The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method

Computational Mechanics (Impact Factor: 2.53). 04/2011; 47(4):395-408. DOI: 10.1007/s00466-010-0550-9


A modeling method for flapwise and chordwise bending vibration analysis of rotating pre-twisted Timoshenko beams is introduced.
In the present modeling method, the shear and the rotary inertia effects on the modal characteristics are correctly included
based on the Timoshenko beam theory. The kinetic and potential energy expressions of this model are derived from the Rayleigh–Ritz
method, using a set of hybrid deformation variables. The equations of motion of the rotating beam are derived from the kinetic
and potential energy expressions introduced in the present study. The equations thus derived are transmitted into dimensionless
forms in which main dimensionless parameters are identified. The effects of dimensionless parameters such as the hub radius
ratio, slenderness ration, etc. on the natural frequencies and modal characteristics of rotating pre-twisted beams are successfully
examined through numerical studies. Finally the resonance frequency of the rotating beam is evaluated.

KeywordsModal characteristics–Pre-twisted rotating Timoshenko beams–Hybrid deformation variables–Dimensionless parameters–Rayleigh–Ritz method

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    • "Yoo et al. [12] studied the stability of a cantilever beam, which is attached to an axial oscillating base undergoing periodic impulsive force with Kane's method. Zhu [13] investigated the vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method. Stoykov and Ribeiro [14] studied the vibration of rotating 3D beams by the p-version finite element method. "
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    ABSTRACT: Analytical solutions for the vibration of a beam with axial force subjected to generalized support motion are obtained in this paper. The finite element method (FEM) is introduced to validate the analytical solution obtained by an analytical approach. The dynamic responses of clamped–clamped, pinned–pinned and clamped–pinned beams with axial tension or compression are obtained via analytical approach and FEM. Comparing results show that the analytical approach is effective. The analytical analysis shows that the resonance will occur in general when the oscillatory frequency of transverse motion or rotation of any support end is equal to the natural frequency of the beam. Moreover, several cases in which the resonance disappears even if the frequencies of support excitations are equal to the natural frequencies of the beam are detected and are validated by the FEM solution.
    Journal of Sound and Vibration 03/2015; 338. DOI:10.1016/j.jsv.2014.11.004 · 1.81 Impact Factor
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    • "Thus β is zero at the tip and β is at a maximum value at the hub. The effect of the blade structural pre-twist is therefore accounted for in this model following an approach developed by Zhu [19]. "
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    ABSTRACT: In-plane vibrations of wind turbine blades are of concern in modern multi-megawatt wind turbines. Today׳s turbines with capacities of up to 7.5 MW have very large, flexible blades. As blades have grown longer the increasing flexibility has led to vibration problems. Vibration of blades can reduce the power produced by the turbine and decrease the fatigue life of the turbine. In this paper a new active control strategy is designed and implemented to control the in-plane vibration of large wind turbine blades which in general is not aerodynamically damped. A cable connected active tuned mass damper (CCATMD) system is proposed for the mitigation of in-plane blade vibration. An Euler–Lagrangian wind turbine model based on energy formulation has been developed for this purpose which considers the structural dynamics of the system and the interaction between in-plane and out-of-plane vibrations and also the interaction between the blades and the tower including the CCATMDs. The CCATMDs are located inside the blades and are controlled by an LQR controller. The turbine is subject to turbulent aerodynamic loading simulated using a modification to the classic Blade Element Momentum (BEM) theory with turbulence generated from rotationally sampled spectra. The turbine is also subject to gravity loading. The effect of centrifugal stiffening of the rotating blades has also been considered. Results show that the use of the proposed new active control scheme significantly reduces the in-plane vibration of large, flexible wind turbine blades.
    Journal of Sound and Vibration 11/2014; 333(23):5980-6004. DOI:10.1016/j.jsv.2014.05.031 · 1.81 Impact Factor
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    • "The parameters , , and are the second area moments of inertia and the second area products of inertia of a cross section of the blade respectively. The effect of the blade structural pre-twist is therefore accounted for in this model following Zhu [3]. The term is the in-plane modal stiffness of the tower/nacelle and is the out-of-plane modal stiffness of the tower/nacelle. "
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    ABSTRACT: The aim of this paper is to develop an active structural control scheme to control wind turbine nacelle/tower out-of-plane vibration. An active tuned mass damper (ATMD) is designed an placed inside the turbine nacelle. An Euler Lagrangian wind turbine model based on energy formulation is developed for this purpose, which considers the structural dynamics of the system and the interaction between in-plane and out-of-plane vibrations. Also, the interaction between the blades and the tower including the ATMD is considered. The wind turbine is subjected to gravity and turbulent aerodynamic loadings. A three-dimensional (3D) model of a wind turbine foundation is designed and analysed in the finite element geotechnical code PLAXIS. The rotation of the foundation is measured and used to calculate a rotational spring constant for use in wind turbine models to describe the soil-structure interaction (SSI) between the wind turbine foundation and the underlying soil medium. Damage is induced in the soil medium by a loss in foundation stiffness. The active control scheme is shown to reduce nacelle/tower vibration when damage occurs.
    Key Engineering Materials 07/2013; 569(2013):660-667. DOI:10.4028/ · 0.19 Impact Factor
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