# Dynamics of Microbeams under Multi-Frequency Excitations

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## Abstract

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## 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

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**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