Effect of supply flow rate on performance of pneumatic non-contact gripper using vortex flow

Experimental Thermal and Fluid Science 79 (2016) 91–100

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Effect of supply flow rate on performance of pneumatic non-contact gripper using vortex flow Jianghong Zhao, Xin Li ⇑ State Key Lab of Fluid Power Transmission and Control, Zhejiang University, 38 Zheda-Road, Hangzhou, Zhejiang 310027, PR China

a r t i c l e

i n f o

Article history: Received 11 June 2015 Received in revised form 12 June 2016 Accepted 18 June 2016 Available online 30 June 2016 Keywords: Vortex gripper Performance indexes Suction force Suspension region Suspension stiffness Noncontact handling

a b s t r a c t The vortex gripper is a new kind of non-contact gripper, which generates negative pressure by blowing compressed air into a vortex cup through two tangential nozzles. It can provide an adequate suction force for handling a workpiece without contact. This gripper can avoid the disadvantages of traditional gripping devices, such as inducing mechanical scratches, local stress concentration, frictional static electricity, and blots on the workpiece. In this study, we experimentally and theoretically investigated the effect of supply flow rate on the performance of the vortex gripper. First, we proposed three performance indexes for evaluating the properties of the vortex gripper: the maximum force, suspension region, and suspension stiffness. Then, we obtained a series of F–h (i.e., the suction force against the spacing between the gripper and the workpiece) curves at different supply flow rates and the pressure distributions at the surface of the workpiece for different values of spacing h. Based on the experimental data, we analyzed the effect of the supply flow rate on the maximum force, and by nondimensionalization of the F–h curves, the changes in the suspension region were assessed. Furthermore, we proposed an additive method of pressure distribution and deduced a simplified theoretical formula for suspension stiffness. In addition, from the perspective of the suspension stability of the workpiece, we evaluated the physical significance of the slope of the F–h curves after nondimensionalization. The findings of this study could help researchers to comprehend the operation of the vortex gripper and provide guidance for implementing the vortex gripper in practical applications. Ó 2016 Elsevier Inc. All rights reserved.

1. Introduction On automatic production lines, the workpiece frequently needs to be gripped and transported. Thus Fantoni et al. provided a review of grippers and robotic hands in automated production processes [1]. Mechanical paws and rubber suction cups are the most commonly used end-effectors, but because they have to make contact with the workpiece, they may cause some damage to it. For instance, they may induce mechanical scratches, local stress concentration, frictional static electricity, or blots on the workpiece. Such damage is usually fatal to precision workpieces such as LCD glass substrates, and silicon wafers [2,3]. Furthermore, in the food and pharmaceutical industries, contact between the end-effector and the workpiece may cause contamination, reducing the quality of the products. In addition, usually they need control strategies which are sometimes very complicated. In order to solve these problems, researchers have developed a variety of noncontact handling devices. For instance, Rawal et al. developed a noncontact ⇑ Corresponding author. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.expthermflusci.2016.06.020 0894-1777/Ó 2016 Elsevier Inc. All rights reserved.

end effector for handling of bakery products [4]; Li and Kagawa proposed a noncontact gripper using swirl vanes [5]; Ozcelik et al. designed a noncontact end-effector for handling of garments and evaluated the results of handling various materials [6,7]; Davis et al. developed an end effector based on the Bernoulli principle for handling sliced fruit and vegetables [8]. Among them, the pneumatic non-contact gripper, which uses air as the force transmission medium, is widely used. It does not produce a magnetic field or need feedback control. In addition, it has a simple construction and is easy to maintain. The vortex gripper, a new kind of non-contact gripper, was proposed recently. It generates negative pressure and suction force by using a high-speed vortex airflow and can grip the workpiece without any contact. Compared with traditional pneumatic non-contact grippers (e.g., Bernoulli gripper [8,9]), the vortex gripper has the advantage of low gas consumption. As a result, many related researches have been carried out and reported in recent years. In 2008, Li et al. experimentally investigated the fundamental characteristics of the vortex gripper, namely the pressure distribution and the suction force [10]. They deduced that there exists a very small space under the vortex gripper in which the suction force is

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positively associated with the spacing between the gripper and the workpiece, i.e., the slope of the F–h curve is positive. Thus, the workpiece can be stably suspended at a certain distance under the gripper. In another research, Li et al. analyzed the velocity and pressure fields inside the vortex gripper in detail using numerical fluid dynamics calculations, which revealed the relationship between flow velocity and pressure distribution [11]. Iio et al. observed the form of the vortex flow in an underwater environment [12]. Moreover, Wu et al. obtained experimental data of velocity distribution using visual equipment and the particle image velocimetry (PIV) method and proposed several empirical formulae for the velocity distribution of vortex flow [13]. However, all these researches were focused on the description and analysis of the physical phenomena. There have been no studies on the operation parameters of the vortex gripper, which are more important from the user’s point of view. In particular, the supply flow rate of the compressed air is a significant operation parameter, as it is the only parameter that users can adjust because others (e.g., the design parameters) have been determined already. Therefore, it is important to understand how the performance of the vortex gripper changes as the supply flow rate varies. Based on these considerations, we experimentally and theoretically studied the effect of supply flow rate on the performance of the vortex gripper. We first proposed three performance indexes for evaluating the characteristics of the vortex gripper, and then we analyzed the effect of the supply flow rate on each of these performance indexes. Table 1 is the nomenclature of the alphabetic characters that we used in this paper. 2. Proposed performance indexes for vortex gripper 2.1. Principle of vortex gripper As shown in Fig. 1, the basic structure of the vortex gripper includes a cylindrical vortex chamber and two tangential nozzles, which are processed on the circular wall of the chamber. Compressed air blows into the vortex chamber through the tangential

Table 1 Nomenclature. Symbol

Quantity

SI Unit

d F Fmax g h hB hC hmax H H1 H2 kB 0 kB L m p p1 p2 Q r, z R1 R2 ua ur ur

Diameter of tangential nozzles Suction force Maximum force Acceleration due to gravity Spacing between gripper and workpiece Spacing at stable suspension position Diameter of tangential nozzles Optimum spacing Height of vortex chamber Height of upper part of vortex chamber Height of lower part of vortex chamber Suspension stiffness Suspension stiffness after nondimensionalization

mm N N m/s2 mm mm mm mm mm mm mm N/mm mm1

Distance between nozzle and central point Mass of workpiece Pressure Pressure distribution dominated by vortex flow Pressure distribution dominated by gap flow Supply flow rate Cylindrical coordinates Radius of vortex chamber Radius of annular skirt Circumferential velocity Radial velocity The average radial velocity Air density Gripping coefficient Coefficient of viscosity

mm kg Pa Pa Pa L/min (ANR) – mm mm m/s m/s m/s kg/m3 – Pas

q g l

Vortex gripper h

Tangential nozzles Vortex Q chamber

Work piece

Top view

Front view

Fig. 1. Schematic of vortex gripper.

nozzles and forms a high-speed vortex flow along the internal face of the chamber. Similar to a tornado, the centrifugal force produced by the vortex flow pushes the air in the center towards the peripheral region of the chamber, and thus a negative pressure zone with rarefied air is created at the center. As a result, a suction force will be applied on a workpiece placed under the gripper, which can then be picked up. In addition, as compressed air is constantly supplied through the nozzles into the vortex chamber, the airflow will continually vent through the gap between the gripper and the workpiece. This exhaust flow ensures that there will be no contact between the gripper and the workpiece.

2.2. Performance indexes To study the application of the vortex gripper in non-contact handling, we propose three performance indexes based on the suction force characteristic curve (F–h curve): maximum force, suspension region, and suspension stiffness. In this section, we elaborate on the definition and physical meaning of these three indexes. (1) Maximum force The study by Li et al. [10] reported that the suction force F changes when the spacing between the vortex gripper and workpiece, h, changes. Fig. 2 shows a typical F–h curve for a given supply condition. When h is small, the gripper generates a repulsive force on the workpiece. As h increases, the repulsive force reduces to zero, and then a suction force is generated. After the suction force reaches a maximum, it decreases gradually. The maximum of the curve, marked by A, corresponds to the maximum suction force Fmax, which is generated at the spacing of hmax. Therefore, the gripper can pick up a workpiece whose weight is less than Fmax, that is,

F

Suspension region

A

Fmax kB B

C

mg

Experimental force Weight of work piece

hB hmax

hC

Fig. 2. The definition of performance index.

h

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Fmax reflects the adsorption capacity of the vortex gripper. For a given supply condition, a larger Fmax would be more advantageous. (2) Suspension region Let us suppose that the gripper picks up a workpiece of mass m whose weight is marked by the red line in Fig. 2. The red line and the F–h curve intersect at two points, point B and point C, and the spacing at these two points are marked by hB and hC, respectively. Although the suction force is equal to the weight of the workpiece both at point B and point C, we can observe that the slope of the F– h curve is positive at point B, whereas it is negative at point C. This means the stable suspension position of the workpiece can only be point B and not point C. To illustrate this, let us consider the situation in which the vortex gripper handles a workpiece at point B. If the workpiece is exposed to a tiny interference and strays from point B, it will go back to point B after the interference fades away because of the positive slope. Next, let us consider the case in which the workpiece is suspended at point C. If the workpiece gets closer to the gripper than point C because of an external interference, i.e., h decreases, the suction force will increase and become greater than the weight of the workpiece. As a result, the workpiece will continue to move to point B, where it will be stably suspended. On the other hand, if the workpiece moves away from the gripper, i.e., h increases, the suction force will decrease and become less than the weight of the workpiece, causing the workpiece to fall down. Therefore, point B is defined as the stable suspension position, and point C is defined as the boundary suspension position. Moreover, the region h < hC is designated as the suspension region. As long as the workpiece is handled within the suspension region, it can reach a stable levitation position. For the effective operation of the vortex gripper, a larger suspension region would be more advantageous. This is because when the mechanical hand moves the gripper close to a workpiece, it has to make sure that the workpiece is handled within the suspension region during the process of picking it up. If the suspension region is not large enough, a manipulator with high positional accuracy would be required, and the gripper and the workpiece would have to be kept strictly parallel to avoid contact with or damage to the workpiece.

to an external interference, e.g., a downdraft in the clean room or an inertial force due to acceleration/deceleration during the gripping process, it will vibrate about the equilibrium point. In order to prevent the workpiece from falling down, the vibration amplitude should be as small as possible. Improving the stiffness at the suspension point, i.e., the suspension stiffness, is an effective means of suppressing the vibration amplitude. The suspension stiffness kB refers to the slope of the F–h curve, i.e., dF/dh, at point B. Therefore, the larger kB is, the smaller will be the offset spacing dh caused by the interference of the external force dF, which means the state of suspension of the workpiece will be more stable. 3. Experimental methods 3.1. Test vortex gripper According to the principle described in Section 2.1, we designed and manufactured a vortex gripper as shown in Fig. 3. The specific dimensions of the test gripper are listed in Table 2. The basic structure of the gripper includes the vortex chamber of radius 10 mm and height 2 mm, the annular skirt of radius 25 mm set outside the chamber, and two nozzles of diameter 0.5 mm, which are tangential to the internal face of the chamber. 3.2. Experimental setup for measuring suction force In order to obtain the experimental F–h curves accurately, we set up the measuring apparatus as shown in Fig. 4. The vortex gripper is fixed by the locating mechanism to keep it parallel to the surface of the plate. The locating mechanism consists of locating pins, a spring, and a spring stopper. The plate is fixed in the horizontal direction by the air guide and held in the vertical direction by the force sensor fixed on the sliding base at the bottom. The air guide has high linearity and ensures that the plate is able to move in the vertical direction without any friction. The up and down movement of the plate is realized by adjusting the location of the sliding base fixed on the sliding track using the feeding bolt. Thus, the spacing between the gripper and the workpiece, h, can be easily Table 2 Size of vortex gripper (unit: mm).

(3) Suspension stiffness As previously mentioned, the workpiece will stably levitate at point B after the gripper picks it up. If the workpiece is exposed

d

R1

R2

H

L

H1

H2

0.5

10

25

2

9.2

0.5

1.5

d

H1

Tangential nozzle Front view

d

Vortex chamber Annular skirt

L

H

H2

R2 R1

Top view Fig. 3. Schematic structure and photograph of vortex gripper used in this study.

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Compressed air Spring

Spring stopper

Spring stopper Spring

Stable base

Locating pins Vortex gripper h

Locating pins Plate

F

Vortex gripper Plate Air guide

G0 Laser meter

Stable base Air guide Laser meter

G0-F Sliding shaft

Force sensor

Force sensor Sliding base & sliding track Feeding bolt Feeding bolt Fig. 4. Schematic and photograph of experimental setup for measuring suction force.

controlled by adjusting the feeding bolt. The laser meter fixed under the plate can observe the location of the plate in the vertical direction to determine the value of h in real time, and the force sensor can obtain the difference between the weight of the plate G0 and the suction force of the gripper F. As G0 is a known quantity, we can evaluate the force F easily. During the experiment, the following steps are performed: (1) adjust the feeding bolt to make the lower surface of the gripper contact the upper surface of the plate; (2) set down the locating pins and make sure that the ends of the pins contact the top of the vortex gripper; (3) lock the clamps to hold the pins tightly; (4) compress the spring and tighten the spring stopper so that the gripper is pulled up to the stable base; (5) set the laser meter to zero; (6) adjust the feeding bolt so that the laser meter displays a value of approximately 0.1 mm (i.e., h = 0.1 mm); (7) open the gas switch to provide a certain flow rate of compressed air to the gripper; (8) adjust the value of h, increasing it very gradually to 1.5 mm, and record the values measured by the force sensor and the laser meter using a data acquisition (DAQ) card. Thus, by this method, the F–h curve can be obtained for a certain supply flow rate. To obtain the F–h curves at different supply flow rates, the above steps are repeated by adjusting the flow rate of the compressed air.

the sliding part and surface of the stationary table are flat and smooth. During the experiment, the following steps are performed: (1) place two pieces of steel sheet of the same known thickness symmetrically between the gripper and the measuring table (the steel sheet is not drawn in Fig. 5); (2) set down the three locating pins and make sure that the ends of the pins contact the top of the vortex gripper; (3) lock the clamps to hold the pins tightly; (4) compress the spring and tighten the spring stopper so that the gripper is pulled up to the stable base; (5) remove the steel sheet slowly and carefully to create a spacing between the measuring table and the gripper equal to the thickness of the steel sheet; (6) open the gas switch to provide a certain flow rate of compressed air to the gripper; (7) move the sliding bar very slowly to avoid the influence of the sliding bar on the airflow inside the gripper; (8) continuously record the position of the pressure tap at the sliding bar using the displacement sensor and the pressure at the pressure tap through the pressure port using the pressure sensor. Thus, by changing the thickness of the steel sheet, the pressure distribution at the surface of the measuring table can be obtained for any spacing between the measuring table and the gripper. 4. Results and discussion

3.3. Experimental setup for measuring pressure distribution 4.1. Effect of flow rate on maximum force In order to analyze the working of the vortex gripper in detail, we need to measure the pressure distribution at the surface of the workpiece. Therefore, we set up the measuring apparatus, as shown in Fig. 5, which consists of a stable base, locating mechanism, measuring table, displacement sensor, and pressure sensor. The measuring table, shown in Fig. 5(c) has two main parts: the stationary table and the sliding bar. The stationary table is stable and simulates the workpiece, whereas the sliding bar is used to measure the pressure at the pressure tap, which is positioned on the surface of the stationary table through the pressure port. The sliding bar is located in the middle of the stationary table, and

4.1.1. 1Condition for maximum force Fig. 6 shows the F–h curve for the supply flow rate of Q = 5.0 L/ min (Atmosphere Normale de Reference, thereafter, ANR). We can see that the suction force changes as the spacing varies. To illustrate this trend in more detail, we measured the pressure distribution at the surface of the workpiece with h set at 0.15, 0.18, 0.20, 0.30, 0.40, 0.75, and 1.50 mm, and the results are shown in Fig. 7. We can observe from the curves of the pressure distribution that the airflow inside the vortex gripper is composed of the vortex flow inside the vortex chamber (r < R1) and the gap flow between the

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Compressed air Spring stopper Spring

Spring stopper

Spring

Locating pins

Stable base

Stable base

Locating pins Vortex gripper

Vortex gripper Sliding bar

h

Measuring table Displacement sensor

Measuring table

(a)

(b) Pressure tap Pressure port

Displacement sensor 60 mm

Sliding bar Stationary table

Pressure sensor

(c) Fig. 5. Experimental setup for measuring pressure distribution: (a) schematic structure of apparatus, (b) photograph of apparatus, and (c) schematic of measuring table.

Accordingly, we can simplify the Navier–Stokes (N–S) equation to the following form:

30

Fmax -2 F [10 N]

20

@p u2 @u2 ¼ q a þ l 2r @r r @z

10

0

-10 0

h max 0.5

1

1.5

h [mm] Fig. 6. F–h curve of vortex gripper at Q = 5 L/min (ANR).

annular skirt and the workpiece (R1 < r < R2). The vortex flow forms a cupped pressure distribution in the vortex chamber influenced by the centrifugal force, whereas the gap flow generates a pressure distribution under the annular skirt dominated by both the viscidity and inertia of the airflow. Because the vortex flow is surrounded by the gap flow, the pressure distribution produced by the vortex flow superimposes on the pressure distribution due to the gap flow. Moreover, we can observe from the figure that the shape of the pressure distribution inside the chamber remains almost the same when h is in the range 0.15–0.35 mm, which corresponds to the region with positive slope in the F–h curve. This implies that the flow regime of the vortex flow is independent of the spacing h. However, the pressure under the annular skirt changes significantly as h varies. Therefore, we infer that the analysis of the flow regime of the gap flow is the key to the analysis of the pressure distribution inside the vortex gripper. For flow inside a narrow gap, r the viscosity term l @u usually plays the dominant role in influenc@z2 ing the pressure distribution @p . In addition, the airflow has a large @r circumferential velocity in the gap between the annular skirt and the plate. Thus, the inertial effect of the circumferential velocity 2 q ura will also affect the pressure distribution considerably.

ðR1 < r < R2 Þ

ð1Þ

Next, we analyze the pressure distributions shown in Fig. 7 in more detail. When the spacing is very small (i.e., h = 0.15 mm), the viscous friction between the airflow and the surfaces of the gap becomes very large. Therefore, the large viscosity term in Eq. (1) leads to a high positive pressure distribution, resulting in a high-pressure situation inside the vortex chamber. The force applied by the vortex gripper on the workpiece is repulsive in this case. With the gradual increase in spacing (i.e., from 0.15 mm to 0.20 mm), the radial velocity component of the gap flow, ur, r decreases, resulting in a decrease in its distribution @u . Accordingly, @z both the viscosity of the gap flow and its resulting positive pressure distribution decrease, and the pressure distribution inside the vortex chamber changes from positive to negative. Thus, the force applied by the vortex gripper on the workpiece becomes a suction force and increases gradually. During this process, because the effect of the change in spacing on the flow regime of the vortex flow inside the chamber is very small, the shape of the pressure distribution inside the chamber remains almost the same. When the spacing further increases to 0.3 mm and 0.35 mm, the viscosity effect of the gap flow is further weakened, and the inertial effect of the circumferential velocity in Eq. (1) increases, causing the pressure distribution of the gap flow to gradually change from positive to negative. For example, when h = 0.35 mm, the pressure between the annular skirt and the workpiece is completely negative, which makes the pressure inside the chamber decrease to a minimum and the suction force on the workpiece reach a maximum. However, with the continued increase in spacing (i.e., h > 0.35 mm), the pressure distribution gradually returns to the atmospheric pressure. We can also see that there is an obvious pressure fluctuation. This may be because the air in the peripheral region of the gripper enters the chamber and transfers external interference to the flow regime of the vortex flow. In addition, the suction force will decrease continuously during this process.

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h = 0.15 mm

h = 0.18 mm

h = 0.20 mm

h = 0.30 mm

400 200 0 -200 -400

p [Pa (G)]

-600 -800 gap flow vortex flow gap flow gap flow vortex flow gap flow gap flow vortex flow gap flow gap flow vortex flow gap flow -1000 400 h = 0.75 mm h = 1.50 mm h = 0.35 mm h = 0.40 mm 200 0 -200 -400 -600 -800 gap flow vortex flow gap flow gap flow vortex flow gap flow gap flow vortex flow gap flow gap flow vortex flow gap flow -1000 R2(25) R1(10) 0 R1(10) R2(25) R2(25) R1(10) 0 R1(10) R2(25) R2(25) R1(10) 0 R1(10) R1(10) 0 R1(10)

r [mm]

r [mm]

r [mm]

r [mm]

Fig. 7. Pressure distribution at the surface of workpiece for Q = 5.0 L/min (ANR).

4.1.2. Relationship between maximum force and supply flow rate We obtained the F–h curves for ten different supply flow rates (3.0, 4.0, . . . 11.0, and 12.0 L/min (ANR)) as shown in Fig. 8. We can see that all the curves have the same variation trend, i.e., for a certain h, the larger the flow rate, the larger is the suction force. In order to clarify the relationship between the maximum force and the supply flow rate, we plotted the maximum force against the supply flow rate in Fig. 9. By fitting a quadratic curve (shown by the red broken line) to the experimental data, we obtained a correlation coefficient of 0.9996, which is very close to 1. Therefore, the maximum force has a quadratic relationship to the supply

Q = 3.0 L/min (ANR) Q = 4.0 L/min (ANR) Q = 5.0 L/min (ANR) Q = 6.0 L/min (ANR) Q = 7.0 L/min (ANR) Q = 8.0 L/min (ANR) Q = 9.0 L/min (ANR) Q = 10.0 L/min (ANR) Q = 11.0 L/min (ANR) Q = 12.0 L/min (ANR)

140 120 100 -2 F [10 N]

80 60 40 20 0 -20 -40 0

0.4

0.8

1.2

h [mm] Fig. 8. F–h curves of vortex gripper at different supply flow rates.

150 Maximum force Fitted curve

F max [10 -2 N]

Thus, as h increases, the decrease in the viscosity effect and increase in the inertial effect of the gap flow cause the pressure distribution to change continually from positive to negative, and thus the suction force has a rising trend. However, after the spacing crosses a certain optimum value, the air in the peripheral region of the gripper enters the chamber, changing the increasing trend of the suction force to a decreasing trend. The suction force corresponding to this optimum spacing, i.e., hmax, is the maximum force Fmax. For instance, when Q = 5.0 L/min, hmax is 0.35 mm, and Fmax is 23.88  102 N.

100

50

0

3

6

9

12

Q [L/min (ANR)] Fig. 9. Maximum force of vortex gripper against supply flow rate.

flow rate. We know that the circumferential velocity is the main component of the vortex flow and is determined by the supply flow rate. The square of the circumferential velocity determines the centrifugal force, which not only dominates the pressure distribution inside the chamber but also determines the pressure distribution between the annular skirt and the workpiece when the spacing is equal to hmax. As the suction force can be obtained by the areal integral of the pressure distribution, we can easily comprehend the quadratic relationship between the supply flow rate and the maximum force. Furthermore, we plot the optimum spacing hmax against the supply flow rate in Fig. 10. We can notice that, as the supply flow rate increases, the optimum spacing corresponding to the maximum force decreases gradually. We know from our previous computational fluid dynamics (CFD) analysis that the attraction of the negative pressure between the annular skirt and the workpiece will cause the air outside the gripper to enter the chamber, changing the increasing trend of the suction force to a decreasing trend [11]. The increase in the supply flow rate will increase the attraction of the negative pressure, leading to the occurrence of the maximum force at a smaller spacing. Therefore, hmax corresponding to

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0.4

0.8 0.6

0.3

F/Fmax [-]

hmax [mm]

0.4

0.2

0.1

0.2

0.7

0

0.65

-0.2

Q = 3.0 L/min (ANR) Q = 4.0 L/min (ANR) Q = 5.0 L/min (ANR) Q = 6.0 L/min (ANR) Q = 7.0 L/min (ANR) Q = 8.0 L/min (ANR) Q = 9.0 L/min (ANR) Q = 10.0 L/min (ANR) Q = 11.0 L/min (ANR) Q = 12.0 L/min (ANR)

0.6

-0.4 0.55

-0.6

0

3

6

9

0.5 0.4

-0.8

12

0.6

0.8

1

1.2

-1

Q [L/min (ANR)]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

h [mm]

Fig. 10. Optimum spacing of vortex gripper against supply flow rate.

Fig. 12. F–h curves at different supply flow rates after nondimensionalization.

the maximum force decreases as the supply flow rate gradually increases. 1

hB

4.2. Suspension region

hC

During actual use of the vortex gripper, its maximum force will be determined by the supply flow rate. In order to ensure the safety of handling process, the weight of the workpiece should be smaller than the maximum force offered by the gripper, i.e., mg = gFmax, where g is the gripping coefficient and usually 50% < g < 70%. Then, the suspension region is determined by the intersection of the line corresponding to gFmax and the F–h curve, i.e., the point C. For the F–h curve at a certain supply flow rate, if we replace F by F/Fmax as shown in Fig. 11, then the suspension region is determined by point C, which is the intersection of the line corresponding to g and the dimensionless F–h curve. Therefore, we can transform the F–h curves at different supply flow rates, shown in Fig. 8, into curves with the same order of magnitude by a similar dimensionless processing of the vertical axis, which will facilitate our discussion of the influence of the supply flow rate on the suspension region. The processed curves are plotted in Fig. 12. We can see from Fig. 12 that the dimensionless curves have the same variation tendency, although the supply flow rate is not the same. In addition, the curves show a slight translation to the left as the flow rate increases. For a given g, e.g., g = 60% marked by the red broken line in Fig. 12, we can obtain the spacing of the stable suspension position and the boundary suspension position from the intersections of the broken line and the F–h curves (hB and hC, respectively). The results are plotted in Fig. 13, in which we can observe that both hB and hC decrease gradually with the increase in supply flow rate. In addition, the decrease is faster at lower flow rates (3.0–6.0 L/min), whereas both hB and hC tend to be constant at flow rates larger than 6.0 L/min. This implies that the relationship between the dimensionless force and the spacing

0.6 0.4 0.2 0 6

3

9

A

Fmax B

Fig. 13. hB and hC at different supply flow rates after nondimensionalization.

is an inherent characteristic of the vortex gripper and is hardly dependent on the supply flow rate, especially at larger flow rates. This inference greatly facilitates the description to the characteristics of the vortex gripper and the prediction of the F–h curves at different supply flow rates. Moreover, it could simplify the production and selection of products for both producers and users. 4.3. Suspension stiffness 4.3.1. Calculation of suspension stiffness When the workpiece is picked up by the gripper, it will be suspended at point B. Thus, the suspension stiffness at point B, kB, is very important for the suspension stability of the workpiece. According to the definition of stiffness, kB is equal to dF when dh h = hB, i.e., the slope of the F–h curve at point B. However, the vor-

hB

hmax

Suspension region

A

1 C

B

hC

h

12

Q [L/min (ANR)]

F/Fmax

F

Fmax

h [mm]

0.8

hB

C

hmax

Suspension region

Fig. 11. Nondimensionalization of suction force.

hC

h

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J. Zhao, X. Li / Experimental Thermal and Fluid Science 79 (2016) 91–100

tex flow in the vortex chamber is a very complicated threedimensional flow, and we have been unable to formulate a better theoretical description for it. As a result, we cannot calculate the F–h curve accurately or obtain the slope of the curve theoretically. Therefore, in order to analyze the characteristics of the suspension stiffness in more detail, we proposed a simplified calculation method as follows. According to the F–h curve shown in Fig. 2, the spacing of the stable suspension position, hB, is in the region h < hmax. We also know that in the region h < hmax, the shape of the pressure distribution inside the vortex chamber remains the same despite the change in h, whereas the pressure distribution between the annular skirt and the workpiece changes significantly as h varies. Therefore, as shown in Fig. 14, the pressure distribution p(r) is the superposition of a part that is not affected by h (denoted by p1(r)) and another part that is affected by h (denoted by p2(r)). We can observe that the curves of p1(r) coincide completely when h is set as 0.15 mm, 0.18 mm, and 0.20 mm, whereas the p2(r) curves show significant variation. As the suction force F applied by the gripper on the workpiece is equal to the areal integral of the pressure, we have

Z

R2

F¼

pðrÞ  2prdr

0

Z

R2

¼

p1 ðrÞ  2prdr 

Z

0

R2

p2 ðrÞ  2prdr

ð2Þ

0

Differentiating with respect to h, we obtain

hR i hR i R R d 0 2 p1 ðrÞ  2prdr d 0 2 p2 ðrÞ  2prdr dF  ¼ dh dh dh

ð3Þ

Because p1(r) is independent of h, we can simplify Eq. (3) as follows:

dF ¼ dh

hR i R d 0 2 p2 ðrÞ  2prdr

Integrating Eq. (7) with respect to h with the boundary condition of p = 0 when r = R2, we can obtain a subsection function for p2(r) as follows:

p2 ðrÞ ¼

: 6lQ3 ðln R2  ln rÞ ph

ð8Þ

ðR1 6 r 6 R2 Þ

By substituting Eq. (8) in Eq. (4), we can obtain:

dF 9lQ ðR22  R21 Þ ¼ 4 dh h

ð9Þ

Fig. 15 shows the comparison of the experimental and computed results when Q is set as 5.0, 8.0, and 11.0 L/min (ANR). The experimental results were obtained by difference calculations of the experimental data shown in Fig. 8, whereas the computed results, shown by black broken lines, were obtained directly from Eq. (9). Because dF/dh is proportional to Q in Eq. (9), the theoretical curves of dF/dh show an upward movement as the supply flow rate increases, similar to the experimental results. This implies that the simplified calculation method adequately represents the variation tendency of dF/dh with respect to h and Q. Thus, we can easily obtain the suspension stiffness kB by substituting the spacing at the stable suspension position, hB, in Eq. (9). Furthermore, we discuss the suspension stability of the workpiece, which is dominated by the inherent frequency of the system consisting of the gripper and the workpiece. The inherent frequency is related to kB/m, and the following derivation demonstrates that kB/m can be represented by the slope of the F–h curve after nondimensionalization at the stable suspension position, i.e., 0 0 kB . As shown in Fig. 16, kB is defined as 0

kB ¼

 dðF=F max Þ  dh h¼hB

ð10Þ

ð4Þ

dh

By this simplification, we can avoid the discussion of p1(r), which is dominated by a complicated airflow, and only concentrate on p2(r), which is dominated by the gap flow. For p2(r), we can use the simplified N–S equation to describe the viscous flow in the region R1 6 r 6 R2 as follows:

400

dF/dh [N/mm]

@p @ 2 ur ¼l 2 @r @z

ð5Þ

We know that the average of ur (i.e.,ur ) can be got as follows:

Q ur ¼ 2prh

Q = 5 L/min(ANR) Q = 8 L/min(ANR) Q = 11 L/min(ANR) Equation (9)

300

200

100

ð6Þ 0

From Eqs. (5) and (6), we can obtain:

ðR1 6 r 6 R2 Þ

0.14

0.16

0.18

0.2

0.22

0.24

h [mm]

ð7Þ

Fig. 15. Slopes of F–h curves at different supply flow rates.

p [Pa (G)]

@p 6lQ ¼ @r prh3

8 < 6lQ3 ðln R2  ln R1 Þ ð0 6 r 6 R1 Þ ph

600

600

600

400

400

400

200

200

200

=

0 -200

-200

-400

h = 0.15 mm h = 0.18 mm h = 0.20 mm

-600 R2(25)

+

0

R1(10)

0

r [mm] p(r)

R1(10)

R2(25)

-400

h = 0.15 mm h = 0.18 mm h = 0.20 mm

-600 R2(25)

R1(10)

0

R1(10)

R2(25)

0 -200 -400

h = 0.15 mm h = 0.18 mm h = 0.20 mm

-600 R2(25)

r [mm] p1(r) Fig. 14. Decomposition of pressure distribution at Q = 5.0 L/min (ANR).

R1(10)

0

r [mm] p2(r)

R1(10)

R2(25)

J. Zhao, X. Li / Experimental Thermal and Fluid Science 79 (2016) 91–100

F/Fmax 1

for different spacing h. Based on the experimental data, we obtained the following conclusion:

Suspension region

A kB’

C

B

hB

hC

hmax

h 0

Fig. 16. Definition of kB .

Table 3 0 kB (unit: mm1) at different supply flow rates (unit: L/min (ANR)). Q 0 kB

3 8.52

4 8.39

5 8.33

6 8.33

7 8.48

8 8.49

9 8.38

10 8.55

11 8.41

12 8.39

To ensure the safety of the handling process, the weight of the workpiece that is picked up by the gripper should be smaller than the maximum force, i.e., mg = gFmax, in which the gripping coefficient is smaller than one. Then, we can transform Eq. (10) to the following form: 0

kB ¼



g dF  

mg dh

¼ h¼hB

99

g kB

ð11Þ

g m 0

Eq. (11) indicates that kB is only related to kB and m. In addition, we can observe from Fig. 12 that the curves show a slight translation to the left as the flow rate increases. For g equal to 0.6, we 0 obtained the values of kB at different supply flow rates, as shown in Table 3. The data in the table implies that the maximum error 0 in the values of kB as the flow rate varies is less than 3%. Therefore, 0 we can infer kB is almost constant with respect to the supply flow 0 rate. In other words, kB is an inherent characteristic of the vortex gripper that is independent of the supply flow rate. Similar to the spring–mass system, the gripper and the workpiece, which has a mass of m, form a second-order system for a given physical dimen0 sion of the gripper. As a result, kB represents the inherent frequency of the system, which is independent of the change in supply flow rate.

(1) The suction force F depends on the spacing between the workpiece and the gripper, h, and the F–h curve at a given supply flow rate is convex, i.e., there is an optimum spacing hmax at which the suction force is maximum Fmax. Furthermore, according to experimental results, as the supply flow rate increases, Fmax has an approximately quadratic relationship to Q, whereas hmax has a slight decreasing tendency. (2) We nondimensionalized the F–h curves by replacing F with F/Fmax. It was found that the dimensionless F–h curves of different supply flow rates have the same variation tendency, which implies that the dimensionless F–h curve is an inherent characteristic of the gripper. We also deduced that the increase in flow rate causes the dimensionless F–h curves to have a slight translation to the left, thus decreasing the suspension region slightly. However, such variation is only noticeable when the flow rate is very small; all the curves almost overlap beyond a certain value of the flow rate. (3) We decomposed the pressure distribution into two parts: the vortex flow part, which is not affected by changes in h, and the gap flow part, which is very sensitive to changes in h. Based on this decomposition, we proposed a simplified calculation method for the suspension stiffness, and the stiffness calculated by this method was found to be very close to the experimental value. In addition, we analyzed the physical significance of the slope of the dimensionless 0 F–h curve at the stable suspension position, i.e., kB . The 0 expression for kB contains the ratio of the stiffness to the mass of the workpiece, i.e., the intrinsic frequency. There0 fore, kB can reflect the stability of the system formed by the gripper and the workpiece. However, for practical use of the vortex gripper in automatic production lines, there are still multiple design parameters of the gripper that need to be set and analyzed besides the supply flow rate, such as the height of the vortex gripper, number of nozzles, and diameters of the chamber and annular skirt. These design parameters will also influence the performance of the vortex gripper, and further experiments and researches are necessary to investigate their effects.

Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 51375441) and the Science Fund for Creative Research Groups of National Natural Science Foundation of China (No. 51221004).

5. Conclusion and future work The vortex gripper is a new kind of non-contact gripper, which generates a suction force as a result of a high-speed vortex flow. It can effectively avoid the disadvantages of traditional gripping device, such as mechanical scratches, local stress concentration, frictional static electricity, and blots on the workpiece. Thus, it is used on the production lines of precision workpieces such as semiconductor wafers. In this study, we experimentally and theoretically investigated the effect of the supply flow rate, a very important operation parameter of the vortex gripper, on the performance of the vortex gripper. First, we proposed three performance indexes for evaluating the properties of the vortex gripper: maximum force, suspension region, and suspension stiffness. Then, we obtained a series of F–h curves at different supply flow rates and the pressure distribution at the surface of workpiece

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