Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations

Abstract

The multi-frequency excitation of a microbeam, resting on a nonlinear foundation, is investigated and the governing equation of motion of the microbeam system is developed. The viscoelastic-type foundation is considered by assuming nonlinear parameters for both Pasternak and Winkler coefficients. The well-known Galerkin approach is utilized to discretize the governing equation of motion and to obtain its nonlinear ordinary differential equations. The multiple time-scales method is employed to study the multi-frequency excitation of the microbeam. Furthermore, the resonant conditions due to the external excitation as well as the combination resonances for the first two modes are investigated. The influences of different parameters, namely the Pasternak and Winkler coefficients, the position of the applied force and the geometrical factors on the frequency response of the system are examined.

Keywords

Nonlinear vibration microbeams multi-frequency excitation micro-scale effect Winkler and Pasternak foundation

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References

Abbasion

Rafsanjani

Avazmohammadi

et al. (2009) Free vibration of microscaled Timoshenko beams. Applied Physics Letters 95(14): Art. No. 143122.

Abdel-Rahman

Younis

Nayfeh

(2002) Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of Micromechanics and Microengineering 12(6): 759–766.

Askari H (2014) Nonlinear vibration and chaotic motion of uniform and non-uniform carbon nanotube resonators. MASc thesis, University of Ontario Institute of Technology, Oshawa, Canada, 2014.

Askari

Esmailzadeh

Zhang

(2014a) Nonlinear vibration analysis of nonlocal nanowires. Composites Part B: Engineering 67: 607–613.

Askari

Saadatnia

Esmailzadeh

et al. (2014b) Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations. Journal of Sound and Vibration 333(22): 5817–5835.

Batra

Porfiri

Spinello

(2008) Vibrations of narrow microbeams predeformed by an electric field. Journal of Sound and Vibration 309(3–5): 600–612.

Chao

PCP

Chiu

Liu

(2008) DC dynamic pull-in predictions for a generalized clamped-clamped micro-beam based on a continuous model and bifurcation analysis. Journal of Micromechanics and Microengineering 18(11): Article No. 115008.

Chin

C-M

Nayfeh

(1997) Three-to-one internal resonances in hinged-clamped beams. Nonlinear Dynamics 12(2): 129–154.

Dehrouyeh-Semnani

Mostafaei

Nikkhah-Bahrami

(2016) Free flexural vibration of geometrically imperfect functionally graded microbeams. International Journal of Engineering Science 105: 56–79.

10.

Diba

Esmailzadeh

Younesian

(2014) Nonlinear vibration analysis of isotropic plate with inclined part-through surface crack. Nonlinear Dynamics 78(4): 2377–2397.

11.

Farokhi

Ghayesh

(2015a) Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. International Journal of Mechanical Sciences 90: 133–144.

12.

Farokhi

Ghayesh

(2015b) Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. International Journal of Engineering Science 91: 12–33.

13.

Farokhi

Ghayesh

Amabili

(2013) Nonlinear resonant behavior of microbeams over the buckled state. Applied Physics A 113(2): 297–307.

14.

Zhang

(2013) Free vibration analysis of a microbeam subjected to a symmetric electrostatic field. Mechanics of Advanced Materials and Structures 20(4): 257–263.

15.

Ghayesh

Amabili

(2014) Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam. Composites Part B: Engineering 60: 371–377.

16.

Ghayesh

Farokhi

(2015a) Chaotic motion of a parametrically excited microbeam. International Journal of Engineering Science 96: 34–45.

17.

Ghayesh

Farokhi

(2015b) Coupled longitudinal-transverse-rotational behaviour of shear deformable microbeams. Composites Part B: Engineering 77: 319–328.

18.

Ghayesh

Farokhi

(2015c) Nonlinear dynamics of microplates. International Journal of Engineering Science 86: 60–73.

19.

Ghayesh

Farokhi

(2016) Parametric instability of microbeams in supercritical regime. Nonlinear Dynamics 83(3): 1171–1183.

20.

Ghayesh

Farokhi

Alici

(2015) Subcritical parametric dynamics of microbeams. International Journal of Engineering Science 95: 36–48.

21.

Ghayesh

Farokhi

Amabili

(2013) Coupled nonlinear size-dependent behaviour of microbeams. Applied Physics A 112(2): 329–338.

22.

Ghayesh

Farokhi

Amabili

(2014) In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Composites Part B: Engineering 60: 423–439.

23.

Ghorbanpour Arani

Haghparast

Ghorbanpour Arani

(2016) Size-dependent vibration of double-bonded carbon nanotube-reinforced composite microtubes conveying fluid under longitudinal magnetic field. Polymer Composites 37(5): 1375–1383.

24.

Grover

(2013) Transverse vibrations in micro-scale viscothermoelastic beam resonators. Archive of Applied Mechanics 83(2): 303–314.

25.

Guo

(2013) Thermo-elastic dissipation of microbeam resonators in the framework of generalized thermo-elasticity theory. Journal of Thermal Stresses 36(11): 1156–1168.

26.

Han

Zhang

Wang

(2015) Static bifurcation and primary resonance analysis of a MEMS resonator actuated by two symmetrical electrodes. Nonlinear Dynamics 80(3): 1585–1599.

27.

Hassanpour

Cleghorn

Esmailzadeh

et al. (2007) Vibration analysis of micro-machined beam-type resonators. Journal of Sound and Vibration 308(1-2): 287–301.

28.

Hassanpour

Cleghorn

Esmailzadeh

et al. (2011) Experimental measurement of resonance frequencies of asymmetric micro-bridge resonators. Journal of Intelligent Material Systems and Structures 22(2): 127–136.

29.

Jaber

Ramini

Younis

(2016) Multifrequency excitation of a clamped–clamped microbeam: Analytical and experimental investigation. Microsystems & Nanoengineering 2: 16002.

30.

Jamshidifar

Askari

Fidan

(2017) Parameter identification and adaptive control of carbon nanotube resonators. Asian Journal of Control 19(2): 1–10.

31.

Jia

Yang

Kitipornchai

et al. (2010) Free vibration of geometrically nonlinear micro-switches under electrostatic and Casimir forces. Smart Materials and Structures 19(11): Art. No. 115028.

32.

Lei

Guo

et al. (2016) Size-dependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity. AIP Advances 6(10): Art. No. 105202.

33.

Tang

Roukes

(2007) Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications. Nature Nanotechnology 2(2): 114–120.

34.

Nayfeh

Mook

(1995) Nonlinear Oscillations, New York: Wiley-VCH Verlag GmbH.

35.

Ramezani

Alasty

Akbari

(2006) Effects of rotary inertia and shear deformation on nonlinear free vibration of microbeams. Journal of Vibration and Acoustics 128(5): 611–615.

36.

Ross

Cabal

Trauernicht

et al. (2005) Temperature-dependent vibrations of bilayer microbeams. Sensors and Actuators A: Physical 119(2): 537–543.

37.

Saadatnia Z, Askari H, Younesian D, et al. (2013) Nonlinear vibration analysis and approximate solution of micro-beams. In: Proceedings, International Design Engineering Technical Conferences, Portland, Oregon, USA, 4–7 August 2013.

38.

Shabani

Hatami

Golzar

et al. (2013) Coupled vibration of a cantilever micro-beam submerged in a bounded incompressible fluid domain. Acta Mechanica 224(4): 841–850.

39.

Shafiei

Kazemi

Ghadiri

(2016) Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science 102: 12–26.

40.

Shenas

Malekzadeh

Mohebpour

(2016) Vibrational behavior of variable section functionally graded microbeams carrying microparticles in thermal environment. Thin-Walled Structures 108: 122–137.

41.

Subramanian

Oden

Kennel

et al. (2002) Glucose biosensing using an enzyme-coated microcantilever. Applied Physics Letters 81(2): 385–387.

42.

Trinh

Nguyen

et al. (2016) Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory. Composite Structures 154: 556–572.

43.

Vatankhah

Kahrobaiyan

Alasty

et al. (2013) Nonlinear forced vibration of strain gradient microbeams. Applied Mathematical Modelling 37(18–19): 8363–8382.

44.

Xia

Wang

Yin

(2010) Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. International Journal of Engineering Science 48(12): 2044–2053.

45.

Jia

(2008) Electromechanical-fluidic coupled dynamics for microbeams. Proc. IMechE Part C: J. Mechanical Engineering Science 222(3): 535–548.

46.

Younesian

Marjani

Esmailzadeh

(2013) Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations. Nonlinear Dynamics 71(1–2): 109–120.

47.

Younesian

Sadri

Esmailzadeh

(2014) Primary and secondary resonance analyses of clamped-clamped micro-beams. Nonlinear Dynamics 76(4): 1867–1884.

48.

Younis

Nayfeh

(2003) A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics 31(1): 91–117.