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

Abstract

The application of additive technologies, namely, fused deposition modeling, is a new reality for prototyping gripping devices of industrial robots. However, during 3D printing of holes and nozzle elements, difficulties arise with reducing their diameter. Therefore, this article conducts a comprehensive study of the Bernoulli gripping device prototype with a cylindrical nozzle, manufactured by fused deposition modeling 3D printing. The three main reasons for reducing the diameter of the gripper nozzle after printing were due to the poor-quality model, excessive extrusion of plastic in the middle of the arc printing path, and linear shrinkage of printing material after cooling. The proposed methodology consisted of determining the three coefficients that allowed the determination of the diameter of the designed nozzle. The use of air pressure distributions on the surface of the manipulation object, and lifting forces of gripping devices with different 3D printing layer heights were found. It was experimentally determined that as the height of the printing layer increased, the lifting force decreased. This was due to the formation of swirls due to the increased roughness of the grip surface. It was proven that as the height between the manipulation object and the grip increased, the effect of surface roughness on the lifting force decreased, resulting in an increase in the lifting force. Determination of the rational operating parameters of gripping devices manufactured by 3D printing from the point of view of maximum lifting force, were determined.

1. Introduction

At present, manufacturing additive technology is gaining great popularity [1,2,3,4,5,6,7,8,9,10,11,12,13], and 3D printing is being applied not only to production units but also as a means of production [14]. In particular, the authors of [8] concluded that most companies choose additive manufacturing (AM) to save time and costs, which are most relevant when producing new products and small batches. Only a small percentage of companies have chosen AM for reasons of environmental friendliness, in comparison with classic production methods. In addition, the authors noted an increase in the innovative component for the introduction of AM in companies, and a decrease in the impact of the supply chain on production capacities. Therefore, the introduction of additive manufacturing in various industries is a current topic and further research is warranted. Currently, many researchers are conducting research related to the optimization and improvement of the quality of 3D printing [15,16,17]. As a separate direction, we can single out studies that focus on evaluating the impact of printing parameters on the dimensions [18,19] and surface quality [20] of this prototype.

The use of AM has become widespread in the field of robotics [21,22,23,24,25,26,27,28]. In addition, it should be noted that there is an increasing use of 3D printing in the construction of gripping devices for industrial robots [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51]. The papers [22,23,26,37] describe the process of 3D printing and manufacturing of mechanical gripping devices by FDM printing. Among other important directions is the production of soft gripping devices using additive manufacturing [24,25,27,28,29,30,31,32,33,36,38,39,40,41]. A new important direction of using additive technologies in the manufacture of grippers is the use of magnetic materials during 3D printing [34]. It should be noted that 3D printing is less useful for pneumatic gripping devices of industrial robots. However, with the active development and introduction of flexible materials into 3D printing, it has become possible to print suction cups for pneumatic grips of different shapes [52,53,54,55] (Figure 1).

One promising type of pneumatic gripper is the jet gripper [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71]. Jet grippers have completely different characteristics than vacuum grips due to the effect of raising the force that occurs when the air jet flows around the object of manipulation of the developed nozzle elements [47]. The advantages of jet gripping devices include the high accuracy of basing objects [52,53,54]; the gripping of flexible objects [56,57,58], brittle objects [62,64] and objects heated to high temperatures; good dynamic characteristics [59,61]; design simplicity [60]; and durability. Since these grippers have a high air flow rate, scientists pay considerable attention to optimizing the motion parameters of gripping systems and their dynamics using jet grippers [72,73,74,75,76,77,78,79,80,81,82,83,84,85]. Often, jet grips with a cylindrical and annular nozzle are called Bernoulli gripper devices (BGD), since the description of power characteristics occurs using Bernoulli’s law. Enthusiasts of 3D printing tried to print BGD using fused deposition modeling (FDM) [86] printing technology, but it had low lift and, depending on the plastic blowing system (when 3D printing), there may have been problems with the vision of the nozzle elements. Using another photopolymer (SLA) 3D printing technology, the authors of [87] created a Bernoulli grip with an annular nozzle for tissue capture during laparoscopy. However, FDM 3D printing is the most available and favorable technology in terms of economic feasibility of prototyping and production, to date. Therefore, the study of the technology and the justification of the printing parameters of the nozzle elements of Bernoulli’s gripping devices is a relevant topic.

Little attention has been paid to FDM 3D printing for the manufacture of pneumatic gripping devices of industrial robots, in the analysis of the use of additive manufacturing (AM). The issue of FDM 3D printing holes and nozzle elements of invaders has only been partially considered. At the same time, the thickness of the nozzle wall and shrinkage of the material were considered. The method proposed in this paper will allow a determination of the preliminary diameter of the model nozzle in order to provide the necessary diameter after 3D printing. In particular, no studies have yet been conducted on the effect of printing parameters on the power characteristics of Bernoulli pneumatic gripping devices. However, the 3D printing parameters will critically affect the power characteristics of gripping devices, which suggests that such investigation is in urgent need. Therefore, the authors of this paper have conducted, for the first time, a study of the effect of printing parameters on the power characteristics of Bernoulli gripping devices with a cylindrical nozzle, which allows a justification of the rational parameters of FDM 3D printing of gripping devices with cylindrical nozzle elements and their performance characteristics.

2. Materials and Methods

The principle of the BGD operation was as follows. Compressed air from the Bernoulli chamber (label 1, Figure 2), through the cylindrical nozzle (label 2) with radius r_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).

The performance characteristics of the BGD depend on the parameters of the power supply and those geometric parameters affecting the formation of air flow in the radial gap, in particular, the radius of the nozzle r_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.

During the 3D printing of a BGD using FDM technology [88], deformations of the printing model occur, associated with the printing process itself. Since the main element from the point of view of making a BGD is the gripping device’s nozzle, with 3D printing of small diameter holes by the finished part, many deviations can occur relating to the 3D model quality (STL file), over-extrusion, vertical seam that passes through the hole, total shrinkage of the model and printing parameters (fluidity, printing speed, grade height, etc.). All these listed parameters will affect the diameter of the printed nozzle and its geometry, which will affect the swirls formed from the suboptimal geometry of the nozzle and the lifting force of the Bernoulli gripping devices. The main three reasons for reducing the diameter of the gripping device nozzle after 3D printing were identified as: poor 3D STL model quality, excessive extrusion of plastic in the middle of the arc printing path, and linear shrinkage of the filament material after cooling of the object. The proposed methodology consisted of determining the three coefficients that allow the determination of the diameter of the designed nozzle (Figure 3).

The method of determining the diameter of the designed nozzle was based on the step-by-step determination of the three compensation coefficients of the nozzle diameter. To determine the next coefficient, the already-adjusted nozzle diameter was used, which allowed the rejection of the influence of these effects on each other. The proposed method had two main assumptions: (1) the cylindrical nozzle was placed perpendicular to the printing plane, and, (2) printing took place in a closed-type 3D printer, which prevented the influence of external parameters on the printing process.

3. Results and Discussion

3.1. Coefficient Quality of Model

Before printing the Bernoulli gripping devices, it was necessary to investigate the effect of 3D model parameters on the real geometric parameters obtained during the 3D printing of nozzle elements. When preparing a 3D model for printing, it is stored in STL (standard tessellation language) format. This is the standard format for all software packages (slicers) that prepare the G-code for managing the 3D printer. To prepare the 3D model of Bernoulli gripping devices and other investigated elements, the software package SolidWorks is used. When a model is saved to STL format, a CAD (computer-aided design) 3D model is saved to the 3D model, which is described by triangles of various shapes. In the software package SolidWorks, as in all other software packages, there are two parameters that are responsible for the quality of the stored 3D model in the STL format (chord height and α angular tolerance). Chord height is the maximum distance that software will create between the surface of the original 3D model and the surface of the STL file. Angular tolerance limits the angle between the normals of adjacent triangles. The default value is often 15 degrees, and some programs specify this tolerance as 0 to 1. To study the effect of these parameters on the 3D of hands, we created a separate model consisting of a nozzle element with a diameter of d_n = 6 mm (Figure 4).

The 3D model of the nozzle had a longitudinal element, created to provide uniform cooling of a detail during 3D printing. As for the press, the 3D printer Wanhao Duplicator 6 was used [89], which was equipped with blowing of a detail only on one hand. In addition, the longitudinal member was conveniently used to contain nozzle parameters during measurement, and denote printing parameters for further investigation. To illustrate the effect of parameters, we considered the effect of changing the parameters, chord height and angular tolerance, on the geometry of the STL model (Figure 5).

Figure 5 clearly shows that the number of triangles in the STL file describes the 3D model of the nozzle, which was more influenced by angular tolerance. Since the maximum variation in the number of triangles from which the STL file was formed was answered by a chord height of 0.03 mm, the nozzle models were later printed (Figure 4) with this chord height value. Printing was performed with this constant parameter, as shown in

Table 1. Three-dimensional printing parameters.

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

. The preparation of the G-code for printing was performed using the Wanhao Cura 18.04 software package, and the printing itself was conducted with a closed thermal camera.

Dependencies were defined on the effect of the angular tolerance parameter on the diameter of the printed nozzle at the given CAD model diameter of 6 mm (Figure 6). Measurements were carried out with a caliper, with an accuracy of 0.005 mm in different places 10 times, for each of the five samples with the same model parameters.

From Figure 6, it is apparent that when the angular tolerance decreased to 0.5 degrees, the nozzle diameter increased to an average of 5.71 mm, which was 0.13 mm less than at an angular tolerance of 30 degrees. This was due to the fact that at an angular tolerance of 30 degrees, the number of triangles from which the STL file will consist, was 264; and at 0.5 degrees, the number of triangles was 7236. In addition, it can be seen that there were average nozzle diameter values that matched, for example, at angular tolerances of 20 and 30 degrees, and at 6 and 7 degrees. This trend was due to the fact that at these angular tolerance values, the number of triangles that made up the STL model was equal to or different by only a few triangles. The G-code created by the Cura program in preparation for printing the nozzle with STL files at an angular tolerance of 30 and 0.5 degrees, was considered. In particular, from the entire program code, a third layer was selected → the extruder center path when printing the outer wall of the nozzle opening → and for this trajectory, the extreme instructions for moving the extruder along the Y axis (

Table 2. G-code of extreme points on the Y axis of the internal trajectory.

Angular Tolerance	30 Degrees	0.5 Degrees
Ymax	G1 X107.774 Y103.162 E33.61291	G1 X107.347 Y103.199 E33.59496
Ymin	G1 X107.384 Y96.802 E33.76570	G1 X107.429 Y96.802 E33.75383
Ymax − Ymin	6.36	6.399

).

Under ideal conditions, the diameter of the printed nozzle should be equal to:

$$d=Y_{max}-Y_{min}-w_e,$$

(1)

where Y_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).

For G-codes obtained for an angular tolerance of 30 degrees, the diameter was d = 5.96 mm, and for an angular tolerance of 0.5 degrees, then d = 5.999 mm. From this, it was concluded that when creating an STL file for 3D printing, it is not recommended to use the standard parameters that are embedded into the CAD software. The parameters must be configured manually, and the angular tolerance selected as the minimum value to ensure the maximum quality of nozzle and other elements of the model. This will avoid use of the nozzle diameter compensation factor in the future. In case it is not possible, in the STL model, to adjust the angular tolerance and/or chord height to increase the quality of the 3D printing, it is recommended to increase the diameter of the nozzle hole using the STL model quality compensation factor:

$$k_{STL}=d_n-\left(\frac{\left(Y_{max}-Y_{min}\right)+\left(X_{max}-X_{min}\right)}{2}\right)+w_e,$$

(2)

where d_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.

Another solution to the poor quality STL model of the finished product to be printed is the use of Arc Welder Plugin [91], which allows the Slicers program to replace linear interpolation commands G1 with arc interpolation commands G2 and G3. Using this plugin has the advantages of reducing the size of the G-code file by replacing many linear commands with one arc, in addition to a smooth and fast extruder movement during printing. However, the use of this plugin has been little studied, and at this stage does not work stably with STL models of different quality.

3.2. Coefficient Overexstrusion under Arc Path Motion

In the 3D printing of Bernoulli grippers, nozzle elements with a diameter of less than 8 mm are normally used. This is due to a critical increase in the consumption characteristics of compressed air with large nozzle diameters of gripping devices. When the 3D printer extrudes a layer of plastic in a straight line, it evenly distributes it on both sides of the center of the extruder nozzle (Figure 7), however, when moving along an arc path, an undesirable effect occurs and more material is presented inside the arc than necessary, and, vice versa, less material on the outside of the arc (Figure 8).

This problem (Figure 8) when printing holes is usually solved experimentally, or simplified equations or plugins are used, proposed by manufacturers of various software. For example, resource RepRap [92] offers the Arc Compensation algorithm, which calculates the decrease in the diameter of the hole by comparing the areas, and fills the hole with an extruder in a straight and arc path. In accordance with this algorithm, an equation was proposed to calculate the radius of passage of the center of the extruder, taking compensation into account:

$$r=\frac{w_e+\sqrt{w_e^2+4R^2}}{2},$$

(3)

where R is the radius of the required hole.

However, this algorithm was calculated for only one pass and did not allow the determination of the motion parameters of the extruder when it was necessary to form the hole wall as a result of several passes. Therefore, we proposed a method for calculating the displacement of the inner and outer radius of the extruded plastic layer when 3D printing along an arc path, as shown in Figure 9.

When the extruder passes along an arc trajectory with a radius r of the path rdφ for the same time, the areas S_o and S_i are filled with plastic:

$$S_o=\left(r_2-r\right)d\phi \cdot r+\frac{1}{2}{(r_2-r)}^{2}d\phi =\frac{1}{2}\left(r_2^2-r^2\right)d\phi ,$$

(4)

$$S_i=r_1\left(r-r_1\right)d\phi +\frac{1}{2}{(r-r_1)}^{2}d\phi =\frac{1}{2}\left(r^2-r_1^2\right)d\phi ,$$

(5)

where r₁ is the internal radius of the extruded arc, r is the radius of movement of the center of the extruder, and r₂ is the external radius of the extruded arc.

Since plastic is evenly extruded through the extruder on both sides relative to its radius of motion r, and the outer and inner area of the extruded plastic is filled at the same time, these areas are level:

$$S_o=S_i\Rightarrow 2r^2=r_1^2+r_2^2.$$

(6)

The total area of the extruded plastic when moving in an arc is equal:

$$S_t=S_o+S_i=\frac{1}{2}\left(r_2^2-r_1^2\right)d\phi .$$

(7)

If the extruder moves with some speed v, then:

$$v=r{\frac{d\phi }{dt}}_{}{\Rightarrow }_{}d\phi =\frac{vdt}{r},$$

(8)

$$S_t=\frac{1}{2}\left(r_2^2-r_1^2\right)\frac{vdt}{r},$$

(9)

Therefore, with uniform movement of the extruder along a rectilinear trajectory, the area which the extruder fills is equal to:

$$S_l=w_{e}vdt,$$

(10)

where v is the extruder speed, t is the extruder move time, and w_e is the extrusion width.

During 3D printing, the printer extrudes the same amount of plastic, both in rectilinear and arc motions, so that the areas S_t and S_l can be equated and a dependence can be obtained:

$$2w_{e}r=r_2^2-r_1^2.$$

(11)

Using dependences (6) and (11), a system of equations is obtained:

$$\left\{\begin{array}{c}{r}_2^2+r_1^2=2r^2;\\ r_2^2-r_1^2=2w_{e}r.\end{array}\right.$$

(12)

From the system of Equation (12), we can determine two radii by specifying the third. For example, setting the radius of the trajectory of the extruder r we find:

$$\begin{array}{l}{r}_2=\sqrt{r\left(r+w_e\right)},\\ r_1=\sqrt{r\left(r-w_e\right)}.\end{array}$$

(13)

If we specify the radius r₁ = R, then we find Equation (3), and if we specify the radius r₂ for the radius r we obtain:

$$r=-\frac{w_e}{2}+\sqrt{\frac{w_e^2}{4}+{\left(r_2\right)}^2}.$$

(14)

However, the above model describes the radius offset of only one layer of extruded plastic, and FDM printing practically never uses walls in one layer of material. Therefore, we proposed Algorithm 1 for calculating the arc compensation coefficient, determining the necessary increase in the hole radius to provide a given radius during design. In this case, during the operation of Algorithm 1, the displacement of the inner layer of the hole for different wall thickness was calculated.

Algorithm 1 The displacement of the inner layer of the hole for different wall thickness
1:	INPUT: r1, x, we
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:	$k_{Arc}\leftarrow 2\cdot \left(c-x\right)$

Using Algorithm 1, the effect of the radius r of the printed cylindrical nozzle and the width of the wall x of this nozzle on the arc compensation factor of the nozzle radius were calculated (Figure 10).

As shown in Figure 10, the nozzle radius compensation factor was critical only when the nozzle radius was less than 1 mm, and practically did not change after reaching a wall thickness of 2 mm. This was due to the fact that at a radius of more than 3 mm, the coefficient of compensation of the radius of one layer of extruded plastic was less than 0.01 mm. The effect of nozzle wall thickness x on nozzle hole diameter in Algorithm 1, at a mm radius, was considered (Figure 11).

To compare the effect of the reduction in the diameter of the printing nozzle from the arc motion of the extruder and the experimental data on the zero point, the diameter at the wall thickness was set as x = 0.4 mm (Figure 11a). Using Algorithm 1, it was possible to derive the value of the change in the diameter of the nozzle hole formed due to the arc motion of the extruder, and compare the obtained data with the experimental ones (Figure 12).

As shown in Figure 12, the change in the diameter of the print nozzle depended on the arc motion of the extruder by only 31…33% and this percentage depended only on the radius and thickness of the nozzle wall. This was because material shrinkage had the greatest effect on the diameter of the print nozzle. Therefore, we now will further consider the issue of shrinkage, taking into account the proposed arc compensation coefficient.

3.3. Coefficient Shrinkage of Material

Many publications have been devoted to the definition of shrinkage of material in 3D printing, namely, in [93,94,95,96] and others. However, in these works, everything comes down to the authors’ proposal of a general shrinkage coefficient for the entire model. However, as previously defined for holes such as nozzles, there are additional factors affecting the reduction in the size of the holes obtained during 3D printing. Therefore, for further shrinkage calculations, both the maximum quality settings of the 3D model and the arc compensation factor were used. As such, all of those factors that affect the reduction in size of the 3D print nozzle were discarded, to obtain a shrinkage value dependent on the print parameters and material parameters. The volume shrinkage value was calculated using the equation:

$${\epsilon }_v=\frac{V_0-V_{end}}{V_0}\cdot 100\%,$$

(15)

where V₀ is the initial wall volume of the designed nozzle, and V_end is the final wall volume of the printed nozzle.

The average linear shrinkage of the part was determined by the equation:

$${\epsilon }_l=\frac{L_0-L_{end}}{L_0}\cdot 100\%,$$

(16)

where L₀ 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.

Experimental research methods are usually used to determine linear shrinkage. To further investigate the effect of shrinkage on the reduction in the size of the print nozzle, we printed nozzles using the elements in Figure 5 and the parameters given in

Table 3. Experimental data for the calculation of shrinkage in 3D printing.

No	d (mm)	x (mm)	we (mm)	d+kArc  (mm)	dend (mm)	V0 (mm2)	Vend (mm2)	εv (%)	εl (%)
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

.

The studies showed that the d_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).

As shown in Figure 13, volume shrinkage for almost all filament manufacturers was close to 1%, and ranged ±0.2%. However, linear shrinkage, for which model sizing is carried out during 3D printing, had a large deviation for the various filament manufacturers, reaching up to 3%. Large shrinkage in 3D Plast and Wanhao samples may have been due to the use of non-high-quality raw materials, or process disruption. However, the linear shrinkage of filaments may have been due to variations between batches of the same manufacturer. Therefore, in order to ensure the most accurate 3D printing, it was necessary to determine the linear shrinkage factor for each batch and calculate the shrinkage compensation factor of the diameter of the printing hole:

$$k_{shr}=d\left(\frac{100}{100-{\epsilon }_l}-1\right)$$

(17)

.

For the above PLA filaments of various manufacturers, the diameter of the 3D nozzle model was calculated to provide the nominal diameter of the print nozzle of the gripping devices according to the equation:

$$d=d_n+k_{STL}+k_{Arc}+k_{shr}$$

(18)

.

For a nozzle with a diameter of 6 mm, wall thickness of 1.6 mm, using Wanhao filament, and with the maximum quality of a 3D model, the diameter of the 3D nozzle model was d = 6.325 mm, which was the largest for all filaments for which the experiment was conducted. For the calculated nozzle diameter, gripping devices were printed at different printing heights, from 0.05 to 0.2 mm, in increments of 0.05 mm. The diameter of the 3D printed nozzle was 6 ± 0.01 mm, which lay within the accuracy of the measuring instruments. The obtained results of the study make it possible to assert the adequacy of the proposed model for calculating the diameter of holes of small diameters using FDM 3D printing. Using the proposed model, Bernoulli grippers with nozzles less than 8 mm in diameter can be printed with a maximum accuracy of the printing hole diameter.

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

Bernoulli grippers use compressed air to provide lifting force, and their operation has been described above (Figure 2). Based on the design of the gripping devices, when the nozzle elements are 3D printed, the nozzle geometry will have the greatest influence on the power characteristics of the printed gripping device, which will not be smooth, since it is formed by a build-up of plastic with a certain layer height. Even compared with the classic production of Bernoulli grippers, 3D printing, delightfully, will ensure a greater roughness of the end surface of the grip. To determine the effect of the geometry of the nozzle surface on the power characteristics, nozzle elements were printed with their subsequent splitting along the nozzle axis. This made it possible to survey the section of nozzle elements on a microscope at 100 times magnification (Figure 14).

It can be seen from Figure 14 that there was some deviation in the surface geometry that formed the nozzle wall. To determine the average value of performances of each of the layers, the following equation was used:

$$c_{\mathrm{mid}}=\frac{c_{\max }+c_{\min }}{2}$$

(19)

.

Most often, for FDM 3D printing, the diameter of the extruder nozzle is 0.4 mm. When printing with a 0.4 mm nozzle, to ensure optimal printing parameters, a range of printing layer heights would be selected, from 0.05 to 0.2 mm. Therefore, our next studies were carried out for this range of printing layer heights, and the average value of the performance of the layers was examined (

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 hp (mm)	Cmax (mm)	Cmin (mm)	Cmid (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

).

As can be seen from the experimental studies in Table 4, the average performance of the layers during 3D printing was half the height of the printing layer. Using the data on the geometry of the shape of the nozzle element of the gripping devices during printing, it was possible to conduct 3D modeling of the gripping devices for further investigation (Figure 15).

3.5. Description of the Finite Element Method for BGD Research

The mathematical model of air in the radial interval between the interacting surfaces of a BGD and an OM was based on the Navier–Stokes’s equation average according to Reynolds (Reynolds-averaged Navier–Stokes equations) (RANS) [100,101]. The SST-model of turbulence [102] and the gamma-model of laminar and turbulent transition [103] were used for modeling. The main equations for describing these models are presented in Appendix B.

Numerical modeling was carried out in the environment of computing hydro-gas dynamics by Ansys-CFX with the use of the SST gamma-model of turbulence. For carrying out calculations, the program environments in the settlement area were constructed with an unstructured differential grid (Figure 16).

The total number of nodes in the calculation area depended on the outer radius of the gripper r₁ 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.

Using the sonicTurbFoam solver (for turbulent streams of the compressed gases moving with sound and supersonic speeds), the value of the distribution of teak, speed, and force over the surfaces of the studied model, was obtained.

3.6. Methods of Experimental Research

To obtain experimental results of the pressure distribution on the surface of the object of manipulation during gripping with the help of Bernoulli gripping devices with nozzle elements of different parameters, an experimental setup was developed (Figure 18).

The scheme of work of the experimental installation is shown in Figure 19. The experimental setup (Figure 18) consisted of: 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 (m₁, m₂, m₃), 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.

The scheme shows a pneumatic diagram of the supply of compressed air to the chamber of the Bernoulli gripping device—1. After supplying compressed air to the chamber of the gripper—5 on the surface of the object of manipulation—6, zones of excess and negative pressure were formed and the object of manipulation was attracted to micrometers m₁, m₂ and m₃. This provided a pre-set with a micrometer (m₁, m₂ and m₃) 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

To determine the effect of the height of the gripper print layer on its power characteristics, the manipulation object gripping process was simulated in the Ansys-CFX software environment using the finite element method presented above. In particular, an arc compensation algorithm was used to obtain a nozzle with a diameter of 6 mm during 3D printing and the use of Wanhao plastic. At the same time, the height of the printing layer changed from 0.05 mm to 0.2 mm with a pitch of 0.05 mm, which made it possible to determine the distribution of pressure on the surface of the object of manipulation (Figure 20).

As can be seen from the results of the simulation shown in Figure 20, grips with different layer heights during 3D printing had different rarefaction zones. In all other zones, the pressure distribution was almost equal for all gripping devices with different 3D printing layer heights, which led to different lifting forces. As can be seen from Figure 20, the vacuum zone increased as the height of the layer increased. This was due to the fact that, compared with the classic gripper in FDM-printed grippers, the edge of the nozzle was a rounded edge (Figure 21).

It can be seen from Figure 21 that the rounded edge of the air flowed smoothly around the narrowing zone of the flow, thereby preserving the kinetic energy of the air flow. Having a large kinetic energy, the air flow created a wide vacuum zone, thereby, increasing the lifting force of the Bernoulli gripping devices (Figure 22). However, the height of the print layer cannot be thoughtlessly increased—this may result in poor sintering of the layers, so it is recommended that the height of the layer does not exceed 50% of the diameter of the extruder nozzle. In our case, when printing with a 0.4 mm nozzle, the height of the printing layer did not exceed 0.2 mm.

Obviously, from Figure 22, the lifting force increased as the height of the printing layer increased at different feed pressures, which was a great advantage. However, since this was only a process simulation, it did not take into account the many processes that occur when gripping manipulation objects, such as the error of the hole geometry, the roughness of the active surface of the gripping devices, and the properties of the gripping material and the manipulation object. Since all these properties cannot be taken into account in the simulation, an experimental study was carried out on the effect of layer height in 3D printing. For this, according to the above experimental research method, the distribution of air over the surface of the object manipulation was formed by the Bernoulli gripping device (Figure 23).

As can be seen from Figure 23, by rounding the edge of the nozzle as a result of 3D printing, the pressure in the area opposite the nozzle decreased as the height of the nozzle h increased. Deflection of nozzle roundness and surface roughness critically affected the size of the vacuum zone. In FDM 3D printing, surface roughness of the invaders in the rarefaction zone produced swirls, causing pressure surges and a reduction in the length of the rarefaction zone. For further analysis of the experiment, the lifting forces of the Bernoulli grippers were determined (Figure 24) by integrating the pressure distribution on the surface of the OM:

$$F=2\pi {\displaystyle \underset{0}{\overset{r_g}{\int }}\left(p_a-p\right)rdr}.$$

(20)

As can be seen from Figure 24, the maximum lifting force was achieved by using gripping devices printed at the layer height of h = 0.05 mm. This was due to the fact that at this layer height, the minimum roughness of the active grip surface was provided and the effect of rounding the printed nozzle was present. Additionally, at inlet pressure 300 kPa, the greatest deviations of lifting force were observed, as a bigger pressure of turbulence critically influenced the lifting force. However, depending on the height of the gap h_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).

Figure 25 shows a clear relationship between the height of the gap h_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).

As seen in Figure 26, the most linear force characteristic had a gripper that was printed at a layer height of h = 0.15 mm. However, the highest lifting force compared with the classic aluminum grip was a BGD which was printed at a layer height of h = 0.05 mm. For a BGD printed at a layer height of h = 0.2 mm, the lifting force decreased at 300 kPa pressure, which was caused by the formation of local turbulence zones through the surface roughness of the gripping devices. For further analysis, the BGD of a printed layer height of h = 0.05 mm was considered, which showed the highest power characteristics compared with other 3D printers. An experimental study of the effect of BGD input pressure (h = 0.05 mm) on its power characteristics at different contact heights between OM and GD was undertaken (Figure 27).

Figure 27 shows that as the supply pressure increased, the lifting force increased. In particular, low power characteristics were noted at a gap of 0.15 mm to 0.25 mm, due to the increased effect of the roughness of the active surface of the Bernoulli gripping devices on its lifting force. In addition, focus should be placed on the operating range of the supply pressure. For this Bernoulli gripping device with a layer height of 0.05 mm, the operating range started from 200 kPa. From the point of view of operation of the gripping device, it was obviously necessary to provide the most linear dependence of the supply pressure and the lifting force. The most linear, and providing the maximum lifting force, was a working gap between the OM and GD, from 0.3 mm to 0.4 mm. Therefore, if low contact with the object manipulation is necessary, it is necessary to choose friction elements with a height of 0.3…0.4 mm to ensure the maximum lifting force.

4. Conclusions

The article presents a comprehensive study of the prototype of Bernoulli gripping devices with a cylindrical nozzle, manufactured by FDM 3D printing. In order to ensure an appropriate nozzle diameter of the Bernoulli gripping devices when 3D printing, a methodology was developed to determine the diameter of the CAD model so that after printing the gripping devices, the diameter of the nozzle was planned. The three main reasons for reducing the diameter of the gripping device nozzle after 3D printing were identified as being due to poor 3D STL model quality, excessive extrusion of plastic in the middle of the arc printing path, and linear shrinkage of filament material after object cooling. The proposed methodology consisted of determining the three coefficients that allowed the determination of the diameter of the designed nozzle.

It was determined that excessive extrusion of plastic in the middle of the arc printing path only affected the nozzle diameter when the diameter was less than 8 mm. Given the arc compensation coefficient, it was possible to determine the linear and volume shrinkage of several PLA filaments of various companies with an accuracy range of ±0.2%. According to the offered methodology of compensation of diameter of a nozzle, 3D printing of a Bernoulli gripping device with a cylindrical nozzle was carried out with a diameter of 6 mm and by changing the printing layer height from 0.05 to 0.2 mm. It was proved that the height of the printing layer did not affect the diameter of the cylindrical nozzle manufactured by FDM 3D printing, and the diameter error was ±0.02 mm.

Proposed methodology and an installation for experimental studies of air pressure distribution over the surface of the object manipulation were constructed. It was determined that as the height of the printing layer increased, the lifting force decreased due to the swirls formed through the surface roughness of the manipulation object. It was proved that as the height between the object of manipulation and the gripper increased, the effect of surface roughness on the lifting force decreased, which led to its increase. It was found that the maximum lifting force was provided when printing the grip with a layer height of 0.05 mm, and a height of 0.4 mm between the object of manipulation and the grip. The complex power characteristics of the gripper were determined with a printing layer height of 0.05 mm, at the change of supply pressure from 150 to 300 kPa, and a change of height between the object of manipulation and the grip from 0.15 to 0.4 mm. It was established that the operating range of the feed pressure for a gripper with a printing layer height of 0.05 mm started from 200 kPa, and the rational height of the friction elements at low contact with the object of manipulation was from 0.3 to 0.4 mm.

This study has certain limitations in relation to the assumptions. One limitation concerns the orientation of the nozzle during 3D printing, which in this study was required to be oriented perpendicular to the table. The second limitation concerns the design of the 3D printer, namely, a closed-type printer was used to ensure a constant temperature and uniform shrinkage of the part. In the future, it is planned to conduct 3D printing and research of gripping devices using nozzle elements of various geometric shapes and printing in different orientations. Such a study will simplify the technology of manufacturing gripping devices and ensure their maximum power characteristics using additive technologies. Therefore, this article serves as a basis for introducing and manufacturing more complex geometry of nozzle elements and holes using FDM and SLA 3D printing.