | 1.540 | 5 |

**Table 4.**Experimental data of calculation of average value of layer protrusions on 3D printing of nozzle elements of grippers.

No. | Layer Height for 3D Printing h_{p} (mm) | C_{max} (mm) | C_{min} (mm) | C_{mid} (mm) |
---|---|---|---|---|

1 | 0.05 | 0.03 | 0.01 | 0.02 |

2 | 0.10 | 0.06 | 0.04 | 0.05 |

3 | 0.15 | 0.11 | 0.05 | 0.08 |

4 | 0.20 | 0.14 | 0.07 | 0.1 |

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## Share and Cite

**MDPI and ACS Style**

Mykhailyshyn, R.; Duchoň, F.; Mykhailyshyn, M.; Majewicz Fey, A.
Three-Dimensional Printing of Cylindrical Nozzle Elements of Bernoulli Gripping Devices for Industrial Robots. *Robotics* **2022**, *11*, 140.
https://doi.org/10.3390/robotics11060140

**AMA Style**

Mykhailyshyn R, Duchoň F, Mykhailyshyn M, Majewicz Fey A.
Three-Dimensional Printing of Cylindrical Nozzle Elements of Bernoulli Gripping Devices for Industrial Robots. *Robotics*. 2022; 11(6):140.
https://doi.org/10.3390/robotics11060140

**Chicago/Turabian Style**

Mykhailyshyn, Roman, František Duchoň, Mykhailo Mykhailyshyn, and Ann Majewicz Fey.
2022. "Three-Dimensional Printing of Cylindrical Nozzle Elements of Bernoulli Gripping Devices for Industrial Robots" *Robotics* 11, no. 6: 140.
https://doi.org/10.3390/robotics11060140