# Dynamics of Microbeams under Multi-Frequency Excitations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Response to Two Harmonic Sources

#### 2.1. Problem Formulation

#### 2.2. Case I: Primary Resonance

#### 2.3. Case II: Secondary Resonances

## 3. Case III: Response to Three Harmonic Sources

## 4. Experimental Case Study

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Clark, J.R.; Hsu, W.T.; Abdelmoneum, M.A.; Nguyen, C.C. High-Q UHF micromechanical radial-contour mode disk resonators. J. Microelectromech. Syst.
**2005**, 14, 1298–1310. [Google Scholar] [CrossRef] - Wang, J.; Ren, Z.; Nguyen, C.C. 1.156-GHz self-aligned vibrating micromechanical disk resonator. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2004**, 51, 1607–1628. [Google Scholar] [CrossRef] [PubMed] - Xie, Y.; Li, S.S.; Lin, Y.W.; Ren, Z.; Nguyen, C.T.C. 1.52-GHz micromechanical extensional wine-glass mode ring resonators. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2008**, 55, 890–907. [Google Scholar] [PubMed] - Nguyen, C.T.-C. MEMS technology for timing and frequency control. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2007**, 54, 251–270. [Google Scholar] [CrossRef] [PubMed] - Wong, A.-C.; Nguyen, C.T.-C. Micromechanical mixer-filters (“mixlers”). J. Microelectromech. Syst.
**2004**, 13, 100–112. [Google Scholar] [CrossRef] - Fedder, G.K. CMOS-MEMS resonant mixer-filters. In Proceedings of the IEEE International Electron Devices Meeting IEDM Technical Digest, Washington, DC, USA, 5–7 December 2005; pp. 274–277.
- Chen, F.; Brotz, J.; Arslan, U.; Lo, C.C.; Mukherjee, T.; Fedder, G.K. CMOS-MEMS resonant RF mixer-filters. In Proceedings of the 18th IEEE International Conference on Micro Electro Mechanical Systems (MEMS 2005), Miami Beach, FL, USA, 30 January–3 February 2005; pp. 24–27.
- Koskenvuori, M.; Tittonen, I. Improvement of the conversion performance of a resonating multimode microelectromechanical mixer-filter through parametric amplification. IEEE Electron Device Lett.
**2007**, 28, 970. [Google Scholar] [CrossRef] - Elnagar, A.M.; El-Bassiouny, A.F. Response of self-excited three-degree-of-freedom systems to multifrequency excitations. Int. J. Theor. Phys.
**1992**, 31, 1531–1548. [Google Scholar] [CrossRef] - Uranga, A.; Verd, J.; Lopez, J.L.; Teva, J.; Abadal, G.; Torres, F.; Barniol, N. Fully integrated MIXLER based on VHF CMOS-MEMS clamped-clamped beam resonator. Electron. Lett.
**2007**, 43, 452–454. [Google Scholar] [CrossRef] - Hecht, D. Multifrequency acoustooptic diffraction. IEEE Trans. Sonics Ultrason.
**1977**, 24, 7–18. [Google Scholar] [CrossRef] - Hammad, B.K. Natural Frequencies and Mode Shapes of Mechanically Coupled Microbeam Resonators with an Application to Micromechanical Filters. Shock Vib.
**2014**, 2014, 939467. [Google Scholar] [CrossRef] - El-Bassiouny, A.F.; Abd El-Latif, G.M. Resonances in nonlinear structure vibrations under multifrequency excitations. Phys. Scr.
**2006**, 74, 410. [Google Scholar] [CrossRef] - Ramini, A.; Ibrahim, A.I.; Younis, M.I. Mixed frequency excitation of an electrostatically actuated resonator. Microsyst. Technol.
**2016**, 22, 1967–1974. [Google Scholar] [CrossRef] - Jaber, N.; Ramini, A.; Hennawi, Q.; Younis, M.I. Wideband MEMS resonator using multifrequency excitation. Sens. Actuators A Phys.
**2016**, 242, 140–145. [Google Scholar] [CrossRef] - Jaber, N.; Ramini, A.; Younis, M.I. Multifrequency excitation of a clamped–clamped microbeam: Analytical and experimental investigation. Microsyst. Nanoeng.
**2016**, 2, 16002. [Google Scholar] [CrossRef] - Ilyas, S.; Ramini, A.; Arevalo, A.; Younis, M.I. An experimental and theoretical investigation of a micromirror under mixed-frequency excitation. J. Microelectromech. Syst.
**2015**, 24, 1124–1131. [Google Scholar] [CrossRef] - Farokhi, H.; Ghayesh, M.H. Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int. J. Eng. Sci.
**2015**, 91, 12–33. [Google Scholar] [CrossRef] - Rashvand, K.; Rezazadeh, G.; Mobki, H.; Ghayesh, M.H. On the size-dependent behavior of a capacitive circular micro-plate considering the variable length-scale parameter. Int. J. Mech. Sci.
**2013**, 77, 333–342. [Google Scholar] [CrossRef] - Farokhi, H.; Ghayesh, M.H. Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. Int. J. Mech. Sci.
**2015**, 90, 133–144. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Farokhi, H.; Amabili, M. In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos. Part B Eng.
**2014**, 60, 423–439. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Farokhi, H. Coupled longitudinal-transverse-rotational behaviour of shear deformable microbeams. Compos. Part B Eng.
**2015**, 77, 319–328. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Farokhi, H.; Alici, G. Subcritical parametric dynamics of microbeams. Int. J. Eng. Sci.
**2015**, 95, 36–48. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Farokhi, H.; Alici, G. Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes. Int. J. Mech. Sci.
**2015**, 103, 247–264. [Google Scholar] [CrossRef] - Younis, M.I. MEMS Linear and Nonlinear Statics and Dynamics; Springer: New York, NY, USA, 2011. [Google Scholar]
- Nayfeh, A.H.; Younis, M.I. Dynamics of MEMS resonators under superharmonic and subharmonic excitations. J. Micromech. Microeng.
**2005**, 15, 1840. [Google Scholar] [CrossRef]

**Figure 1.**Schematic for the microbeam under direct current (DC) and two alternating current (AC) loads.

**Figure 2.**(

**a**) Variation of the static deflection of the microbeam tip with the DC voltage; (

**b**) Variation of the natural frequency of the microbeam with the DC voltage.

**Figure 3.**Frequency response curve near beam resonance, ${V}_{\mathrm{DC}}=5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=2\text{}\mathrm{V}$, $\text{}{V}_{\mathrm{AC}2}=0$, $c=0.1$.

**Figure 4.**Multi frequency excitation (

**a**) Lower amplitude: ${V}_{\mathrm{DC}}=2.5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=0.5\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=906\text{}\mathrm{MHz}$, ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, $c=0.05$; (

**b**) Higher amplitude: ${V}_{\mathrm{DC}}=3.5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=906\text{}\mathrm{MHz}$, $c=0.05$.

**Figure 5.**Frequency response curves near superharmonic resonances for ${V}_{\mathrm{DC}}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=0.35\text{}\mathrm{V}$ (

**a**) Single source excitation at ${V}_{\mathrm{AC}2}=0$; (

**b**) two-source excitation at ${V}_{\mathrm{AC}2}=0.1\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=7.5\text{}\mathrm{kHz}$.

**Figure 6.**Frequency response curves near superharmonic resonances and two source excitation at ${V}_{\mathrm{DC}}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=0.35\text{}\mathrm{V}$ (

**a**) ${V}_{\mathrm{AC}2}=0.15\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=7.5\text{}\mathrm{kHz}$; (

**b**) ${V}_{\mathrm{AC}2}=0.25\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=7.5\text{}\mathrm{KHz}$.

**Figure 7.**Subharmonic frequency response curve for ${V}_{\mathrm{DC}}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=0.15\text{}\mathrm{V}$ and (

**a**) Single source excitation, ${V}_{\mathrm{AC}2}=0$; (

**b**) two source excitation at ${V}_{\mathrm{AC}2}=0.01\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=15\text{}\mathrm{kHz}$.

**Figure 8.**Subharmonic frequency response curve for ${V}_{\mathrm{DC}}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=0.15\text{}\mathrm{V}$, (

**a**) ${V}_{\mathrm{AC}2}=0.013\text{}\mathrm{V}$; (

**b**) ${V}_{\mathrm{AC}2}=0.017\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=15\text{}\mathrm{kHz}$.

**Figure 9.**Frequency response curve near beam primary resonance at ${V}_{\mathrm{DC}}=5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=2\text{}\mathrm{V}$, ${V}_{\mathrm{AC}2}=0$, and $c=0.1$.

**Figure 10.**Multifrequency response near the beam primary resonance due to three-source excitation at ${V}_{\mathrm{DC}}=5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=0.7\text{}\mathrm{V}$, ${V}_{\mathrm{AC}2}=5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}3}=5\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=4\text{}\mathrm{MHz}$, ${\mathrm{\Omega}}_{3}=8\text{}\mathrm{MHz}$, $c=0.1$.

**Figure 11.**Multi-frequency excitation off-resonance: ${V}_{\mathrm{DC}}=3\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, ${V}_{\mathrm{AC}3}=6\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=885\text{}\mathrm{MHz}$, ${\mathrm{\Omega}}_{3}=890\text{}\mathrm{MHz}$, $c=0.1$.

**Figure 12.**Multi-frequency excitation off-resonance: ${V}_{\mathrm{DC}1}=5\text{}\mathrm{V}$, ${V}_{\mathrm{DC}2}=0\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=1.5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, ${V}_{\mathrm{AC}3}=8.5\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=887.72\text{}\mathrm{MHz}$, ${\mathrm{\Omega}}_{3}=887.92\text{}\mathrm{MHz}$, $c=0.1$, amplified response due to higher loads.

**Figure 13.**Multi-frequency excitation off-resonance: ${V}_{\mathrm{DC}}=5\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=1\text{}\mathrm{V}$, ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, ${V}_{\mathrm{AC}3}=8\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{2}=887.72\text{}\mathrm{MHz}$, ${\mathrm{\Omega}}_{3}=887.92\text{}\mathrm{MHz}$, $c=0.1$, the two peaks merge as one combined peak.

**Figure 14.**(

**a**) Top view of the fabricated microbeam; (

**b**) experimental setup used for testing the microelectromechanical systems (MEMS) devices.

**Figure 15.**Frequency sweep results near the primary resonance of the first mode at ${V}_{\mathrm{DC}}=3\text{}\mathrm{V}$ (

**a**) different ${V}_{\mathrm{AC}}$ values; (

**b**) ${V}_{\mathrm{AC}}=2\text{}\mathrm{V}$.

**Figure 16.**Resonances using two-source excitation near the first mode at ${V}_{\mathrm{DC}}=3\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=2\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{1}=$ Swept, ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, (

**a**) ${\mathrm{\Omega}}_{2}=$ as shown; (

**b**) ${\mathrm{\Omega}}_{2}=1\text{}\mathrm{kHz}$.

**Figure 17.**Resonances using three-source excitation near the first mode at ${V}_{\mathrm{DC}}=3\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=2\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{1}=$ Swept, ${\mathrm{\Omega}}_{2}=1\text{}\mathrm{kHz}$ (

**a**) ${V}_{\mathrm{AC}2}=\mathrm{as}\text{}\mathrm{shown}$, ${V}_{\mathrm{AC}3}=\mathrm{as}\text{}\mathrm{shown}$, ${\mathrm{\Omega}}_{3}=5\text{}\mathrm{kHz}$; (

**b**) ${V}_{\mathrm{AC}2}=6\text{}\mathrm{V}$, ${V}_{\mathrm{AC}3}=6\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{3}=2\text{}\mathrm{kHz}$.

**Figure 18.**Resonances using four-source excitation near the first mode at ${V}_{\mathrm{DC}}=3\text{}\mathrm{V}$, ${V}_{\mathrm{AC}1}=2\text{}\mathrm{V}$, ${\mathrm{\Omega}}_{1}=\mathrm{Swept}$, ${V}_{\mathrm{AC}2}=\mathrm{as}\text{}\mathrm{shown}$, ${\mathrm{\Omega}}_{2}=1\text{}\mathrm{kHz}$, ${V}_{\mathrm{AC}3}=\mathrm{as}\text{}\mathrm{shown}$, ${\mathrm{\Omega}}_{3}=3\text{}\mathrm{kHz}$, ${V}_{\mathrm{AC}4}=\mathrm{as}\text{}\mathrm{shown}$, ${\mathrm{\Omega}}_{4}=5\text{}\mathrm{kHz}$.

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

Young’s Modulus ($E$) | $160\text{}\mathrm{Gpa}$ |

Density ($\mathsf{\rho}$) | $2332\text{}\mathrm{Kg}/{\mathrm{m}}^{3}$ |

Beam length (L) | $2\text{}\mathsf{\mu}\mathrm{m}$ |

Beam width ($b$) | $200\text{}\mathrm{nm}$ |

Beam thickness ($h$) | $158\text{}\mathrm{nm}$ |

Gab ($d$) | $75\text{}\mathrm{nm}$ |

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

$E$ | $82.7\text{}\mathrm{Gpa}$ |

$\mathsf{\rho}$ | $1400\text{}\mathrm{Kg}/{\mathrm{m}}^{3}$ |

${c}_{\mathrm{non}}$ | $0.00289$ |

L | $500\text{}\mathsf{\mu}\mathrm{m}$ |

$b$ | $50\text{}\mathsf{\mu}\mathrm{m}$ |

h | $3\text{}\mathsf{\mu}\mathrm{m}$ |

$d$ | $3\text{}\mathsf{\mu}\mathrm{m}$ |

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

Young’s Modulus ($E$) | $160\text{}\mathrm{Gpa}$ |

Density ($\mathsf{\rho}$) | $2332\text{}\mathrm{Kg}/{\mathrm{m}}^{3}$ |

Beam length ($L$) | $1.85\text{}\mathsf{\mu}\mathrm{m}$ |

Beam width ($b$) | $200\text{}\mathrm{nm}$ |

Beam thickness ($h$) | $180\text{}\mathrm{nm}$ |

Gab ($d$) | $75\text{}\mathrm{nm}$ |

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

Non-dimensional axial force (Nnon) | $76.3$ |

Flexural regidity (EI) | 2.93 × 10^{−11} N·m^{2} |

Damping ratio | 6 × 10^{−4} |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ibrahim, A.; Jaber, N.; Chandran, A.; Thirupathi, M.; Younis, M. Dynamics of Microbeams under Multi-Frequency Excitations. *Micromachines* **2017**, *8*, 32.
https://doi.org/10.3390/mi8020032

**AMA Style**

Ibrahim A, Jaber N, Chandran A, Thirupathi M, Younis M. Dynamics of Microbeams under Multi-Frequency Excitations. *Micromachines*. 2017; 8(2):32.
https://doi.org/10.3390/mi8020032

**Chicago/Turabian Style**

Ibrahim, Alwathiqbellah, Nizar Jaber, Akhil Chandran, Maloth Thirupathi, and Mohammad Younis. 2017. "Dynamics of Microbeams under Multi-Frequency Excitations" *Micromachines* 8, no. 2: 32.
https://doi.org/10.3390/mi8020032