Keywords: beam, bending vibrations, rotary inertia, meshless methods, DFE formulation.

Numerous structural configurations such as helicopter, turbine and compressor blades, spinning spacecraft, satellites, and also rotating shafts and linkages, at least for the first few modes of free vibrations, are modeled as axially loaded beams or beam assemblies. The free vibration analysis of axially loaded beams has been investigated by many researchers. The outcome of these studies has been different beam theories and various analytical and numerical analysis tools. For approximate solutions one may discretize the system by either the lumped-mass method or one of the methods based on the assumed deformation shapes. The latter category includes the Rayleigh-Ritz method, the Galerkin method, and the Finite Element Method (FEM) where wing structure element matrices are evaluated from assumed fixed shape functions (like polynomials). A generalized linear eigenvalue problem then results and one can evaluate the natural frequencies and modes of vibration of beams and beam structures.

Alternatively, the Dynamic Stiffness Matrix (DSM) method can be used to evaluate the natural modes of vibration of the beam structure. Obviously the method gives more accurate results because it exploits the exact member theory. A generalized nonlinear eigenvalue problem then results. But it implies, in many cases, mathematical procedures that are difficult to deal with, and are often limited to special cases.

The Dynamic Finite Element (DFE) approach in vibration analysis of Euler- Bernoulli beams and beam assemblies is well established [1,2]. The DFE formulation has proven to produce the accurate solutions for lateral bending, coupled flexural-torsional vibration of beams, and for structures composed of such beam elements. This method, presented as an intermediate approach, combines the generality of the well-known "weighted residual method" procedure, as used in the FEM, and the high precision provided by DSM method. The weighting functions and shape functions are evaluated referring to the appropriate exact DSM formulation. The eigenvalue problem resulting from this method is also a nonlinear one. It has been shown that the DFE approach can be advantageously extended to more complex cases which distinguishes this method from the DSM method [2].

The aim of this investigation is to extend the DFE methodology to the vibration analysis of axially loaded Timoshenko beams by including first the rotary inertia effects in the beam governing differential equation. In order to further simplify the problem at this stage, the shear deformation, damping, and ant changes in the geometric and structural properties are neglected (i.e., uniform and homogeneous beam). It is demonstrated that the proposed Dynamic Finite Element method, in this case, leads to the same "exact" DSM formulation where only ONE uniform beam element can be used to converge on any number of natural frequency with any desired accuracy. That is why the DFE method, in this case, is considered to be a "meshless" formulation. Based on the proposed dynamic shape functions, the principle of virtual work (PVW) and the weighted residual method, it would be possible to extend the DFE method to more complex cases including shear deformation, damping effects and variable geometric and structural parameters.

1

Hashemi, S.M., Richard, M.J., Dhatt, G., "A Bernoulli-Euler stiffness approach for vibrational analysis of spinning linearly tapered beams", ASME Paper No. 97-GT-500, in "42nd ASME Gas Turbine and Aeroengine Congress", 1998.

2

Hashemi, S.M., Richard, M.J., Dhatt, G., "A new dynamic finite element (DFE) formulation for lateral free vibrations of Euler-Bernoulli spinning beams using trigonometric shape functions", Journal of Sound and Vibration, 220(4), 601-624, 1999. doi:10.1006/jsvi.1998.1922

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