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Technical Brief

# Mass Sensitivity of Nonuniform Microcantilever Beams

[+] Author and Article Information
Sajal Sagar Singh

Mechanical and Aerospace Engineering,
Kandi 502285, India

Prem Pal

Department of Physics,
Kandi 502285, India

Ashok Kumar Pandey

Mechanical and Aerospace Engineering,
Kandi 502285, India
e-mail: ashok@iith.ac.in

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 28, 2016; final manuscript received June 22, 2016; published online August 8, 2016. Assoc. Editor: Miao Yu.

J. Vib. Acoust 138(6), 064502 (Aug 08, 2016) (7 pages) Paper No: VIB-16-1056; doi: 10.1115/1.4034079 History: Received January 28, 2016; Revised June 22, 2016

## Abstract

Microelectromechanical systems (MEMS) based cantilever beams have been widely used in various sensing applications. Previous studies have aimed at increasing the sensitivity of biosensors by reducing the size of cantilever beams to nanoscale. However, the influence of nonuniform cantilever beams on mass sensitivity has rarely been investigated. In this paper, we discuss the mass sensitivity with respect to linear and nonlinear response of nonuniform cantilever beam with linear and quartic variation in width. To do the analysis, we use the nonlinear Euler–Bernoulli beam equation with harmonic forcing. Subsequently, we derive the mode shape corresponding to linear, undamped, free vibration case for different types of beams with a tip mass at the end. After applying the boundary conditions, we obtain the resonance frequencies corresponding to various magnitudes of tip mass for different kinds of beams. To do the nonlinear analysis, we use the Galerkin approximation and the method of multiple scales (MMS). Analysis of linear response indicates that the nondimensional mass sensitivity increases considerably by changing the planar geometry of the beam as compared to uniform beam. At the same time, sensitivity further increases when the nonuniform beam is actuated in higher modes. Similarly, the frequency shift of peak amplitude of nonlinear response for a given nondimensional tip mass increases exponentially and decreases quadratically with tapering parameter, α, for diverging and converging nonuniform beam with quartic variation in width, respectively. For the converging beam, we also found an interesting monotonically decreasing and increasing behavior of mass sensitivity with tapering parameter α giving an extremum point at $α=0.5$. Overall analysis indicates a potential application of the nonuniform beams with quartic converging width for biomass sensor.

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## References

Ekinci, K. L. , Yang, Y. T. , and Roukes, M. L. , 2004, “ Ultimate Limits to Inertial Mass Sensing Based Upon Nanoelectromechanical Systems,” J. Appl. Phys., 95(5), pp. 2682–2689.
Zalalutdinov, M. , Ilic, B. , Czaplewski, D. , Zehnder, A. , Craighead, H. G. , and Parpia, J. M. , 2000, “ Frequency-Tunable Micromechanical Oscillator,” Appl. Phys. Lett., 77(20), pp. 3287–3289.
Pandey, A. K. , Venkatesh, K. P. , and Pratap, R. , 2000, “ Effect of Metal Coating and Residual Effect on the Resonant Frequency of MEMS Resonators,” Sadhana, 34(4), pp. 651–662.
Kambali, P. N. , Swain, G. , and Pandey, A. K. , 2016, “ Frequency Analysis of Linearly Coupled Modes of MEMS Arrays,” ASME J. Vib. Acoust., 138(2), p. 021017.
Kambali, P. N. , and Pandey, A. K. , 2015, “ Nonlinear Response of a Microbeam Under Combined Direct and Fringing Field Excitation,” ASME J. Comput. Nonlinear Dyn., 10(5), p. 051010.
Kim, I. K. , and Lee, S. I. , 2013, “ Theoretical Investigation of Nonlinear Resonances in a Carbon Nanotube Cantilever With a Tip-Mass Under Electrostatic Excitation,” J. Appl. Phys., 114(10), p. 104303.
Kacem, N. , Arcamone, J. , Perez-Murano, F. , and Hentz, S. , 2010, “ Dynamic Range Enhancement of Nonlinear Nanomechanical Resonant Cantilevers for Highly Sensitive NEMS Gas/Mass Sensor Applications,” J. Micromech. Microeng., 20(4), p. 045023.
Balachandran, B. , and Magrab, E. , 2009, Vibrations, Cengage Learning, Toronto, ON, Canada.
Chaterjee, S. , and Pohit, G. , 2009, “ A Large Deflection Model for the Pull-In Analysis of Electrostatically Actuated Microcantilever Beams,” J. Sound Vib., 322(4–5), pp. 969–986.
Anderson, T. J. , Nayfeh, A. H. , and Balachandran, B. , 1996, “ Experimental Verification of the Importance of the Nonlinear Curvature in the Response of a Cantilever Beam,” ASME J. Vib. Acoust., 118(1), pp. 21–27.
Anderson, T. J. , Balachandran, B. , and Nayfeh, A. H. , 1994, “ Nonlinear Resonances in a Flexible Cantilever Beam,” ASME J. Vib. Acoust., 116(4), pp. 480–484.
Blevins, R. D. , and Plunkett, R. , 1980, “ Formulas for Natural Frequency and Mode Shape,” ASME J. Appl. Mech., 47(2), pp. 461–462.
Mabie, H. H. , and Rogers, C. B. , 1964, “ Transverse Vibrations of Doubletapered Cantilever Beams With End Support and With End Mass,” J. Acoust. Soc. Am., 36(3), pp. 463–469.
Mabie, H. H. , and Rogers, C. B. , 1974, “ Transverse Vibrations of Doubletapered Cantilever Beams With End Support and With End Mass,” J. Acoust. Soc. Am., 55(5), pp. 986–991.
Lau, J. H. , 1984, “ Vibration Frequencies of Tapered Bars With End Mass,” ASME J. Appl. Mech., 51(1), pp. 179–181.
Auciello, N. M. , and Nole, G. , 1998, “ Vibrations of a Cantilever Tapered Beam With Varying Section Properties and Carrying a Mass at the Free End,” J. Sound Vib., 214(1), pp. 105–119.
Wang, H. C. , 1967, “ Generalized Hypergeometric Function Solutions on Transverse Vibration of a Class of Nonuniform Beams,” ASME J. Appl. Mech., 34(3), pp. 702–708.
Williams, F. W. , and Banerjee, J. R. , 1985, “ Flexural Vibration of Axially Loaded Beams With Linear or Parabolic Taper,” J. Sound Vib., 99(1), pp. 121–138.
Abrate, S. , 1995, “ Vibration of Non-Uniform Rods and Beams,” J. Sound Vib., 185(4), pp. 703–716.
Wang, C. Y. , 2013, “ Vibration of a Tapered Cantilever of Constant Thickness and Linearly Tapered Width,” Arch. Appl. Mech., 83(1), pp. 171–176.
Singh, S. S. , Pal, P. , and Pandey, A. K. , 2015, “ Pull-In Analysis of Non-Uniform Microcantilever Beams Under Large Deflection,” J. Appl. Phys., 118(20), p. 204303.
Clark, R. L. , Burdisso, R. A. , and Fuller, C. R. , 1993, “ Design Approaches for Shaping Polyvinylidene Fluoride Sensors in Active Structural Acoustic Control (ASAC),” J. Intell. Mater. Syst. Struct., 4(3), pp. 354–365.

## Figures

Fig. 1

(a) Transverse vibration and axial stretching of a cantilever beam under uniformly distributed load. (b) The beam geometry used in our analysis. α = 0 corresponds to a uniform beam, α>0 corresponds to diverging beam, and α<0 corresponds to converging beams. Furthermore, n = 1 implies beam with linear variation in width, and n = 4 implies beam with quartic variation in width.

Fig. 2

Variation of nondimensional frequency versus tapering parameter for (a) linear diverging beam, (b) linear converging beam, (c) quartic diverging beam, (d) quartic converging beam, (e) variation of nondimensional mass sensitivity with tapering ratio for different types of tapering, and (f) variation of nondimensional mass sensitivity versus mode number for uniform, μ=0.0, and nonuniform beam with quartic converging beams, α=−0.3 and α=−0.6

Fig. 3

(a) Numerical and approximate solution for uniform beam, (b) converging beam with linear tapering, (c) converging beam with quartic tapering, (d) nonlinear response showing the shift in frequency due to added mass, (e) the frequency shift due to added mass at various α for converging beam with linear variation in width, (f) diverging beam with linear variation in width, (g) converging beam with quartic variation in width, and (h) diverging beam with quartic variation in width

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