28-02-2013, 02:07 PM
Non-linear vibration of Euler-Bernoulli beams
Non-linear vibration.pdf (Size: 5.25 MB / Downloads: 107)
Abstract
In this paper, variational iteration (VIM) and parametrized
perturbation (PPM) methods have been used to investigate
non-linear vibration of Euler-Bernoulli beams subjected to
the axial loads. The proposed methods do not require small
parameter in the equation which is difficult to be found for
nonlinear problems. Comparison of VIM and PPM with
Runge-Kutta 4th leads to highly accurate solutions.
INTRODUCTION
The demand for engineering structures is continuously increasing. Aerospace vehicles, bridges,
and automobiles are examples of these structures. Many aspects have to be taken into consideration
in the design of these structures to improve their performance and extend their life. One
aspect of the design process is the dynamic response of structures. The dynamics of distributedparameter
and continuous systems, like beams, were governed by linear and nonlinear partialdifferential
equations in space and time. It was difficult to find the exact or closed-form solutions
for nonlinear problems. Consequently, researchers were used two classes of approximate
solutions of initial boundary-value problems: numerical techniques [28, 31], and approximate
analytical methods [2, 26]. For strongly non-linear partial-differential, direct techniques, such
as perturbation methods, were not utilized to solve directly the non-linear partial-differential
equations and associated boundary conditions. Therefore first partial-differential equations are
discretized into a set of non-linear ordinary-differential equations using the Galerkin approach
and the governing problems are then solved analytically in time domain.
DESCRIPTION OF THE PROBLEM
Consider a straight beam on an elastic foundation with length L, a cross-section A, a mass
per unit length , moment of inertia I , and modulus of elasticity E that subjected to an axial
force of magnitude ˜ F as shown in Fig. 1. It is assumed that the cross-sectional area of the
beam is uniform and its material is homogenous. The beam is also modeled according to the
Euler-Bernoulli beam theory. Planes of the cross sections remain planes after deformation,
straight lines normal to the mid-plane of the beam remain normal, and straight lines in the
transverse direction of the cross section do not change length. The first assumption ignores
the in plane deformation.
RESULTS AND DISCUSSIONS
The behavior of ψ(A, t) obtained by VIM and PPM at α = π and β = 0.15 is shown in Figs. 2
and 3. Influence of coefficients β and α on frequency and amplitude has been investigated and
plotted in Figs. 4 and 5, respectively. The comparison of the dimensionless deflection versus
time for results obtained from VIM, PPM and Runge-Kutta 4th order has been depicted in
Fig. 6 for α = π and β = 0.15, with maximum deflection at the center of the beam equal to
five (A=5). The solutions are also compared for t=0.5 in Table 1. It can be observed that
there is an excellent agreement between the results obtained from VIM and PPM with those
of Runge-Kutta 4th order method [1].
CONCLUSIONS
In this paper, nonlinear responses of a clamped-clamped buckled beam are investigated. Mathematically,
the beam is modeled by a partial differential equation possessing cubic non-linearity
because of mid-plane stretching. Governing non-linear partial differential equation of Euler-
Bernoulli’s beam is reduced to a single non-linear ordinary differential equation using Galerkin
method. Variational Iteration Method (VIM) and Paremetrized Perturbation Method (PPM)
have been successfully used to study the non-linear vibration of beams. The frequency of both
methods is exactly the same and transverse vibration of the beam center is illustrated versus
amplitude and time. Also, the results and error of these methods are compared with Runge-
Kutta 4th order. It is obvious that VIM and PPM are very powerful and efficient technique
for finding analytical solutions. These methods do not require small parameters needed by
perturbation method and are applicable for whole range of parameters. However, further research
is needed to better understanding of the effect of these methods on engineering problems
especially mechanical affairs.