# Bidirectional Linear Motion by Travelling Waves on Legged Piezoelectric Microfabricated Plates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}-actuated stator and a 4 × 4 mm

^{2}slider. Due to the low amplitude of the generated waves, preloads and voltages as high as 30 N and 80 V, respectively, were required, though speeds of more than 1 m/s, with minimum steps of 25 nm, were obtained.

^{2}robot, based on an array of electrothermally actuated and manually attached silicon legs, was able to move a 3 mg slider at a speed of 8 µm/s with a maximum payload of 40 mg in just one direction. Ebefors et al.’s work [24] relied on electrothermal actuation to develop a hybrid silicon–polyimide walker, capable of reaching speeds of 6 mm/s at 18 V. In a more recent work, a 9 × 9 mm

^{2}monolithic conveyor based on tilted air jets demonstrated the ability to move 2 mg objects within the device plane [25]. However, positional precision was not reported for this complex system that required four high-pressure-controlled electrovalves.

## 2. Device Design

## 3. Materials and Methods

## 4. Results and Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the device design. A bridge structure of length L and width W consisting of a silicon substrate with thickness ${t}_{s}$, covered by an AlN piezoelectric layer of thickness tp. Two symmetrically disposed metallic electrodes were placed closed to the edges, starting at a distance ${l}_{1}$ and ending at a distance ${l}_{2}$.

**Figure 2.**Mode shapes involved in the travelling waves (TW) generation obtained from the finite element method (FEM) analysis. (

**a**) Modes (4,0) and (5,0) in the low frequency (LF) range, (

**b**) modes (10,0) and (11,0) in the intermediate frequency (IF) range and (

**c**) modes (19,0) and (20,0) in the high frequency (HF) range. Resonant frequency is also included. Colour bar represents the normalised modal displacement in the Z-axis.

**Figure 3.**Second derivatives ${\phi}^{\u2033}$ (solid lines) of the modes involved in the TW generation in the (

**a**) low frequency (LF) range, (

**b**) intermediate frequency (IF) range and (

**c**) high frequency (HF) range. The positions of the zeros (dotted vertical lines) and optimum patch design (green area) are also indicated. A double-headed arrow indicates the patch based on the zeros of the second derivative.

**Figure 4.**Resultant envelopes from the simulations with optimal patches and patches deduced from the approach based on the zeros of the second derivative in the (

**a**) LF range, (

**b**) IF range and (

**c**) HF range. The dotted vertical lines indicate a centred 50% of the length.

**Figure 5.**Photographs of the fabricated devices. (

**a**) Silicon dice containing two HF designs, wire bonded to a printed circuit board (PCB), (

**b**) detail of the legs attached and (

**c**) the experimental setup, with a gold-patterned slider on top of a legged TW motor, and constrained to a lane by four glass pieces.

**Figure 6.**Measured conductance of three different fabricated devices: (

**a**) LF, (

**b**) IF and (

**c**) HF. Modal identification was deduced from laser Doppler vibrometer by accounting for the number of nodal lines. The dashed vertical line indicates the driving frequency.

**Figure 7.**TW magnitude (per unit of applied voltage) and phase for different devices along the bridge length. Measurement (solid lines) and calculation (dotted line) in (

**a**) LF, (

**b**) IF and (

**c**) HF devices. The dashed vertical lines indicate a centred 50% of length.

**Figure 8.**Measured results for the 3 mg slider on the LF device with 4 legs. (

**a**) Slider position and (

**b**) deduced velocities during the experiment with 2 V excitation signals at 19.3 kHz. The phase difference between the signals of the patches was alternated between 90° and −90° every second.

**Figure 9.**Results from the kinetic characterization of the LF device: (

**a**) average speed of the sliders versus their mass for a voltage amplitude of 6 V and (

**b**) average speed of the 23 mg slider at different excitation amplitudes.

**Figure 10.**Study of the minimum displacement of the slider for 10 sinusoidal cycles of 10 V at 19 kHz. (

**a**) Discrete steps on the direction of propagation of the TW (X-axis) and the orthogonal one (Y-axis) and (

**b**) estimated step height.

Motor | Size | Actuation | Actuation | Payload | Speed | Positional Resolution | Reference |
---|---|---|---|---|---|---|---|

2D bidirectional | 60 × 15 mm^{2} | Piezoelectric | 80 V | 80 N | 1 m/s | 25 nm | [22] |

1D unidirectional | 10 × 10 mm^{2} | Electrothermal | 18 V | 40 mg | 8 µm/s | N.A. | [23] |

1D unidirectional | 15 × 5 mm^{2} | Electrothermal | 18 V | 2.5 g | 6 mm/s | N.A. | [24] |

2D bidirectional | 9 × 9 mm^{2} | Jet air | 20 kPa | 2 mg | N.A. | N.A. | [25] |

Rotational bidirectional | Ø3 mm | Piezoelectric | PZT, 5 V | 15 mg | 1500 rpm | N.A. | [27] |

1D unidirectional | Array, individual size 112 × 16 µm^{2} | Piezoelectric | PZT, 5 V | 2 mg | 1 mm/s | N.A. | [28] |

Parameter | Symbol | Value | Unit | Reference |
---|---|---|---|---|

Silicon density (kg/m^{3}) | ${\rho}_{s}$ | 2329 | kg/m^{3} | [31] |

Silicon Young’s modulus (GPa) | ${E}_{s}$ | 169 | GPa | [32] |

AlN density (kg/m^{3}) | ${\rho}_{s}$ | 3260 | kg/m^{3} | [31] |

AlN Young’s modulus (GPa) | ${E}_{s}$ | 370 | GPa | [33] |

AlN piezoelectric coefficient | ${d}_{31}$ | 1.28 | pm/V | [30] |

**Table 3.**Figures of merit deduced from the TW envelopes of Figure 4 for the two types of patches under study for the three frequencies considered. The values for the design parameters l

_{1}and l

_{2}are also given.

Frequency Range | LF | IF | HF | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameter | l_{1}(mm) | l_{2}(mm) | SWR | <TW> (nm/V) | l_{1}(mm) | l_{2}(mm) | SWR | <TW> (nm/V) | l_{1}(mm) | l_{2}(mm) | SWR | <TW> (nm/V) |

Optimum patch | 1.00 | 2.20 | 1.28 | 5.69 | 1.40 | 2.40 | 1.18 | 1.51 | 1.20 | 2.70 | 1.40 | 0.44 |

Zeros-based patch | 0.83 | 3.16 | 1.52 | 8.05 | 0.33 | 1.25 | 1.05 | 1.46 | 0.18 | 0.66 | 1.07 | 0.41 |

**Table 4.**Device parameters calculated from the electrical impedance measurements: resonant frequency, quality factor and electrical conductance. The mid-frequency of each working range, calculated from the measured resonant frequencies is also indicated. Simulated frequencies from Figure 2 are given in parentheses.

Device Parameter | LF Range | IF Range | HF Range | |||
---|---|---|---|---|---|---|

(4,0) Mode | (5,0) Mode | (10,0) Mode | (11,0) Mode | (19,0) Mode | (20,0) Mode | |

Q-factor, Q | 541 | 567 | 385 | 406 | 120 | 84 |

Motional conductance, $\Delta G$ ($\mathsf{\mu}\mathrm{S}$) | 0.094 | 0.21 | 0.30 | 0.16 | 1.12 | 2.02 |

Resonant frequency, f_{r} (kHz) | 14.7 (13.9) | 23.9 (23.0) | 101 (102) | 122 (125) | 399 (388) | 441 (431) |

Mid-frequency, f_{drive} (kHz) | 19.3 (18.5) | 112 (114) | 420 (410) |

Device Design | LF | IF | HF | ||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | f_{drive} (kHz) | SWR | <TW> (nm/V) | f_{drive} (kHz) | SWR | <TW> (nm/V) | f_{drive} (kHz) | SWR | <TW> (nm/V) |

Simulation | 18.5 | 1.28 | 5.69 | 114 | 1.18 | 1.51 | 410 | 1.40 | 0.44 |

Experiment | 19.3 | 1.46 | 6.01 | 112 | 1.55 | 1.68 | 420 | 3.35 | 0.42 |

**Table 6.**Summary of the figures of merit of the different designs, each of them actuated at the three frequencies.

${\mathit{f}}_{\mathit{d}\mathit{r}\mathit{i}\mathit{v}\mathit{e}}$(kHz) | 19.3 | 112 | 420 | |||
---|---|---|---|---|---|---|

Device Design | SWR | <TW> (nm/V) | SWR | <TW> (nm/V) | SWR | <TW> (nm/V) |

LF | 1.46 | 6.01 | 2.5 | 1.64 | N.A. | |

IF | 1.68 | 5.42 | 1.55 | 1.68 | N.A. | |

HF | 1.77 | 7.01 | 3.86 | 1.63 | 3.35 | 0.42 |

Slider (Thickness) | Length (mm) | Width (mm) | Mass (mg) |
---|---|---|---|

Silicon (40 µm) | 10 | 2 | 3 |

Glass (200 µm) | 11 | 2.4 | 11 |

Silicon (200 µm) | 17.5 | 2.8 | 23 |

Silicon (500 µm) | 12.9 | 2.36 | 35 |

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**MDPI and ACS Style**

Ruiz-Díez, V.; Hernando-García, J.; Toledo, J.; Ababneh, A.; Seidel, H.; Sánchez-Rojas, J.L.
Bidirectional Linear Motion by Travelling Waves on Legged Piezoelectric Microfabricated Plates. *Micromachines* **2020**, *11*, 517.
https://doi.org/10.3390/mi11050517

**AMA Style**

Ruiz-Díez V, Hernando-García J, Toledo J, Ababneh A, Seidel H, Sánchez-Rojas JL.
Bidirectional Linear Motion by Travelling Waves on Legged Piezoelectric Microfabricated Plates. *Micromachines*. 2020; 11(5):517.
https://doi.org/10.3390/mi11050517

**Chicago/Turabian Style**

Ruiz-Díez, Víctor, Jorge Hernando-García, Javier Toledo, Abdallah Ababneh, Helmut Seidel, and José Luis Sánchez-Rojas.
2020. "Bidirectional Linear Motion by Travelling Waves on Legged Piezoelectric Microfabricated Plates" *Micromachines* 11, no. 5: 517.
https://doi.org/10.3390/mi11050517