Browse > Article
http://dx.doi.org/10.12989/scs.2018.26.4.453

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)
Publication Information
Steel and Composite Structures / v.26, no.4, 2018 , pp. 453-461 More about this Journal
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
non-linear vibration; analytical solution; beam vibration; fourth-order Runge-Kutta method;
Citations & Related Records
Times Cited By KSCI : 8  (Citation Analysis)
연도 인용수 순위
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   DOI
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.   DOI
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.   DOI
4 Babilio, E. (2013), "Dynamics of an axially functionally graded beam under axial load", Eur. Phys. J. Special Topics, 222(7), 1519-1539.   DOI
5 Babilio, E. (2014), "Dynamics of functionally graded beams on viscoelastic foundation", Int. J. Struct. Stabil. Dyn., 14(8), 1440014.   DOI
6 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.   DOI
7 Szekrenyes, A. (2015), "A special case of parametrically excited systems: Free vibration of delaminated composite beams", Eur. J. Mech. - A/Solids, 49, 82-105.   DOI
8 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.   DOI
9 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.   DOI
10 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.   DOI
11 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.   DOI
12 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.
13 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.   DOI
14 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.
15 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.   DOI
16 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.   DOI
17 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.   DOI
18 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.   DOI
19 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.   DOI
20 Bayat, M. and Pakar, I. (2017b), "Analytical study on non-natural vibration equations", Steel Compos. Struct., Int. J., 24(6), 671-677.
21 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.
22 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.
23 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.   DOI
24 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.   DOI
25 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.   DOI
26 He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 211-212.   DOI
27 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.   DOI
28 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.
29 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.   DOI
30 He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Physics Letters A, 374(23), 2312-2314.   DOI
31 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.   DOI
32 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.   DOI
33 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.   DOI
34 Kapania, R.K. and Goyal, V.K. (2002), "Free vibration of unsymmetrically laminated beams having uncertain ply orientations", AIAA Journal, 40(11), 2336-2340.   DOI
35 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.   DOI
36 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.   DOI
37 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.   DOI
38 Lewandowski, R. (1987), "Application of the Ritz method to the analysis of nonlinear free vibrations of beams", J. Sound Vib., 114(1), 91-101.   DOI
39 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.
40 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.   DOI
41 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.   DOI
42 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.   DOI
43 Kapania, R.K. and Raciti, S. (1989), "Nonlinear vibrations of unsymmetrically laminated beams", AIAA, 27(2), 201-210.   DOI
44 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.   DOI
45 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.
46 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.   DOI
47 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.   DOI
48 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.   DOI
49 Sheikholeslami, M. and Ganji, D.D. (2013), "Heat transfer of Cu-water nanofluid flow between parallel plates", Powder Technol., 235, 873-879.   DOI
50 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.   DOI
51 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.   DOI
52 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.   DOI
53 Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58(11), 2131-2133.   DOI
54 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.   DOI