Contactless transportation systems based on near-field acoustic levitation have the benefit of compact design and easy control which are able to meet the cleanliness and precision demands required in precision manufacturing. However, the problems involved in contactless positioning and transporting cylindrical objects have not yet been addressed. This paper introduces a contactless transportation system for cylindrical objects based on grooved radiators. A groove on the concave surface of the radiator produces an asymmetrical pressure distribution which results in a thrusting force to drive the levitator horizontal movement. The pressure distribution between the levitator and the radiator is acquired by solving the Reynolds equation. The levitation and the thrusting forces are obtained by integrating the pressure and the pressure gradient over the concave surface, respectively. The predicted results of the levitation force agree well with experimental observations from the literature. Parameter studies show that the thrusting force increases and converges to a stable value as the groove depth increases. An optimal value for the groove arc length is found to maximize the thrusting force, and the thrusting force increases as the groove width, the radiator vibration amplitude, and the levitator weight increase.

## I. INTRODUCTION

Handling and transporting micro-electromechanical systems components or semiconductors is a challenging task due to their fragility and surface-sensitive characteristics.^{1} Therefore, contactless manipulation is necessary in many situations.^{2} Near-field acoustic levitation (NFAL) is a contactless technology that has attracted widespread attention in recent years. Compared to other contactless technologies, such as air cushion^{3} or magnetic levitation,^{4} the NFAL has two distinct advantages: (1) compact operation system and (2) no material restrictions for the levitator.^{5} In 1975, when Whymark^{6} investigated the ultrasonic levitation trapping small objects at the nodal positions of the standing wave field, he also found that a planar disk made of brass hovers above a piston operating at a vibration frequency of 20 kHz. This levitation phenomenon is referred to as NFAL. Hereafter, many researchers studied the mechanism and application of the NFAL. Hashimoto *et al.*^{7} presented a numerical model in which the radiation surface vibrates harmonically in the in-phase mode. Based on this model, they derived a relationship between the levitation distance, the vibration amplitude, and the weight per unit area. Later, Hu *et al.*^{8} pointed out that the sound field between the radiator and the levitator produces two forces which act on the bottom of the levitator. The first one is the acoustic radiation force that levitates the object. The second one is the acoustic viscous force that transports or rotates the object. These two forces play different roles in various applications. The acoustic radiation force is the main concerning force in contactless positioning or squeeze film bearing.^{9,10} In these applications, the acoustic viscous force is unwanted in order to guarantee stability of the levitator. However, in contactless transportation^{11} or ultrasonic motor^{12} applications, both forces are important.

The NFAL is mainly categorized based on the property of the vibration wave. Two major types of the NFAL are (1) the traveling wave type levitation and (2) the standing wave type levitation. Correspondingly, two methods are generally used to generate an acoustic viscous force in the NFAL.^{13} The first method uses a traveling wave sound field in the fluid between the vibrator and the levitator. This corresponds to the traveling wave type acoustic levitation. For example, Hashimoto *et al.*^{14} used two longitudinal vibration systems to excite the traveling wave along a beam. A 7.6 g duralumin plate was able to be transported along this beam. Based on this transportation system, other investigators studied its transportation characteristics^{15,16} or expanded its application field, e.g., for different cross-sections of the levitator.^{17} In order to acquire a better transportation performance, Li *et al.*^{18} introduced several engraved grooves on the surface of the levitator. An obvious characteristic of these traveling wave type-based transportation systems^{14–18} is that they require a guide rail to produce vibration. Therefore, Koyama *et al.*^{19} proposed a self-running sliding stage that produces a traveling wave vibration by itself. This stage not only produces self-levitation but also generates a thrusting force that produces forward motion.

The second method uses the asymmetric standing wave field to produce a gradient in the sound pressure field in the fluid domain. This corresponds to the standing wave type acoustic levitation. For instance, Hu *et al.*^{20} used a wedge-shaped stator to generate the standing wave vibration. The levitator can be driven to move in the horizontal direction due to the gradient in the stator vibration amplitude along the stator's length. Based on the structure from this study, Hu *et al.*^{21} developed a plate-shaped stator with triangular grooves to build a transportation system for long distance contactless transportation. These transportation systems^{20,21} with a single vibration source cannot ensure the consistent motion velocity of the levitator because the vibration distribution is not uniform. Wei *et al.*^{22} employed two longitudinal vibration sources to build a transportation system based on the standing wave type. These two vibration sources generate two non-parallel squeeze films with a pressure gradient between them. Similar to Ref. 19, Chen *et al.*^{23} reported a self-running and self-floating actuator based on the standing wave type. This actuator uses its coupled resonant vibration mode to produce an asymmetric standing wave field. Aono *et al.*^{24} proposed inserting the levitator between two opposing vibration sources to increase the levitator's holding force. This method can be used to acquire a high thrusting force for standing wave type transportation systems. Yet, this method decreases the load-carrying capacity of the transportation system since the two levitation forces oppose each other. Thus, this method is only suitable for light objects. However, the abovementioned transportation systems^{14–23} can only carry levitators with planar surfaces. Since some precision components, e.g., in microsystems, exhibit cylindrical surfaces, the problem of contactless transport and positioning remains.

Previous investigation^{25} regarding the NFAL stability problem revealed that an engraved groove on the levitator can break the symmetry of the sound field, which produces a restoring force. Therefore, this study deals with introducing a groove on the concave surface of the radiator and its stabilizing effect on the problem of contactless transport of cylindrical components. This modified radiator generates the levitation force and the thrusting force simultaneously. A transportation system composed of multiples of these modified radiators can levitate and move a cylindrical object. The pressure distribution between the radiator and the levitator is described by the non-linear Reynolds equation, which considers the horizontal movement of the levitator. The eight-point discrete method is used to solve the discontinuous film thickness problem, which arises due to the groove. The levitation force and the thrusting force are calculated by integrating the pressure and the pressure gradient, respectively. Numerical predictions of the levitation force agree well with experimental results from the literature.^{26} Finally, the effects of the groove parameters, the radiator vibration amplitude, and the levitator weight on the thrusting force are studied.

This paper is organized as follows. Section II describes the working principle and the transportation system design. Section III presents the governing equations and the solution process for the pressure distribution. Sections IV and V discuss the pressure distribution of the squeeze film and the parameter study, respectively.

## II. WORKING PRINCIPLE

In the NFAL, increasing the vibration amplitude enhances the load-carrying capacity of the levitation system.^{27} The Langevin transducer is suitable for heavy loads since it produces high intensity vibration.^{28} A typical Langevin transducer^{26} mainly consists of five parts: the bolt, the back mass, the four piezoelectric transducer (PZT) plates, the horn, and the radiator with the concave surface, as shown in Fig. 1(a). Owing to the piezoelectric effect, the four PZT plates will generate the same frequency vibration along the through-thickness direction when a sinusoidal voltage with frequency *f* is applied. The horn transmits and amplifies the vibration produced by the PZT plates to the radiator. Since the radiator with the concave surface vibrates continuously, the air film between the concave surface and the outer surface of the cylindrical object is squeezed. Thus, the levitation force is generated by the squeeze film.

In the case of a smooth concave surface, there is a symmetric pressure distribution in the axial direction.^{29} Therefore, no axial transport velocity is generated and the levitator's hovering position remains unchanged. Symmetry of the pressure distribution is broken by introducing a groove on the concave surface,^{25} as shown in Fig. 1(b). The groove's geometric parameters are prescribed by its width $lg$, its depth $hg$, and its arc length $\theta g$. Figure 1(c) is presented to clearly depict the location of the groove within the radiator. According to the gas-film lubrication theory,^{30} the established pressure gradient in the axial direction produces thrusting force $Ft$. Furthermore, levitation force $Fl$ caused by the squeeze effect counteracts the weight of levitator $W$. Therefore, this method not only realizes the levitator suspension, but also drives its horizontal movement without any additional equipment.

Using multiple Langevin transducers creates a contactless transportation system for cylindrical objects, as shown in Fig. 2. This system can meet the demands for long-distance transportation. The levitator's transportation trajectory can be controlled by distributing transducers, by adjusting the radiator vibration amplitude, and by the orientation of the grooved surface. Changing the position of the groove can, for example, be used to slow down or stop the levitator at the end of the transportation system, cf. Fig. 2.

## III. GOVERNING EQUATIONS AND NUMERICAL SOLUTION

### A. Reynolds equation

The analytic model for one radiator is shown in Fig. 3. The center point, inner radius, and width of the radiator are denoted as $O$, $R$, and $L$, respectively. The center point and outer radius of the levitator are defined as $Ol$ and $r$, respectively. Since the air film between the levitator and the radiator is symmetric in the circumferential direction and no other force exists to drive a rotation of the levitator, the levitator rotation is not considered. Further, the aerodynamic effect is neglected and therefore, only the squeeze effect exists in the air film. Hence, both center points are located on the *x* axis. The eccentricity displacement $e$ is defined as the distance between $O$ and $Ol$. The difference between $R$ and $r$ is denoted as the nominal clearance $c$. Since the levitator is simultaneously supported by several radiators (cf. Fig. 2), the levitator's inclination in the *xOz*-plane is neglected. In addition, the levitator is assumed to be a rigid body and well balanced.

The Reynolds equation is widely used to describe the pressure distribution of the squeeze film in the NFAL.^{9,31} Some assumptions are introduced here to deduce the Reynolds equations. First, air is assumed to behave in a compressible manner and is treated as Newtonian fluid.^{32} Second, since the film thickness is much smaller than the radiator's radius, the squeeze film is assumed to function an isothermal film^{33} and the pressure gradient in film thickness direction is neglected.^{15} Third, since the transportation speed is usually low, around several centimeters per second,^{20,34} the Reynolds number is lower than the upper limit value for laminar flow.^{33} Therefore, the airflow in the squeeze film is treated as laminar flow. Finally, the magnitude of fluid inertia and body forces are smaller than the viscous force^{32} and they are neglected. The axial displacement of the levitator relative to its initial position is denoted as $u$. Based on these assumptions, the expression of the dimensionless Reynolds equation considering the movement of the levitator in the *z*-direction is^{25,29}

in which $P=p/pa$ and $H=h/c$ are the dimensionless air film pressure and air film thickness, respectively. $T=\omega \u22c5t=2\pi f\u22c5t$ is the dimensionless form of time *t* and the dimensionless width position is $Z=z/R$. $pa$ denotes the pressure of the ambient air, $h$ is the thickness of the air film, and the squeeze number is $\sigma =12\mu a\omega R2/pac2$. The coefficients $\Lambda =6\mu au\u0307R/pac2$ and $\alpha =\rho au\xa8R/2pa$ represent the effects of the levitator movement speed $u\u0307$ and its acceleration $u\xa8$ on the pressure. $\mu a$ and $\rho a$ are the viscosity coefficient and the density of air.

### B. Boundary conditions and film thickness

As shown in Fig. 3, the arc length of the concave surface of the radiator is defined as $\theta a$. Due to the continuity of the pressure at the interface to the ambient air, the pressure on the boundary of the squeeze film meets the following conditions:

Since the radial displacement of the radiator can be treated as uniform,^{26} the normalized mode shape $V(\theta ,z)$ equals 1. It is assumed that the existence of the groove does not affect the radiator's vibration amplitude distribution in order to conveniently conduct the parameter study. This assumption is reasonable if the dimension of the groove is much smaller than that of the radiator. Referring to Figs. 1(b) and 3, the grooved domain includes the front ($z=L/2$), the rear ($z=L/2\u2212lg$), the right ($\theta =\pi /2+\theta g/2$), and the left ($\theta =\pi /2\u2212\theta g/2$) boundaries. Thus, the squeeze film thickness is expressed as

where $\xi $ is the maximum value of the radial vibration displacement.

### C. Solution process

The computational domain is the area of the radiator's concave surface, as shown on the left in Fig. 4. This domain is divided into a grid of $m\xd7n$ sections, in which *m* defines the discretization in *θ*- and *n* in *z*-directions, i.e., $\Delta \theta =\theta a/m$ and $\Delta z=L/n$. The film thickness becomes discontinuous at the edges of the groove domain. Therefore, the partial differentials $(\u2202H/\u2202\theta )i,j$ and $(\u2202H/\u2202Z)i,j$ in Eq. (1) are unsolvable.

In the field of gas lubrication, the eight-point discrete method is adopted to solve this discontinuous film thickness problem.^{35,36} All nodes on the four boundary lines (i.e., $z=\xb1L/2$ and $\theta =\pi /2\xb1\theta a/2$) meet the boundary conditions in Eqs. (2) and (3). The inside node $Gi,j$ is surrounded by the area $\Omega ij$, which is spanned by the four red dashed lines $\Gamma ij,1\u22124$, as seen on the right in Fig. 4. The adjacent eight points $Nij,1\u22128$ in the eight-point discrete grid are evenly distributed on the four perimeter lines $\Gamma ij,1\u22124$.

By integrating Eq. (1) over $\Omega ij$, the dimensionless rate of airflow passing through $\Omega ij$ can be expressed as

The surface integral on the left side and the second term on the right side of Eq. (5) can be transformed to a closed line integral by using Green's theorem,

The path of line integration runs counterclockwise, which means from $\Gamma ij,1$ to $\Gamma ij,4$. Since the projections of $\Gamma ij,2$ and $\Gamma ij,4$ on the *θ*-axis equal zero, the first term on the left side of Eq. (6) reduces to

The film thickness of each perimeter line is approximately equal to the arithmetic average film thickness of the two corresponding points which lie on the perimeter line^{35} such as $H3(\Gamma ij,1)=(Hi\u22121/4,j+1/23+Hi+1/4,j+1/23)/2$. Subsequently, using the center difference method and the trapezoidal rule, the right side of Eq. (7) can be expressed as

All terms in Eq. (6) can be expanded in the same form as Eq. (8). Thus, the nonlinear differential equation that describes the relation between the pressure and the film thickness over time is stated as

The Newton–Raphson method is used to linearize Eq. (9),^{37} which acquires the pressure distribution in the computational domain. The mean levitation force $F\xafl$ and the mean thrusting force $F\xaft$ in one vibration period follow as^{11}

Under the steady conditions, the mean levitation force counterbalances the weight of the levitator, i.e., $F\xafl=W$. The mean thrusting force is mainly influenced by four variables: film thickness $h$, pressure gradient in *z*-direction $\u2202p/\u2202z$, levitator movement speed $u\u0307$, and its acceleration $u\xa8$.

## IV. PRESSURE DISTRIBUTION AND VALIDATION

A basic design of this radiator is used to quantify the effect of the groove on the pressure distribution. The parameters for the basic design and its operating condition are listed in Table I. In addition to this basic setup, transportation velocity $u\u0307$ and acceleration $u\xa8$ are assumed to be zero to conveniently analyze the effect of the groove parameters on the thrusting force by using control variates. The influence of the mesh size on the numerical results has been checked in previous studies,^{25,29} which is not displayed here for the sake of brevity. A mesh resolution of $m\xd7n=60\xd740$ has been identified for sufficient results with respect to calculation time and convergence. All parameters remain as listed in Table I, if not specified elsewhere throughout the following studies.

Parameter . | Value . | Parameter . | Value . |
---|---|---|---|

Vibration amplitude $\xi $ | 9 μm | Vibration frequency f | 20 kHz |

Radiator concave radius R | 10 mm | Levitator outer radius r | 9.96 mm |

Nominal clearance c | 40 μm | Radiator width L | 20 mm |

Radiator arc length $\theta a$ | 120° | Groove width $lg$ | 3 mm |

Groove depth $hg$ | 1 mm | Groove arc length $\theta g$ | 80° |

Air dynamic viscosity $\mu a$ | 1.81 × 10^{−5 }Pa s | Air pressure $pa$ | 1.013 × 10^{5} Pa |

Air density $\rho a$ | 1.204 kg/m^{3} | Weight of the levitator $W$ | 0.5 N |

Parameter . | Value . | Parameter . | Value . |
---|---|---|---|

Vibration amplitude $\xi $ | 9 μm | Vibration frequency f | 20 kHz |

Radiator concave radius R | 10 mm | Levitator outer radius r | 9.96 mm |

Nominal clearance c | 40 μm | Radiator width L | 20 mm |

Radiator arc length $\theta a$ | 120° | Groove width $lg$ | 3 mm |

Groove depth $hg$ | 1 mm | Groove arc length $\theta g$ | 80° |

Air dynamic viscosity $\mu a$ | 1.81 × 10^{−5 }Pa s | Air pressure $pa$ | 1.013 × 10^{5} Pa |

Air density $\rho a$ | 1.204 kg/m^{3} | Weight of the levitator $W$ | 0.5 N |

Since the radiator vibrates continuously, the film thickness and pressure vary over time. When the squeeze system reaches a steady condition, the transient film thickness and pressure are periodic functions in time with vibration frequency *f*. The dimensionless pressure distributions of the radiator's basic configurations with a smooth and a grooved surfaces at the onset time of one steady period are shown in Fig. 5(a). The corresponding distributions of the dimensionless pressure gradient are displayed in Fig. 5(b).

For the radiator configuration with a smooth surface, the pressure distribution is symmetric with respect to the *xOz*- and the *xOy*-planes, see top of Fig. 5(a). Therefore, only levitation force $Fl$ is produced and no thrusting force acts on the levitator. For the grooved configuration, corresponding to the bottom of Fig. 5(a), the pressure value in the groove domain equals the ambient pressure, i.e., $P=1$. This results from the contact between groove domain and ambient air such that the squeeze effect is not evidenced in the groove domain. Hence, symmetry of the pressure distribution about the *xOy*-plane is broken due to the existence of the groove. The asymmetric pressure distribution in the *z*-direction produces thrusting force $Ft$, and the available domain with squeeze effect is smaller than for the smooth surface configuration. Thus, given the same weight of the levitator, the dimensionless maximum pressure value in the grooved condition ($Pmax=1.1443$) is higher than that in the smooth condition ($Pmax=1.1333$).

Figure 5(b) shows that the pressure gradient value is antimetric in the *z*-direction for the smooth surface configuration. Since the positive part and the negative part are of equal magnitude [$\Sigma (\u2202P/\u2202Z)=0$], the thrusting force equals zero, which corresponds to the results presented in Fig. 5(a). However, the balance between the positive part and the negative part is changed by the groove. The pressure gradient value in the groove domain equals zero. The negative part dominates in this case [$\Sigma (\u2202P/\u2202Z)=\u22120.3813$], which means that the thrusting force points toward the negative *z*-direction.

The variation in thrusting force $Ft$ in one steady period is shown in Fig. 6. The value of the thrusting force alternates between negative and positive. The maximum negative thrusting force absolute value is 1.6355 mN, which is larger than the positive absolute value 1.3341 mN. Therefore, a mean thrusting force of $F\xaft$ $F\xaft$ = –0.1182 mN is generated, since negative contributions for the thrusting force dominate.

The measurement method of the levitation force stated in Ref. 26 is represented by the simplified diagram shown in Fig. 7(a). When the levitation system reaches a steady condition, the levitation force generated by the squeeze film is counteracting the external load which contains the weight of the levitator and two objects. Therefore, the value of the levitation force is adjusted by changing the weight of the two objects. At the same time, the levitation height is measured by the displacement sensor. Consequently, experimental data which describes the relationship between the levitation height and the levitation force is obtained. In order to validate the proposed calculation method, numerical calculation results are compared to with these experimental data from Ref. 26. Some calculation parameters are adjusted compared to the basic configuration. The adjusted parameters are listed in Table II. All other parameters remain as prescribed for the basic configuration.

Parameter . | Value . | Parameter . | Value . |
---|---|---|---|

Vibration amplitude $\xi $ | 10.8 μm | Vibration frequency f | 16.11 kHz |

Radiator concave radius R | 10.015 mm | Levitator outer radius r | 9.985 mm |

Nominal clearance c | 30 μm | Radiator arc length $\theta a$ | 110° |

Total mesh grid number $m\xd7n$ | $55\xd740$ | Radiator width L | 20 mm |

Parameter . | Value . | Parameter . | Value . |
---|---|---|---|

Vibration amplitude $\xi $ | 10.8 μm | Vibration frequency f | 16.11 kHz |

Radiator concave radius R | 10.015 mm | Levitator outer radius r | 9.985 mm |

Nominal clearance c | 30 μm | Radiator arc length $\theta a$ | 110° |

Total mesh grid number $m\xd7n$ | $55\xd740$ | Radiator width L | 20 mm |

Figure 7(b) shows the comparison between numerical calculation and experimental results in terms of the levitation force. The levitation force decreases with increasing levitation height $c\u2212e$. This is due to the weakening of the squeeze effect for higher levitation heights.^{7} The calculated results match well with the experimental data.

## V. PARAMETRIC ANALYSIS

For a given radiator, the thrusting force is mainly determined by four variables as mentioned in Sec. III C. The first two variables, i.e., the film thickness and the pressure gradient, are mainly affected by the groove parameters, the radiator vibration amplitude, and the levitator weight. The other two variables, i.e., levitator movement velocity and its acceleration, are assumed to be zero as mentioned in Sec. IV. In the following parametric study, only the absolute values of the forces are presented, since the directions of the thrusting forces are the same.

### A. Groove depth

According to Li *et al.*,^{38} the groove depth has an obvious influence on the squeeze effect. In order to determine the effect of the groove depth on the thrusting force, a configuration with three different radiator vibration amplitudes ($\xi $=8, 9, and 10 *μ*m) is analyzed for varying groove depths. Figure 8 displays the variation of the thrusting force with the groove depth for all three vibration amplitude configurations. The thrusting force obviously increases when the groove depth is lower than 0.9 mm. This phenomenon occurs because the magnitude of the groove depth approximates the magnitude of the nominal clearance in low groove depth. Thus, the squeeze effect still exists in the groove domain, which in turn leads to a deviation in the pressure in the groove domain from the ambient pressure. The squeeze effect weakens with increasing groove depth. Finally, the thrusting force converges to a constant value when the groove depth exceeds 0.9 mm. Therefore, the groove depth is set to 1 mm in this study to acquire a high thrusting force. This trend is visible for different amplitudes. The relationship between the thrusting force and the amplitude will be discussed in Sec. V D.

### B. Groove arc length

Figure 9 presents the relation between thrusting force and groove arc length for three different vibration amplitudes, namely, $\xi =$ 8, 9, and 10 *μ*m. The thrusting force initially increases and then decreases as the groove arc length increases. For a groove arc length of $\theta g=0\xb0$, the concave surface is a smooth surface. In this case, the thrusting force equals zero as discussed in Sec. IV. For a groove arc length of $\theta g=120\xb0$, the concave surface becomes a new smooth surface with the width $L\u2212lg$. This new surface is also symmetric in the *z*-direction. Thus, the thrusting force is also zero. As the groove arc length increases, the available domain with the squeeze effect decreases. At a constant levitation force, the maximum pressure value and the pressure gradient increase while the film thickness decreases. According to Eq. (10), the thrusting force is simultaneously affected by the pressure gradient and the film thickness. Therefore, there is an optimum value for the groove arc length. Since the high-pressure air is located in the area near the middle line $\theta =90\xb0$ in the *θ*-direction as shown in Fig. 5(a), the influence of the groove on the pressure gradient and the film thickness attenuates as the groove arc length increases. Thus, the variation of the thrusting force is steeper for small groove arc lengths ($\theta g<40\xb0$) than for large groove arc lengths ($\theta g>40\xb0$). The optimal groove arc length falls on the low-value side and is nearly 40° for the given parameters and operating condition.

### C. Groove width

$W$ $\theta g$ $hg$ In this subsection, the groove arc length is set to $\theta g=$ 40°. Figure 10 shows the relationship between thrusting force and groove width for three different vibration amplitudes ($\xi =$ 8, 9, and 10 *μ*m). The thrusting force increases with increasing groove width. This result agrees with the numerical and experimental results presented by Liu *et al.*,^{25} who used an engraved groove levitator to replace the levitator's misalignment. However, the available domain with the squeeze effect decreases if the groove width increases. Thus, the levitator cannot be suspended when the groove width is too great. Consequently, the study of the thrusting force is only meaningful within a small range of the groove width.

### D. Vibration amplitude and levitator weight

The change in vibration amplitude is assumed to stem from changes in the driving voltage,^{39} while the vibration frequency remains unchanged in order to keep the same vibration mode shape of the radiator. In this subsection, the groove arc length is set to $\theta g=$ 40°. Figure 11 depicts the thrusting force as a function of the vibration amplitude at four different weights of the levitator, namely, *W = *0.4, 0.5, 0.6, and 0.7 N. The thrusting force increases as the weight increases, which is consistent with the results of Refs. 13 and 25. A higher levitator weight indicates a smaller levitation height at a constant vibration amplitude. According to Li *et al.*,^{16} the thrusting force increases under these conditions. Additionally, the thrusting force follows a positive linear trend with increasing vibration amplitude. This is due to the fact that the squeeze effect is enhanced by increasing vibration amplitude.^{25,40}

## VI. CONCLUSIONS

This paper introduced the design of a contactless transportation system for cylindrical objects. The levitation and transport are based on near-field acoustic levitation. The thrusting force for the transported items is generated by the radiator that has an engraved groove. Theoretical analysis of the influence of the groove on the pressure distribution is also presented. The nonlinear Reynolds equation which describes the relationship between the pressure and the motion of the levitator has been solved by combining the eight-point discrete method and the Newton–Raphson method. The levitation force and the thrusting force are calculated from the solved pressure distribution. The calculated levitation force agrees well with similar results in the literature.^{26} Moreover, the effects of groove depth, groove arc length, and groove width on the thrusting force have been investigated to determine suitable groove parameters. The predicted results yield the following conclusions:

The thrusting force initially increases and then remains at a stable value as the groove depth increases.

There is an optimal groove arc length to maximize the thrusting force.

Increasing the groove width can improve the transportation performance within a small range of groove width.

Increasing the vibration amplitude of the radiator can increase the thrusting force.

A higher levitator weight can lead to higher thrusting forces.

Future work addresses the manufacture of a prototype for this transportation system. Furthermore, precisely controlling the motion trajectory of the levitator by adding a control algorithm is planned. This paper and our continuing research supply a helpful guide for designing a practical contactless transportation system for cylindrical objects in industrial applications.

## ACKNOWLEDGMENT

The first author acknowledges the support provided by the China Scholarship Council (CSC) (Grant No. 201808340068).