# A Rotational Gyroscope with a Water-Film Bearing Based on Magnetic Self-Restoring Effect

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## Abstract

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## 1. Introduction

## 2. Structure Design

#### 2.1. Mechanical Structure

_{mod}is added to the rotor through the electric path. Deionized water in the cavity of the lower supporting pillar forms a thin water film as a bearing between the rotor ball surface and the ball bowl when the rotor rotates at a high spinning speed. SHS is fabricated on the rotor ball to decrease sliding friction further and guarantees the stable spinning motion and precessional motion (with suitable damping characteristic) of the rotor. Air bearing will produce a much lower sliding friction to the rotor, while it is mainly used as journal bearing and is not suitable for the designed gyroscope. Another factor that prevents the application of air bearing is that such a low damp will lead to a long dynamic adjusting time to precessional motion when sensing input angular speed. A detection electrode plate is fixed by the lower supporting pillar 100 μm to the lower surface of rotor disk (at zero state), forming four tilt induced capacitors with it.

#### 2.2. Fabrication of SHS on the Rotor Ball

_{3}O

_{4}. A magnetic rotor rotated at 120 rpm in the solution to accelerate the reaction. Afterwards, the ball was cleaned in distilled water and dried in N

_{2}for 1.5 h. Finally, the ball was immersed in an ethanol solution of fluorinated silane (0.5 wt %) for half an hour and dried in a vacuum oven at 120 °C for an hour. Then, SHS was fabricated. SEM images of Fe substrate, fabricated Fe

_{3}O

_{4}nanosheets with a small image at a higher resolution, are shown in Figure 2a,b. Nanosheets with the thickness of 50–60 nm are uniformly distributed on the ball surface (Figure 2b). To evaluate wettability of the material surface conveniently, water contact angles (CAs) of carbon steel sheet of the same material (Figure 2c) and fabricated SHS on it with the same process technology (Figure 2d) are measured. Thus, SHS with the same water CA of 167° is fabricated on a rotor ball.

## 3. Operational Principle

_{i}from the water-film and the support force F from the upper supporting pillar ($\sum {N}_{i}=-F$). Gravity is neglected here. Input angular speed at the direction of the gyroscope stator plane will produce a proportional Coriolis torque, which deflects the rotor disk from the stator plane. When the rotor is deflected, a magnetic self-restoring torque M

_{c}produces, having the tendency of dragging the rotor disk back to the stator plane. During the rotor precessional motion, a damping torque M

_{d}appears at the interface between the rotor ball and the water-film bearing. Rotor disk deflects under Coriolis inertial torque M

_{G}, self-restoring torque M

_{c}, damping torque M

_{d}, and inertial torque M

_{I}. Rotor disk will be balanced at a certain position with a deflection angle of φ, which is small in value for the large self-restoring coefficient. Differential tilt induced capacitance pairs detect deflection angles α, β at two perpendicular directions (Figure 3a). A force and torque analysis diagram is shown as Figure 3b, in which M

_{sr}and M

_{sd}represent actuating torque and damping torque, respectively. Close-loop driving scheme adopted ensures a stable spinning motion of the rotor during precessional motion, namely, ensuring M

_{sr}equals M

_{sd}.

#### 3.1. Self-Restoring Effect of Rotor

_{0}is the permeability of vacuum, S is the effective overlap area of the magnetic flux to the stator pole, which is a function of δ, g is the gap between the rotor disk and the stator pole, r, R are outer radius of the rotor disk and inner radius of the stator pole. Then, total stored magnetic energy in the gap is AW

_{m}, where A is a parameter related to dimensions. Thus, restoring torque M

_{c}is calculated as:

_{c}(δ = 90°) is proportional to the rotor deflection angle φ under the condition that φ is of small value ($\mathrm{sin}\varphi \approx \varphi ,\text{\hspace{0.17em}}\mathrm{cos}\varphi \approx 1$) with the coefficient $C=A\frac{{B}^{2}}{2{\mu}_{0}}S\frac{Rr}{R-r}$. Therefore, under certain parameters of the stator, increasing rotor radius r and improving magnetic flux density B will result in a larger restoring coefficient C. Rotor is highly magnetized parallel to around 1 tesla at the rotor plane. A stronger magnetic field will cause the rotor disk to be attracted tightly to the stator pole, which prevents the rotor from stable driving. In addition, tight adhesion between the rotor ball and the ball bowl will destroy the superhydrophobic surface on the rotor ball. Restoring torques with different radius (r values) under deflection angles from −1° to 0° are simulated in ANSYS Maxwell (16.0, Ansys, Canonsburg, PA, USA) (Figure 5). It can be seen that restoring coefficient C increases with the rotor radius r, while, when r increases from 5.4 mm to 5.5 mm, linearity deteriorates. Rotor radius is designed as 5.4 mm finally. Because of spinning motion of the magnetic rotor, coefficient C is a periodic function of δ with the period of 180°. Magnetic self-restoring torque M

_{c}, under rotor deflection angle φ of 1°, for different δ with the interval of 6° is simulated in ANSYS Maxwell. Output data is put in Matlab (R2016a, MathWorks, Natick, MA, USA) and fitted by a method of sum of sine (Figure 6). Expression of the fitted curve is:

_{c}(T = 2π/0.03484 = 180.3) is approximately equal to 180, and maximum, minimum M

_{c}are obtained when rotor deflection axis is perpendicular (δ = 90°) and parallel (δ = 0°) to magnetization direction. For the proposed gyroscope, angle value (0.03484δ + 4.633) can be expressed as ($4\pi ft+\psi $), where f is the frequency of rotor spinning motion, t is the time, and Ψ is the phase difference between initial state and maximum restoring torque state. Thus, self-restoring coefficient C is a period function with the period of $1/2f$, expressed as: (382.86 + 145.4 sin (4πft + Ψ)) μNm/°, namely, (0.0219 + 0.0083 sin (4πft + Ψ)) (Nm/rad).

#### 3.2. Dragging Torque of Water-Film Bearing to Spinning and Precessional Motion

_{s}between the water film and the SHS of the rotor is expressed as:

_{L}is the tangential stress limit, which is small in amplitude and omitted, η is the viscosity of the water film, b is the average slip length, and v

_{s}is the slip speed. Thus, during spinning motion, drag torque T

_{fs}exerted by the water-film bearing on the rotor ball is calculated as:

_{r}(θ) is the distance from the integral position to axis of spinning motion, and R

_{r}is the radius of the rotor ball. τ

_{s}is related to integral position, which is a function of θ. SHS turns the contact with the water film to a Cassie state with air bubbles within nanosheets between solid–liquid interface, thus decreasing integral area S in (6) and reducing drag torque T

_{fs}. During precessional motion, damping torque M

_{d}exerted on the rotor ball by the water-film bearing is calculated as:

_{s}

_{1}represents tangential stress during precessional motion, and R

_{d}(θ) is the distance from the integral position to axis of precessional motion, both of which are functions of θ, ω

_{p}is the precessional angular speed. Value of the expression in brackets is constant for the gyroscope and, thus, damping torque M

_{d}is proportional to ω

_{p}.

#### 3.3. Sensing Principle

_{0}Y

_{0}Z

_{0}and XYZ coordinate (without spin motion). For the rated spinning speed is as high as 10,000 rpm, nutation, which is proportional to angular momentum H in frequency and inversely proportional to angular momentum H in amplitude, is neglected. Thus, inertial torque of rotational motion M

_{I}is neglected. Then, according to force and torque analysis diagram of Figure 3b, gyroscope dynamic balance equations in X

_{0}, Y

_{0}directions are as below:

_{x(y)}, D

_{x(y)}, ω

_{x(y)}are the electromagnetic elastic coefficient, damping coefficient, input angular speeds in X

_{0}(Y

_{0}) directions, respectively, and deflection angles α, β are of small values. For symmetrical structure of the gyroscope, let C = C

_{x}= C

_{y}, D = D

_{x}= D

_{y}. When ω

_{x}, ω

_{y}are step inputs with amplitude of A

_{x}, A

_{y}, the solutions of Labels (8) and (9) give:

_{x}/A

_{y}). Equations (10) and (11) indicate that the rotor axis will rotate from the null position (where α = 0, β = 0) at the angular speed of CH/(H

^{2}+ D

^{2}) and converge with the time constant of $({H}^{2}+{D}^{2})/CD$ to a static position with deflection angles of α = (H/C)A

_{x}and β = (H/C)A

_{y}. As reasoned in Section 3.1, self-restoring coefficient C is a function C(t) expressed in the form of C

_{1}+ C

_{2}sin (4πft + Ψ) (C

_{1}= 0.0219, C

_{2}= 0.0083). Thus, Equations (10) and (11) are modified as:

_{2}is 0.0083 and f is 167 in the design. Thus,

_{1}= 0.0219, C

_{2}= 0.0083) as below:

_{x}, ω

_{y}are impulse inputs with amplitude of A

_{x}, A

_{y}, the solutions of (8) and (9) give:

_{1}H/(H

^{2}+ D

^{2}). Sinusoidal self restoring coefficient C has no influence on impulse response.

#### 3.4. Differential Capacitance Detection and Signal Processing

_{0}and Y

_{0}directions. Differential capacitance detection is applied. Detection electrode plate with four poles and the rotor disk form four tilt induced capacitors, with radially opposite ones forming a differential pair. Schematic diagram of two differential pairs, ${C}_{{X}^{+}}$ and ${C}_{{X}^{-}}$, ${C}_{{Y}^{+}}$ and ${C}_{{Y}^{-}}$, testing angles of α and β are shown in Figure 7. Four initial capacitances with no angular speed inputs are ${\mathrm{C}}_{{X}_{0}^{+}}$, ${\mathrm{C}}_{{X}_{0}^{-}}$, ${\mathrm{C}}_{{\mathrm{Y}}_{0}^{+}}$, and ${\mathrm{C}}_{{\mathrm{Y}}_{0}^{-}}$. Four capacitances are approximately the same and main errors come from fabrication. For the symmetry of the structure, one differential pair of ${C}_{{X}^{+}}$ and ${C}_{{X}^{-}}$ is analyzed here. According to [33],

_{mod}(2 V, 15 kHz) is added to the upper surface of the rotor ball. Currents go through the tilt induced capacitor pair (${C}_{{X}^{+}}$, ${C}_{{X}^{-}}$), differential amplifier, proportional amplifier and obtain the output voltage V

_{o}

_{1}as shown in Figure 9a. Tilt induced capacitors C

_{f+}, C

_{f−}are adjusted to satisfy:

_{o1}at initial state (with no angular speed input) is zero. The amplification factor of the proportional amplifier is adjusted to be G

_{op3}, and then the output of V

_{o1}is expressed as:

_{o1}is proportional to deflecting angle α. Similarly, the amplitude of the output voltage of the other differential pair ${C}_{{Y}^{+}}$ and ${C}_{{Y}^{-}}$ is proportional to deflecting angle β. Differential signal V

_{o1}is then processed as Figure 9b. DC component in the signal is removed by high-pass filter first and then the signal is demodulated by multiplying modulation signal V

_{mod}. Finally, low-pass filter, with cut-off frequency of 10 Hz, is added to obtain low-frequency output voltage signal.

## 4. Measurement and Discussion

^{−5}, respectively, with calculated H of 2.784 × 10

^{−4}. Lower amplitude of C than theoretical value of 0.0219 calculated in Section 3.1 is because of factors such as manufacture error and magnetic flux leakage.

## 5. Conclusions

_{c}, proportional to rotor deflection angle φ, is produced, which balances Coriolis torque. With spinning motion of the rotor, M

_{c}changes in sinusoidal waveform, and the average restoring coefficient is 0.02017 Nm/rad. Water-film bearing adopted in the design, as a substitute to complicated rotor suspension system in MSGs and ESGs, provides stable rotor support with low friction. To further decrease sliding friction, an SHS with water CA of 167° is fabricated on the rotor ball, which increases rated spinning speed from 8970 rpm to 10,084 rpm. Differential capacitance detection is adopted by installing four sectorial poles on a plate under the rotor disk. Simulation by Maxwell indicates that detection linearity is ideal when rotor deflection angle is under 1°, which limits measurement range of the proposed gyroscope to −30°/s–30°/s. Spectral analysis to output voltage signal was done to confirm cut-off frequency of LPF and 333 Hz noise component was observed, which is produced by magnetic self-restoring effect of the rotor. Static measurement parameters and dynamic parameters are acquired through experiments, with the low bias stability of only 0.5°/h. Excellent performance is the result of the special structure design with a water-film bearing together with the utilization of magnetic self-restoring effect of the rotor.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Saukoski, M.; Aaltonen, L.; Halonen, K.A.I. Zero-rate output and quadrature compensation in vibratory MEMS gyroscopes. IEEE Sens. J.
**2007**, 7, 1639–1651. [Google Scholar] [CrossRef] - Elsayed, M.; Nabki, F.; Sawan, M.; El-Gamal, M. A 5 V Gyroscope with 3 aF/°/s Sensitivity, 0.6°/√hr Mechanical Noise and Drive-Sense Crosstalk Minimization. In Proceedings of the 2011 International Conference on Microelectronics (ICM), Hammamet, Tunisia, 19–22 December 2011; pp. 1–5. [Google Scholar]
- Williams, C.B.; Shearwood, C.; Mellor, P.H.; Yates, R.B. Modelling and testing of a frictionless levitated micromotor. Sens. Actuators
**1997**, 61, 469–473. [Google Scholar] [CrossRef] - Shearwood, C.; Ho, K.Y.; Williams, C.B.; Gong, H. Development of a levitated micromotor for application as a gyroscope. Sens. Actuators
**2000**, 82, 85–92. [Google Scholar] [CrossRef] - Zhang, W.P.; Chen, W.Y.; Zhao, X.L.; Wu, X.S.; Liu, W.; Huang, X.G.; Shao, S.Y. The study of an electromagnetic levitating micromotor for application in a rotating gyroscope. Sens. Actuators
**2006**, 132, 651–657. [Google Scholar] [CrossRef] - Liu, W.; Chen, W.Y.; Zhang, W.P.; Huang, X.G.; Zhang, Z.R. Variable-capacitance micromotor with levitated diamagnetic rotor. Electron. Lett.
**2008**, 44, 681–683. [Google Scholar] [CrossRef] - Liu, K.; Zhang, W.P.; Liu, W.; Chen, W.Y.; Li, K.; Cui, F.; Li, S.P. An innovative micro-diamagnetic levitation system with coils applied in micro-gyroscope. Microsyst. Technol.
**2010**, 16, 431–439. [Google Scholar] [CrossRef] - Dauwalter, C.R.; Ha, J.C. A high performance magnetically suspended MEMS spinning wheel gyroscope. In Proceedings of the PLANS 2004—Position Location and Navigation Symposium, Monterey, CA, USA, 26–29 April 2004; pp. 70–77. [Google Scholar]
- Dauwalter, C.R.; Ha, J.C. Magnetically suspended MEMS spinning wheel gyroscope. IEEE Aerosp. Electron. Syst. Mag.
**2005**, 20, 21–26. [Google Scholar] [CrossRef] - Tang, J.; Xiang, B.; Zhang, Y. Dynamic characteristics of the rotor in a magnetically suspended control moment gyroscope with active magnetic bearing and passive magnetic bearing. ISA Trans.
**2014**, 53, 1357–1365. [Google Scholar] [CrossRef] [PubMed] - Torti, R.; Gondhalekar, V.; Tran, H.; Selfors, B.; Bart, S.; Maxwell, B. Electrostatically suspended and sensed micromechanical rate gyroscope. In Proceedings of the SPIE’s International Symposium on Optical Engineering and Photonics in Aerospace Sensing, Orlando, FL, USA, 13 July 1994; pp. 27–38. [Google Scholar]
- Esashi, M. Saving energy and natural resource by micro-nanomachining. In Proceedings of the Fifteenth IEEE International Conference on Micro Electro Mechanical Systems, Las Vegas, NV, USA, 24 January 2002; pp. 220–227. [Google Scholar]
- Murakoshi, T.; Endo, Y.; Fukatsu, K.; Nakamura, S.; Esashi, M. Electrostatically levitated ring-shaped rotational gyroscope/accelerometer. Jpn. J. Appl. Phys.
**2003**, 42, 2468–2472. [Google Scholar] [CrossRef] - Nakamura, S. MEMS inertial sensor toward higher accuracy & multi-axis sensing. In Proceedings of the 2005 IEEE Sensors, Irvine, CA, USA, 30 October–3 November 2005; pp. 939–942. [Google Scholar]
- Kraft, M.; Evans, A. System level simulation of an electrostatically levitated disk. In Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems, San Diego, CA, USA, 27–29 March 2000; pp. 27–29. [Google Scholar]
- Kraft, M.; Farooqui, M.M.; Evans, A.G.R. Modelling and design of an electrostatically levitated disk for inertial sensing applications. J. Micromech. Microeng.
**2001**, 11, 423–427. [Google Scholar] [CrossRef] - Gindila, M.V.; Kraft, M. Electronic interface design for an electrically floating micro-disk. J. Micromech. Microeng.
**2003**, 13, S11–S16. [Google Scholar] [CrossRef] - Kukharenka, E.; Farooqui, M.M.; Grigore, L.; Kraft, M.; Hollinshead, N. Electroplating moulds using dry film thick negative photoresist. J. Micromech. Microeng.
**2003**, 13, 95–98. [Google Scholar] [CrossRef] - Damrongsak, B.; Kraft, M. A micromachined electrostatically suspended gyroscope with digital force feedback. In Proceedings of the 2005 IEEE Sensors, Irvine, CA, USA, 30 October–3 November 2005; pp. 401–404. [Google Scholar]
- Damrongsak, B. Design and simulation of a micromachined electrostatically suspended gyroscope. In Proceedings of the Seminar on MEMS Sensors and Actuators, London, UK, 28 April 2006; pp. 267–272. [Google Scholar]
- Xia, D.Z.; Chen, S.L.; Wang, S.R. Development of a prototype miniature silicon microgyroscope. Sensors
**2009**, 9, 4586–4605. [Google Scholar] [CrossRef] [PubMed] - Loveday, P.W.; Rogers, C.A. The influence of control system design on the performance of vibratory gyroscopes. J. Sound Vib.
**2002**, 255, 417–432. [Google Scholar] [CrossRef] - Cui, J.; Guo, Z.Y.; Zhao, Q.C.; Yang, Z.C. Force rebalance controller synthesis for a micromachined vibratory gyroscope based on sensitivity margin specifications. J. Microelectromech. Syst.
**2011**, 20, 1382–1394. [Google Scholar] [CrossRef] - Yoxall, B.E.; Chan, M.; Harake, R.S.; Pan, T.; Horsley, D.A. Rotary Liquid Droplet Microbearing. J. Microelectromech. Syst.
**2012**, 21, 721–729. [Google Scholar] [CrossRef] - Sun, G.; Liu, T.; Sen, P.; Shen, W.; Gudeman, C.; Kim, C. Electrostatic side-drive rotary stage on liquid-ring bearing. J. Microelectromech. Syst.
**2014**, 23, 147–156. [Google Scholar] [CrossRef] - Takei, A.; Matsumoto, K.; Shomoyama, I. Capillary motor driven by electrowetting. Lab Chip
**2010**, 10, 1781–1786. [Google Scholar] [CrossRef] [PubMed] - Kulik, V.M.; Semenov, B.N.; Boiko, A.V.; Seoudi, B.M.; Chun, H.H.; Lee, I. Measurement of dynamic properties of viscoelastic materials. Exp. Mech.
**2009**, 49, 417–425. [Google Scholar] [CrossRef] - Keyes, D.; Abernathy, F. A model for the dynamics of polymers in laminar shear flows. J. Fluid Mech.
**1987**, 185, 503–522. [Google Scholar] [CrossRef] - Rabin, Y.; Zielinska, B.J.A. Scale-dependent enhancement and damping of vorticity disturbances by polymers in elongational flow. Phys. Rev. Lett.
**1989**, 63, 512–516. [Google Scholar] [CrossRef] [PubMed] - Dubief, Y.; White, C.M.; Terrapon, V.E.; Shaqfeh, E.S.G.; Moin, P.; Lele, S.K. On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech.
**2004**, 514, 271–280. [Google Scholar] [CrossRef] - Chen, D.; Liu, X.; Zhang, H.; Li, H.; Weng, R.; Li, L.; Zhang, Z. Friction reduction for a rotational gyroscope with mechanical support by fabrication of a biomimetic superhydrophobic surface on a ball-disk shaped rotor and the application of a water film bearing. Micromachines
**2017**, 8, 223. [Google Scholar] [CrossRef] - Spikes, H.; Granick, S. Equation for slip of simple liquids at smooth solid surface. Langmuir
**2003**, 19, 5065–5071. [Google Scholar] [CrossRef] - Li, H.; Liu, X.; Weng, R.; Zhang, H. Micro-Angle Tilt Detection for the Rotor of a Novel Rotational Gyroscope with 0.47’’ Resolution. Front. Inf. Technol. Electron. Eng.
**2016**, 18, 591–598. [Google Scholar] [CrossRef]

**Figure 2.**SEM images of (

**a**) untreated rotor ball surface and (

**b**) the rotor ball surface with fabricated nanosheets; Optical photos of a water droplet on (

**c**) untreated carbon steel sheet surface and (

**d**) SHS.

**Figure 5.**Restoring torques with different radius (r values) under deflection angles from −1° to 0°.

**Figure 11.**(

**a**) linear fitting of output voltages under input angular speed range of −30°/s to 30°/s; (

**b**) log-log plot of Allan deviation versus averaging time.

**Figure 12.**(

**a**) photograph of the rate table with the proposed rotational gyroscope and a Micro-electromechanical Systems (MEMS) quartz vibratory gyroscope on it for dynamic characteristic test; (

**b**) impulse responses of the proposed rotational gyroscope and a MEMS quartz vibratory gyroscope.

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## Share and Cite

**MDPI and ACS Style**

Chen, D.; Liu, X.; Zhang, H.; Li, H.; Weng, R.; Li, L.; Rong, W.; Zhang, Z.
A Rotational Gyroscope with a Water-Film Bearing Based on Magnetic Self-Restoring Effect. *Sensors* **2018**, *18*, 415.
https://doi.org/10.3390/s18020415

**AMA Style**

Chen D, Liu X, Zhang H, Li H, Weng R, Li L, Rong W, Zhang Z.
A Rotational Gyroscope with a Water-Film Bearing Based on Magnetic Self-Restoring Effect. *Sensors*. 2018; 18(2):415.
https://doi.org/10.3390/s18020415

**Chicago/Turabian Style**

Chen, Dianzhong, Xiaowei Liu, Haifeng Zhang, Hai Li, Rui Weng, Ling Li, Wanting Rong, and Zhongzhao Zhang.
2018. "A Rotational Gyroscope with a Water-Film Bearing Based on Magnetic Self-Restoring Effect" *Sensors* 18, no. 2: 415.
https://doi.org/10.3390/s18020415