Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques

Abstract

In this paper, nonlinear responses of a clamped-clamped buckled beam are investigated. Two efficient and easy mathematical techniques called He's Variational Approach and Laplace Iteration Method are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics.

Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

“…Table 1 and Table 2, a comparison is made between the presented method and four different methods, with semi-analytical approach by Barari et al (2011), with a semi-analytical method by Nikkar et al (2014), with semi-analytical approach by Johnson et al (2014) and with numerical method RK-4. It is seen that the computed values of deflection at the center of the beam for different values of maximum amplitude of oscillation are in strong agreement with the results by these methods.…”

“…In Table 3 Barari et al (2011). Error-III is obtained by taking the existing solution form the method by Nikkar et al (2014). In all of these calculations the errors are shown for three different amplitude of oscillation of beam over a wide range of time.…”

“…From Table 3, it can be clearly seen that the results by present method converges closely to other results in literature and are in strong agreement. (5) Table 2: Differences between the time marching solution of equation (5) Figure 3, time marching deflection response over a wide range of time is shown where the result obtained from present method is graphically compared with the result computed from semianalytical approach by Nikkar et al (2014). In Figure 4, the time marching response of deflection by present method is compared with the solution of deflection response of center of the beam by Barari et al (2011).…”

“…Moreover, it is observed that that the accumulated error decreases with an increase in the time step size for both the classical RK-4 and present method. Barari et al 2011, Nikkar et al 2014and Johnson et al 2014 in literature and also with classical numerical scheme RK-4. It is worthy of noting that the present scheme does not involve iterative time integration like other methods in literature and thus no associated accumulative errors.…”

“…Nonlinear vibrations of the Euler-Bernoulli beam has been investigated by Barari et al (2011) using the Variational iteration and parameterized perturbation methods. Nikkar et al (2014) studied the nonlinear vibration response of Euler-Bernoulli beam by approximate analytical techniques where they utilized Variational He's approach and Laplace iteration technique in order to solve the respective nonlinear governing equations. Pakar and Bayat (2013) applied a max-min approach, also by He, to investigate the nonlinear vibrational response of Euler-Bernoulli beams subjected to an axial load.…”