48
N. Herisanu et al.
5 Conclusions
In this work, we present a reliable procedure, namely the OAFM to analytically solve
a nonlinear differential equation of transversal vibration of hinged–hinged flexible
beam. Using Galerkin’s method and suitable transformations, the governing equation
is transformed into a nonlinear differential equation. Our technique is valid even if the
nonlinear differential equation does not contain any small or large parameters. Our
construction is completely different in comparison with any other procedures known
in the literature. We refer to the linear operator and to the auxiliary functions which
contain several convergence-control parameters C i , which ensure a fast convergence
of the solution after the first iterations. This procedure is effective, explicit, and very
accurate and can be applied to other nonlinear dynamical systems.
References
1. M. Pakdemirli, A comparison of two perturbation methods for vibrations of systems with
quadratic and cubic nonlinearities. Mech. Res. Commun. 21(2), 203–208 (1994)
2. A. Nayfeh, W. Lacarbonara, On the discretization of distributed-parameter systems with
quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997)
3. I.S. Son, Y. Uchiyama, W. Lacarbonara, H. Yabuno, Simply supported elastic beams under
parametric excitation. Nonlinear Dyn. 53, 129–138 (2008)
4. M.H. Ghayesh, S. Balar, Non-linear parametric and stability analysis of two dynamic models
of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850–2859 (2010)
5. A. Abe, Accuracy improvement of the method of multiple scales for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Math. Probl.Eng. Art ID
890813 (2010)
6. J.S. Peng, Y. Lui, J. Yang, A semianalytical method for nonlinear vibration of Euler-Bernoulli
beams with general boundary conditions. Math. Probl. Eng. Art ID 591786 (2010)
7. H. Ding, G.C. Zhang, L.Q. Chen, Supercritical equilibrium solutions of axially moving beams
with hybrid boundary conditions. Mech. Res. Commun. 38, 52–56 (2011)
8. J.L. Huang, R.K.L. Su, W.H. Li, S.H. Chen, Stability and bifurcation of an axially moving
beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471–485 (2011)
9. W. Zhang, X.W. Feng, W.Z. Jean, Local bifurcations and codimension-3 degenerate bifurcations
of quintic nonlinear beam under parametric excitation. Chaos, Solitons Fractals 24, 977–998
(2005)
10. H.M. Sedighi, K.H. Shirazi, J. Zare, An analytical solution of transversal oscillation of quintic
non-linear beam with homotopy analysis method. Int. J. Non-Linear Mech. 47, 777–784 (2012)
11. M. Bayat, I. Pokar, On the approximate analytical solution to non-linear oscillation systems.
Shock Vib. 20, 43–52 (2013)
12. A.A. Al-Qaisia, M.H. Hamdan, On nonlinear frequency veering and mode localizations of a
beam with geometric imperfection resting on elastic foundation. J. Sound Vib. 332, 4641–4655
(2013)
13. M. Bayat, I. Pokar, L. Cveticanin, Nonlinear vibration of stringer shell by means of extended
Hamiltonian approach. Arch. Appl. Mech. 84, 43–50 (2014)
14. M. Poorjamshidian, J. Sheiki, S.M. Moghadas, M. Nakhaie, Nonlinear vibrations analysis of
the beam carrying a moving mass using modified homotopy. J. Solid Mech. 6, 389–396 (2014)
15. C.M. Wang, H. Zhang, N. Challamel, Y. Xiong, Buckling of nonlocal columns with allowance
for selfweight. J. Eng. Mech. 142, 04016037 (2016)
N. Herisanu et al.
5 Conclusions
In this work, we present a reliable procedure, namely the OAFM to analytically solve
a nonlinear differential equation of transversal vibration of hinged–hinged flexible
beam. Using Galerkin’s method and suitable transformations, the governing equation
is transformed into a nonlinear differential equation. Our technique is valid even if the
nonlinear differential equation does not contain any small or large parameters. Our
construction is completely different in comparison with any other procedures known
in the literature. We refer to the linear operator and to the auxiliary functions which
contain several convergence-control parameters C i , which ensure a fast convergence
of the solution after the first iterations. This procedure is effective, explicit, and very
accurate and can be applied to other nonlinear dynamical systems.
References
1. M. Pakdemirli, A comparison of two perturbation methods for vibrations of systems with
quadratic and cubic nonlinearities. Mech. Res. Commun. 21(2), 203–208 (1994)
2. A. Nayfeh, W. Lacarbonara, On the discretization of distributed-parameter systems with
quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997)
3. I.S. Son, Y. Uchiyama, W. Lacarbonara, H. Yabuno, Simply supported elastic beams under
parametric excitation. Nonlinear Dyn. 53, 129–138 (2008)
4. M.H. Ghayesh, S. Balar, Non-linear parametric and stability analysis of two dynamic models
of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850–2859 (2010)
5. A. Abe, Accuracy improvement of the method of multiple scales for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Math. Probl.Eng. Art ID
890813 (2010)
6. J.S. Peng, Y. Lui, J. Yang, A semianalytical method for nonlinear vibration of Euler-Bernoulli
beams with general boundary conditions. Math. Probl. Eng. Art ID 591786 (2010)
7. H. Ding, G.C. Zhang, L.Q. Chen, Supercritical equilibrium solutions of axially moving beams
with hybrid boundary conditions. Mech. Res. Commun. 38, 52–56 (2011)
8. J.L. Huang, R.K.L. Su, W.H. Li, S.H. Chen, Stability and bifurcation of an axially moving
beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471–485 (2011)
9. W. Zhang, X.W. Feng, W.Z. Jean, Local bifurcations and codimension-3 degenerate bifurcations
of quintic nonlinear beam under parametric excitation. Chaos, Solitons Fractals 24, 977–998
(2005)
10. H.M. Sedighi, K.H. Shirazi, J. Zare, An analytical solution of transversal oscillation of quintic
non-linear beam with homotopy analysis method. Int. J. Non-Linear Mech. 47, 777–784 (2012)
11. M. Bayat, I. Pokar, On the approximate analytical solution to non-linear oscillation systems.
Shock Vib. 20, 43–52 (2013)
12. A.A. Al-Qaisia, M.H. Hamdan, On nonlinear frequency veering and mode localizations of a
beam with geometric imperfection resting on elastic foundation. J. Sound Vib. 332, 4641–4655
(2013)
13. M. Bayat, I. Pokar, L. Cveticanin, Nonlinear vibration of stringer shell by means of extended
Hamiltonian approach. Arch. Appl. Mech. 84, 43–50 (2014)
14. M. Poorjamshidian, J. Sheiki, S.M. Moghadas, M. Nakhaie, Nonlinear vibrations analysis of
the beam carrying a moving mass using modified homotopy. J. Solid Mech. 6, 389–396 (2014)
15. C.M. Wang, H. Zhang, N. Challamel, Y. Xiong, Buckling of nonlocal columns with allowance
for selfweight. J. Eng. Mech. 142, 04016037 (2016)