# Modeling a Pull-In Instability in Micro-Machined Hybrid Contactless Suspension

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hybrid Suspension

#### 2.1. Fabrication

#### 2.2. Operating Principle

#### 2.3. Preliminary Experimental Results

^{−1}is the vacuum permittivity, ${\epsilon}_{r}$ is the relative permittivity (for air ${\epsilon}_{r}\approx 1$) and h is the space between an electrode’s plane and the equilibrium point of the proof mass.

## 3. Analytical Model

#### The Accelerometer Equation of Motion

## 4. Static Pull-In Instability

## 5. Dynamic Pull-In Instability

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

$AM$ | Amplitude modulator |

$DRIE$ | Deep reactive-ion etching |

$NS$ | Negative stiffness |

$SIO$ | Silicon-on-Insulator |

$\mu $-CS | Micro-machined Contactless Suspensions |

$\mu $-ECS | Micro-machined Electrostatic Suspensions |

$\mu $-MCS | Micro-machined Magnetic Suspensions |

$\mu $-HCS | Micro-machined Hybrid Suspensions |

## Appendix A

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**Figure 1.**The hybrid suspension: (

**a**) the prototype glued to a PCB. The scaled up image at the bottom right corner shows the alignment and the SU-8 post for spacing (the top electrode structure is not connected); (

**b**) the exploded view; (

**c**) the electrodes patterned at the bottom (right) and top (left); electrode structures: 1, generating negative stiffness; 2, sensing displacement; 3, feedback electrodes; (

**d**) a view of the aligning electrode and coil structures from the rear (Pyrex glass) of the device.

**Figure 2.**The bottom electrode structure fabricated by using a Si wafer with an SU-8 layer of 30 $\mathsf{\mu}\mathrm{m}$ in thickness: (

**a**) the front side of the structure; (

**b**) the rear of the structure.

**Figure 3.**The prototype under experimental test: (

**a**) the device is fixed on a PCB (front side); (

**b**) the interfacial electronics (rear side); (

**c**) top, bottom and coil structures are connected to the PCB (scaled image); (

**d**) measurements of force against displacement.

**Figure 4.**Schematic diagram for modeling the hybrid contactless suspension: (

**a**) ${u}_{1}$ and ${u}_{2}$ are the potentials applied to the top and bottom electrodes, respectively; h is the space between an electrode’s plane and the equilibrium point of the proof mass; ${h}_{l}$ is the levitation height between the plane formed by the upper turn of the coils and the equilibrium point of the proof mass; ${i}_{el}$ and ${i}_{es}$ are the eddy currents corresponding to the maximum current density; (

**b**) coordinate frames and generalized coordinates to define the position of the disc-shaped proof mass around the origin: ${q}_{v}$, ${q}_{l}$, $\alpha $ and $\beta $ are the generalized coordinates corresponding to vertical, lateral and angular displacements, respectively.

**Figure 5.**Bifurcation diagram: (

**a**) dashed red lines show the evolution of the bifurcation map depending on constant D ($\kappa =1.0$); solid lines depict the evolution of the bifurcation map depending on spacing $\kappa =h/{h}_{l}$ ($D=2.0$); (

**b**) comparison of the quasi-exact and reduced models for $D=2.0$, $\kappa =1.0$ and $\xi =0.07$ (the relative error is less than 2%).

**Figure 6.**Static and dynamic bifurcation diagrams: solid black lines correspond to unphysical stagnation.

Parameters of the Prototype | ||

Diameter of the proof mass | ($\mathrm{m}\mathrm{m}$) | 3.2 |

Thickness of the proof mass | ($\mathsf{\mu}\mathrm{m}$) | 30 |

Levitation height | ($\mathsf{\mu}\mathrm{m}$) | 150 |

Spacing | ($\mathsf{\mu}\mathrm{m}$) | 50 |

Results of Measurements | ||

Stiffness ($U=0$) | ($\mathrm{N}{\mathrm{m}}^{-1}$) | 0.043 |

Stiffness ($U=11$ V) | ($\mathrm{N}{\mathrm{m}}^{-1}$) | 0.03 |

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**MDPI and ACS Style**

Poletkin, K.V.; Korvink, J.G. Modeling a Pull-In Instability in Micro-Machined Hybrid Contactless Suspension. *Actuators* **2018**, *7*, 11.
https://doi.org/10.3390/act7010011

**AMA Style**

Poletkin KV, Korvink JG. Modeling a Pull-In Instability in Micro-Machined Hybrid Contactless Suspension. *Actuators*. 2018; 7(1):11.
https://doi.org/10.3390/act7010011

**Chicago/Turabian Style**

Poletkin, Kirill V., and Jan G. Korvink. 2018. "Modeling a Pull-In Instability in Micro-Machined Hybrid Contactless Suspension" *Actuators* 7, no. 1: 11.
https://doi.org/10.3390/act7010011