Dynamic stiffness matrix method for axially moving micro-beam

  • Received : 2012.10.08
  • Accepted : 2012.12.05
  • Published : 2012.12.25


In this paper the dynamic stiffness matrix method was used for the free vibration analysis of axially moving micro beam with constant velocity. The extended Hamilton's principle was employed to derive the governing differential equation of the problem using the modified couple stress theory. The dynamic stiffness matrix of the moving micro beam was evaluated using appropriate expressions of the shear force and bending moment according to the Euler-Bernoulli beam theory. The effects of the beam size and axial velocity on the dynamic characteristic of the moving beam were investigated. The natural frequencies and critical velocity of the axially moving micro beam were also computed for two different end conditions.


  1. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H. and Rahaeifard, M. (2010), "On the size-dependent behaviour of functionally graded micro-beams", Mater. Design, 31(5), 2324-2329.
  2. Banerjee, J.R. (2003), "Free vibration of sandwich beams using the dynamic stiffness method", Compos. Struct., 81(18-19), 1915-1922.
  3. Banerjee, J.R. (2004), "Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam", J. Sound Vib., 270(1-2), 379-401.
  4. Banerjee, J.R. (2012), "Free vibration of beams carrying spring-mass systems-A dynamic stiffness approach", Compos. Struct., 104-105, 21-26.
  5. Banerjee, J.R. and Gunawardana, W.D. (2007), "Dynamic stiffness matrix development and free vibration analysis of a moving beam", J. Sound Vib., 303(1-2), 135-143.
  6. Capron, M.D. and Williams, F.W. (1988), "Exact dynamic stiffness for an axially loaded uniform timoshenko member embedded in an elastic medium", J. Sound Vib., 124(3), 453-466.
  7. Chong, A.C.M. and Lam, D.C.C. (1999a), "Strain gradient plasticity effect in indentation hardness of polymers", J. Mater. Res., 14(10), 4103-4110.
  8. Chong, A.C.M., Yang, F., Lam, D.C.C. and Tong, P. (2001), "Torsion and bending of micron-scaled structures", J. Mater. Res., 16(4), 1052-1058.
  9. Eisenberger, M., Abramovich, H. and Shulepov, O. (1995), "Dynamic stiffness analysis of laminated beams using a first order shear deformation theory", Compos. Struct., 31(4), 265-271.
  10. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710.
  11. Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W., (1994), "Strain gradient plasticity: theory and experiments", Acta Metall. Mater., 42(2), 475-487.
  12. Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2012), "Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory", J. Sound Vib., 331(1), 94-106.
  13. Koiter, W.T. (1964), "Couple stresses in the theory of elasticity, I and II", Proc. K. Ned. Akad. Wet. (B) 67, 17-44.
  14. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", Int. J. Eng. Sci., 46(5), 427-437.
  15. Lam, D.C.C. and Chong, A.C.M. (1999b), "Indentation model and strain gradient plasticity law for glassy polymers", J. Mater. Res., 14(9), 3784-3788.
  16. Lee, U. and Jang, I. (2007), "On the boundary conditions for axially moving beams", J. Sound Vib., 306(3-5), 675-690.
  17. Lee, U. and Oh, H.M. (2005), "Dynamics of an axially moving viscoelastic beam subject to axial tension", Int. J. Solids Struct., 42(8), 2381-2398.
  18. Lee, U., Kim, J. and Oh, H.M. (2004), "Spectral analysis for the transverse vibration of an axially moving Timoshenko beam", J. Sound Vib., 271(3-5), 685-703.
  19. Lim, C.W., Li, C. and Yu, J.L. (2009), "The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams", Interact. Multiscale Mech., 2(3), 223-233.
  20. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solids, 56(12), 3379-3391.
  21. Murphy, K.D. and Zhang, Y. (2000), "Vibration and stability of a cracked translating beam", J. Sound Vib., 237(2), 319-335.
  22. Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16(11), 2355-2359.
  23. Poole, W.J., Ashby, M.F. and Fleck, N.A. (1996), "Micro-hardness of annealed and work-hardened copper polycrystals", Scripta Mater., 34(4), 559-564.
  24. Simsek, M. (2010), "Dynamic analysis of an embedded microbeam carrying a moving micro particle based on the modified couple stress theory", Int. J. Eng. Sci., 48(12), 1721-1732.
  25. Sreeram, T.R. and Sivaneri, N.T. (1998), "FE-analysis of a moving beam using Lagrangian multiplier method", Int. J. Solids Struct., 35(28-29), 3675-3694.
  26. Viola, E., Ricci, P. and Aliabadi, M.H. (2007), "Free vibration analysis of axially loaded cracked Timoshenko beam structures using the dynamic stiffness method", J. Sound Vib., 304(1-2), 124-153.
  27. Wang, L., Chen, J.S. and HU, H.Y. (2009), "Radial basis collocation method for dynamic analysis of axially moving beams", Interact. Multiscale Mech., 2(4), 333-352.
  28. Williams, F.W. and Kennedy, D. (1987), "Exact dynamic member stiffness for a beam on an elastic foundation", Earthq. Eng. Struct. D., 15(1), 133-136.
  29. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743.

Cited by

  1. Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium vol.18, pp.6, 2016,
  2. Size-Dependent Vibration of Axially Moving Viscoelastic Micro-Plates Based on Sinusoidal Shear Deformation Theory vol.09, pp.02, 2017,