Deflection Assumption Research Articles

Comparison of linear and non-linear blade model predictions in Bladed to measurement data from GE 6MW wind turbine

The length and flexibility of wind turbine blades are increasing over time. Typically, the dynamic response of the blades is analysed using linear models of blade deflection, enhanced by various ad-hoc non-linear correction models. For blades undergoing large deflections, the small deflection assumption inherent to linear models becomes less valid. It has previously been demonstrated that linear and nonlinear blade models can show significantly different blade response, particularly for blade torsional deflection, leading to load prediction differences. There is a need to evaluate how load predictions from these two approaches compare to measurement data from the field.In this paper, time domain simulations in turbulent wind are carried out using the aero-elastic code Bladed with linear and non-linear blade deflection models. The turbine blade load and deflection simulation results are compared to measurement data from an onshore prototype of the GE 6MW Haliade turbine, which features 73.5m long LM blades. Both linear and non-linear blade models show a good match to measurement turbine load and blade deflections. Only the blade loads differ significantly between the two models, with other turbine loads not strongly affected. The non-linear blade model gives a better match to the measured blade root flapwise damage equivalent load, suggesting that the flapwise dynamic behaviour is better captured by the non-linear blade model. Conversely, the linear blade model shows a better match to measurements in some areas such as blade edgewise damage equivalent load.

A general in situ probe spacing correction method for dual probe heat pulse sensor

The dual probe heat pulse (DPHP) method is gaining popularity for measuring soil thermal properties. However, because of the fact that the DPHP measured heat capacity (c) of soil is hyper-sensitive to probe spacing variation, probe deflection causes large error in the measured c. To deal with probe deflection, recently, Liu et al. (2013) has proposed an in situ probe spacing correction method, but, their method was based on the inline deflection assumption. Wen et al. (2015) found that for non-inline deflected DPHP sensors, the method of Liu et al. (2013) provided poor reduction of the error in c. To cope with the non-inline deflection, in this study, we introduce a new DPHP sensor design by using three thermistors within the same temperature probe. A related probe spacing correction method for non-inline deflected probes was also presented. Both numerical simulation and experiment were conducted to test the new model. We define θ as the inclination angle of the probe deviation from the vertical direction. For experiment of the outward deflection (5.8°<θ<6.8°), the error in c is in the range from −23% to −76%; after inline in situ correction, the error decreases to the range of 12% to 44%; after using our non-inline in situ correction, the error is between 1% and 8%. Our three-dimensional finite element numerical simulation also demonstrates that, compared with the method of Liu et al. (2013), the new sensor and correcting method can significantly eliminate errors in c caused by probe deflections. The new DPHP sensor design and the non-inline deflection model have the potential to replace the method of Liu et al. (2013), and become the standard method for correcting probe spacing in situ.

Flutter of thermally buckled angle-ply laminates with variable fiber spacing

Abstract This study presents the first aerothermoelastic analysis of angle-ply laminates with variable fiber spacing. Based on the Von Karman large deflection assumptions and quasi-steady supersonic aerodynamic theory, the effects of fiber distribution and temperature gradient on the thermal postbuckling, vibration, and flutter behaviors of angle-ply laminates subjected to aerodynamic force and thermal stress are discussed using the finite element method. The numerical results reveal that fiber redistribution can efficiently increase the critical buckling temperature, natural frequencies and flutter boundary. A new boundary along which the eigenvalue drops to zero is observed, and the chaotic phenomenon is eased due to temperature gradient.

3D hydroelastic analysis of very large floating bodies over variable bathymetry regions

The coupled-mode model developed by Belibassakis and Athanassoulis (J Fluid Mech 531:221–249, 2005) is extended and applied to the hydroelastic analysis of three-dimensional large floating bodies of shallow draft lying over variable bathymetry regions. The method is also applicable to the problem of wave interaction with ice sheets of small thickness. A general bathymetry is assumed, characterized by a continuous depth function, joining two regions of constant, but possibly different, depth. We consider the scattering problem of harmonic incident surface waves, under the combined effects of variable bathymetry and a floating elastic plate of orthogonal planform shape. Under the assumption of small-amplitude waves and plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin elastic-plate theory. To consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. Numerical results concerning floating structures over flat and inhomogeneous seabed are presented, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic responses are discussed.

An analytical procedure for transient response determination of annular FSDT and CPT nanoplates via nonlocal elasticity theory

In this paper, an analytical procedure is introduced for transient response determination of annular nanoplates under impulsive transverse load and radial stress. The equations of motion are extracted using the Hamilton’s principle by small deflection assumption and considering the Eringen nonlocal elasticity theory in conjunction with the first order shear deformation theory as the displacement field. These equations which are a system of partial differential equations with variable coefficients are solved using the perturbation technique and the eigenfunction expansion method. The results are compared with those obtained by the classical plate theory and finite elements method and the effects of nonlocal parameter, radial stress and geometries on the response are studied.

A free vibration analysis of toroidal composite shells in free space

The main aim of this paper is to present a dynamic analysis of toroidal shell structures in free space. The simplified thin shell theory, based on the typical small deflection assumption, but enriched by the transverse shear terms, was used. The constituting material is not isotropic but a multilayer composite angle-ply laminate, with non-uniform thickness.A numerical procedure based on the classical Rayleigh–Ritz method, was utilized. The dynamic independent variables were expressed in terms of a double Fourier series expansion as function of the azimuth and internal meridian circular angle. This allowed to determine easily the strain and kinetic energy expressions, and consequently the stiffness and mass matrices.Then, the numerical problem was simplified to the classical generalized eigenvalue problem relation by differentiating the whole energetic functional, as in the Rayleigh–Ritz method. The solutions were determined by using an appropriate algorithm. The high frequency vibration modes were considered and their peculiarities with respect to the low frequency modes were pointed out. The dependence of the obtained results on the laminate winding angle was also considered.Comparisons with the other authors results were necessary to validate the utilized numerical approach.

Nonlinear aeroelastic modelling for wind turbine blades based on blade element momentum theory and geometrically exact beam theory

Due to the increasing size and flexibility of large wind turbine blades, accurate and reliable aeroelastic modelling is playing an important role for the design of large wind turbines. Most existing aeroelastic models are linear models based on assumption of small blade deflections. This assumption is not valid anymore for very flexible blade design because such blades often experience large deflections. In this paper, a novel nonlinear aeroelastic model for large wind turbine blades has been developed by combining BEM (blade element momentum) theory and mixed-form formulation of GEBT (geometrically exact beam theory). The nonlinear aeroelastic model takes account of large blade deflections and thus greatly improves the accuracy of aeroelastic analysis of wind turbine blades. The nonlinear aeroelastic model is implemented in COMSOL Multiphysics and validated with a series of benchmark calculation tests. The results show that good agreement is achieved when compared with experimental data, and its capability of handling large deflections is demonstrated. Finally the nonlinear aeroelastic model is applied to aeroelastic modelling of the parked WindPACT 1.5 MW baseline wind turbine, and reduced flapwise deflection from the nonlinear aeroelastic model is observed compared to the linear aeroelastic code FAST (Fatigue, Aerodynamics, Structures, and Turbulence).

Aerothermoelastic analysis of composite laminates with variable fiber spacing

This study presents the effects of variable fiber spacing on the thermal postbuckling, vibration, and flutter behaviors of composite laminates subjected to aerodynamic force and thermal stress. A 54 degree-of-freedom high order triangular plate element is developed based on the Von Karman large deflection assumptions and quasi-steady supersonic aerodynamic theory. The numerical results reveal that the redistribution of fibers can dramatically increase the critical buckling temperature, and efficiently increase the natural frequencies and flutter boundary. The sequence of the natural mode may be altered and the frequency coalescence pair may change. The stability boundary is different and some specific phenomena are discussed.

Research on Semi-Rigid Connections Calculation of Laminated Wood Frame Beam-Column

Generally laminated wood frame beam-column uses the additional steel bolts to connect, its performance is a semi-rigid characteristics under the loads. In the small deflection assumption conditions, this paper is based on the theory of semi-rigid connection, derived the semi-rigid rod end stiffness matrix and moment calculation formula of laminated wood structural frame beam-column element under the arbitrary loads. By SAP2000 the type of flexible coefficient is adopted to establish finite element numerical analysis based on 1# wood frame structure-building of science and technology R&amp;D center in Hebei province. The result shows that each component is basically the same with the original design internal forces while the flexible coefficient is range of 0.41 to 0.82, the framework of semi-rigid connections in the actual calculation should be considered. The internal force calculation method is proposed, which provide a basis for wood design calculation that takes into account the semi-rigid connections.

Simplified buckling and ultimate strength analysis of composite plates in compression

The work presented here concerns the ultimate strength predictions of simply supported, square plates of laminated composite material subjected to uniaxial in-plane compressive load. Plates having a range of thicknesses and initial geometric imperfections have been investigated. Several models are established, each based on first order shear deformation theory and assumption of small deflections. The approaches give reasonable but somewhat conservative estimates for the thicker plates considered, while for the thinner plates, neglect of the post-buckling behaviour makes the results very conservative. It will be necessary to use a large deflection plate theory for some of the models to realise their full potential.

Stiction of Flexural MEMS Structures

A variational method using the principle of virtual work (PVW) is presented to formulate the problem of the microcantilever stiction. Compared with the Rayleigh–Ritz method using the arc-shaped or S-shaped deflection, which prescribes the boundary conditions and thus the deflection shape of a stuck cantilever beam, the new method uses the matching conditions and constraint condition derived from PVW and minimization of the system free energy to describe the boundary conditions at the contact separation point. The transition of the beam deflection from an arc-shape-like one to an S-shape-like one with the increase of the beam length is shown by the new model. The (real) beam deflection given by this new model deviates more or less from either an arc-shape or an S-shape, which has significant impact on the interpretation of experimental data. The arc-shaped or S-shaped deflection assumption ignores the beam bending energy inside the contact area and the elastic energy due to the beam/substrate contact, which is inappropriate as shown by this study. Furthermore, the arc-shaped or S-shaped deflection only approximately describes the deflection shape of a stuck beam with zero external load and obviously, the external load changes the beam deflection. The Rayleigh–Ritz method using the arc-shaped or S-shaped deflection assumption in essence can only be used to tell approximately whether stiction occurs or not. Rather than assuming a certain deflection shape and by incorporating the external load, the new method offers a more general and accurate study not only on the microcantilever beam stiction but also on its de-adherence.

A precise model for the shape of an adhered microcantilever

Abstract A variational method using the principle of virtual work (PVW) is presented to formulate the problem of the microcantilever stiction. Compared with the Rayleigh–Ritz method using the arc-shaped or S-shaped deflection, which prescribes the boundary conditions and thus the deflection shape of a stuck cantilever beam, the new method uses the matching conditions and constraint condition derived from PVW and minimization of the system free energy to describe the boundary conditions at the contact separation point. The transition of the beam deflection from an arc-shape-like one to an S-shape-like one with the increase of the beam length is shown by the new model. The (real) beam deflection given by this new model deviates more or less from either an arc-shape or an S-shape, which has significant impact on the interpretation of experimental data. The arc-shaped or S-shaped deflection assumption ignores the beam bending energy inside the contact area and the elastic energy due to the beam/substrate contact, which is inappropriate as shown by this study. Furthermore, the arc-shaped or S-shaped deflection only approximately describes the deflection shape of a stuck beam with zero external load and obviously, the external load changes the beam deflection. The Rayleigh–Ritz method using the arc-shaped or S-shaped deflection assumption in essence can only be used to tell approximately whether stiction occurs or not. Rather than assuming a certain deflection shape and by incorporating the external load, the new method offers a more general and accurate study not only on the microcantilever beam stiction but also on its de-adherence.

On the Dynamics of the Micro-Ring Driven by Traveling Bias Voltages

There is no literature mentioned the modeling of the microstructures subjected to traveling electrostatic forces. This paper is the first time to present an analytical approach to investigate the dynamics of a micro-ring structure driven by the traveling bias voltage. The traveling electrostatic forces may come from the sequentially-actuated actuating electrodes arranged around the flexible ring. A linearized distributed model considering the electromechanical coupling effect is derived based on the small deflection assumption. According to the analytical results, the stiffness of the micro-ring will be softened periodically with the traveling speed of the driving voltage and the variation increases with the increasing of the voltage.

Nonlinear Transient Response of Laminated Composite Shells

This paper first compares the writers’ results of static and dynamic analyses of plates, cylindrical and spherical shells employing four-, eight-, and nine-noded elements with different integration rules with those of earlier investigators and including some of the recent composite theories. Thereafter, the nonlinear transient responses of laminated composite cylindrical and spherical shell panels with cutouts are investigated taking up additional examples that are yet to appear in the published literature. For these, the finite-element model is employed using eight-noded C0 continuity, an isoparametric quadrilateral element considering von Karman large deflection assumptions. In the time integration, the Newmark average acceleration method is used in conjunction with a modified Newton–Raphson iteration scheme. Important conclusions with respect to nonlinear transient responses are summarized for cylindrical and spherical shells with and without cutouts.

Two-material optimal design for nonlinear elastica

The work considers an optimal design problem in the context of nonlinear elastica. More specifically, we deal with finding the best way of mixing fixed amounts of two different elastic materials, so as to minimize the tip deflection of a cantilever beam loaded on its free extreme under the assumption of large deflections. Applying an optimality criteria method to the relaxed problem, simulations give us numerical evidence that the original design problem admits classical solutions (i.e. there is no microstructure) and those are the same as the respective ones for the case of small deflections.

Critical static loads calculations in finite element method of three-layered annular plates

This paper presents the results of critical, static loads calculations of three-layered, annular plates with a soft core. The plate with slidably clamped edges, uniformly loaded with compressive stress on the inner edges of outer layers and of the symmetric cross-section structure is the subject of consideration. In the description of deformation of layers, the assumption of the equal deflections of plate layers has or has not been used. The calculations were carried out for several plate models built of finite elements using the system ABAQUS. The analysis of the distribution of values of critical loads, depending on various thicknesses of plate core, different values of facing thicknesses and on two kinds of core foam material of different stiffnesses, indicates some essential results, which are important in the plate stability problems. Among them the observation of possible area of too high values of critical loads shows the limitation of usage of rather universally applied assumption of equal layers deflections in cross-section deformation of the plates with thick core. This conclusion seems to be particularly important for designers.

Topology optimization for synthesis of contact-aided compliant mechanisms using regularized contact modeling

A topology optimization technique for systematically designing contact-aided compliant mechanisms (CCM) is presented in this paper. A CCM is a single piece elastic body that uses intermittent contacts in addition to elastic deformation to transmit force and motion. Contact interactions give rise to interesting nonlinear and nonsmooth behaviors even under the small deflection assumption made in this work. The difficulties associated with the nondifferentiability inherent in the CCM systematic synthesis problem are circumvented by using a regularized contact model. This model uses a smooth approximation of the unilateral displacement constraints that are used to model contact interactions. The use of a regularized contact model in the underlying state problem makes it possible to use efficient smooth optimization algorithms for the systematic synthesis of CCMs. The formulation of the design problem for CCMs, sensitivity analysis, and solution methodology are presented. The paper includes CCM designs that exhibit nonsmooth motion and force transmission characteristics, which are not possible or practical with compliant mechanisms that do not use intermittent contact interactions.

The effect of extension/twist anisotropy on compression buckling in cylindrical shells

The study of axial compression buckling of isotropic cylinders has received much attention by various researchers over the years. It is commonly acknowledged that the presence of minute imperfections significantly reduces potential buckling loads in comparison with classical linear predictions. This approach, of including geometric imperfections, has been extended by a significant, yet fewer, number of researchers to composite cylindrical shells. As the current study shows, imperfections may not be the only major factor for the discrepancy between experimental and linear buckling loads. Flexural/twist anisotropy, present in most balanced, symmetric laminates with angle-ply layers, is shown to play a significant role in reducing buckling loads from those predicted by classical analysis. Indeed, the assumption of deflections in the form of a double sine series appears to be questionable for such laminates. A recently reported classically linear analysis that included the effects of flexural/twist coupling on buckling loads has been further developed to include the effects of extension/twist coupling. It is shown that buckling loads can be improved by making the laminate antisymmetric rather than symmetric, for the class of quasi-isotropic laminates, whilst retaining a spiral mode shape.

Anisotropy-Induced Spiral Buckling in Compression-Loaded Cylindrical Shells

The study of axial compression buckling of isotropic cylinders has received much attention by various researchers over the years. It is commonly acknowledged that the presence of minute imperfections reduces potential buckling loads significantly in comparison with classical linear predictions. This approach has been extended by a significant, yet fewer, number of researchers to composite cylindrical shells. It is shown that imperfections may not he the only major factor for the discrepancy between experimentally obtained buckling loads and those predicted from linear bifurcation theory with orthotropic properties. Flexural/twist anisotropy, present in most balanced, symmetric laminates with angle ply layers is shown to play a significant role in reducing buckling loads from those classically predicted. Indeed, the assumption of deflections in the form of a double sine series appears to be questionable for such laminates. A previously unreported classical linear analysis including the effect of flexural/twist coupling is developed. Backed up by detailed comparison with finite element studies, it is shown that buckling loads can be reduced by up to 30% for a class of quasi-isotropic laminates and is accompanied by a change in mode form from doubly periodic to spiral in nature.

Nonlinear finite element analysis of composite shell under impact

Large deflection dynamic responses of laminated composite cylindrical shells under impact are analyzed by the geometrically nonlinear finite element method based on a generalized Sander’s shell theory with the first order transverse shear deformation and the von-Karman large deflection assumption. A modified indentation law with inelastic indentation is employed for the contact force. The nonlinear finite element equations of motion of shell and an impactor along with the contact laws are solved numerically using Newmark’s time marching integration scheme in conjunction with Akay type successive iteration in each step. The ply failure region of the laminated shell is estimated using the Tsai-Wu quadratic interaction criteria. Numerical results, including the contact force histories, deflections and strains are presented and compared with the ones by linear analysis. The effect of the radius of curvature on the composite shell behaviors is investigated and discussed.