Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear vibration of unsymmetrical laminated composite beam on elastic foundation

  • Pakar, I. (Mashhad Branch, Islamic Azad University) ;
  • Bayat, M. (Roudehen Branch, Islamic Azad University) ;
  • Cveticanin, L. (Faculty of Technical Sciences, Novi Sad, University of Novi Sad)
  • Received : 2017.09.25
  • Accepted : 2017.12.07
  • Published : 2018.02.25
⟨ Previous Next ⟩

Abstract

In this paper, nonlinear vibrations of the unsymmetrical laminated composite beam (LCB) on a nonlinear elastic foundation are studied. The governing equation of the problem is derived by using Galerkin method. Two different end conditions are considered: the simple-simple and the clamped-clamped one. The Hamiltonian Approach (HA) method is adopted and applied for solving of the equation of motion. The advantage of the suggested method is that it does not need any linearization of the problem and the obtained approximate solution has a high accuracy. The method is used for frequency calculation. The frequency of the nonlinear system is compared with the frequency of the linear system. The influence of the parameters of the foundation nonlinearity on the frequency of vibration is considered. The differential equation of vibration is solved also numerically. The analytical and numerical results are compared and is concluded that the difference is negligible. In the paper the new method for error estimation of the analytical solution in comparison to the exact one is developed. The method is based on comparison of the calculation energy and the exact energy of the system. For certain numerical data the accuracy of the approximate frequency of vibration is determined by applying of the suggested method of error estimation. Finally, it has been indicated that the proposed Hamiltonian Approach gives enough accurate result.

Keywords

References

  1. Alkayem, N.F., Cao, M., Zhang, Y., Bayat, M. and Su, Z. (2017), "Structural damage detection using finite element model updating with evolutionary algorithms: a survey", Neural Comput. Appl., pp. 1-23. DOI: https://doi.org/10.1007/s00521-017-3284-1
  2. Arikoglu, A. and Ozkol, I. (2005), "Solution of boundary value problems for integro-differential equations by using transform method", Appl. Math. Comput., 168(2), 1145-1158. https://doi.org/10.1016/j.amc.2004.10.009
  3. Azrar, L., Benamar, R. and White, R.G. (1999), "A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes. Part I: general theory and application to the single mode approach to free and forced vibration analysis", J. Sound Vib., 224(2), 183-207. https://doi.org/10.1006/jsvi.1998.1893
  4. Babilio, E. (2013), "Dynamics of an axially functionally graded beam under axial load", Eur. Phys. J. Special Topics, 222(7), 1519-1539. https://doi.org/10.1140/epjst/e2013-01942-8
  5. Babilio, E. (2014), "Dynamics of functionally graded beams on viscoelastic foundation", Int. J. Struct. Stabil. Dyn., 14(8), 1440014. https://doi.org/10.1142/S0219455414400148
  6. Baghani, M., Jafari-Talookolaei, R.A. and Salareih, H. (2011), "Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic", Appl. Math. Model., 35(1), 130-138. https://doi.org/10.1016/j.apm.2010.05.012
  7. Bambill, D.V., Rossit, C.A., Rossi, R.E., Felix, D.H. and Ratazzi, A.R. (2013), "Transverse free vibration of non uniform rotating Timoshenko beams with elastically clamped boundary conditions", Meccanica, 48(6), 1289-1311. https://doi.org/10.1007/s11012-012-9668-5
  8. Basu, D. and Kameswara Rao, N.S.V. (2013), "Analytical solutions for Euler-Bernoulli beam on visco-elastic foundation subjected to moving load", Int. J. Numer. Anal. Meth. Geomech., 37(8), 945-960. https://doi.org/10.1002/nag.1135
  9. Bayat, M. and Pakar, I. (2017a), "Accurate semi-analytical solution for nonlinear vibration of conservative mechanical problems", Struct. Eng. Mech., Int. J., 61(5), 657-661. https://doi.org/10.12989/sem.2017.61.5.657
  10. Bayat, M. and Pakar, I. (2017b), "Analytical study on non-natural vibration equations", Steel Compos. Struct., Int. J., 24(6), 671-677.
  11. Bayat, M., Pakar, I. and Cveticanin, L. (2014), "Nonlinear free vibration of systems with inertia and static typa cubic nonlinearities: An analytical approach", Mechanism and Machine Theory, 77, 50-58. https://doi.org/10.1016/j.mechmachtheory.2014.02.009
  12. Bayat, M., Pakar, I. and Bayat, M. (2016), "Nonlinear vibration of rested Euler-Bernoulli beams on linear elastic foundation using Hamiltonian approach", Vibroengineering PROCEDIA, 10, 89-94.
  13. Bayat, M., Pakar, I. and Cao, M.S. (2017), "Energy based approach for solving conservative nonlinear systems", Earthq. Struct., Int. J., 13(2), 131-136.
  14. Cheng, C.J., Chiu, S.W., Cheng, C.B. and Wu, J.Y. (2012), "Customer lifetime value prediction by a Markov chain based data mining model: Application to an auto repair and maintenance company in Taiwan", Scientia Iranica, 19(3), 849-855. https://doi.org/10.1016/j.scient.2011.11.045
  15. Civalek, O. (2006), "Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation", J. Sound Vib., 294(4), 966-980. https://doi.org/10.1016/j.jsv.2005.12.041
  16. Civalek, O. (2013), "Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches", Compos. Part B: Eng., 50, 171-179. https://doi.org/10.1016/j.compositesb.2013.01.027
  17. Clementi, F., Demeio, L., Mazzilli, C.E.N. and Lenci, S. (2015), "Nonlinear vibrations of non-uniform beams by the MTS asymptotic expansion method", Continuum. Mech. Thermodyn., 27(4-5), 703-717. https://doi.org/10.1007/s00161-014-0368-3
  18. Ding, H., Chen, L.Q. and Yang, S.P. (2012), "Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load", J. Sound Vib., 331(10), 2426-2442. https://doi.org/10.1016/j.jsv.2011.12.036
  19. Fang, J. and Zhou, D. (2015), "Free vibration analysis of rotating axially functionally graded-tapered beams using Chebyshev-Ritz method", Mater. Res. Innov., 19(sub5), 1255-1262.
  20. Ghasemi, A.R. and Mohandes, M. (2016), "The effect of finite strain on the nonlinear free vibration of a unidirectional composite Timoshenko beam using GDQM", Adv. Aircr. Spacecr. Sci., 3(4), 379-397. https://doi.org/10.12989/aas.2016.3.4.379
  21. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 211-212. https://doi.org/10.1515/IJNSNS.2008.9.2.211
  22. He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Physics Letters A, 374(23), 2312-2314. https://doi.org/10.1016/j.physleta.2010.03.064
  23. He, P., Liu, Z.S. and Li, C. (2013), "An improved beam element for beams with variable axial parameters", Shock Vib., 20(4), 601-617. https://doi.org/10.1155/2013/708910
  24. Jafari-Talookolaei, R.A., Salareih, H. and Kargarnovin, M.H. (2011), "Analysis of large amplitude free vibrations of unsymmetrically laminated composite beams on a nonlinear elastic foundation", Acta Mechanica, 219(1-2), 65-75. https://doi.org/10.1007/s00707-010-0439-x
  25. Jamshidi, N. and Ganji, D.D. (2010), "Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire", Current Appl. Phys., 10(2), 484-486. https://doi.org/10.1016/j.cap.2009.07.004
  26. Kapania, R.K. and Goyal, V.K. (2002), "Free vibration of unsymmetrically laminated beams having uncertain ply orientations", AIAA Journal, 40(11), 2336-2340. https://doi.org/10.2514/2.1573
  27. Kapania, R.K. and Raciti, S. (1989), "Nonlinear vibrations of unsymmetrically laminated beams", AIAA, 27(2), 201-210. https://doi.org/10.2514/3.10082
  28. Lenci, S. and Clementi, F. (2012a), "Effects of shear stiffness, rotatory and axial inertia, and interface stiffness on free vibrations of a two-layer beam", J. Sound Vib., 331(24), 5247-5267. https://doi.org/10.1016/j.jsv.2012.07.004
  29. Lenci, S. and Clementi, F. (2012b), "On flexural vibrations of shear deformable laminated beams", Proceedings of ASME 2012 International Mechanical Engineering Congress and Exposition, Houston, TX, USA, November, pp. 581-590.
  30. Lenci, S., Clementi, F. and Mazzilli, C.E.N. (2013), "Simple formulas for the natural frequencies of non-uniform cables and beams", Int. J. Mech. Sci., 77, 155-163. https://doi.org/10.1016/j.ijmecsci.2013.09.028
  31. Lenci, S., Clementi, F. and Warminski, J. (2015), "Nonlinear free dynamics of a two-layer composite beam with different boundary conditions", Meccanica, 50(3), 675-688. https://doi.org/10.1007/s11012-014-9945-6
  32. Lewandowski, R. (1987), "Application of the Ritz method to the analysis of nonlinear free vibrations of beams", J. Sound Vib., 114(1), 91-101. https://doi.org/10.1016/S0022-460X(87)80236-5
  33. Navarro, H.A. and Cveticanin, L. (2016), "Amplitude-frequency relationship obtained using Hamiltonian approach for oscillators with sum of non-integer order nonlinearities", Appl. Math. Comput., 291, 162-171.
  34. Nguyen, N.H. and Lee, D.Y. (2015), "Bending analysis of a single leaf flexure using higher-order beam theory", Struct. Eng. Mech., Int. J., 53(4), 781-790. https://doi.org/10.12989/sem.2015.53.4.781
  35. Poloei, E., Zamanian, M. and Hosseini, S.A.A. (2017), "Nonlinear vibration analysis of an electrostatically excited micro cantilever beam coated by viscoelastic layer with the aim of finding the modified configuration", Struct. Eng. Mech., Int. J., 61(2), 193-207. https://doi.org/10.12989/sem.2017.61.2.193
  36. Pradhan, S.C. and Murmu, T. (2009), "Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method", J. Sound Vib., 321(1-2), 342-362. https://doi.org/10.1016/j.jsv.2008.09.018
  37. Ramana, P.V. and Prasad, B.R. (2014), "Modified Adomian Decomposition Method for Van der Pol equations", Int. J. Non-Linear Mech., 65, 121-132. https://doi.org/10.1016/j.ijnonlinmec.2014.03.006
  38. Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., Int. J., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065
  39. Sheikholeslami, M. and Ganji, D.D. (2013), "Heat transfer of Cu-water nanofluid flow between parallel plates", Powder Technol., 235, 873-879. https://doi.org/10.1016/j.powtec.2012.11.030
  40. Sheikholeslami, M. and Ganji, D.D. (2015), "Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM", Comput. Method. Appl. Mech. Eng., 283, 651-663. https://doi.org/10.1016/j.cma.2014.09.038
  41. Sheikholeslami, M. and Ganji, D.D. (2016), "Nanofluid hydrothermal behavior in existence of Lorentz forces considering Joule heating effect", J. Molecul. Liquids, 224, 526-537. https://doi.org/10.1016/j.molliq.2016.10.037
  42. Sheikholeslami, M., Ellahi, R., Ashorynejad, H.R., Domairry, G. and Hayat, T. (2014), "Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium", J. Comput. Theor. Nanosci., 11(2), 486-496. https://doi.org/10.1166/jctn.2014.3384
  43. Sheikholeslami, M., Ganji, D.D. and Rashidi, M.M. (2016), "Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model", J. Magnet. Magnet. Mater., 416, 164-173. https://doi.org/10.1016/j.jmmm.2016.05.026
  44. Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58(11), 2131-2133. https://doi.org/10.1016/j.camwa.2009.03.056
  45. Singh, G., Rao, V. and Iyengar, N.G.R. (1991), "Analysis of the nonlinear vibrations of unsymmetrically laminated composite beams", AIAA, 29(10), 1727-1804. https://doi.org/10.2514/3.10796
  46. Singh, G., Rao, G.V. and Iyengar, N.G.R. (1992), "Nonlinear bending of thin and thick unsymmetrically laminated composite beams using refined finite element model", Comput. Struct., 42(4), 471-479. https://doi.org/10.1016/0045-7949(92)90114-F
  47. Szekrenyes, A. (2015), "A special case of parametrically excited systems: Free vibration of delaminated composite beams", Eur. J. Mech. - A/Solids, 49, 82-105. https://doi.org/10.1016/j.euromechsol.2014.07.003
  48. Wang, L., Ma, J., Li, L., Peng, J. (2013a), "Three-to-one resonant responses of inextensional beams on the elastic foundation", ASME J. Vib. Acoust., 135(1), 011015. https://doi.org/10.1115/1.4007019
  49. Wang, L., Ma, J., Peng, J. and Li, L. (2013b), "Large amplitude vibration and parametric instability of inextensional beams on the elastic foundation", Int. J. Mech. Sci., 67, 1-9. https://doi.org/10.1016/j.ijmecsci.2012.12.002
  50. Wang, L., Ma, J., Yang, M., Li, L. and Zhao, Y. (2013c), "Multimode dynamics of inextensional beams on the elastic foundation with two-to-one internal resonances", J. Appl. Mech., 80(6), 061016. https://doi.org/10.1115/1.4023694
  51. Wang, L., Ma, J., Zhao, Y. and Liu, Q. (2013d), "Refined modeling and free vibration of inextensional beams on the elastic foundation", J. Appl. Mech., 80(4), 041026. https://doi.org/10.1115/1.4023032
  52. Xu, L. (2010), "Application of Hamiltonian approach to an oscillation of a mass attached to a stretched elastic wire", Math. Comput. Appl., 15(5), 901-906.
  53. Yu, Y.P., Wu, B.S. and Lim, C.W. (2012), "Numerical and analytical approximations to large post-buckling deformation of MEMS", Int. J. Mech. Sci., 55(1), 95-103. https://doi.org/10.1016/j.ijmecsci.2011.12.010
  54. Zenkour, A.M., Allam, M.N.M. and Sobhy, M. (2010), "Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak's elastic foundations", ActaMechanica, 212, 233-252.

Cited by

  1. Thermo-mechanical behavior of porous FG plate resting on the Winkler-Pasternak foundation vol.9, pp.6, 2018, https://doi.org/10.12989/csm.2020.9.6.499