Structural Engineering and Mechanics

  • 제72권2호
  • /
  • Pages.229-244
  • /
  • 2019
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Flapwise and non-local bending vibration of the rotating beams

  • Mohammadnejad, Mehrdad (Department of Civil Engineering, Birjand University of Technology) ;
  • Saffari, Hamed (Department of Civil Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman)
  • 투고 : 2018.07.15
  • 심사 : 2019.05.30
  • 발행 : 2019.10.25
⟨ 이전 논문 다음 논문 ⟩

초록

Weak form integral equations are developed to investigate the flapwise bending vibration of the rotating beams. Rayleigh and Eringen nonlocal elasticity theories are used to investigate the rotatory inertia and Size-dependency effects on the flapwise bending vibration of the rotating cantilever beams, respectively. Through repetitive integrations, the governing partial differential equations are converted into weak form integral equations. The novelty of the presented approach is the approximation of the mode shape function by a power series which converts the equations into solvable one. Substitution of the power series into weak form integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of the non-trivial solution for resulting system of equations. Accuracy of the proposed method is verified through several numerical examples, in which the influence of the geometry properties, rotatory inertia, rotational speed, taper ratio and size-dependency are investigated on the natural frequencies of the rotating beam. Application of the weak form integral equations has made the solution simpler and shorter in the mathematical process. Presented relations can be used to obtain a close-form solution for quick calculation of the first five natural frequencies of the beams with flapwise vibration and non-local effects. The analysis results are compared with those obtained from other available published references.

키워드

참고문헌

  1. Adhikari, S., Murmu, T. and McCarthy, M.A. (2013), "Dynamic finite element analysis of axially vibrating nonlocal rods", Finite Element Anal. Design, 63, 42-50, http://dx.doi.org/10.1016/j.finel.2012.08.001.
  2. Aranda-Ruiz, J., Loya, J. and Fernandez-Saez, J. (2012), "Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory", J. Compos. Struct., 94, 2990-3001, https://doi.org/10.1016/j.compstruct.2012.03.033
  3. Aksencer, T., Aydogdu, M. (2011), "Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory", Physica E Low-dimensional Syst. Nanostruct., 43, 954-959, https://doi.org/10.1016/j.physe.2010.11.024
  4. Aydogdu, M. (2009), "Axial vibration of the nanorods with the nonlocal continuum rod model", Physica E Low-dimensional Syst. Nanostruct., 41, 861-864, https://doi.org/10.1016/j.physe.2009.01.007
  5. Banerjee, J.R. and Jackson, D.R. (2013), "Free vibration of a rotating tapered Rayleigh beam: A dynamic stiffness method of solution", J. Comput. Struct., 124, 11-20, https://doi.org/10.1016/j.compstruc.2012.11.010
  6. Behera, L. and Chakraverty, S. (2015), "Application of Differential Quadrature method in free vibration analysis of nanobeams based on various nonlocal theories", Comput. Math. Appl., 69, 1444-1462, https://doi.org/10.1016/j.camwa.2015.04.010
  7. Chang, T.P. (2013), "Axial vibration of non-uniform and nonhomogeneous nanorods based on nonlocal elasticity theory", Appl. Math. Comput., 219, 4933-4941. https://doi.org/10.1016/j.amc.2012.11.059
  8. Dehrouyeh-Semnani, A.M. (2015), "The influence of size effect on flapwise vibration of rotating microbeams", International J. Eng. Sci., 94, 150-163, https://doi.org/10.1016/j.ijengsci.2015.05.009
  9. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218, 7406-7420, https://doi.org/10.1016/j.amc.2011.12.090.
  10. Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013), "Vibration analysis of Euler-Bernoulli nanobeams by using finite element method", Appl. Math. Model., 37, 4787-4797, https://doi.org/10.1016/j.apm.2012.10.016
  11. Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.A. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Comput., 229, 283-295, https://doi.org/10.1016/j.amc.2013.12.072
  12. Emam, S.A. (2013), "A general nonlocal nonlinear model for buckling of nanobeams", Appl. Math. Model., 37, 6929-6939, https://doi.org/10.1016/j.apm.2013.01.043
  13. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88, https://doi.org/10.1016/j.compstruct.2012.09.030
  14. Eltaher, M.A., Khater, M.E., Emam, Samir A. (2016), "A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams", Appl. Math. Model., 40(5-6), 4109-4128, https://doi.org/10.1016/j.apm.2015.11.026
  15. Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013), "Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams", Compos. Struct., 99, 193-201, https://doi.org/10.1016/j.compstruct.2012.11.039
  16. Eltaher, M.A., Abdraboh, A.M. and Almitani, K.H. (2018), "Resonance frequencies of size dependent perforated nonlocal nanobeam", Microsyst. Technol., 24(9), 3925-3937, https://doi.org/10.1007/s00542-018-3910-6
  17. Eltaher, M.A., Abdelrahman, A.A., Al-Nabawy, A., Khater, M., and Mansour, A. (2014), "Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position", Appl. Math. Comput., 235, 512-529, https://doi.org/10.1016/j.amc.2014.03.028
  18. Eltaher, M.A., Hamed, M.A., Sadoun, A.M., and Mansour, A. (2014), "Mechanical analysis of higher order gradient nanobeams", Appl. Math. Comput., 229, 260-272, https://doi.org/10.1016/j.amc.2013.12.076
  19. Ganesh, R. and Ganguli, R. (2013), "Stiff string approximations in Rayleigh-Ritz method for rotating beams", Appl. Math. Comput., 219, 9282-9295, https://doi.org/10.1016/j.amc.2013.03.017
  20. Ghannadpour, S.A.M., Mohammadi, B. and Fazilati, J. (2013), "Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method", Compos. Struct., 96, 584-589, https://doi.org/10.1016/j.compstruct.2012.08.024
  21. Huang, C. L., Lin, W. Y. and Hsiao, K. M. (2010), "Free vibration analysis of rotating Euler beams at high angular velocity", J. Comput. Struct., 88, 991-1001, https://doi.org/10.1016/j.compstruc.2010.06.001
  22. Huang, Y.; Li, X.-F. (2010), "A new approach for free vibration of axially functionally graded beams with non-uniform crosssection", J. Sound Vib. 329(11), 2291-2303, https://doi.org/10.1016/j.jsv.2009.12.029
  23. Kim, H., Yoo, H.H. and Chung, J. (2013), "Dynamic model for free vibration and response analysis of rotating beams", J. Sound Vib., 332, 5917-5928, https://doi.org/10.1016/j.jsv.2013.06.004
  24. Li, L., Zhang, D.G. and Zhu, W.D. (2014), "free vibration analysis of a rotating hub-functionally graded material beam system with the dynamic stiffening effect", J. Sound Vib., 333, 1526-1541, https://doi.org/10.1016/j.jsv.2013.11.001
  25. Lee, H.L. and Chang, W.J. (2010), "Surface and small-scale effects on vibration analysis of a nonuniform nanocantilever beam", Physica E Low-dimensional Syst. Nanostruct., 43, 466-469, https://doi.org/10.1016/j.physe.2010.08.030
  26. Murmu, Y. L., Adhikari, S. and Friswell, M.I. (2013), "Dynamic characteristics of damped viscoelastic nonlocal EulereBernoulli beams", Europe. J. Mech. A/Solids, 42, 125-136, https://doi.org/10.1016/j.euromechsol.2013.04.006
  27. Murmu, T. and Pradhan, S.C. (2009), "Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory", Physica E Low-dimensional Syst. Nanostruct., 41, 1451-1456, https://doi.org/10.1016/j.physe.2009.04.015
  28. Mohammadnejad, M. (2015), "A new analytical approach for determination of flexural, axial and torsional natural frequencies of beams", Struct. Eng. Mech., 55(3), 655-674, https://doi.org/10.12989/sem.2015.55.3.655
  29. Nguyen, N.T., Kim, N.I., Lee, J. (2015), "Mixed finite element analysis of nonlocal Euler-Bernoulli nanobeams", Finite Element Anal. Design, 106, 65-72, https://doi.org/10.1016/j.finel.2015.07.012
  30. Narendar, S. (2012), "Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanotube with consideration of transverse shear deformation and rotary inertia", Appl. Math. Comput., 219, 1232-1243, https://doi.org/10.1016/j.amc.2012.07.032
  31. Niknam, H. and Aghdam, M.M., "A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation" Compos. Struct., http://dx.doi.org/10.1016/j.compstruct.2014.09.023.
  32. Phadikar, J.K. and Pradhan, S.C. (2010), "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Comput. Mater. Sci., 49, 492-499, https://doi.org/10.1016/j.commatsci.2010.05.040
  33. Pradhan, S.C. and Murmu, T. (2010), "Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever", Physica E: Low-dimensional Systems and Nanostructures, 42, 1944-1949, https://doi.org/10.1016/j.physe.2010.03.004
  34. Salehipour, H., Nahvi, H. and Shahidi, A. R. (2014), "Exact analytical solution for free vibration of functionally graded micro/nano plates via three-dimensional nonlocal elasticity", Physica E Low-dimensional Syst. Nanostruct., http://dx.doi.org/10.1016/j.physe.2014.10.001.
  35. Saffari, H., Mohammadnejad, M., and Bagheripour,MH. (2012), "Free vibration analysis of non prismatic beams under variable axial forces", Struct. Eng. Mech., 43(5), 561-582, http://dx.doi.org/10.12989/sem.2012.43.5.561
  36. Saffari, H. and Mohammadnejad, M. (2015), "On the application of weak form integral equations to free vibration analysis of tall structures", Asian J. Civil Eng. (BHRC), 16(7), 977-999.
  37. Salehipour, H., Shahidi, A.R. and Nahvi, H. (2015), "Modified nonlocal elasticity theory for functionally graded materials", International J. Eng. Sci., 90, 44-57, https://doi.org/10.1016/j.ijengsci.2015.01.005
  38. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", J. Eng. Sci., 52, 56-64, https://doi.org/10.1016/j.ijengsci.2011.11.011
  39. Thai, H.T., Vo, T.P. (2012), "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams", J. Eng. Sci., 54, 58-66, https://doi.org/10.1016/j.ijengsci.2012.01.009
  40. Tang, A.-Y., Li, X.-F., Wu, J.-X. and Lee, K.Y. (2015), "Flapwise bending vibration of rotating tapered Rayleigh cantilever beams", J. Construct. Steel Res., 112, 1-9, https://doi.org/10.1016/j.jcsr.2015.04.010