42
N. Herisanu et al.
solutions based on discretization via the Galerkin method are contrasted with direct
application of the multiple scales method.
The nonlinear characteristics of the parametric resonance of simply supported
elastic beams are investigated by Son et al. [3] considering the instability in the lowest
mode, whereas Ghayesh and Balar [4] considered the motion of the parametric vibrations of an axially moving Timoshenko beam taking into consideration two nonlinear
models. Abe [5] proposed an accuracy improvement to the multiple scales method for
nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Based on differential quadrature method, Peng et al. [6] proposed a new
semianalytic method for the geometrically nonlinear vibration of Euler–Bernoulli
beams with different boundary conditions. The supercritical equilibrium solutions
of an axially moving beam supported by sleeves with torsion springs are analyzed
by Ding et al. [7]. The supercritical transport speed ranger, equilibria, and critical
speeds of axially moving beams with hybrid boundary conditions are calculated
from a nonlinear integro–partial–differential equation. Huang et al. [8] studied the
fundamental and subharmonic resonances of an axially moving beam subject to periodic lateral force excitation. The incremental harmonic balance method was used to
evaluate the nonlinear dynamic behavior.
Local bifurcations of a nonlinear beam under parametric excitation are studied
by Zhang et al. [9], and Sedighi et al. [10] presented an analytical solution for vibrations of quintic nonlinear beam. Bayat and Pokar [11] implemented the variational
approach for the Duffing equation with constant coefficients and for a restrained
uniform beam carrying an intermediate lumped mass. Al-Qaisia and Hamdan [12]
investigated the effect of an initial geometric imperfection on the in-plane nonlinear
natural frequencies of an elastic Euler–Bernoulli beam resting on a Winkler elastic
foundation. Bayat et al. [13] studied the nonlinear free vibration of a cylindrical shell
by introducing the extended version of the Hamiltonian approach. Poorjamshidian
et al. [14] presented the nonlinear vibration of a simple-supported flexible beam
with constant velocity carrying a moving mass. The response time of the beam is
obtained by means of a combination of the homotopy method and traditional perturbation. Wang et al. [15] considered the buckling of Euler–Bernoulli columns coupled
with Eringen’s nonlocal theory under a tip load and uniformly distributed axial load.
Analytical solutions for this type of clamped-free beam were derived.
The main objective of the present work is to propose an accurate procedure to
obtain an analytical solution for vibrations of a beam with quintic nonlinearity. The
optimal auxiliary function method (OAFM) is applied to find analytical approximate
solutions of the governing nonlinear differential equations. An example is given
which shows that our technique is simple, easy to use, and very accurate. This method
is independent of the presence of small or large parameters and is based on the
construction and determination of the linear operator and of the auxiliary functions,
combined with a convenient way to optimally control the convergence of the solution.
An accurate and explicitly analytical solution is obtained using only the first iteration.
N. Herisanu et al.
solutions based on discretization via the Galerkin method are contrasted with direct
application of the multiple scales method.
The nonlinear characteristics of the parametric resonance of simply supported
elastic beams are investigated by Son et al. [3] considering the instability in the lowest
mode, whereas Ghayesh and Balar [4] considered the motion of the parametric vibrations of an axially moving Timoshenko beam taking into consideration two nonlinear
models. Abe [5] proposed an accuracy improvement to the multiple scales method for
nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Based on differential quadrature method, Peng et al. [6] proposed a new
semianalytic method for the geometrically nonlinear vibration of Euler–Bernoulli
beams with different boundary conditions. The supercritical equilibrium solutions
of an axially moving beam supported by sleeves with torsion springs are analyzed
by Ding et al. [7]. The supercritical transport speed ranger, equilibria, and critical
speeds of axially moving beams with hybrid boundary conditions are calculated
from a nonlinear integro–partial–differential equation. Huang et al. [8] studied the
fundamental and subharmonic resonances of an axially moving beam subject to periodic lateral force excitation. The incremental harmonic balance method was used to
evaluate the nonlinear dynamic behavior.
Local bifurcations of a nonlinear beam under parametric excitation are studied
by Zhang et al. [9], and Sedighi et al. [10] presented an analytical solution for vibrations of quintic nonlinear beam. Bayat and Pokar [11] implemented the variational
approach for the Duffing equation with constant coefficients and for a restrained
uniform beam carrying an intermediate lumped mass. Al-Qaisia and Hamdan [12]
investigated the effect of an initial geometric imperfection on the in-plane nonlinear
natural frequencies of an elastic Euler–Bernoulli beam resting on a Winkler elastic
foundation. Bayat et al. [13] studied the nonlinear free vibration of a cylindrical shell
by introducing the extended version of the Hamiltonian approach. Poorjamshidian
et al. [14] presented the nonlinear vibration of a simple-supported flexible beam
with constant velocity carrying a moving mass. The response time of the beam is
obtained by means of a combination of the homotopy method and traditional perturbation. Wang et al. [15] considered the buckling of Euler–Bernoulli columns coupled
with Eringen’s nonlocal theory under a tip load and uniformly distributed axial load.
Analytical solutions for this type of clamped-free beam were derived.
The main objective of the present work is to propose an accurate procedure to
obtain an analytical solution for vibrations of a beam with quintic nonlinearity. The
optimal auxiliary function method (OAFM) is applied to find analytical approximate
solutions of the governing nonlinear differential equations. An example is given
which shows that our technique is simple, easy to use, and very accurate. This method
is independent of the presence of small or large parameters and is based on the
construction and determination of the linear operator and of the auxiliary functions,
combined with a convenient way to optimally control the convergence of the solution.
An accurate and explicitly analytical solution is obtained using only the first iteration.