Elasticity solution for free vibration of laminated beams

Abstract Based on the two-dimensional theory of elasticity, a new approach combining the state space method and the differential quadrature method is presented in this paper for freely vibrating laminated beams. Applying the differential quadrature method to the state space formulations along the axial direction of the beam, new state equations about state variables at discrete points are obtained. Using matrix theory, the solution can be easily derived, which can very conveniently deal with the continuity conditions. Frequency equation governing the free vibration of laminated beams is then derived and the natural frequencies are obtained. No other assumption on deformations and stresses along the thickness direction is introduced, so that the present method is efficient for laminated beams with arbitrary thickness. It also can cope with arbitrary boundary conditions without applying Saint-Venant’s principle. Numerical examples of multi-layered beams and sandwich beams are performed. Results are verified by comparing them with the published results obtained from various finite element methods and shear beam theories.

[1]  K. M. Ahmed,et al.  Free vibration of curved sandwich beams by the method of finite elements , 1971 .

[2]  Anthony M. Waas,et al.  Effects of shear deformation on buckling and free vibration of laminated composite beams , 1997 .

[3]  M. Pandey,et al.  Differential quadrature method in the buckling analysis of beams and composite plates , 1991 .

[4]  Fuh-Gwo Yuan,et al.  A higher order finite element for laminated beams , 1990 .

[5]  Tarun Kant,et al.  Analytical solution to the dynamic analysis of laminated beams using higher order refined theory , 1997 .

[6]  R. Knops,et al.  Three-Dimensional Problems of the Theory of Elasticity , 1967, The Mathematical Gazette.

[7]  J. Banerjee.  Frequency equation and mode shape formulae for composite Timoshenko beams , 2001 .

[8]  A. S. Bassiouni,et al.  Dynamic analysis for laminated composite beams , 1999 .

[9]  Y. P. Tseng,et al.  In-plane vibration of laminated curved beams with variable curvature by dynamic stiffness analysis , 2000 .

[10]  Buckling of Sandwich Cylinders Under Axial Load , 1962 .

[11]  Y. K. Cheung,et al.  Bending and vibration of multilayer sandwich beams and plates , 1973 .

[12]  Yogesh M. Desai,et al.  Free vibrations of laminated beams using mixed theory , 2001 .

[13]  K. Chandrashekhara,et al.  Free vibration of composite beams including rotary inertia and shear deformation , 1990 .

[14]  K. Chandrashekhara,et al.  Free vibration of composite beams using a refined shear flexible beam element , 1992 .

[15]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[16]  Moshe Eisenberger,et al.  Dynamic stiffness analysis of laminated beams using a first order shear deformation theory , 1995 .

[17]  F. J. Plantema,et al.  Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates and Shells , 1966 .

[18]  K. Ahmed.  Dynamic analysis of sandwich beams , 1972 .

[19]  Robert E. Miller,et al.  A new finite element for laminated composite beams , 1989 .

[20]  J. N. Reddy,et al.  On the Solutions to Forced Motions of Rectangular Composite Plates , 1982 .

[21]  L. S. Teoh,et al.  The vibration of beams of fibre reinforced material , 1977 .

[22]  C. Shu,et al.  APPLICATION OF GENERALIZED DIFFERENTIAL QUADRATURE TO SOLVE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 1992 .