Free Vibration Analysis of Delaminated Beam with Stochastic Parameters

Document Type : Research Article

Authors

1 Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran

2 Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

Abstract

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.

Keywords

Main Subjects

References

[1] J.T.S. Wang, Y.Y. Liu, J.A. Gibby, Vibration of split beams, Journal of Sound and Vibration, 84(4) (1982)491-502.
[2] P.M. Mujumdar, S. Suryanarayan, Flexural vibrations of beams with delaminations, Journal of Sound and Vibration, 125(3) (1988) 441-461.
[3] M.H. Shen, J.E. Grady, Free vibrations of delaminated beams, AIAA Journal, 30(5) (1992) 1361-1370.
[4] R.A. Jafari-Talookolaei, M.H. Kargarnovin, M.T.Ahmadian, On the dynamic response of a delaminated composite beam under the motion of an oscillating mass,Journal of Composite Materials, 46(22) (2012) 2863-2877.
[5] M.H. Kargarnovin, M.T. Ahmadian, R.A. Jafari-Talookolaei, M. Abedi, Semi-analytical solution for the free vibration analysis of generally laminated composite Timoshenko beams with single delamination,Composites: Part B, 45 (2013) 587-600.
[6] E. Nikolaidis, D.M. Ghiocel, S. Singhal, Engineering Design Reliability Handbook, CRC Press, United States of America, 2005.
[7] S.A. Ramu, R. Ganesan, Stability analysis of a stochastic column subjected to stochastically distributed loadings using the finite element method, Finite Element in Analysis and Design, 11 (1992) 105-115.
[8] J. Cheng, R. Xiao, Probabilistic free vibration analysis of beams subjected to axial loads, Advances in Engineering Software, 38 (2007) 31-38.
[9] R. Ganesan, V.K. Kowda, Buckling of composite beamcolumns with stochastic properties, Journal of Reinforced Plastics and Composites, 24 (2005) 513-543.
[10] S. Irani, S. Sazesh, Random vibration of cantilever tapered beam under stochastic excitation, Modares Mechanical Engineering, 13 (2013) 138-145 (In Persian).
[11] A.A. Alizadeh, H. Mirdamadi, Free vibration and divergence instability of pipes conveying fluid with uncertain structural parameters, Modares Mechanical Engineering, 15 (2015) 247-254 (In Persian).
[12] E. Vanmarche, Random Fields- Analysis and Synthesis, The MIT Press, United States of America, 1983.
[13] A.D. Kiureghian, J.B. Ke, The stochastic finite element method in structural reliability, Probabilistic Engineering Mechanics, 3 (1988) 83-91.
Amirkabir Journal of Mechanical Engineering
Volume 49, Issue 4
March 2018
Pages 731-742

Files

Share

How to cite

Statistics