A Semi-analytical Solution for Flexural Vibration of Micro Beams Based on the Strain Gradient Theory

Authors

1 Sharif University of Technology Mechanical Engineering Department

2 Sharif University of Technology Mechanical Engineering Department

Abstract

In this paper, the flexural free vibrations of three dimensional micro beams are investigated based on strain gradient theory. The most general form of the strain gradient theory which contains five higher-order material constants has been applied to the micro beam to take the small-scale effects into account. Having considered the Euler-Bernoulli beam model, governing equations of motion are written by utilizing the Hamilton’s principle. Then, the state-space solution technique is used to find some solutions for natural frequencies of the beam under various boundary conditions. The numerical results show that the resonant frequencies are significantly dependent on the length scale parameter of the micro beam. The less the non-dimensional length scale is, the more deviation appears between results obtained for natural frequencies of micro shaft by strain gradient theory and classical continuum theory. Moreover, except for a micro shaft which is simply supported at both ends, the extra type of boundary conditions emerges from using strain gradient theory significantly affects the results.

Keywords

Main Subjects


1 .Aifantis, E.C, Strain gradient interpretation of size effects. Int. J. Fract. 95(1999) 299-314.

2 .Ansari, R. Gholami, R. Faghih Shojaei, M. Mohammadi, V. Sahmani, S., Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Compos. Struct. 100 (2013) 385-397.

3 .Ansari, R. Gholami, R. Sahmani, S, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos. Struct. 94(2011)221-228.

 

4 .Asghari. M, Kahrobaiyan, M. H, Ahmadian M.T,  A nonlinear Timoshenko beam formulation based on the modified couple stress theory. International Journal of Eng. Sci. 48 (2010a) 1749-1761.

5 .Asghari.M. Ahmadian, M.T, Kahrobaiyan, M.H, Rahaeifard, M, On the size-dependent behavior of functionally graded micro-beams. Materials & Design 31,(2010b) 2324-2329.

6 .Asghari. M, Kahrobaiyan, M.H, Nikfar. M, Ahmadian. M.T, A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech. 223 (2012) 1233-1249.

7 .Beskou. S.P, Polyzos. D, Beskos. D.E, Dynamic analysis of gradient elastic flexural beams. Struct. Eng. Mech. Material 15, (2003a)705-716.

8 .Beskou. S.P, Tsepoura. K.G, Polyzos. D,  Bending and stability analysis of gradient elastic beams. Int. J. Solid Struct. 40 (2003b) 385-400.

9 .Boer. M.P.D, Luck. D.L, Ashurst. W.R, High-performance surface-micromachined inchworm actuator. J. Microelectromech. System 13(2004) 63-74.

10 .Danesh. V, Asghari. M,  Analysis of micro-rotating disks based on the strain gradient elasticity. Acta Mech.(2013).

11 .Fleck.  N.A, Muller. G.M, Ashby. M.F, Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia 42, (1994) 475-487.

12 .Ghayesh. M.H, Amabili. M, Farokhi, H, Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63 (2013) 52-60.

13 .Hall. N.A, Okandan. M, Degertekin. F.L, Surface and bulk-silicon-micromachined optical displacement sensor fabricated with the Swift-Lite process. J. Microelectromechanic System 15 (2006) 770-776.

14 .Kahrobaiyan. M.H, Asghari. M, Ahmadian. M.T,  Strain gradient beam element. Finite Elements in Analysis and Design 68 (2013) 63-75.

15 .Kahrobaiyan. M.H,  Asghari. M, Rahaeifard.  M,  Ahmadian.  M.T, A nonlinear strain gradient beam formulation. Int. J. Eng. Sci. 49 (2011) 1256-1267.

16 .Kahrobaiyan. M.H, Rahaeifard. M, Tajalli. S.A, Ahmadian. M.T, A strain gradient functionally graded Euler–Bernoulli beam formulation. Int. J. Eng. Sci. 52(2012) 65-76.

17 .Kang. X  and Xi. Z.W, Size effect on the dynamic characteristic of a micro beam based on cosserat theory. J. Mech. Strength 29(2007) 1-4.

18 .Ke. L-L, Wang. Y-S, Yang. J, Kitipornchai. S, Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. 50(2012) 256-267.

19 .Kong. S, Zhou. S, Nie. Z, Wang. K, The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int. J. Eng. Sci. 46 ( 2008) 427-437.

20 .Kong. S, Zhou. S, Nie. Z, Wang. K, Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int. J. Eng. Sci. 47(2009) 487-498.

21 .Lam. D.C.C, Yang. F, Chong. A.C.M, Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solid 51(2003) 1477-1508.

22 .Lun. F.Y, Zhang. P, Gao. F.B, Jia. H.G, Design and fabrication of micro-optomechanical vibration sensor. Microfabrication Tech. 120 (2006) 61-64.

23 .McFarland. A.W, Colton. J.S, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, J. Micromechanic Microengi. 15(2005) 1060-1067.

24 .Mindlin. R.D and Eshel. N.N, On first strain-gradient theories in linear elasticity, Int. J. Solid Struct. 4 (1968)109-124.

25 .Moser. Y, Gijs.  M.A.M, Miniaturized flexible temperature sensor. J. Microelectromech. Sys. 16 (2007) 1349-1354.

26 .Ramezani. S, A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory. Int. J. Non-Linear Mech. 47(2012) 863-873.

27 .Roy. S.F.S and Mehregany. M, Determination of Young’s modulus and residual stress of electroless nickel using test structures fabricated in a new surface micromachining process. Microsystem Tech. 2(1996) 92-96.

28 .Wang. B, Zhao. J,  Zhou. S, A microscale Timoshenko beam model based on strain gradient elasticity theory. Euro J. Mech. A/Solid 29 (2010) 591-599.

29 .Wang. WL, Hu. S.J, Modal response and frequency shift of the cantilever in a noncontact atomic force microscope. App. Physic Letters 87(2005) 18350-18356.

30 .Xia. W, Wang. L, Yin. L, Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. Int. J. Eng. Sci. 48(2010) 2044-2053.