Research Papers

Frequency Shifts of Micro and Nano Cantilever Beam Resonators Due to Added Masses

[+] Author and Article Information
Adam Bouchaala

Physical Sciences and Engineering Division,
King Abdullah University of Science
and Technology,
Thuwal 23955-9600, Saudi Arabia

Ali H. Nayfeh

Department of Engineering Science
and Mechanics,
Virginia Polytechnic Institute and State
University, MC 0219,
Blacksburg, VA 24061;
Department of Mechanical Engineering,
University of Jordan,
Amman 11942, Jordan

Mohammad I. Younis

Physical Sciences and Engineering Division,
King Abdullah University of Science
and Technology,
Thuwal 23955-9600, Saudi Arabia
e-mail: Mohammad.Younis@kaust.edu.sa

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 28, 2015; final manuscript received February 19, 2016; published online May 25, 2016. Assoc. Editor: M. Porfiri.

J. Dyn. Sys., Meas., Control 138(9), 091002 (May 25, 2016) (9 pages) Paper No: DS-15-1599; doi: 10.1115/1.4033075 History: Received November 28, 2015; Revised February 19, 2016

We present analytical and numerical techniques to accurately calculate the shifts in the natural frequencies of electrically actuated micro and nano (carbon nanotubes (CNTs)) cantilever beams implemented as resonant sensors for mass detection of biological entities, particularly Escherichia coli (E. coli) and prostate specific antigen (PSA) cells. The beams are modeled as Euler–Bernoulli beams, including the nonlinear electrostatic forces and the added biological cells, which are modeled as discrete point masses. The frequency shifts due to the added masses of the cells are calculated for the fundamental and higher-order modes of vibrations. Analytical expressions of the natural frequency shifts under a direct current (DC) voltage and an added mass have been developed using perturbation techniques and the Galerkin approximation. Numerical techniques are also used to calculate the frequency shifts and compared with the analytical technique. We found that a hybrid approach that relies on the analytical perturbation expression and the Galerkin procedure for calculating accurately the static behavior presents the most computationally efficient approach. We found that using higher-order modes of vibration of micro-electro-mechanical-system (MEMS) beams or miniaturizing the sizes of the beams to nanoscale leads to significant improved frequency shifts, and thus increased sensitivities.

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Grahic Jump Location
Fig. 1

Schematic of an electrically actuated cantilever microbeam with an added point mass

Grahic Jump Location
Fig. 2

The tip displacement of a cantilever microbeam

Grahic Jump Location
Fig. 3

Variation of the first natural frequency with the DC voltage

Grahic Jump Location
Fig. 4

Frequency shift as a function of the mass position for the (a) first, (b) second, and (c) third modes

Grahic Jump Location
Fig. 5

Frequency shift as a function of DC voltage

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Fig. 6

Schematic of an electrically actuated CNT with an added point mass

Grahic Jump Location
Fig. 7

The static tip displacement of a cantilever CNT



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