Study of Coupled Lateral-Torsional Free Vibrations of Laminated Composite Beam: Analytical Approach
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##### Study of Coupled Lateral-Torsional Free Vibrations of Laminated Composite Beam: Analytical Approach

Authors: S.H. Mirtalaie, M.A. Hajabasi

Abstract:

In this paper, an analytical approach is used to study the coupled lateral-torsional vibrations of laminated composite beam. It is known that in such structures due to the fibers orientation in various layers, any lateral displacement will produce a twisting moment. This phenomenon is modeled by the bending-twisting material coupling rigidity and its main feature is the coupling of lateral and torsional vibrations. In addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies. Then, the governing differential equations are derived using the Hamilton-s principle and the mathematical model matches the Timoshenko beam model when neglecting the effect of bending-twisting rigidity. The equations of motion which form a system of three coupled PDEs are solved analytically to study the free vibrations of the beam in lateral and rotational modes due to the bending, as well as the torsional mode caused by twisting. The analytic solution is carried out in three steps: 1) assuming synchronous motion for the kinematic variables which are the lateral, rotational and torsional displacements, 2) solving the ensuing eigenvalue problem which contains three coupled second order ODEs and 3) imposing different boundary conditions related to combinations of simply, clamped and free end conditions. The resulting natural frequencies and mode shapes are compared with similar results in the literature and good agreement is achieved.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1078739

References:

[1] R.B. Abarcar, P.F. Cunniff, "The vibration of cantilever beams of fiber reinforced material," Journal of Composite Materials, vol. 6, pp. 504- 517, 1972.
[2] E.H. Mansfield, A.J. Sobey, "The fibre composite helicopter blade, part 1: stiffness properties, part 2: prospects for aeroelastic tailoring," Aero Quart., vol. 30, pp. 413-49, 1979.
[3] K.K. Teh, C.C. Huang, "The vibrations of generally orthotropic beams, a finite element approach," J. Sound. Vib., vol. 62, pp. 195-206, 1979.
[4] K. Chandrashekhara, K. Krishanamurthy, S. Roy, "Free vibration of composite beams including rotatory inertia and shear deformation," Composite Structures, vol. 14, pp. 269-279, 1990.
[5] H. Abramovich, "Shear deformation and rotatory inertia effects of vibrating composite beams," Composite Structures, vol. 20, pp. 165-173, 1992.
[6] V. Yildirim, "Rotary inertia, axial and shear deformation effects on the in-plane natural frequencies of symmetric cross-ply laminated circular arches," J. Sound. Vib., vol. 224, no. 4, pp. 575-589, 1999.
[7] L.S. Teoh, C. C. Huang, "The vibration of beams of fibre reinforced material," J. Sound Vib. vol. 51, pp. 467-73, 1977.
[8] J.R. Banerjee, F. W. Williams, "Exact dynamics stiffness matrix for composite Timoshenko beams with applications," J. Sound Vib., vol. 194, no. 4, pp. 573-585, 1996.
[9] L.C. Bank, C.H. Kao, "Dynamic response of composite beam, in: D. Hui, J.R. Vinson, (Eds.), Recent Advances in the Macro- and Micro- Mechanics of Composite Material Structures," AD 13, The Winter Annual Meeting of the 1975 ASME, Chicago, IL, November-December 1977.
[10] J.R. Banerjee, "Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method," Computers and Structures, vol. 69, pp. 197-208, 1998.
[11] M.O. Kaya, O. Ozdemir Ozgumus, "Flexural-torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM," J. Sound Vib., vol. 306, pp. 495- 506, 2007.
[12] J.R. Banerjee, "Frequency equation and mode shape formulae for composite Timoshenko beams," Composite Structures, vol. 51, pp. 381- 388, 2001.