Finite element model for vibration analysis of pre-twisted Timoshenko beam

Abstract

A new linearly pre-twisted Timoshenko beam finite element, which has two nodes and four-degrees-offreedom per node, is developed and subsequently used for coupled bending-bending vibration analysis of pre-twisted beams with uniform rectangular cross-section. First, displacement functions based on two coupled displacement fields (the polynomial coefficients are coupled through consideration of the differential equations of equilibrium) are derived for pre-twisted beams whose flexural displacements are coupled in two planes. This approach helps to reduce the number of nodal variables. Next, the stiffness and mass matrices of the finite element model are obtained by using the energy expressions. Finally, the natural frequencies of pre-twisted Timoshenko beams are obtained and compared with previously published theoretical and experimental results to confirm the accuracy and efficiency of the present model. Excellent agreement is found with the previous studies. Also, the new pre-twisted Timoshenko beam element has good convergence characteristics. r

Modeling of small-scale structures and the extension to the analysis of micro tool's dynamics

“…Using the results of free vibration analysis of spinning beams available in the literature, numerical comparison is carried out using the prediction from the proposed method and that based on the CFEM. Tables 4-1 When presenting results for Tables 4-1 The comparison of the predictions from the present study with those of Lin et al [167] and Yardimoglu et al [168] is given in In all of the comparisons made in this first layer of validation, a good agreement between the natural frequencies predicted from the SFEM and previous studies is found.…”

On the vibration and stability of spinning axially loaded pre-twisted Timoshenko beams

“…The system Eq. (13) represents the solution to the following two eigenvalue problems: (i) free vibration and (ii) buckling stability.…”

“…In comparison with results between linearly and nonlinearly pre-twisted beams, the deviations increased as increasing pretwist angle. By treating the blade as a pre-twisted Timoshenko beam, the vibration equations of motion of the pre-twisted blade were developed using different techniques [6][7][8][9][10][11][12][13] to study the effects of geometric aspects, rotary inertia and shear deformation on the lateral frequencies of the blade. The variational method was utilized by Carnegie [6] to derive the bending-bending-torsion equations of motion for a pre-twisted cantilever blade allowing for the shear deformation and rotary inertia effects.…”

“…The effects of shear deformation and rotary inertia on the natural frequencies of a twisted beam are same as those of the corresponding straight beam. Based on coupled displacement fields, a new finite element model was developed and used by Yardimoglu and Yildirim [13] to determine the natural frequencies of pre-twisted rectangular Timoshenko beams. The authors indicated that the new pre-twisted Timoshenko beam element possesses good convergence characteristics.…”

Parametric instability of twisted Timoshenko beams with localized damage

“…The effect of the twist angle on the natural frequencies and mode shapes of the cantilever turbine blade was dealt with by Sabuncu [23] based on Euler-Bernoulli beam theory. The bending vibration behavior of the twisted blade was studied by Lin et al [24], Banerjee [25], and Yardimoglu and Yildirim [26] to examine the influence of geometric aspects, rotary inertia and shear deformation on the lateral frequencies using Timoshenko beam theory. The dynamic behavior of a twisted Timoshenko beam rotating about an axis normal to its longitudinal axis was analyzed by Rao and Gupta [27], and Sabuncu and Evran [28] to study the effect of rotational speed on the natural frequency and stability of the blade.…”