Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method

Abstract The free vibration analysis of axially loaded composite Timoshenko beams is carried out by using the dynamic stiffness matrix method. This is accomplished by developing an exact dynamic stiffness matrix of a composite beam with the effects of axial force, shear deformation and rotatory inertia taken into account, i.e. it is for an axially loaded composite Timoshenko beam. The theory includes the (material) coupling between the bending and torsional modes of deformations which is usually present in laminated composite beams due to ply orientation. An analytical expression for each of the elements of the dynamic stiffness matrix is derived by rigorous application of the symbolic computing package reduce . Use of such expressions leads to substantial savings in computer time when compared with numerical methods usually adopted in the absence of such expressions. The application of the dynamic stiffness matrix is demonstrated by investigating the free vibration characteristics of an example composite beam for which some comparative results are available. The solution technique used to yield the natural frequencies is that of the Wittrick–Williams algorithm. The effects of axial force, shear deformation and rotatory inertia on the natural frequencies are demonstrated. The theory developed has applications to composite wings and helicopter blades.

[1]  P. Cunniff,et al.  The Vibration of Cantilever Beams of Fiber Reinforced Material , 1972 .

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

[3]  Omri Rand,et al.  Free vibration of thin-walled composite blades , 1994 .

[4]  Frederic W. Williams,et al.  Exact Buckling and Frequency Calculations Surveyed , 1983 .

[5]  H. Abramovich,et al.  Free vibrations of non-symmetric cross-ply laminated composite beams , 1994 .

[6]  W. H. Wittrick.  General sinusoidal stiffness matrices for buckling and vibration analyses of thin flat-walled structures , 1968 .

[7]  Franklin Y. Cheng,et al.  Dynamic Matrix of Timoshenko Beam Columns , 1973 .

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

[9]  Erian A. Armanios,et al.  Free Vibration Analysis of Anisotropic Thin-Walled Closed-Section Beams , 1994 .

[10]  Haim Abramovich,et al.  Shear deformation and rotary inertia effects of vibrating composite beams , 1992 .

[11]  L. Librescu,et al.  Free vibration and aeroelastic divergence of aircraft wings modelledas composite thin-walled beams , 1991 .

[12]  F. W. Williams,et al.  Natural frequencies of frames with axially loaded Timoshenko Members , 1973 .

[13]  Frederic Ward Williams,et al.  EXACT DYNAMIC STIFFNESS MATRIX FOR COMPOSITE TIMOSHENKO BEAMS WITH APPLICATIONS , 1996 .

[14]  T. A. Weisshaar,et al.  Vibration Tailoring of Advanced Composite Lifting Surfaces , 1985 .

[15]  John Fitch,et al.  Solving Algebraic Problems with Reduce , 1985, J. Symb. Comput..

[16]  J. R. Banerjee,et al.  Free vibration of composite beams - An exact method using symbolic computation , 1995 .

[17]  E. Smith,et al.  Formulation and evaluation of an analytical model for composite box-beams , 1991 .

[18]  V. Ramamurti,et al.  Structural dynamic analysis of composite beams , 1990 .

[19]  T. M. Wang,et al.  Vibrations of frame structures according to the Timoshenko theory , 1971 .

[20]  J. S. Fleming,et al.  Dynamics in engineering structures , 1973 .

[21]  Mark V. Fulton,et al.  Free-Vibration Analysis of Composite Beams , 1991 .

[22]  C. Sun,et al.  Vibration analysis of laminated composite thin-walled beams using finite elements , 1991 .

[23]  E. H. Mansfield,et al.  The Fibre Composite Helicopter Blade , 1979 .

[24]  V. Koloušek,et al.  Anwendung des Gesetzes der virtuellen Verschiebungen und des Reziprozitätssatzes in der Stabwerksdynamik , 1941 .

[25]  Prabhat Hajela,et al.  Free vibration of generally layered composite beams using symbolic computations , 1995 .

[26]  F. W. Williams,et al.  A GENERAL ALGORITHM FOR COMPUTING NATURAL FREQUENCIES OF ELASTIC STRUCTURES , 1971 .