Nonlinear Vibration of a Microbeam on a Winkler Foundation and Subjected to an Axial Load using Modified Couple Stress Theory

Authors

1 Associate Professor, Mechanical Engineering Department, Parand Branch, Islamic Azad University, Parand, Iran

2 Mechanical Engineering Department, Parand Branch, Islamic Azad University, Parand, Iran

Abstract

In this paper, nonlinear vibration analysis of a microbeam on Winkler type of foundation and subjected to an axial compressive load on its both ends is investigated. The partial differential governing equation of motion in transverse direction for the Euler-Bernoulli microbeam considering the Hook’s law based on the couple stress theory and applying the Hamilton principle is derived. Using Galerkin method, the partial differential equation of vibration in lateral direction is converted to an ordinary differential equation and then is analytically solved using the He’s nonlinear method to obtain frequency response of the microbeam. Effect of changes of various parameters such as geometrical size scale of microbeam, stiffness of Winkler foundation and axial compressive load on the nonlinear and linear natural frequencies are all investigated. It is seen that by increasing the axial compressive load, the dimensionless ratio of nonlinear frequency to linear frequency increases and the value of this ratio for the pinned-pinned microbeam is greater than the one for a clamped-clamped microbeam.

Keywords

References

 [1] Nayfeh AH, Mook DT (1995) Nonlinear oscillation. Wiley, New York, USA.
[2] Nayfeh AH, Pai PF (2004) Linear and nonlinear structural mechanics.Wiley, New Jersey, USA.
[3] Reissner E (1972) On one dimensional finite strain beam theory, the plane problem. J Appl Mech Tech Phy 23(5): 795-894.
[4] Simo JC (1985) A finite strain beam formulation, the three dimensional dynamic problem, part I, computational methods. Appl Mech Eng 49: 55-70.
[5] Simo JC, Vu-quoc L (1986) A three dimensional finite-strain rod model. part II, Ccmputational aspects, computational methods. Appl Mech Eng 58: 79-116.
[6] Crespodasilva MRM (1988) Nonlinear flexural-flexural-torsional-extensional dynamics of beams-I. Int J Solids Struct 24(12): 1225-1234.
[7] Crespodasilva MRM (1988) Nonlinear flexural-flexural-torsional-extensional dynamics of beams-II, responses analysis. Int J Solids Struct 24(12): 1235-1242.
[8] Singh G, Rao GV, and Iyengar NGR (1990) Re-investigation of large-amplitude free vibrations of beams using finite element. J Sound Vib 143(2): 351-355.
[9] Lewandowski R (1994) Nonlinear free vibration of beams by the finite element and continuation method. J Sound Vib 170(5): 577-593.
[10] Foda MA (1999) Influence of shear deformation and rotary inertia on nonlinear free vibration of beam with pinned ends. Comput Struct 71: 663-670.
[11] Ribeiro P, Petyt M (1999) Nonlinear vibration of beams with internal resonance by the hierarchical and finite element method. J Sound Vib 244(4) 591-624.
[12] Patel BP, Ganapathi M (2001) Nonlinear torsional vibration and damping analysis of sandwich beams. J Sound Vib 240(2): 385-393.
[13] Luczko J (2002) Bifurcations and internal resonances in space curved Rrods. Comput Methods Appl Mech Eng 191: 3271-3296.
[14] Attard MM (2003) Finite strain beam-theory. Int J Solids Struct 40(17): 4563-4584.
[15] Agrawal S, Chakraborty A, Gopaluhrishnan S, (2006) Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solution. Compos Struct 72: 91-104.
[16] Mata P, Oller S, Barbat AH (2008) Dynamic analysis of beam structures considering geometric and constitutive nonlinearity, computational Mmethod. Appl Mech Eng 197: 857-878.
[17] Gupta RK, Balou GJ, Janardhan GR, Rao GV (2009) Relatively simple finite element formulation for the large amplitude free vibrations of uniform beam. Finite Elem Anal Des 45: 624-631.
[18] Stoykov S, Ribeiro P (2010) Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross sections, the influence of warping, shear deformation and longitudinal displacement. Int J Mech Sci 52(11): 1505-1521.
[19] Ke L-L (2011) Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 93: 342-350.
[20] Simsek M, Yurtcu HH (2013) Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Composite Structures 97: 378-386.
[21] Simsek M, Reddy JN (2013) A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos Struct 101: 47-58.
[22] Simsek M (2014) Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method. Compos Struct 112: 264-272.
[23] Zhang B, He Y, Liu D, Gan Z, Shen L (2013) A novel size-dependent functionally graded curved mircobeam model based on the strain gradient elasticity theory. Compos Struct 106: 374-392.
]24[ عطار ع، طهماسبی پور م، دهقان محمد (1397) بررسی تاثیر پارامترهای هندسی بر جابه‌جایی خارج از صفحه میکروتیر پیزوالکتریکی با سطح مقطع T شکل. مجله علمی پژوهشی مکانیک سازه‌ها و شاره‌ها 9-1 :(4)8.
]25[ سام‌پور س، معین خواه ح، رحمانی ح (1398) حل تحلیلی پاسخ گذرای غیرخطی میکروتیر ویسکوالاستیک با تحریک الکتریکی بر اساس تئوری الاستیسیته ریز قطبی. مجله علمی پژوهشی مکانیک سازه‌ها و شاره‌ها 138-125 :(3)9.
[26] Mamaghani AE, Khadem SE, Bab S (2016) Vibration control of a pipe conveying fluid under external periodic excitation using a nonlinear energy sink. Nonlinear Dyn 86(3): 1761-1795.
[27] Mamaghani AE, Khadem SE, Bab S (2018) Irreversible passive energy transfer of an immersed beam subjected to a sinusoidal flow via local nonlinear attachment. Int J Mech Sci 138: 427-447
[28] Ebrahimi-Mamaghani A, Khadem SE (2016) Vibration analysis of a beam under external periodic excitation using anonlinear energy sink. Modares Mechanical Engineering 16(9): 186-194. (in Persian)
[29] Ebrahimi-Mamaghani A, Sotudeh-ghrebagh R, Zarghami R, Mostoufi N (2019) Dynamics of two-phase flow in vertical pipes. J Fluid Struct 87: 150-173.
[30] Ebrahimi-Mamaghani A, Mirtalebi SH, Ahmadian MT, Mostoufi N (2020) Magneto-mechanical stability of axially functionally graded supported nanotubes. Mater Res Express 6(12): 1250c5.
[31] Mirtalebi SH, Ebrahimi-Mamaghani A, Ahmadian MT, Mostoufi N (2019) Vibration control and manufacturing of intelligibly designed axially functionally graded cantilevered macro/micro-tubes. IFAC-PapersOnLine 52(10): 382-387.
Journal of Solid and Fluid Mechanics
Volume 10, Issue 4
December 2021
Pages 181-194

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