Free Vibration Analysis of Delaminated Beam with Stochastic Parameters

In this article, the random vibration analysis of a delaminated beam is performed for the first time by considering the thicknesswise location of the delamination and the Young’s modulus as the stochastic parameters. First, the delaminated beam is divided into four intact sub-beams. Then by introducing a beam element and based on the beam’s classical theory, the kinetic and potential energies of each sub-beam are derived. The considered higher order element has three nodes, one at each end and one at the midpoint and each node has two degrees of freedom, namely, deflection and slope. Using the mentioned energies, the stiffness and mass matrixes of the element are obtained. Next, by assembling the above stated matrices and considering the continuity conditions for adjoining elements at the delamination boundaries, the total stiffness and mass matrices are obtained. In employing the continuity conditions, the deflection and slope of these elements are taken to be equal. At the end, by applying the boundary conditions the governing differential equations of motion are obtained in matrix form. Then by modeling the stochastic parameters as random fields, the governing deterministic differential equation of the system is transformed into a stochastic differential equation. The continuous random fields are discretized by mid-point and local average discretization methods. Finally using the Monte Carlo simulation method in each iteration loop, each stochastic differential equation is transformed into a deterministic differential equation. For free vibration analysis, the eigenvalue problem is solved to investigate the frequencies and mode shapes of the system. Consequently, having the eigenpairs of the system, the statistical properties of free vibration characteristics of the beam such as expected values, standard deviations and probability density functions are obtained and the effect of different parameters of the beam and delamination are studied. Also in order to verify the obtained equations and the written computer programs, the deterministic frequencies of the beam are compared with other results and very good agreement is observed.