Aeroelastic response and stability analysis of composite rotor blades in forward flight

The aeroelastic phenomena of a composite rotor blade are investigated using a finite element method. The aeroelastic equations of motion of the rotor blade are formulated using a large deflection-type beam model that has no artificial restrictions on the magnitudes of displacements and rotations due to the degree of nonlinearity. The sectional elastic constants of a composite box beam including warping deformations are determined from the refined cross-sectional finite element method. The nonlinear, periodic blade steady response is obtained using a time finite element method on full finite element equation. The aeroelastic stability of a rotor blade is investigated by linearizing the dynamics of the rotor blade about the nonlinear equilibrium position. Numerical results of the steady equilibrium deflections and the aeroelastic modal damping are presented for various configurations of composite rotor blades and are compared with those obtained from a previously published modal analysis using a moderate deflection-type beam theory.     

单闭室复合材料薄壁梁的结构阻尼

研究单闭室复合材料薄壁梁的结构阻尼特性。基于变分渐进法（VAM) 和Hamilton原理,分别建立薄壁梁的截面力位移关系和运动方程;采用Galerkin法对薄壁梁进行自由振动分析;在获得薄壁梁振动模态矢量的基础上,根据最大应变能理论,对薄壁梁的模态阻尼性能进行预测,并且将阻尼预测的结果与现有的有限元计算结果进行对比,验证了本文阻尼分析模型的有效性。进一步针对周向均匀刚度配置(CUS)和周向反对称刚度配置（CAS）两种构型复合材料薄壁箱形梁以及一个翼型截面梁,进行阻尼计算,揭示了纤维铺层角和截面宽高比等参数的影响     

A hybrid force/stiffness matrix method for the analysis of thin-walled composite frames

A novel method for the analysis of frames constructed of thin-walled members of anisotropic composite materials is presented in this paper. The method accounts for non-isotropic coupling effects that exist in composite material beams due to the anisotropy of the composite material laminates that form the thin-walled cross section. The method also accounts for warping effects known to be significant in thin-walled members. The analysis is performed by the direct stiffness matrix method utilizing a new approach that divides each thin-walled member of the frame into one-dimensional warping-beam superelements and non-warping conventional beam elements. The element stiffness matrices for these two one-dimensional beam elements are obtained by a numerical procedure that is based on the classical force method analysis. The stiffness matrices of both beam elements are 12 × 12 matrices corresponding to the six degrees of freedom per node required for conventional space frame analysis. The remarkable feature of this representation is that warping is accounted for without introducing additional degrees of freedom to account for the bimoment and warping twist in the members. This is accomplished by use of the warping-beam superelement that linearizes the regions of non-uniform torsion in the thin-walled beam. Examples of space frame structures constructed of thin-walled composite material I-beams are presented to demonstrate the method. Results of analyses using the proposed method are compared with those obtained from two-dimensional finite element models.     

Dynamic analysis and damping of composite structures embedded with viscoelastic layers

In recent years, it has been found that composites co-cured with viscoelastic materials can enhance the damping capacity of a composite structural system with little reduction in stiffness and strength. Because of the anisotropy of the constraining layers, the damping mechanism of co-cured composites is quite different from that of conventional structures with metal constraining layers. This paper presents an analysis of the dynamic properties of multiple damping layer, laminated composite beams with anisotropic stiffness layers, by means of the finite element-based modal strain energy method. ANSYS 4.4A finite element software has been used for this study. The variation of resonance frequencies and modal loss factors of various beam samples with temperature is studied. Some of these results are compared with the closed-form theoretical results of an earlier published work. For obtaining optimium dynamic properties, the effects of different parameters, such as layer orientation angle and compliant layering, are studied. Also, the effect of using a combination of different damping materials in the system for obtaining stable damping properties over a wide temperature range is studied.     

压电悬臂梁的动力学建模与主动控制

为计及压电层的刚度特性,基于一阶剪切变形理论推导压电层合梁的抗弯刚度,由Hamilton变分原理建立压电层合梁的有限元模型,采用模态叠加方法对有限元模型降阶。在应变最大处配置制动器和传感器,基于鲁棒极点配置算法对低阶系统设计状态反馈,配置极点至期望位置。把针对低阶系统设计的控制器作用于原系统以实现振动主动控制。数值算例验证了这种力学建模方法和控制器设计方法的有效性。     

Investigation of Functionally Graded and Viscoelastic Layers Effects on the Dynamic Behavior of Functionally Graded Electro-Rheological Sandwich Beams

Vibration characteristics of functionally graded electro-rheological (FGER) sandwich beams are investigated. While a vast majority of studies have been reported about functionally graded material (FGM) or electrorheological fluids (ERF) composite beams, few, if any, works are conducted about FGER models. In order to validate the present finite element formulation of the FGER beam model, the results of the developed finite element (FE) model are compared with the results of an experimental test on a fabricated ERF composite beam. The effects of FGM volume fraction index, electric field, and thickness of the viscoelastic core are studied on the natural frequencies and modal loss factors of the FGER beam.     

弹性耦合对直升机复合材料桨叶稳定性的影响

研究了弹性耦合对复合材料桨叶动特性和气弹稳定性的影响, 所采用的结构模型考虑了剪切变形、剖面面外翘曲变形和复合材料弹性耦合。推导出同时考虑剪切和翘曲影响的小应变、中等变形梁的应变2位移关系,并构造出21 个自由度梁单元, 应用Hamilton 原理推导出桨叶运动的有限元方程。在此基础上, 对三种不同构型的复合材料桨叶进行固有频率计算和气弹稳定性分析。计算结果表明: 尽管复合材料弹性耦合对桨叶固有频率的影响非常小, 但却改变了固有振型分布, 使桨叶挥舞-摆振-扭转运动之间存在耦合; 弹性耦合对桨叶气弹稳定性有很大的影响; 正的挥舞2扭转耦合使得摆振一阶稳定性增加, 负的挥舞2扭转耦合却使摆振一阶稳定性下降。      

Vibration analysis of beams with non‐local foundations using the finite element method

In this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.     

Aeroelastic analysis of bearingless rotors with a composite flexbeam

An aeroelastic analysis of bearingless rotors is investigated using large deflection beam theory in hover and forward flight. The bearingless configuration consists of a single flexbeam with a wrap-around torque tube and pitch links located at the leading edge and trailing edge of the torque tube. The outboard main blade, flexbeam, and torque tube are all assumed as an elastic beam undergoing arbitrary large displacements and rotations, that are discretized into beam finite elements. In the bearingless rotors, a flexbeam has various sections made of laminate. The sectional elastic constants of a composite flexbeam, including the warping deformations, are determined from a refined cross-sectional finite element method. Numerical results of the static deflections and the aeroelastic modal damping are presented for various configurations of a composite flexbeam and are compared with previously published experimental results and theoretical values obtained from a modal analysis using a moderate deflection-type beam theory.     

Coupled deflection analysis of thin-walled Timoshenko laminated composite beams

For the deflection analyses of thin-walled Timoshenko laminated composite beams with the mono- symmetric I-, channel-, and L-shaped sections, the stiffness matrices are derived based on the solutions of the simultaneous ordinary differential equations. A general thin-walled composite beam theory considering shear deformation effect is developed by introducing Vlasov’s assumptions. The shear stiffnesses of thin-walled composite beams are explicitly derived from the energy equivalence. The equilibrium equations and force-deformation relations are derived from energy principles. By introducing 14 displacement parameters, a generalized eigenvalue problem that has complex eigenvalues and multiple zero eigenvalues is formulated. Polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to the equilibrium equations. Then the displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to nonzero and zero eigenvalues, respectively. Finally, the stiffness matrices are evaluated by applying the member force-displacement relations to the displacement functions. In addition, the finite beam element formulation based on the classical Lagrangian interpolation polynomial is presented. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with the finite element results using the isoparametric beam elements and the detailed three-dimensional analysis results using the shell elements of ABAQUS. Particularly the effects of shear deformations on the deflection of thin-walled composite beams with the mono-symmetric I-, channel-, and L-shaped sections with various lamination schemes are investigated.     

梁杆结构二阶效应分析的一种新型梁单元

推导了一种计及梁杆二阶效应的新型两结点梁单元。首先依据插值理论构造了三结点Euler-Bernoulli梁单元的位移场:使用五次Hermite插值函数建立梁单元的侧向位移场,二次Lagrange插值函数建立梁单元的轴向位移场,进而由非线性有限元理论推导了单元的线性刚度矩阵和几何刚度矩阵,然后使用静力凝聚方法消除三结点梁单元中间结点的自由度,从而得到一种考虑轴力效应的新型两结点梁单元。实例分析表明,此新型梁单元具有很高的计算精度,使用此单元进行梁杆结构分析可获得相当准确的二阶位移和内力。     

基于Layerwise理论的共固化粘弹阻尼复合材料动特性分析

共固化粘弹性复合材料兼具结构承载和阻尼减振功能。针对传统的混合单元法在应用于粘弹性夹层复合材料结构阻尼性能分析时存在着前处理困难、计算规模大、精度低以及难以考虑正交各向异性铺层自身损耗能力的缺点,推导了一种基于Layerwise离散层理论的四节点四边形复合材料层合板单元,并利用直接复特征值解法建立了共固化粘弹性复合材料结构的阻尼性能分析方法。将该方法应用于不同的阻尼结构,分析结果与文献中已公开结果和混合单元法的计算结果进行了对比验证。结果表明,基于离散层理论的层合板单元具有计算精度高、前处理建模简单和计算规模小的优点,可有效应用于复杂共固化粘弹性复合材料结构的阻尼性能分析和设计。     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

Multi-material topology optimization of laminated composite beam cross sections

This paper presents a novel framework for simultaneous optimization of topology and laminate properties in structural design of laminated composite beam cross sections. The structural response of the beam is evaluated using a beam finite element model comprising a cross section analysis tool which is suitable for the analysis of anisotropic and inhomogeneous sections of arbitrary geometry. The optimization framework is based on a multi-material topology optimization model in which the design variables represent the amount of the given materials in the cross section. Existing material interpolation, penalization, and filtering schemes have been extended to accommodate any number of anisotropic materials. The methodology is applied to the optimal design of several laminated composite beams with different cross sections. Solutions are presented for a minimum compliance (maximum stiffness) problem with constraints on the weight, and the shear and mass center positions. The practical applicability of the method is illustrated by performing optimal design of an idealized wind turbine blade subjected to static loading of aerodynamic nature. The numerical results suggest that the proposed framework is suitable for simultaneous optimization of cross section topology and identification of optimal laminate properties in structural design of laminated composite beams.     

基于弹性-黏弹性对应原理的复合材料层合板模态阻尼预测:理论及其有限元实现

提出一种利用通用有限元软件求解复合材料结构模态阻尼的有限元方法。该方法基于扩展弹性-黏弹性对应原理,定义出具有频率依存性的黏弹性复合材料复刚度矩阵,并借助ABAQUS提供的二次开发接口UMAT将其编入求解器中,结合复特征值法求解任意铺层层合板的模态阻尼。与已有的理论方法相比,本模型的计算结果更为接近实验数据。从而验证了本文提出的数值分析方法的有效性和精确性,为利用ABAQUS软件分析各向异性材料阻尼提供了一条有效途径。     

阻尼矩阵与刚度矩阵的一种直接修正方法

提出了一种利用复模态测量数据同时修正有限元阻尼与刚度矩阵的有效方法。借助于矩阵的奇异值分解得到了满足动力方程的最小修正矩阵。该方法有一个简洁的表达式,修正过程简单而且容易实现,数值试验表明修改后的结构参数能准确地同试验值吻合。     

Coupling effects in bending, buckling and free vibration of generally laminated composite beams

A closed form expression to determine the effective flexural modulus of a laminated composite beam is developed and presented in this contribution. This effective flexural modulus is applied to the bending, buckling and free vibration response of generally laminated composite beams with various boundary supports. The expression was developed using the combination of the Euler–Bernoulli beam and classical lamination theory. In addition the results of an extensive finite element analysis are used to validate the analytical model. The comparison of the analytical results, the finite element results and the experimental results showed good correlation. It is also observed that coupling response is an important variable that must be included in the computation of the effective flexural stiffness of generally laminated beam.     

含分层损伤复合材料加筋层合板的动承载能力

采用有限元方法研究了含穿透分层损伤复合材料加筋层合板的动力响应和承载能力。根据复合材料层合板一阶剪切理论, 推导了复合材料层合板单元的刚度阵和质量阵列式;同时采用Adams 应变能法与Rayleigh阻尼模型相结合的方法, 构造了相应的阻尼阵列式;为了防止在低阶模态中分层处出现的上、下子板不合理的嵌入现象, 建立了含分层损伤复合材料加筋层合板动力分析中分层分析模型和虚拟界面联接模型。并采用Tsai提出的刚度退化准则和动力响应分析的精细积分法, 对在动荷载作用下含分层损伤复合材料加筋层合板结构进行了破坏和承载能力分析。通过典型算例分析, 分别讨论了外载频率、分层深度、筋的位置以及破坏过程中刚度退化对含损伤复合材料加筋层合板动力响应特征和承载能力的影响, 得到了一些具有理论和工程价值的结论。     

Simple Efficient Smart Finite Elements for the Analysis of Smart Composite Beams

This paper is concerned with the development of new simple 4-noded locking-alleviated smart finite elements for modeling the smart composite beams. The exact solutions for the static responses of the overall smart composite beams are also derived for authenticating the new smart finite elements. The overall smart composite beam is composed of a laminated substrate conventional composite beam, and a piezoelectric layer attached at the top surface of the substrate beam. The piezoelectric layer acts as the actuator layer of the smart beam. Alternate finite element models of the beams, based on an “equivalent single layer high order shear deformation theory”, and a “layer-wise high order shear deformation theory”, are also derived for the purpose of investigating the required number of elements across the thickness of the overall smart composite beams. Several cross-ply substrate beams are considered for presenting the results. The responses computed by the present new “smart finite element model” excellently match with those obtained by the exact solutions. The new smart finite elements developed here reveal that the development of finite element models of smart composite beams does not require the use of conventional first order or high order or layer-wise shear deformation theories of beams. Instead, the use of the presently developed locking-free 4-node elements based on conventional linear piezo-elasticity is sufficient.