A new sensitivity analysis for structural optimization of composite rotor blades

This paper presents the mathematical details of the formulation for the sensitivity derivatives for the structural dynamic, aeroelastic stability, and response characteristics of a rotor blade in hover and forward flight. The formulation is denoted by the term semi-analytical approach, because certain derivatives are evaluated by a finite difference scheme. Using this formulation, sensitivity derivatives for structural dynamic and aeroelastic stability characteristics were evaluated for a composite rotor blade. Useful conclusions regarding the relative merits of the semi-analytical approach, when compared to a pure finite difference approach, are obtained. In addition, influence of ply orientation angles on the vibratory hub loads in forward flight is also considered.     

Vibration and flutter of stiff-inplane elastically tailored composite rotor blades

Aeroelastic response, blade and hub loads, and shaft-fixed aeroelastic stability is investigated for a helicopter with elastically tailored stiff-inplane composite rotor blades. A free wake model for nonuniform rotor inflow is integrated with a recently developed finite-element-based aeroelastic analysis for helicopters with tailored composite blades. Pitch-flap and pitch-lag elastic couplings, introduced through the anisotropy of the plies in the blade spar, have a significant effect on the dynamic elastic torsion response. Positive and negative pitch-flap couplings reduce vertical hub shear forces approximately 20% in the high vibration transition flight regime, however, negative pitch-flap elastic coupling significantly increases inplane hub shear forces at all flight speeds. The influence of pitch-flap, pitch-lag, and extension-torsion elastic couplings on the rotating frame blade bending moments is small. Ply-induced composite couplings have a powerful effect on blade stability in both hover and forward flight. Positive pitch-flap, positive pitch-lag, and positive extension-torsion couplings each have a stabilizing effect on lag mode damping. Negative pitch-lag coupling has a strong destabilizing effect on blade lag stability, resulting in a mild instability at moderate flight speeds.     

旋翼/机身非线性气弹耦合配平及稳定性分析

根据Hamilton原理,采用中等变形梁理论,将桨叶离散为15自由度梁单元,用准定常气动模型建立旋翼/刚性机身耦合的有限元非线性方程,用时间有限元法进行气弹耦合配平计算,得到桨叶和机身运动的周期解．在此基础上,引入Peters动态入流模型分析耦合系统的稳定性．并研制相应的计算程序,可用于桨叶响应、桨叶和桨毂载荷、旋翼操纵等方面的分析计算．算例分析结果与相关文献吻合较好,且同时满足桨叶响应和配平方程的收敛性要求．     

大型水平轴风力机叶片气动弹性计算

给出了一种考虑几何非线性的大型风力机静、 动气动弹性一体化计算方法．采用涡尾迹方法进行风力机气动载荷计算．建立风力机风轮的三维壳模型．沿周向平均风力机叶片载荷并加载到结构模型进行非线性静气动弹性分析．基于动力学小扰动假设, 在静平衡构型下进行动力学线性化, 计算风轮固有振动特性．继而结合非定常涡尾迹方法计算风力机动气动弹性响应．计算了NH 1500叶片考虑几何非线性的静气动弹性位移和动气动弹性响应．结果表明,大型风力机叶片几何非线性较为明显地减小静气动弹性位移,同时降低动气动弹性的响应幅值．大型风力机气动弹性响应计算需要考虑几何非线性     

An optimization procedure for the design of prop-rotors in high speed cruise including the coupling of performance,aeroelastic stability,and structures

An optimization procedure is developed to address the complex problem of designing prop-rotors in high speed cruise. The objectives are maximization of the aerodynamic efficiency in high speed cruise and minimization of the total rotor weight. Constraints are imposed on aeroelastic stability in cruise and rotor thrust. An isotropic box beam is used to model the principal load carrying member in the blade. Design variables include blade sweep and twist distributions, rotational velocity in cruise, and the box beam wall thickness. Since the optimization problem is associated with multiple design objectives, the problem is formulated using a multiobjective formulation technique known as the Kreisselmeier-Steinhauser function approach. The optimization algorithm is based on the method of feasible directions. A hybrid approximate analysis technique is used to reduce the computational expense of using exact analyses for every function evaluation within the optimizer. The results are compared to two reference rotors, unswept and swept. The optimum result shows significant improvements in the propulsive efficiency in cruise and reductions in the rotor weight without loss of aeroelastic stability or thrust, when compared to the reference unswept rotor. The swept reference rotor is initially unstable and the optimization procedure has been successful in producing a blade design which is fully stable with significant improvements in efficiency and blade weight. Off-design studies performed indicate that the optimum rotor maintains high propulsive efficiency over a wide range of operating conditions.     

气动、结构、载荷相协调的大型风电叶片自主研发进展

从叶片设计的3个关键环节（气动设计、结构设计和载荷评估）出发,对叶片自主研发进展进行了总结分析．在气动设计方面,概述了计算流体动力学（computational fluid dynamics, CFD）方法、涡方法和叶素动量（blade element momentum, BEM）方法,并依据工程中广泛应用的BEM方法,指出了低风速区风电叶片的解决思路;在结构设计方面,简要概述了基于梁模型的传统设计分析方法,分析了其在大型复合材料叶片薄壳结构上的不足,并对有限元方法（finite element method, FEM）在叶片结构分析中的应用进展进行了介绍;在载荷评估方面,介绍了其对叶片和整机其它部件的影响,阐述了载荷预估方面的工作进展．然后,通过分析3个关键环节之间的相互关系,得到如下结论:建立气动、结构和载荷相协调的叶片优化设计体系,才能真正满足高效低成本的需要．最后,指明了需要进一步研究的主要方向,即高效低载翼型研究,结构非线性有限元分析,气动-结构耦合研究,设计标准制定．最终目标是建立适合中国风资源特点的叶片研发体系,推动我国风电产业发展.     

Dynamic modeling and analysis of wind turbine drivetrain considering the effects of non-torque loads

With the increase of the rotor diameter and the deterioration of operating conditions, modern wind turbines suffer from more and more significant time-varying non-torque loads, which increases the burden of turbine structures especially the gearbox. Based on an aeroelastic loose coupling approach and assembly of the finite element method, an integrated drivetrain coupling analysis model including blade module, aerodynamic module, and gearbox module is established in this study. This proposed model is validated by comparing the calculation results with previous literature. Taking National Renewable Energy Laboratory 5-MW wind turbine as the research object, the gearbox vibration responses, gear meshing forces and bearing forces under non-torque loads caused by blade gravity, wind shear (WS), tower shadow (TS) and yawed inflow are studied in detail. Results show that the y-direction displacements of the gearbox, sun gear 1, sun gear 2 and gear are larger than those in x-direction because F_y or M_x generated by blade gravity, WS and TS dominates the non-torque loads. The non-torque loads lead to a non-uniform planet load sharing especially for the planetary gear stage 1. Because of the fluctuations of non-torque loads, not only the rotation frequencies of the corresponding carrier but also the multiple frequencies of the carrier 1 are observed in the frequency spectrums. The non-torque loads are mainly borne by carrier 1 bearings. Except for the blade gravity, the bearing forces caused by other unsteady inflows have obvious fluctuations.     

Follower loads for high order finite elements

Finite element modelling of hydrostatic compaction where the applied pressure acts normal to the deformed surface requires a geometric nonlinear formulation and follower load terms [1, 5, 7]. These concepts are applied to high order [6] (p-FEM) elements with hierarchic shape functions. Applying the blending function method allows to precisely describe curved boundaries on coarse meshes. High order elements exhibit good performance even for high aspect ratios and strong distortion and therefore allow an efficient discretization of thin-walled structures. Since high order finite elements are less prone to locking effects a pure displacement-based formulation can be chosen. After introducing the basic concept of the p-version the application of follower loads to geometrically nonlinear high order elements is presented. For the numerical solution the displacement based formulation is linearized yielding the basis for a Newton-Raphson iteration. The accuracy and performance of the high order finite element scheme is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

The Vibrational Behavior of Bladed Disks in Consideration of Friction Damping and Contact Elasticity

Rotating turbine blading is subjected to fluctuating gas forces during operation that cause blade vibrations. One of the main tasks in the design of turbomachinery blading is the reduction of the vibration amplitudes of the blades to avoid high resonance stresses that could damage the blading. The vibration amplitudes of the blades can be reduced significantly to a reasonable amount by means of friction damping devices such as underplatform dampers. In the case of blade vibrations, relative displacements between the friction damping devices and the neighboring blades occur and friction forces are generated that provide additional damping to the structure due to the dry friction energy dissipation. In real turbomachinery applications, spatial blade vibrations caused by a complex blade geometry and distributed excitation forces acting on the airfoil accur. Therefore, a three dimensional model including an appropriate spatial contact model to predict the generalized contact forces is necessary to describe the vibrational behavior of the blading with sufficient accuracy, see [1] and [2]. In this paper the contact model presented in [2] is extended to include also local deformations in the contacts between underplatform dampers and the contact surfaces of the adjacent blades. The additional elasticity in the contact influences the resonance frequency of the coupled bladed disk assembly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.     

Thermo-mechanical modeling of turbine blades taking into account structural defects

In turbine blades of aero-engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non-physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non-penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node-to-segment approach proposed in [3] is extended to ten-node tetrahedral elements with quadratic shape functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aeroelastic analysis of large horizontal wind turbine baldes?

A nonlinear aeroelastic analysis method for large horizontal wind turbines is described. A vortex wake method and a nonlinear ?nite element method (FEM) are coupled in the approach. The vortex wake method is used to predict wind turbine aero-dynamic loads of a wind turbine, and a three-dimensional (3D) shell model is built for the rotor. Average aerodynamic forces along the azimuth are applied to the structural model, and the nonlinear static aeroelastic behaviors are computed. The wind rotor modes are obtained at the static aeroelastic status by linearizing the coupled equations. The static aeroelastic performance and dynamic aeroelastic responses are calculated for the NH1500 wind turbine. The results show that structural geometrical nonlinearities signi?cantly reduce displacements and vibration amplitudes of the wind turbine blades. Therefore, structural geometrical nonlinearities cannot be neglected both in the static aeroelastic analysis and dynamic aeroelastic analysis.     

A mathematical model for horizontal axis wind turbine blades

A mathematical model describing the nonlinear vibration of horizontal axis wind turbine (HAWT) blades is proposed in this paper. The system consists of a rotating blade and four components of deformation including longitudinal vibration (named axial extension), out-of-plane bend (named flap), in-plane/edgewise bend (named lead/lag) and torsion (named feather). It is assumed that the center of mass, shear center and aerodynamic center of a cross section all lie on the chord line, and do not coincide with each other. The structural damping of the blade, which is brought about by materials and fillers is taken into account based on the Kelvin–Voigt theory of composite materials approximately. The equivalent viscosity factor can be determined from empirical data, theoretical computation and experimental test. Gravitational loading and aerodynamic loading are considered as distributed forces and moments acting on blade sections. A set of partial differential equations governing the coupled, nonlinear vibration is established by applying the generalized Hamiltonian principle, and the current model is verified by previous models. The solution of equations is discussed, and examples concerning the static deformation, aeroelastic stability and dynamics of the blade are given.     

Nonlinear Vibrations with Friction: A Multiharmonic Approach

Turbine blades are subjected to vibrations caused by static and dynamic loads. To avoid damage, the arising dynamic stresses can be reduced by generating friction contacts between the blades leading to additional damping. The resulting nonlinear equations of motion of the blade dynamics can be solved by the Multiharmonic Balance Method (MHBM), which is described in this paper. With the MHBM, it is possible to consider a multiharmonic excitation as well as a response consisting of several harmonics. In this paper, the MHBM is applied to both a simple 1 DOF nonlinear oscillator and a cyclic model of a blade integrated disk (blisk). (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A triangular shell element for vibration analysis of cambered and twisted fan blades

A ten-node triangular shell element of thirty degrees of freedom was successfully developed and applied to plate and shell static analysis, and free vibrations of rectangular plates with different boundary conditions. The element is also suitable for the vibration analysis of pretwisted and cambered fan blades. Natural frequencies of cambered and untwisted fan blades having a rectangular planform with constant thickness are determined for different shallowness ratio and blade thickness. For different blade-tip twist angles, the element can be applied to obtain the frequencies of twisted blades with constant thickness. Frequencies, resulting from the applied element, are compared to those of experimental results which are available in literature, and to those of shallow shell theory and other finite elements. Good correlations and reasonably accurate results are obtained by using a coarse mesh size.     

On a numerical approach to Stefan-like problems

Summary This paper is devoted to the numerical analysis of a bidimensional two-phase Stefan problem. We approximate the enthalpy formulation byC ⁰ piecewise linear finite elements in space combined with a semi-implicit scheme in time. Under some restrictions related to the finite element mesh and to the timestep, we prove positivity, stability and convergence results. Various numerial tests are presented and discussed in order to show the accuracy of our scheme.This work is supported by the Fonds National Suisse pour la Recherche Scientifique.     

三叶片垂直轴水轮机结构动力响应分析

提出采用改进离散涡和几何精确梁理论混合方法对三叶片垂直轴水轮机进行结构动力响应分析.相比传统的有限元方法,该方法具有求解速度快、建模简单、计算精确等优点.在模态分析中,计算了不同叶片高度下,水轮机叶片和整体的前五阶固有频率,分析了水轮机半径大小和叶片高度对固有频率的影响,结果显示:随着尺寸的增加,叶片和整体固有频率显著减小,整体固有频率更易受到半径大小的影响.在瞬态分析中,考虑了离心载荷和叶片的水动力载荷,得到在工作状况下,旋转一周过程中叶片的最大变形曲线;分析了在不同H/R（叶片高度和半径的比值）的情况下的叶片强度问题,结果显示:当H/R大于3.0时,叶片强度将会失效.     

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates. Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999     

Elastic blade root connection in wind turbine using flexible elements from composites

The aim of an elastic blade root connection with a hub is to decrease loads in the blade root section during wind gusts. Two designs of connection were considered: for the load reduction on the blade in operating regime and for stopped blade unloading under storm wind. In the first case two versions of joint were discussed: the first one — with hinges and U-shaped composite beams, the second one — formed with straight beams oriented in different directions. Both joints have low torsional stiffness in wind direction and much higher stiffnesses around two other axes. Formulas for angular stiffnesses and the methods of obtaining the nonlinear behavior of the joint are presented. The objective of the flexible spar was to allow the blades to bend back out of the wind to reduce loads when the wind turbine was stationary in storm conditions. Calculations supported the feasibility of such a design. With a low torsional stiffness, spar (which can be rigidly connected to the blade) acts as a pitching beam for turbine control. A compound spar design consisting of pultruded bars clamped through specified distance was proposed. Torsional stiffnesses of different types of spars with equal specified bending rigidity were compared.Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 388–400, May–June, 1996.