Coupled Lateral-Torsional Free Vibrations Analysis of Laminated Composite Beam using Differential Quadrature Method
In this paper the Differential Quadrature Method (DQM) is employed to study the coupled lateral-torsional free vibration behavior of the laminated composite beams. In such structures due to the fiber orientations in various layers, the lateral displacement leads to a twisting moment. The coupling of lateral and torsional vibrations is modeled by the bending-twisting material coupling rigidity. In the present study, 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 of the beam. The governing differential equations of motion which form a system of three coupled PDEs are solved numerically using DQ procedure under different boundary conditions consist of the combinations of simply, clamped, free and other end conditions. The resulting natural frequencies and mode shapes for cantilever beam are compared with similar results in the literature and good agreement is achieved.
[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 deformatio,"
Composite Structures, vol. 14, pp. 269-279, 1990.
[5] 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.
[6] H. Abramovich, "Shear deformation and rotatory inertia effects of
vibrating composite beams," Composite Structures, vol. 20, pp. 165-173,
1992.
[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] 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.
[9] 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.
[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] R.E. Bellman, J. Casti, "Differential quadrature and long-term
integration," J. Math. Anal. Appl., vol. 34, pp. 235-238, 1971.
[13] C.W. Bert, S.K. Jang, A.G. Striz, "Two new approximate methods for
analyzing free vibration of structural components," AIAAJ, vol. 26,
pp.612-8, 1988.
[14] C.W. Bert, M. Malik, "Differential quadrature method in computational
mechanics: A review," Applied Mechanics Review, vol. 49, pp. 1-27,
1996.
[15] C. W. Bert, M. Malik, "Differential quadrature method: A powerful new
technique for analysis of composite structures," Composite Structures, vol.
39, pp. 179-89, 1997.
[16] C, Shu, B.E. Richards, "Application of generalized differential quadrature
to solve two-dimensional incompressible Navier Stokes equations," Int J
Numer Meth Fluids, vol. 15, pp. 791-8, 1992.
[17] J.R. Banerjee, "Frequency equation and mode shape formulae for
composite Timoshenko beams," Composite Structures, vol. 51, pp. 381-
388, 2001.
[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 deformatio,"
Composite Structures, vol. 14, pp. 269-279, 1990.
[5] 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.
[6] H. Abramovich, "Shear deformation and rotatory inertia effects of
vibrating composite beams," Composite Structures, vol. 20, pp. 165-173,
1992.
[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] 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.
[9] 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.
[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] R.E. Bellman, J. Casti, "Differential quadrature and long-term
integration," J. Math. Anal. Appl., vol. 34, pp. 235-238, 1971.
[13] C.W. Bert, S.K. Jang, A.G. Striz, "Two new approximate methods for
analyzing free vibration of structural components," AIAAJ, vol. 26,
pp.612-8, 1988.
[14] C.W. Bert, M. Malik, "Differential quadrature method in computational
mechanics: A review," Applied Mechanics Review, vol. 49, pp. 1-27,
1996.
[15] C. W. Bert, M. Malik, "Differential quadrature method: A powerful new
technique for analysis of composite structures," Composite Structures, vol.
39, pp. 179-89, 1997.
[16] C, Shu, B.E. Richards, "Application of generalized differential quadrature
to solve two-dimensional incompressible Navier Stokes equations," Int J
Numer Meth Fluids, vol. 15, pp. 791-8, 1992.
[17] J.R. Banerjee, "Frequency equation and mode shape formulae for
composite Timoshenko beams," Composite Structures, vol. 51, pp. 381-
388, 2001.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:58155", author = "S.H. Mirtalaie and M. Mohammadi and M.A. Hajabasi and F.Hejripour", title = "Coupled Lateral-Torsional Free Vibrations Analysis of Laminated Composite Beam using Differential Quadrature Method", abstract = "In this paper the Differential Quadrature Method (DQM) is employed to study the coupled lateral-torsional free vibration behavior of the laminated composite beams. In such structures due to the fiber orientations in various layers, the lateral displacement leads to a twisting moment. The coupling of lateral and torsional vibrations is modeled by the bending-twisting material coupling rigidity. In the present study, 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 of the beam. The governing differential equations of motion which form a system of three coupled PDEs are solved numerically using DQ procedure under different boundary conditions consist of the combinations of simply, clamped, free and other end conditions. The resulting natural frequencies and mode shapes for cantilever beam are compared with similar results in the literature and good agreement is achieved.
", keywords = "Differential Quadrature Method, Free vibration,
Laminated composite beam, Material coupling.", volume = "6", number = "7", pages = "1266-6", }
