Chordwise Vibration Research Articles

Investigations on bifurcation behavior of wind turbine airfoil response at a high angle of attack

Design load and vibration for parked conditions are gaining in importance for large-scale modern wind turbines with increasing flexibility, especially edgewise vibration when the blade is at a high angle of attack. In this work, flow-induced vibration of the wind turbine airfoil at 90 degrees of attack angle is studied with the fluid-structure interaction (FSI) simulation. The unsteady aerodynamic force due to flow separation and vortex shedding at the high angle of attack causes the chordwise vibration of the airfoil. When the vortex shedding frequency fv gets close to the chordwise natural frequency fn of the airfoil, vortex-induced vibration (VIV) of high amplitude occurs accompanied with the frequency lock-in phenomenon. In the post lock-in regime, it is found that period-3 and torus bifurcation occur successively and the vibration response becomes aperiodic. Dynamic mode decomposition(DMD) technique is used to investigate the mechanism of bifurcation from the perspective of energy balance, through analyzing the vorticity field in the wake and pressure distribution on the airfoil surface. For the certain incoming velocity in the post lock-in regime, since the frequency of the DMD mode f=2fv/3 is close to the natural frequencyfn, both the vibration of frequency 2fv/3 and fv get excited, leading to the onset of bifurcation. The Lissajou curves are obtained through reconstructing the transient pressure of each DMD mode, which indicates that energy transfer mainly exists in modes f=fv. In addition, the reconstructed Lissajou curves based on the leading DMD modes agree well with the original time-domain Lissajou curves.

Free vibration analysis of a rotating single edge cracked axially functionally graded beam for flap-wise and chord-wise modes

This study concerns with the free vibration analyses for centrifugally stiffened cracked beams modelled as functionally graded material in the axial direction. The variation of material properties along the cracked beam is expressed in terms of the power law distribution. Analyses are carried out by the Rayleigh–Ritz method that uses shape functions and energy expressions written for centrifugally stiffened Euler-Bernoulli beams. The influence of the hub radius, angular velocity, crack location to beam length ratio and crack depth to thickness ratio and cross-section size ratio and inhomogeneity on natural frequencies are examined for the beams with the clamped-free boundary condition. The results are verified by using the data available in the open literature and/or the outputs of finite element analyses performed for an axially functionally graded solid beam. The results show that the fundamental frequency parameters reduce with the increasing the ratio of crack depth to thickness. The highest natural frequency droop exists close to the root of the beam for flap-wise and chord-wise vibrations. Another important finding is that reduction of natural frequency parameters in chord-wise modes is larger than that in flap-wise modes when the crack location is varied. This new research should help to improve predictions of the impact of the crack on free vibrations for the rotating cracked axially functionally graded beams.

Detection of Structural Damage in Rotating Beams Using Modal Sensitivity Analysis and Sparse Regularization

Rotating beams are often encountered in the wind turbines and the rotors, and detection of the damages in rotating beams as earlier as possible is central to ensuring the safety and serviceability of practical structures. To this end, a modal sensitivity approach in conjunction with the sparse regularization is proposed in this paper. First, the eigen equations for the flap-wise and chord-wise vibrations of a rotating beam are established upon Hamilton’s principle. Then, damage detection is formulated as a nonlinear least-squares problem that finds the damage coefficients to minimize the error between the measured and calculated data. To solve the nonlinear least-squares problem, the sensitivity method that requires the modal sensitivity analysis is developed. In real applications, damage detection is usually an ill-posed problem and to circumvent the ill-posedness, the sparse regularization is introduced due to the fact that the numbers of actual damage locations are often scarce. Numerical examples are studied and results show that the proposed approach is more accurate than the enhanced sensitivity approach and the flap-wise modal data outperforms the chord-wise modal data in damage detection of rotating beams.

Free vibration analysis of rotating nano-beams for flap-wise, chord-wise and axial modes based on Eringen's nonlocal theory

A mathematical model based on Eringen's nonlocal elasticity theory is presented to analyze free vibration behavior of rotating nano-beams. The model is capable of studying the flap-wise and chord-wise vibrations as well as the axial vibration of rotating nano-beams. The model is based on Euler-Bernoulli beam theory, and incorporates spin-softening and Coriolis effects. Hamilton's principle is employed to derived the governing equations involving virtual displacements, and Ritz method is followed to discretize the governing equations. The governing equations are transformed to an eigenvalue problem in state-space. The stable solutions in frequency domain are indentified by appropriately examining the nature of the complex eigenvalues. The model is validated through some reduced problems available in the literature. The non-dimensional speed-frequency behaviors are presented and discussed for different normalized nonlocal parameters, hub parameters and section aspect ratios. The spin-softening and Coriolis effects are individually illustrated and discussed. The present advanced model for rotating nano-beams is reported for the first time. The present study would help in understanding the dynamics of rotating nano-beams in a comprehensive manner.

Nonlocal longitudinal, flapwise, and chordwise vibrations of rotary doubly coaxial/non-coaxial nanobeams as nanomotors

The advancement in both experiment and theory of rotating nanomotors provides a vital base for succeeding nano-electro-mechanical system technology. This paper deals with nonlocal dynamic modeling of double-nanobeam-systems with coaxial and non-coaxial patterns rotate with a rigid shaft accounting for longitudinal, flapwise, and chordwise vibrations. The full lateral interactions of doubly adjacent nanobeams due to three-dimensional vigorous van der Waals forces are considered in the framework of linear vibrations. Using Hamilton’s principle, the nonlocal equations of motion of the nanosystem are constructed by employing Rayleigh and Timoshenko beam theories. By exploiting the size-dependent mode shapes, the unknown deformation fields are appropriately discretized, the Galerkin approach is applied, and the natural frequencies associated with the longitudinal, in-plane, and out-of-plane motions are evaluated. The roles of various crucial factors, including geometrical properties of the nanosystem, nonlocality and shear effects, angular velocity, radius of the shaft, and intertube distance, in the free dynamic response of the rotating nanosystem are explained and discussed comprehensively. Further, the size-dependent critical angular velocity pertinent to the initial dynamic instability of the nanosystem is also introduced and the effects of influential parameters on this key factor are displayed.

Three-dimensional dynamics of beam-like nanorotors on the basis of newly developed nonlocal shear deformable mode shapes

The longitudinal, chordwise, and flapwise vibrations of nanorotors are going to be examined via nonlocal Euler-Bernoulli and Timoshenko beam models. By exploiting Hamilton’s principle on the basis of the nonlocal constitutive relations, the novel-nonlocal equations of motion that display three-dimensional vibrations of the beam-like nanorotors are constructed and solved for natural frequencies using the Galerkin-based assumed mode methodology. For capturing the frequencies more accurately, the newly developed-size-dependent mode shapes are employed in which the true nonlocal boundary conditions at the free end are satisfied exactly. Accounting for dynamic couplings of the longitudinal and chordwise vibrations, the roles of the angular velocity, nonlocality, length of the nanorotor, and radius of the nanoshaft on the longitudinal, chordwise, and flapwise frequencies are displayed and discussed. Further, the critical angular velocity of the nanorotor by considering the nonlocality is derived and its influential factors are displayed. The crucial role of the shear deformation on the obtained results is also explained methodically.

Simplified modelling and analysis of a rotating Euler-Bernoulli beam with a single cracked edge

The natural frequencies and mode shapes of the flapwise and chordwise vibrations of a rotating cracked Euler-Bernoulli beam are investigated using a simplified method. This approach is based on obtaining the lateral deflection of the cracked rotating beam by subtracting the potential energy of a rotating massless spring, which represents the crack, from the total potential energy of the intact rotating beam. With this new method, it is assumed that the admissible function which satisfies the geometric boundary conditions of an intact beam is valid even in the presence of a crack. Furthermore, the centrifugal stiffness due to rotation is considered as an additional stiffness, which is obtained from the rotational speed and the geometry of the beam. Finally, the Rayleigh-Ritz method is utilised to solve the eigenvalue problem. The validity of the results is confirmed at different rotational speeds, crack depth and location by comparison with solid and beam finite element model simulations. Furthermore, the mode shapes are compared with those obtained from finite element models using a Modal Assurance Criterion (MAC).

Modal characteristics of a rotating flexible beam with a concentrated mass based on the absolute nodal coordinate formulation

A new dynamic model of a rotating flexible beam with a concentrated mass located in arbitrary position is derived based on the absolute nodal coordinate formulation, and its modal characteristics are investigated in this paper. To consider the concentrated mass at an arbitrary location of the beam, a Dirac’s delta function is used to express the mass per unit length of the beam. Based on the proposed dynamic model, the frequency analysis is performed. The nonlinear equation is transformed into the linear one via employing the linear perturbation analysis method. The stiffness matrix of static equilibrium of the system under the deformed condition is obtained, in which the effect of coupling between the longitudinal deformation and transversal deformation is included. This means even if only the chordwise bending equation is solved, the longitudinal vibration effect can be still considered. As we know, once the longitudinal deformation is large, it will significantly affect the chordwise bending vibration. So the proposed model in this paper is more accurate than the traditional dynamic models which are usually lack of the coupling terms between the longitudinal deformation and transversal deformation. In fact, the traditional dynamic models for the chordwise vibration analysis in the existing literature are usually linear due to neglecting the coupling terms, and consequently, they are only suitable for the modal characteristic analysis of a beam under small deformations. In order to get some general conclusions of the natural frequencies and mode shapes, the equation which governs the chordwise bending vibration of the rotating beam is transformed into a dimensionless form. The dynamic model presented in this paper is nonlinear and can be conveniently used to analyze the modal characteristics of a rotating flexible beam with large deformations. To demonstrate the power of the new dynamic model presented in this paper, the dynamic simulations involving the comparisons between the different frequencies obtained using the model proposed in this paper and the models in the existing literature and the investigating in frequency veering and mode shift phenomena are given. The simulation results show that the angular velocity of the flexible beam will give rise to the phenomena of the natural frequency loci veering and the associated mode shift which is verified in the previous studies. In addition, the phenomena of the natural frequency loci veering rather than crossing can be observed due to the changing of the magnitude of the concentrated mass or of the location of the concentrated mass which are found for the first time. Furthermore, there is an interesting phenomenon that the natural frequency loci will veer more than once due to different types of mode coupling between the bending and stretching vibrations of the rotating beam. At the same time, the mode shift phenomenon will occur correspondingly. Additionally, the characteristics of the vibration nodes are also investigated in this paper.

Flap-wise and chord-wise vibrations of axially functionally graded tapered beams rotating around a hub

This paper presents flap-wise and chord-wise flexural vibration analyses for centrifugally stiffened tapered beams made of functionally graded material in axial direction. Functions of material properties varying along beam are defined in terms of the power law distribution. Calculations are conducted by simple computation technique of the Rayleigh–Ritz method that uses simple shape functions and energy expressions written for centrifugally stiffened Euler–Bernoulli beams. Effects of taper ratio, hub radius, angular velocity and non-homogeneity are inspected for the thin beams with several classical boundary conditions. Results given as non-dimensional natural frequencies are validated by the results given in existing literature and/or the outputs of finite element analyses performed for axially functionally graded solid beam. Achievements and limitations of the method are discussed and clearly reflected.

Aspect-Ratio Effects on Aeromechanics of Membrane Wings at Moderate Reynolds Numbers

The aspect ratio of a membrane wing at moderate Reynolds number affects aerodynamic performance as well as membrane deformations. Wind-tunnel experiments are conducted using three different low-aspect-ratio wings (aspect ratios of 1, 1.5, and 2). Aerodynamic performance is determined from force and moment measurements, which are performed using a six-component force transducer. Membrane deformations are obtained using photogrammetry. Mean values and unsteady effects are examined for both aerodynamic performance and membrane deformations. Mean deflection results indicate that lower-aspect-ratio membrane wings show defined U-shape deflections along the span, whereas higher aspect ratios display a progressive rise in deformation to the wing tip. Dominant chordwise vibration modes of the membrane and their spectral content show that lower aspect ratios exhibit higher mode shapes and frequencies, likely caused by increased downwash, which delays the influence of vortex shedding into higher incidences. The frequencies of the dominant modes in membrane motions are found to correlate with the frequencies of lift and drag fluctuations.

Chordwise Bending Vibration Analysis of Functionally Graded Double-Tapered Rotating Beams

Functionally graded materials (FGMs) are designed to achieve a functional performance with gradually variable properties in one or more directions. In this article, free chordwise vibrations of functionally graded double-tapered rotating cantilever beams have been investigated. The functionally graded material of the beam is characterized by continuous variables, Young's modulus, and density that vary along the thickness direction according to a power law. The equations of motion are derived using Lagrange's approach, in which the energy expressions for rotating beam are expressed in terms of hybrid deformation variables. The Rayleigh Ritz method is used to obtain the frequencies of the tapered beam. The effect of the gradient in material proportion, taper ratios, hub radius, and rotational speed on Lagrange's approach with the free chordwise bending vibrations of the tapered beam are investigated and compared with the data in the literature.

Frequency Determination Techniques for Cantilevered Plates with Bending-Torsion Coupling.

Various techniques for solution of the variational statement governing vibration of cantilevered composite plates are compared. The partial Ritz (Kantorovich) method assumes mode shapes only in the chordwise direction and reduces the problem to a set of ordinary differential equations. The important nondimensional parameters for cantilevered plates with bending-torsion coupling then are identified easily, including two that express the relative contribution of the warping stiffness to torsional vibration (/366) and chordwise vibration (£22)- The Rayleigh-Ritz solution, requiring assumed mode shapes in both directions, reveals the fundamental coupling mechanisms affecting the natural mode shapes and frequencies. The finite element method yields direct accurate results, yet masks the fundamental behavior. Experiments were performed to provide data for correlation with the analyses. The various solution techniques provide different levels of insight into the fundamental vibration characteristics of cantilevered plates with bending-torsion coupling, including the importance of chordwise flexure in the torsional vibration modes.

Sequential Quadratic Programming and Dynamic Optimal Design of Rotating Blades*, †

ABSTRACT By studying the Kuhn-Tucker conditions for a structural optimal design problem with multifrequency constraints, we show in this paper that the Optimality Criterion (OC) approach and Mathematical Programming (MP) can be unified. This unification leads to a new approach—Sequential Quadratic Programming (SQP)—which we combine with a technique of temporary relaxation of constraints and application of rational move-limits, based on deviations due to linearization of nonlinear constraints. The SQP algorithm is applied to solve the dynamic optimal design problem of rotating blades with prescribed multifrequency constraints on both flapping and chordwise vibration. A computer program based on the Finite Element Method and explicit design sensitivities is developed and a set of examples is tested. Numerical results for a variety of cross-sectional shapes are presented.