# Control of Spring Softening and Hardening in the Squared Daisy

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## Abstract

**:**

## 1. Introduction

- A MEMS resonator architecture allowing the designer to readily control the type of nonlinearity, that is, yielding either spring hardening or softening;
- An experimental testing methodology allowing the monitoring and control of the hysteresis in the nonlinear resonators; and
- Recommendations for the characterization of nonlinear resonators.

## 2. Materials and Methods

#### Control of the Nonlinearity in MEMS Structures

## 3. Design and Micro-Fabrication Process

#### 3.1. Design

#### 3.2. Fabrication

## 4. Experimental Results

#### 4.1. Description of the Experimental Test Setup

#### 4.2. Description of the Excitation Signals

#### 4.3. Signal Parameters for Each of the Excitation Signals

#### 4.4. Summary of the Measurement Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Presentation of the Mode Shape of the Resonators

Mode Shape | ||
---|---|---|

Simulation | Measurement | |

Variant 1 | ||

Variant 2 |

## Appendix B. Excitation with Continuous Sweep Signals

Excitation Type | Measurements in the: | |
---|---|---|

Time Domain | Frequency Domain | |

Variant 1 $CSF$ | ||

Variant 1 $CSB$ | ||

Variant 2 $CSF$ | ||

Variant 2 $CSB$ |

## Appendix C. Excitation with Pulsed Sweep Signals

Influence of the: | ||
---|---|---|

Amplitude of the Excitation Signal | Order of Excitation of the Frequency | |

Variant 1 | ||

Variant 2 |

## References

- Chen, D.; Wang, Y.; Guan, Y.; Chen, X.; Liu, X.; Xie, J. Methods for Nonlinearities Reduction in Micromechanical Beams Resonators. J. Microelectromech. Syst.
**2018**, 27, 764–773. [Google Scholar] [CrossRef] - Tatar, E.; Mukherjee, T.; Fedder, G. Nonlinearity tuning and its effects on the performance of a MEMS gyroscope. In Proceedings of the 2015 Transducers—18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), Anchorage, AK, USA, 21–25 June 2015; pp. 1133–1136. [Google Scholar] [CrossRef]
- Ouakad, H.M.; Younis, M.I. The dynamic behavior of MEMS arch resonators actuated electrically. Int. J. Non-Lin. Mech.
**2010**, 45, 704–713. [Google Scholar] [CrossRef] - Du, J.; Wei, X.; Ren, J.; Wang, J.; Huan, R. Micromechanical vibration absorber for frequency stability improvement of DETF oscillator. J. Micromech. Microeng.
**2019**, 29. [Google Scholar] [CrossRef] - Ibrahim, R.A. Recent advances in nonlinear passive vibration isolators. J. Sound Vibrat.
**2008**, 314, 371–452. [Google Scholar] [CrossRef] - Hajati, A.; Kim, S.G. Ultra-wide bandwidth piezoelectric energy harvesting. Appl. Phys. Lett.
**2011**, 99, 083105. [Google Scholar] [CrossRef] [Green Version] - Nabavi, S.; Zhang, L. Nonlinear Multi-mode Wideband Piezoelectric MEMS Vibration Energy Harvester. IEEE Sens. J.
**2019**, 4837–4848. [Google Scholar] [CrossRef] - Jia, Y.; Du, S.; Seshia, A.A. Twenty-Eight Orders of Parametric Resonance in a Microelectromechanical Device for Multi-band Vibration Energy Harvesting. Sci. Rep.
**2016**, 6, 30167. [Google Scholar] [CrossRef] [Green Version] - Amirkhan, F.; Robichaud, A.; Ropagnol, X.; Ropagnol, X.; Gratuze, M.; Ozaki, T.; Nabki, F.; Blanchard, F. Active terahertz time differentiator using piezoelectric micromachined ultrasonic transducer array. Opt. Lett.
**2020**, 45, 3589–3592. [Google Scholar] [CrossRef] [PubMed] - Amirkhan, F.; Robichaud, A.; Ropagnol, X.; Ropagnol, X.; Gratuze, M.; Ozaki, T.; Nabki, F.; Blanchard, F. Simulation study of a piezoelectric micromachined ultrasonic transducer as terahertz differentiator. OSA Advanced Photonics Congress (AP) 2020 (IPR, NP, NOMA, Networks, PVLED, PSC, SPPCom, SOF) (2020), paper NoM4C.5. Opt. Soc. Am.
**2020**, NoM4C.5. [Google Scholar] [CrossRef] - Zou, X.; Seshia, A. Non-Linear Frequency Noise Modulation in a Resonant MEMS Accelerometer. IEEE Sens. J.
**2017**, 17, 4122–4127. [Google Scholar] [CrossRef] - Frangi, A.; Guerrieri, A.; Boni, N.; Carminati, R.; Soldo, M.; Mendicino, G. Mode Coupling and Parametric Resonance in Electrostatically Actuated Micromirrors. IEEE Trans. Indust. Electron.
**2018**, 65, 5962–5969. [Google Scholar] [CrossRef] - Tiwari, S.; Candler, R.N. Using flexural MEMS to study and exploit nonlinearities: A review. J. Micromech. Microeng.
**2019**, 29, 083002. [Google Scholar] [CrossRef] - Ilyas, S.; Alfosail, F.K.; Bellaredj, M.L.F.; Younis, M.I. On the response of MEMS resonators under generic electrostatic loadings: Experiments and applications. Nonlinear Dynam.
**2019**, 95, 2263–2274. [Google Scholar] [CrossRef] - Tausiff, M.; Ouakad, H.M.; Alqahtani, H. Global Nonlinear Dynamics of MEMS Arches Actuated by Fringing-Field Electrostatic Field. Arab. J. Sci. Eng.
**2020**, 45, 5959–5975. [Google Scholar] [CrossRef] - Bouchaala, A.; Jaber, N.; Yassine, O.; Shekhah, O.; Chernikova, V.; Eddaoudi, M.; Younis, M.I. Nonlinear-Based MEMS Sensors and Active Switches for Gas Detection. Sensors
**2016**, 16, 758. [Google Scholar] [CrossRef] [Green Version] - Comi, C.; Corigliano, A.; Zega, V.; Zerbini, S. Non linear response and optimization of a new z-axis resonant micro-accelerometer. Mechatronics
**2016**, 40, 235–243. [Google Scholar] [CrossRef] - Feldman, M. Non-linear system vibration analysis using Hilbert transform–I. Free vibration analysis method ‘Freevib’. Mech. Syst. Signal Proces.
**1994**, 8, 119–127. [Google Scholar] [CrossRef] - Feldman, M. The FREEVIB and FORCEVIB Methods. In Hilbert Transform Applications in Mechanical Vibration; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 2011; pp. 181–221. [Google Scholar] [CrossRef]
- Feldman, M. Non-linear system vibration analysis using Hilbert transform–II. Forced vibration analysis method ‘Forcevib’. Mech. Syst. Signal Process.
**1994**, 8, 309–318. [Google Scholar] [CrossRef] - Wei, Y.; Dong, Y.; Huang, X.; Zhang, Z. A Stepped Frequency Sweeping Method for Nonlinearity Measurement of Microresonators. Sensors
**2016**, 16, 1700. [Google Scholar] [CrossRef] [PubMed] - Urasaki, S.; Yabuno, H. Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation. Nonlinear Dynam.
**2020**. [Google Scholar] [CrossRef] - Wei, Y.; Dong, Y.; Huang, X.; Zhang, Z. Nonlinearity measurement for low-pressure encapsulated MEMS gyroscopes by transient response. Mech. Syst. Signal Proces.
**2018**, 100, 534–549. [Google Scholar] [CrossRef] - Gafforelli, G.; Corigliano, A.; Xu, R.; Kim, S. Experimental verification of a bridge-shaped, non-linear vibration energy harvesters. In Proceedings of the 2014 IEEE SENSORS, Valencia, Spain, 2–5 November 2014; pp. 2175–2178. [Google Scholar] [CrossRef]
- Zega, V.; Langfelder, G.; Falorni, L.G.; Comi, C. Hardening, Softening, and Linear Behavior of Elastic Beams in MEMS: An Analytical Approach. J. Microelectromech. Syst.
**2019**, 28, 189–198. [Google Scholar] [CrossRef] - Cho, H.; Jeong, B.; Yu, M.F.; Vakakis, A.F.; McFarland, D.M.; Bergman, L.A. Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities. Int. J. Solids Struct.
**2012**, 49, 2059–2065. [Google Scholar] [CrossRef] [Green Version] - Gratuze, M.; Alameh, A.H.; Nabki, F. Design of the Squared Daisy: A Multi-Mode Energy Harvester, with Reduced Variability and a Non-Linear Frequency Response. Sensors
**2019**, 19, 3247. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Younis, M.I. Introduction to Nonlinear Dynamics. In MEMS Linear and Nonlinear Statics and Dynamics; Younis, M.I., Ed.; Microsystems; Springer: Boston, MA, USA, 2011; pp. 155–249. [Google Scholar] [CrossRef]
- Bao, M.; Yang, H. Squeeze film air damping in MEMS. Sens. Actuators A Phys.
**2007**, 136, 3–27. [Google Scholar] [CrossRef] - Sun, S.; Dai, X.; Feng, Z.; Ding, G.; Zhao, X. Independent nonlinearity tuning of planar spring via geometrical design for wideband vibration energy harvesting. Sens. Actuators A Phys.
**2017**, 267, 393–400. [Google Scholar] [CrossRef] - Podder, P.; Constantinou, P.; Mallick, D.; Amann, A.; Roy, S. Magnetic Tuning of Nonlinear MEMS Electromagnetic Vibration Energy Harvester. J. Microelectromech. Syst.
**2017**, 26, 539–549. [Google Scholar] [CrossRef] - Asadi, K.; Li, J.; Peshin, S.; Yeom, J.; Cho, H. Mechanism of geometric nonlinearity in a nonprismatic and heterogeneous microbeam resonator. Phys. Rev. B
**2017**, 96, 115306. [Google Scholar] [CrossRef] [Green Version] - Piersol, A.G.; Paez, T.L. Harris’ Shock and Vibration Handbook, 6th ed.; McGraw-Hill Handbooks: New York, NY, USA, 2010. [Google Scholar]
- Alameh, A.H.; Gratuze, M.; Nabki, F. Impact of Geometry on the Performance of Cantilever-based Piezoelectric Vibration Energy Harvesters. IEEE Sens. J.
**2019**, 10316–10326. [Google Scholar] [CrossRef] - Robichaud, A.; Deslandes, D.; Cicek, P.V.; Nabki, F. Electromechanical Tuning of Piecewise Stiffness and Damping for Long-Range and High-Precision Piezoelectric Ultrasonic Transducers. J. Microelectromech. Syst.
**2020**, 29, 1189–1198. [Google Scholar] [CrossRef] - Pons-Nin, J.; Gorreta, S.; Dominguez, M.; Blokhina, E.; O’Connell, D.; Feely, O. Design and test of resonators using PiezoMUMPS technology. In Proceedings of the 2014 Symposium on Design, Test, Integration and Packaging of MEMS/MOEMS (DTIP), Cannes, France, 1–4 April 2014; pp. 1–6. [Google Scholar] [CrossRef] [Green Version]
- Alameh, A.H.; Gratuze, M.; Elsayed, M.Y.; Nabki, F. Effects of Proof Mass Geometry on Piezoelectric Vibration Energy Harvesters. Sensors
**2018**, 18, 1584. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cowen, A.; Hames, G.; Glukh, K.; Hardy, B. PiezoMUMPs Design Handbook; MEMSCAP Inc.: Crolles, France, 2014. [Google Scholar]

**Figure 1.**Illustration of the weighted string mechanical system, Reprinted with permission from ref. [33]. Copyright 2010 McGraw-Hill Handbooks.

**Figure 2.**Illustration of one of the available anchoring schemes of the squared daisy MEMS resonator. In this anchoring scheme, the cantilevers 3, 7, 11, and 15 are anchored and shown in green, while the others are free and shown in blue.

**Figure 3.**Influence of the force applied on the deflection of the central proof mass for both SD resonator variants.

**Figure 4.**Simplified overview of the PiezoMUMPs’ fabrication process flow, applied to the fabrication of Variant 2 of the Squared Daisy.

**Figure 5.**Micrograph of the fabricated SD devices: (

**a**) Variant 1 (hardening) and (

**b**) Variant 2 (softening). The different anchoring schemes for each structure can be identified, along with an indication of the vibration measurement point used.

**Figure 8.**Influence of the excitation signal type (CSF, CSB, or PSF) on the frequency behavior when the amplitude of the excitation signal is 20V (

**a**) Variant 1 and (

**b**) Variant 2.

**Figure 9.**Influence of the amplitude and type of the excitation (CSF, CSB, or PSF) on Fmv of (

**a**) Variant 1 and (

**b**) Variant 2.

Variant 1 | Variant 2 | |
---|---|---|

Size of the design ($\mathsf{\mu}\mathrm{m}$ × $\mathsf{\mu}\mathrm{m}$) | 1700 × 1700 | |

Radius of the proof mass, Rm ($\mathsf{\mu}\mathrm{m}$) | 200 | |

Thickness of the substrate ($\mathsf{\mu}\mathrm{m}$) | 400 | |

Thickness of the cantilevers ($\mathsf{\mu}\mathrm{m}$) | 10 | |

Cantilever used as anchors | 1 5 9 13 | 3 7 11 15 |

Free Cantilever | 2 3 4 6 7 8 10 | 1 2 4 5 6 8 9 |

11 12 14 15 16 | 10 12 13 14 16 | |

Resonant frequency (simulation) | 8100 | 4050 |

Nonlinear behavior (simulation) | Hardening | Softening |

Excitation Type | ${\mathit{F}}_{\mathbf{start}}$ (kHz) | ${\mathit{F}}_{\mathbf{end}}$ (kHz) | ${\mathit{T}}_{\mathit{e}}$ (s) | |
---|---|---|---|---|

Variant 1 | $CSF$ | 6 | 10 | 500 |

$CSB$ | 10 | 6 | ||

Variant 2 | $CSF$ | 2 | 6 | 500 |

$CSB$ | 6 | 2 |

${\mathit{F}}_{\mathbf{start}}$ (kHz) | ${\mathit{F}}_{\mathbf{end}}$ (kHz) | ${\mathit{F}}_{\mathbf{res}}$ (Hz) | ${\mathit{T}}_{\mathbf{on}}$ (s) | ${\mathit{T}}_{\mathbf{off}}$ (s) | |
---|---|---|---|---|---|

Variant 1 | 6 | 10 | 10 | 1.06 | 0.53 |

Variant 2 | 2 | 6 |

Excitation Voltage (V) | Variant 1 | Variant 2 | |||||
---|---|---|---|---|---|---|---|

PS | CSF | CSB | PS | CSF | CSB | ||

Maximal Velocity ($\mathrm{m}\mathrm{m}/\mathrm{s}$) | 5 | 90.0 | 93.3 | 93.2 | 20.7 | 17.0 | 42.5 |

10 | 176.6 | 181.50 | 181.7 | 95.0 | 95.7 | 118.0 | |

15 | 234.0 | 261.7 | 234.7 | 109.6 | 110.1 | 193.3 | |

20 | 271.1 | 334.2 | 268.9 | 122.1 | 120.5 | 228.1 | |

${F}_{mv}$ ($\mathrm{Hz}$) | 5 | 7750 | 7722 | 7716 | 4080 | 4197 | 4002 |

10 | 7800 | 7812 | 7797 | 3970 | 3969 | 3854 | |

15 | 7860 | 7929 | 7839 | 3930 | 3942 | 3675 | |

20 | 7910 | 8058 | 7879 | 3920 | 3918 | 3638 | |

Bandwidth ($\mathrm{Hz}$) | 5 | 180 | 185 | 182 | 150 | 213 | 87 |

10 | 180 | 163 | 169 | 160 | 196 | 188 | |

15 | 230 | 234 | 193 | 180 | 156 | 265 | |

20 | 280 | 308 | 221 | 220 | 190 | 294 | |

Quality factor | 5 | 43 | 42 | 42 | 27 | 20 | 46 |

10 | 43 | 48 | 46 | 25 | 41 | 21 | |

15 | 34 | 34 | 41 | 22 | 25 | 14 | |

20 | 28 | 26 | 36 | 18 | 21 | 12 |

**Table 5.**Summary of the performances of Variants 1 and 2 under different excitation when the amplitude of the excitation signal is of 20 $\mathrm{V}$

Excitation Type | Variant 1 | Variant 2 | |
---|---|---|---|

Overestimation of the velocity (%) | $CSF$ | −1.31 | 23.28 |

$CSB$ | 86.81 | −0.81 | |

Overestimation of ${F}_{mv}$ (%) | $CSF$ | 0.05 | 1.87 |

$CSB$ | −7.19 | −0.39 | |

Overestimation of the bandwidth (%) | $CSF$ | −13.64 | 10.00 |

$CSB$ | 33.64 | −21.07 | |

Overestimation of the quality factor (%) | $CSF$ | 16.67 | −7.14 |

$CSB$ | −33.33 | 28.57 | |

FOM ($\mathrm{Hz}\mathrm{m}\mathrm{m}{\mathrm{s}}^{-1}$) | $PSF$ | 26,862 | 75,908 |

$CSF$ | 22,895 | 102,934 | |

$CSB$ | 67,061 | 59,427 | |

Overestimation of the FOM (%) | $CSF$ | 35.60 | −14.77 |

$CSB$ | −21.71 | 149.64 |

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**MDPI and ACS Style**

Gratuze, M.; Alameh, A.-H.; Nabavi, S.; Nabki, F.
Control of Spring Softening and Hardening in the Squared Daisy. *Micromachines* **2021**, *12*, 448.
https://doi.org/10.3390/mi12040448

**AMA Style**

Gratuze M, Alameh A-H, Nabavi S, Nabki F.
Control of Spring Softening and Hardening in the Squared Daisy. *Micromachines*. 2021; 12(4):448.
https://doi.org/10.3390/mi12040448

**Chicago/Turabian Style**

Gratuze, Mathieu, Abdul-Hafiz Alameh, Seyedfakhreddin Nabavi, and Frederic Nabki.
2021. "Control of Spring Softening and Hardening in the Squared Daisy" *Micromachines* 12, no. 4: 448.
https://doi.org/10.3390/mi12040448