Exact Mode Shapes of T-shaped and Overhang-shaped Microcantilevers

Le Tri Dat, Vu Lan, Nguyen Duy Vy


Resonance frequencies and mode shapes of microcantilevers are of important interest in micro-mechanical systems for enhancing the functionality and applicable range of the cantilevers in vibration transducing, energy harvesting, and highly sensitive measurement. In this study, using the Euler-Bernoulli theory for beam, we figured out the exact mode shapes of cantilevers of varying widths such as the overhang- or T-shaped cantilevers. The obtained mode shapes have been shown to significantly deviate from the approximate forms of a rectangular cantilever that are commonly used in mechanics and physics. They were then used to figure out the resonance frequencies of the cantilever. The analytical solutions have been confirmed by using the finite element method simulations with very low deviation. This study suggested a method for correctly obtaining the resonance frequency of microcantilevers with complicated dimensions, such as the doubly clamped cantilever with the undercut, with the overhangs at the clamped positions, or with an attached mass in the middle.


microcantilever; mode shape; analytical method; overhang-shaped; T-shaped

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C. H. Metzger and K. Karrai, Nature 432 (2004) 1002.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc and N. Mavalvala, Phys. Rev. A 74 (2006) 021802(R).

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller and M. D. Lukin, Nat. Phys. 6 (2010) 602–608.

N. D. Vy, L. T. Dat and T. Iida, Appl. Phys. Lett. 109 (2016) 054102.

J. Fritz, M. K. Baller, H. P. Lang, H. Rothuizen, P. Vettiger, E. Meyer, H. J. Guntherodt, C. Gerber and J. K. Gimzewski, Science 288 (2000) 316–318.

B. Ilic, D. Czaplewski, H. G. Craighead, P. Neuzil, C. Campagnolo and C. Batt, Appl. Phys. Lett. 77 (2000) 450–452.

B. Ilic, D. Czaplewski, M. Zalalutdinov, H. G. Craighead, P. Neuzil, C. Campagnolo and C. Batt, J. Vac. Sci. Technol. B - Microelectron. Nanometer. Struct. Process. Meas. Phenom. 19 (2001) 2825–2828.

F. Huber, H. P. Lang, J. Zhang, D. Rimoldi and C. Gerber, Swiss medical weekly 145 (2015) .

C. A. Savran, S. M. Knudsen, A. D. Ellington and S. R. Manalis, Anal. Chem. 76 (2004) 3194–3198.

S.-J. Hyun, H.-S. Kim, Y.-J. Kim and H.-I. Jung, Sens. Actuators B: Chem. 117 (2006) 415–419.

P. Datskos, S. Rajic and I. Datskou, Ultramicroscopy 82 (2000) 49 – 56.

L. T. Dat, H. T. Huy and N. D. Vy, Commun. in Physics 28 (2018) 255.

T. Thundat, R. J. Warmack, G. Y. Chen and D. P. Allison, Appl. Phys. Lett. 64 (1994) 2894–2896.

S. Guillon, D. Saya, L. Mazenq, S. Perisanu, P. Vincent, A. Lazarus, O. Thomas and L. Nicu, Nanotechnology 22 (2011) 245501.

J. A. Plaza, K. Zinoviev, G. Villanueva, M. lvarez, J. Tamayo, C. Domnguez and L. M. Lechuga, Appl. Phys. Lett. 89 (2006) 094109.

S.-D. Kwon, Appl. Phys. Lett. 97 (2010) 164102.

J. E. Sader, Rev. Sci. Instrum. 66 (1995) 4583–4587.

J. E. Sader, J. A. Sanelli, B. D. Adamson, J. P. Monty, X. Wei, S. A. Crawford, J. R. Friend, I. Marusic, P. Mulvaney and E. J. Bieske, Rev. Sci. Instrum. 83 (2012) 103705.

G. Zhang, L. Zhao, Z. Jiang, S. Yang, Y. Zhao, E. Huang, X. Wang and Z. Liu, J. Phys. D: Appl. Phys. 44 (2011) 425402.

N. D. Vy, N. V. Cuong and C. M. Hoang, J. Mecha. 35 (2018) 351–358.

S. Timoshenko and D. Young, Engineering mechanics: Statics, no. v. 1, McGraw-Hill Book Company, Inc., 1937.

DOI: https://doi.org/10.15625/0868-3166/30/3/15080 Display counter: Abstract : 164 views. PDF : 120 views.


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