Non-linear dynamic analysis of a wind turbine blade

ABSTRACT

Wind turbine blades’ dynamic behavior must be well investigated during the design process in order to avoid resonance. Modal analysis and nonlinear dynamic analysis of a rotating wind turbine blade are presented in this paper to study the effects of rotation speed on the natural frequencies and displacements. Accordingly, an enhanced finite element formulation based on uncoupled translational and rotational displacements’ kinetic energy was developed. A twisted wind turbine blade was discretized using beam elements with different cross sections; each has six degrees of freedom per node. At rest, the model results were validated using ABAQUS software. Furthermore, the effects of geometrically non-linear phenomena produced by blade displacement on the vibration characteristics of the rotating blade have been discussed. The blade non-linear problem was solved by applying the Newmark method combined with the Newton-Raphson schema. Results show that the blade natural frequencies and displacements increase under angular rotation speed augmentation. This work highlights the significant sensitivity of the blade vibration characteristics to the geometric non-linearity. Accordingly, a modal and a linear analysis are insufficient to precisely estimate the dynamic behavior of a rotating turbine blade.     

Effects of in-plane constraints on the free vibration of a symmetrically laminated doubly curved panel

Summary. For a symmetrically laminated curved panel, although the stretching-bending anisotropic coupling stiffnesses are zero, but due to presence of the curvature, the in-plane and out-of-plane behaviors of the panel are still coupled, and hence the in-plane constraints at the boundaries have influences on the transverse behavior of the panel. Such effects of in-plane constraints on the free vibration of the symmetrically laminated curved panel are investigated using a modified Galerkin method in this study. Transverse shear deformation of the panel is considered by using first-order shear deformation theory. Numerical results of the symmetric angle-ply and cross-ply laminated panel with simply supported boundary conditions (SS2 and SS3) are presented. Results show that the in-plane boundary constraints have significant effects on the vibration behavior of the symmetrically laminated curved panel, and the effects strongly depend on the radius of curvature, thickness and lamination schemes, etc. Effects of bending-twisting anisotropic coupling of symmetric angle-ply laminate on the vibration behavior are also discussed.     

On designing of flywheels for electromechanical plants

The optimum design of a rotating axially symmetric disk (flywheel) under geometric and strength constraints is examined. As a function of the radial coordinate, the disk thickness should be so selected that a kinetic energy maximum is facilitated for the given rotational speed. This piecewise continuous function constrained from above is determined numerically. The method of penalty functions is used for solving the problem of search for the conditional maximum. The effect of geometrical design constraints on the optimum form of the disk is analyzed.Translated from Problemy Prochnosti, No. 3, pp. 85–88, March, 1991.     

Influence of rotation on vibration characteristics of thick composite disks

Abstract

Thick composite disks are utilized in fast-rotating machines, including turbine disks and flywheels. Dynamic equations of motion for a rotating composite disk have been formulated in a polar coordinate system using Hamilton’s principle, and numerical analysis has been performed by finite element interpolation. The natural frequencies of isotropic and laminated composite disks have been obtained when the rotational speed changes. The effects of transverse shear and rotary inertia on the vibration characteristics of rotating disks have also been investigated.     

风力机叶片载荷谱及疲劳寿命分析

研究了大型风力机玻璃钢叶片载荷谱和疲劳寿命的工程估算方法。运用片条理论分析了影响风力机叶片疲劳寿命的气动载荷分布;应用多体动力学方法,分析了旋转叶片动力刚化效应产生的原因及其对叶片振动模态的影响;根据模态叠加法,计算了叶片在气动力、重力和旋转惯性力等确定性载荷作用下的动应力响应;探讨了玻璃钢叶片的疲劳性能、破坏准则及疲劳寿命估计方法;提出了一种基于 Palmgren Miner 线性累积损伤法则的玻璃钢叶片安全寿命估计方法。通过所设计的 1.5MW 变速变桨距风力机叶片疲劳寿命估计的算例表明,该方法是可靠和实用的。     

旋转几何非线性复合材料薄壁梁的自由振动分析

研究具有几何非线性的旋转复合材料薄壁梁的自由振动。梁的变形引入了Von Kármán几何非线性, 基于Hamilton原理和变分渐进法 (Variational-Asymptotical Method -VMA),导出旋转复合材料薄壁梁的非线性振动偏微分方程组。采用Galerkin法将振动方程离散化为常微分方程组。借助于谐波平衡法 (Harmonic Balance Method -HBM) 建立自由振动的振幅-非线性固有频率关系方程。将上述方程化为非线性特征值问题,采用迭代算法进行求解。将所建立的旋转复合材料薄壁梁非线性自由振动分析模型和计算方法,应用于周向均匀刚度配置(Circumferentially Uniform Stiffness –CUS) 构型复合材料薄壁梁,通过数值计算揭示了纤维铺层角、旋转速度对非线性振动固有频率-振幅关系的影响。     

Vibration based damage detection of rotor blades in a gas turbine engine

This paper describes the problems concerning turbine rotor blade vibration that seriously impact the structural integrity of a developmental aero gas turbine. Experimental determination of vibration characteristics of rotor blades in an engine is very important from fatigue failure considerations. The blades under investigation are fabricated from nickel base super alloy through directionally solidified investment casting process. The blade surfaces are coated with platinum aluminide for oxidation protection. A three dimensional finite element modal analysis on a bladed disk was performed to know the likely blade resonances for a particular design in the speed range of operation. Experiments were conducted to assess vibration characteristics of bladed disk rotor during engine tests. Rotor blade vibrations were measured using non-intrusive stress measurement system, an indirect method of blade vibration measurement utilizing blade tip timing technique. Abnormalities observed in the vibration characteristics of the blade tip timing data measured during engine tests were used to detect the blade damage. Upon disassembly of the engine and subsequent fluorescent penetrant inspection, it was observed that three blades of the rotor assembly were identified to have damaged. These are the blades that exhibited vibration abnormalities as a result of large resonant vibration response while engine tests. Further, fractographic analysis performed on the blades revealed the mechanism of blade failures as fatigue related. The root cause of blade failure is established to be high cycle fatigue from the engine run data history although the blades were put into service for just 6 h of engine operation.     

Vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a high temperature field

Summary. The vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a temperature field is considered. In this context, the blade is modeled as a thin-walled beam mounted on a rigid hub at a presetting angle, rotating with a constant angular velocity, and exposed to a steady temperature field of a prescribed gradient through the blade wall thickness. The effect of the initial twist of the blade is also taken into consideration. Results are presented for two constituents, metal-ceramic based materials that are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution. Numerical results highlighting the effects of the volume fraction in conjunction with those of the presetting and pretwist angles, temperature gradient, rotating speed and hub radius are presented, and pertinent conclusions are outlined.     

基于微动滑移摩擦模型的失谐叶盘系统振动分析

在航空发动机叶-盘系统中,由于叶片失谐导致严重的振动局部化,造成局部叶片的高周疲劳,拟采用增加叶根干摩擦的方法降低失谐叶-盘系统振动幅值,减轻其振动局部化程度。基于微动滑移摩擦模型建立叶根阻尼器干摩擦力本构关系,采用等效椭圆代替阻尼器力与位移函数关系的迟滞回线,获得阻尼器的等效阻尼、等效刚度;将等效阻尼、等效刚度作用于叶-盘系统集中参数模型上,建立单个扇区三自由度整周失谐叶-盘系统动力学方程;采用谐波平衡法对其进行振动特性分析,讨论了叶根干摩擦对失谐叶盘系统振动特性的影响。结果表明:叶根摩擦能显著降低系统共振幅值;有叶根摩擦系统较无叶根摩擦系统的共振峰在频率上延后;增加叶根摩擦可有效降低失谐叶-盘系统的振动局部化程度。     

基于模态理论的叶片动频率计算方法

旋转状态下叶片动频率分布情况是叶片设计以及机组故障诊断必须考虑的重要因素。提出了一种基于模态分析理论,利用实测静态叶片模态参数重构旋转叶片动力方程,从而获得叶片在不同转速下的动频率。该方法可以考虑叶片径高比、叶片安装角、仰角的影响,可以考虑叶片切向和法向振动之间以及多模态之间的耦合,计算结果表明该方法具有很强的实用性。还提出了基于模态理论的叶片动频系数计算公式。最后结合实例进行了计算,并分析了误差来源。     

含静、动力约束的复合材料结构的优化设计

本文利用序列二次规划准解析法,解决了含几何角度变量和厚度变量的复合材料层板的结构优化问题.设计约束包括结构的位移约束,应力约束和固有频率约束.基于虚功原理,对有限元分析中刚度矩阵和几何角度变量成非线性关系的情况,给出了统一的应力,位移约束的灵敏度计算公式,对固有频率的导数计算也作了推导.计算结果表明,序列二次规划准解析法对于解决含静,动力约束的复合材料层板的优化设计问题,具有较高的计算效率;并且,在复合材料层板的优化设计中增加何几角度寻优(即层板直三维空间的最佳几何位置)能获得更为新意的减重效果.      

用动态刚度法分析旋转变截面梁横向振动特性

通过引入动态刚度法分析旋转变截面梁的振动特性。首先基于欧拉-伯努利梁理论给出旋转变截面梁自由振动方程,然后通过动态刚度法推导该旋转梁的动态刚度矩阵,最后运用MATLAB中的fzero函数求解特征值方程得到旋转梁横向振动的固有频率和模态振型。数值计算结果证明了动态刚度法的精度和有效性,同时分析了轮毂半径、转速以及渐变系数对固有频率的影响。     

用单片机实施旋转机械叶片振动监测的间断相位法研究

研究了测量旋转机械叶片振动的间断相位法，提出了测量叶片振动的相对偏移的计算机算法，并将其用８０９８单片机系统实现；本文还对间断相位法在旋转机械叶片振动在线监测中的应用进行了探讨，为实现整级叶片振动在线监测提供了一种可用的手段。     

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

In this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.     

Turbulent interactions for rotating blades and wakes

Modelling and asymptotic analysis of the fully turbulent flow produced on a rotating blade system or cut-disk are described, the cut-disk being a thin disk with a significant amount of solidity removed. This is based on addressing the Reynolds-averaged Navier?CStokes equations in three-dimensional boundary-layer form together with an eddy-viscosity model, with the flow structure being analyzed for high Reynolds numbers. Interaction between blades and wakes plays a substantial part. The relationship between the present three-dimensional flow and the two-dimensional or axisymmetric flow over an isolated blade and a rotating disk is considered and leads to the finding that for the most part the flow on a cut-disk is, to leading order, the same as that on a rotating disk. A physical parameter, the disk solidity, plays an important role linking these two flows. The limiting case of low disk solidity yields the flow past an isolated blade locally, with global wake effects acting to provide the necessary azimuthal periodicity. This low-solidity limit also permits more analytical solutions which show agreement with numerical findings.     

具有轴承不对中故障的柔性非圆截面多转子系统非线性动力学行为

研究了具有轴承不对中故障的柔性非圆截面多转子系统的非线性动力学特性。首先在长轴承、微小不对中量和不平衡量等假设下,利用坐标变换推导出非圆截面多转子-轴承系统的非线性动力学模型。最后采用数值方法,重点分析了系统的非线性振动行为,例如稳态响应、轴心轨迹、Poincaré截面以及最大Lyapunov指数等。结果表明:在较低转速时,转子系统除存在与不平衡故障同步的工频外,还存在由于不对中引起的倍频以及组合频率成分等。随着转速的提高,系统出现倍周期分叉现象,混沌运动等复杂的非线性动力学行为。     

The stability behavior of rotating composite shafts under axial compressive loads

This paper studies the stability behavior of a rotating composite shaft subjected to axial compressive loads using finite element method. The laminated composite shaft is modeled as a Timoshenko shaft by applying the equivalent modulus beam theory. Numerical results correlate well with the reported beam models. The critical speed of the thin-walled composite shaft is dependent on the stacking sequency, the length-radius ratio (L/R) and the boundary conditions. In addition, the effects of the rotational speeds and the disk location on the rotating shafts' stability are examined as well. The critical load of the rotating thin-walled composite shaft can be nearly assessed from the conventional buckling formulation.     

高速定转子均质机转子叶片振动模态分析

采用对接加载模态综合分析法,将定转子均质机关键元件转子系统结构简化为叶-盘系统,对单个转子叶片进行有限单元划分,分析得出了转子转速为4 500 R／MIN时振动频率响应分布趋势;前10阶扩展模态分布具有一定的规律性,在初始4阶模态状态时频率波动小,且与系统固有频率近似;从第5阶模态以后分布波动较大, 且波动幅度很接近,约为20HZ。分析结果有利于定转子均质机关键元件转子系统结构的优化设计。     

旋转偏心质量块式消振电力作动器建模与控制EI北大核心CSCD

直升机等非固定翼飞行器在飞行状态时由桨叶旋转所产生的周期性低频振动会通过刚性机体传递至驾驶舱、航空发动机以及起落架等部位,会造成机体的持续振动,严重时会影响驾驶员的生命安全。提出了旋转偏心质量块式消振电力作动器,并开展了控制方法研究。从理论上推导了旋转偏心质量块式消振电力作动器的输出力模型以及负载转矩模型;提出基于双电机并行独立控制的电力作动器输出力伺服控制策略,在复频域进行了稳定性分析,并针对正弦波非线性负载扰动带来的转矩脉动问题,对双电机并行独立控制策略展开了优化设计,通过负载前馈控制使系统具备良好的鲁棒性和抗干扰性;研制了重量为14 kg的实验样机并完成了优化控制策略前后电力作动器稳态、动态性能的对比验证实验。结果表明,优化控制策略下的作动器输出力动稳态性能满足各项技术指标要求。     

含摩擦与间隙的失谐叶盘系统振动局部化研究

 针对叶片榫头与轮盘榫槽连接处间隙及摩擦,基于叶盘结构典型集中参数模型,建立含干摩擦、间隙的非线性动力学方程,研究失谐叶盘系统振动局部化。在叶身刚度随机失谐下分析叶盘系统对不同耦合刚度的固有特性及共振响应。结果表明,非线性谐调叶盘系统亦出现振动局部化现象。利用振幅放大系数对线性、非线性失谐叶盘系统振动响应局部化研究,振幅放大系数呈失谐阈值现象,且非线性干摩擦、间隙作用会降低失谐系统振动响应局部化程度。而谐调叶盘系统非线性振动,随气流激励力频率变化,系统呈非简谐单周期运动、多周期谐波运动、混沌运动等多种动力学行为。失谐因素存在会使非线性失谐叶盘系统动力学行为更复杂。