Mechanics and finite elements for the damped dynamic characteristics of curvilinear laminates and composite shell structures

Integrated mechanics and a finite element method are presented for predicting the damping of doubly curved laminates and laminated shell composite structures. Damping mechanics are formulated in curvilinear co-ordinates from ply to structural level and the structural modal loss factors are calculated using the energy dissipation method. The modelling of damping at the laminate level is based on first order shear shell theory. An eight-node shell damping finite element is developed. Comparisons of the present model with classical and discrete layer laminate damping theory predictions are shown. Modal damping and natural frequencies of composite plates and open cylindrical shells were measured and correlated with predicted results. Parametric studies illustrate the effect of curvature and lamination on modal damping and natural frequency.     

不同铺设角度下层合板结构参数对声功率影响*

本文研究了不同纤维铺设角度的层合板结构参数对声功率的影响,从而为层合板的低噪声设计提供理论依据。通过分层有限元理论获得层合板结构动力学响应,基于声辐射模态概念分析不同铺设角度下层合板不同铺设方式、宽厚比和弹性模量比对其声功率影响。结果表明,层合板的铺设角度和宽厚比对复合材料层合板结构的声辐射功率影响较大。首先相同铺设角度的层合板,改变弹性模量比,声功率变化不明显;其次改变不同的铺设角度,宽厚比较小的层合板声功率下降的空间更大,更易于声功率的降低。层合板作为结构件时,从降低声功率角度而言总体上对称铺设结构比单向铺设层合板结构有优势,并且相同铺设角度下,反对称铺设层合板可获得更小的辐射声功率。     

Flutter analysis of periodically supported curved panels

This paper presents the one-dimensional axial wave propagation in an infinitely long periodically supported cylindrically curved panel subjected to supersonic airflow. The aerodynamic forces are based on piston theory. For this study the structure is considered as an assemblage of a number of identical cylindrically curved panels each of which will be referred to as a periodic element. A high precision triangular finite element with certain wave boundary conditions (Floquet's principle) is introduced in flutter problems of the proposed structure for the first time. The airflow is assumed in the direction of the straight edges of the panel. It is assumed that the deflection function accounts for a phase lag term only and does not consider any attenuation terms. Aerodynamic damping has been neglected for brevity. For a given geometry a three-dimensional plot related to the phase constant, flutter frequency and pressure parameter has been obtained corresponding to the optimum periodic angle. The “flutter line”(line of instability) has been identified. The limiting values of flutter frequencies and pressure parameters of the “flutter line” are compared with the critical flutter condition of a single curved panel, using two methods—an exact approach and a finite element method. The critical flutter results for multi-supported (1-span, 2-span and 3-span) curved panels are obtained using the band discretization principle.     

Nonlinear flutter analysis of stiffened composite panels in super-sonic flow

The flutter instability of stiffened composite panels subjected to aerodynamic forces in the supersonic flow is investigated. Based on Hamilton's principle,the aeroelastic model of the composite panel is established by using the von Karman large deflection plate theory,piston theory aerodynamics and the quasi-steady thermal stress theory. Then,using the finite element method along with Bogner-Fox-Schmit elements and three-dimensional beam elements,the nonlinear equations of motion are derived. The effect of...     

Stability for Leipholz's Type of Laminated Box Columns with Nonsymmetric Lay-Ups on Elastic Foundation

The stability behavior of the Leipholz's type of laminated box columns with nonsymmetric lay-ups resting on elastic foundation is investigated using the finite element method. Based on the kinematic assumptions consistent with the Vlasov beam theory, a formal engineering approach of the mechanics of the laminated box columns with symmetric and nonsymmetric lay-ups is presented. The extended Hamilton's principle is employed to obtain the elastic stiffness and mass matrices, the Rayleigh damping and elastic foundation matrices, the geometric stiffness matrix due to distributed axial force, and the load correction stiffness matrix accounting for the uniformly distributed nonconservative forces. The evaluation procedures for the critical values of divergence and flutter loads with/without internal and external damping effects are briefly presented. Numerical examples are carried out to validate the present theory with respect to the previously published results. Especially, the influences of the fiber angle change and damping on the divergence and flutter loads of the laminated box columns are parametrically investigated.     

钝前缘梯形翼高超声速风洞颤振试验

为了研究钝前缘翼面的高超声速颤振特性,获得典型翼面高超声速颤振参数以校验非定常气动力和CFD计算,采用具有简单结构动力学特性的钝前缘梯形翼模型,在中国航天空气动力技术研究院FD-07高超声速风洞进行了高超声速风洞颤振试验研究.模型为9 mm厚钝前缘梯形平板翼,采用夹层设计:中间层为钢板,提供模型主要刚度和质量特性;两侧为泡沫,起维形作用.试验模型采用悬臂支撑安装于风洞试验段,试验Mach数分别为4.95和5.95.试验固定Mach数,通过缓慢增加动压以使模型达到颤振临界点,采用小波时频谱分析时域响应,结果显示试验模型发生了弯扭耦合经典颤振.试验采用直接观测法获得了颤振动压、颤振频率和对应的试验密度、总温等颤振相关参数.采用壳单元建立了结构有限元模型,并采用统一升力面理论对模型进行了颤振计算分析,研究了气流密度、结构阻尼、Mach数对颤振计算的影响,并对试验结果与理论计算的偏差进行了讨论.分析认为,计算气流密度、计算结构阻尼、结构建模偏差、试验结果散布特性等因素均会构成计算值和试验值之间的偏差,但即便在计算中考虑上述因素,计算结果与试验值仍存在较大偏差.      

谱有限元数值分析各向异性复合板中导波色散特性

提出谱有限元方法研究层状各向异性复合板中导波的色散特性和波结构。基于三维弹性动力学方程,用有限元方法离散波导截面,波传播方向的位移用简谐波表示,得到了导波色散的特征方程。分析了单层和双层复合板中导波沿不同方向传播的色散特性和波结构,讨论了双层复合板中层厚比对相速度的影响。数值研究结果表明:导波的对称模态沿纤维方向传播时在较宽的频率范围内保持弱色散状态。双层复合板中导波基本模态的相速度在低频时受层厚比的影响较明显,随着频率的增加趋向于相速度较低的材料。数值模拟结果为导波用于复合材料定量无损检测和性能评价提供理论依据。      

Aero-thermo-mechanical characteristics of imperfect shape memory alloy hybrid composite panels

A nonlinear finite element model is provided to predict the static aero-thermal deflection and the vibration behavior of geometrically imperfect shape memory alloy hybrid composite panels under the combined effect of thermal and aerodynamic loads. The nonlinear governing equations are obtained using Marguerre curved plate theory and the principle of virtual work taking into account the temperature-dependence of material properties. The effect of large deflection is included in the formulation through the von Karman nonlinear strain-displacement relations. The thermal load is assumed to be a steady-state constant-temperature distribution, whereas the aerodynamic pressure is modeled using the quasi-steady first-order piston theory. The Newton-Raphson iteration method is employed to obtain the nonlinear aero-thermal deflections, while an eigenvalue problem is solved at each temperature step and static aerodynamic load to predict the free vibration frequencies about the deflected equilibrium position. Finally, the nonlinear deflection and free vibration characteristics of a composite panel are presented, illustrating the effects of geometric imperfection, temperature rise, aerodynamic pressure, boundary conditions and shape memory alloy fiber embeddings on the panel response.     

Active vibration suppression of multilayered plates integrated with piezoelectric fiber reinforced composites using an efficient finite element model

The active vibration suppression of hybrid composite and fiber metal laminate (FML) plates integrated with piezoelectric fiber reinforced composite (PFRC) sensors and actuators is studied for the first time, using an efficient and advanced layerwise plate theory. Unlike the conventional finite elements, the equipotential condition of electroded surfaces of sensors is satisfied exactly and conveniently using a novel concept of electric node. The effective electromechanical properties of the PFRC laminas are computed using a coupled three-dimensional iso-field micromechanical model. Numerical results are presented for both classical constant gain velocity feedback (CGVF) and optimal control strategies. The instability phenomena in CGVF control with conventionally collocated actuator-sensor pairs, and its remedy with a truly collocated arrangement are illustrated. The effect of segmentation of electrodes on the control response is studied. The segmentation of electrodes leads to a multi-input-multi-output (MIMO) configuration. The effects of piezoelectric fiber orientation, volume fraction and dielectric ratio of PFRC on the control response and the actuation/sensing authority are investigated for cantilever and simply supported plates.     

谱有限元模拟正交各向异性黏弹性材料中导波传播特性

研究了导波在正交各向异性黏弹性复合板中传播的色散特性、波结构及功率流密度。基于二维平面运动方程,采用谱有限元方法得到了导波色散的特征方程。分析了正交各向异性黏弹性板中各向异性和黏性对能量速度和波结构的影响,以及基底对导波功率流密度的影响。数值研究结果表明:导波沿纤维方向传播的能量速度大,材料的黏性对速度影响较小,但会减小波结构的幅度;在高频时,基底的存在使两个基本模态的功率流密度分别集中到波导的上下表面,形成弱色散、高衰减及无色散、零衰减的表面波。数值模拟结果为导波用于复合材料定量无损检测和性能评价提供理论依据。      

A damping mechanics model and a beam finite element for the free-vibration of laminated composite strips under in-plane loading

A theoretical framework is presented for predicting the nonlinear damping and damped vibration of laminated composite strips due to large in-plane forces. Nonlinear Green-Lagrange axial strains are introduced in the governing equations of a viscoelastic composite and new nonlinear damping and stiffness matrices are formulated including initial stress effects. Building upon the nonlinear laminate mechanics, a damped beam finite element is developed. Finite element stiffness and damping matrices are synthesized and the static equilibrium is predicted using a Newton-Raphson solver. The corresponding linearized damped free-vibration response is predicted and modal frequencies and damping of the in-plane deflected strip are calculated. Numerical results quantify the nonlinear effect of in-plane loads on structural modal damping of various laminated composite strips. The modal loss-factors and natural frequencies of cross-ply Glass/Epoxy beams subject to in-plane loading are measured and correlated with numerical results.     

Active suppression of panel flutter with piezoelectric actuators using eigenvector orientation method

This paper examines the use of eigenvector orientation method to detect the onset of subsonic and supersonic flutter of panels modeled by finite elements. The accuracy of the eigenvector orientation method for prediction of the flutter boundary (indicated by a gradual loss of orthogonality between two eigenvectors) is demonstrated by using the examples of a swept-back cantilever plate model at subsonic speed and a simply supported plate model at supersonic speed. Piezoelectric layers are assumed to be bonded to the top and bottom surfaces of the simply supported plate in order to provide bending moments to control motions of each finite element. An approach of optimal control design is presented to actively suppress the possible flutter based on linear quadratic regulator theory and the nonlinear modal equations of motions. To illustrate the applicability and effectiveness of using the piezoelectric layers as controllers, several cases are studied and presented. The effects of varying locations of control moments are studied so as to fulfill the objective of adjusting the flutter speed to be within a desirable range. The results illustrate that the control moment manipulation can offset the flutter occurrence and additionally generate a lead time for possibly executing flutter control.     

RELIABILITY ANALYSIS OF SUSPENSION BRIDGES AGAINST FLUTTER

A reliability analysis of suspension bridges against flutter failure is presented using the basic theory of reliability. For the purpose of analysis, uncertainties considered are those arising from the variations in geometric and mechanical properties of bridge, modelling, damping and experimentally obtained flutter derivatives. These uncertainties are incorporated by multiplying the computed flutter wind speed with a number of independent factors, which are considered as random variables. Each factor is assumed to follow log-normal distribution. The wind environment at the site, which may cause flutter failure, is considered as the other uncertainty necessary for computing the reliability against flutter failure. The flutter wind speed for the bridge is determined using a finite element approach and a multimode analysis. The effect of some important parameters such as the mean wind speed at the site, coefficients of variation of the multiplying factors associated with damping, modelling and flutter derivatives on the reliability estimates is investigated. The results of the study show that the reliability against flutter failure is sensitive to the variation of the above parameters.     

宽弦高速跨音风扇颤振特性研究

颤振是航空发动机、燃气轮机等运行安全的重要威胁,但颤振稳定性与流动结构之间的关系尚不清晰。本文使用行波法和影响系数法,对某宽弦复合掠型高速跨音风扇转子的一阶模态进行了颤振特性研究,计算了在100%转速下从堵塞点到近失速点的颤振表现。使用影响系数法时,分析了不同通道数的计算域对气动阻尼计算的影响,并与行波法得到的结果进行了对比。研究了流动结构与叶片表面气动阻尼之间的关系,旨在提高对流动致颤机理的认识。结果表明影响系数法和行波法均能对叶片的气动阻尼进行较好的预测;流动结构方面,激波、激波附面层分离、叶尖泄漏流以及吸力面前缘叶顶附近的非定常压力波动,对叶片的气动阻尼分布有较大的影响。     

Influence of non-linear forces on beam behaviour in flutter conditions

Appropriate researches on non-linear panel flutter behaviour have been already performed by many authors. In most cases the intent of them focuses on the limit cycle determination, with particular interest towards its amplitude versus the flow dynamic pressure. This paper deals first with a study of all the solutions without damping of beam flutter versus the vibration frequency in non-linear post-critical conditions. A numerical model, which takes into account the influence of the non-linear contribution of the structural forces, due to the axial stretching of the beam, has been implemented. A complete analysis of all the possible non-linear solutions without damping leads to the possibility of characterizing the most appropriate conditions for the presence of the post-critical panel flutter limit cycles. Then the complete model, which also takes into account aerodynamic damping, has been utilized, according to the “Piston Theory”, to verify the state evolution of the fluttering damped beam towards the limit cycle, which is very near to the undamped vibrating beam state with minimum amplitude. This convergence test is an interesting aspect of the numerical results.     

Dynamics of a supercritical composite shaft mounted on viscoelastic supports

The damping in a carbon fiber reinforced plastic (CFRP) laminate is greater than that which occurs in most metallic materials. In the supercritical regime, the damping can trigger unstable whirl oscillations, which can have catastrophic effects. The vibrations occurring in a supercritical composite drive shaft are investigated here in order to predict instabilities of this kind. A simply supported carbon/epoxy composite tube mounted on viscoelastic supports is studied, using an approximation of the Rayleigh–Timoshenko equation. The damping process is assumed to be hysteretic. The composite behavior is described in terms of modulus and loss factor, taking homogenized values. The critical speeds are obtained in several analytical forms in order to determine the effects of factors such as the rotatory inertia, the gyroscopic forces, the transverse shear and the supports stiffness. Assuming that the hysteretic damping can be expressed in terms of the equivalent viscous model, the threshold speed is obtained in the form of an analytical criterion. The influence of the various factors involved is quantified at the first critical speed of a subcritical composite shaft previously described in the literature. The influence of the coupling mechanisms on the unsymmetrical composite laminate and the end fittings is also investigated using a finite element model. None of these parameters were found to have a decisive influence in this case. Those having the greatest effects were the transverse shear and the supports stiffness. The effects of the composite stacking sequence, the shaft length and the supports stiffness on the threshold speed were then investigated. In particular, drive shafts consisting only of ±45° or ±30° plies can be said to be generally unstable in the supercritical regime due to their very high loss factors.     

Parameterized aeroelastic modeling and flutter analysis for a folding wing

To investigate the flutter characteristics of a folding wing with different configurations, a parameterized aeroelastic model is proposed. First, a parameterized structural model is established based on the substructure synthesis. Afterwards, the parameterized aerodynamic model is derived for each lifting surface using the so-called Doublet Lattice Method (DLM). The correctness of the resulting aeroelastic model is verified via NASTRAN. Finally, some aeroelastic simulations are performed using the proposed aeroelastic model. The results demonstrate that the flutter characteristics of the folding wing are very sensitive to the folding angle. With increasing folding angle, a transition between two unstable modes occurs. Such a transition results in a sudden change of flutter mode shapes and a jump of critical flutter frequency. Besides, there exists a region of folding angle, where the flutter behavior of the folding wing strongly depends on the structural damping.     

Free vibration of antisymmetric,angle-ply laminated plates including transverse shear deformation by the finite element method

A finite element formulation of the equations governing the laminated anisotropic plate theory of Yang, Norris and Stavsky, is presented. The theory is a generalization of Mindlin's theory for isotropic plates to laminated anisotropic plates and includes shear deformation and rotary inertia effects. Finite element solutions are presented for rectangular plates of antisymmetric angle-ply laminates whose material properties are typical of a highly anisotropic composite material. Two sets of material properties that are typical of high modulus fiber-reinforced composites are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle. The numerical results are compared with the closed form results of Bert and Chen. As a special case, numerical results are presented for thick isotropic plates, and are compared with those for 3-D linear elasticity theory and Mindlin's thick plate theory.     

Analytical prediction of the non-linear response of a self-excited structure

The time-dependent amplitude response of the self-excited harmonium reed vibrating at finite amplitudes is investigated both analytically and experimentally. The analysis contained herein is based on the assumption that all of the significant non-linear forces that act on the reed are of aerodynamic origin and that these forces influence the reed behavior through the system damping. The analysis is carried out, without the aid of empirical techniques, by deriving the induced aerodynamic pressure force as a non-linear function of the amplitude of the reed motion, via an unsteady potential flow field analysis, and then applying the resulting forcing function to the equation of motion of the reed. An approximate solution of the equivalent lumped parameter equation of motion yields functional relationships that predict various observable phenomena, including the limit cycle amplitude. A comparison of the results with experimental data serves to substantiate the analysis. Also presented is a brief discussion of some of the phenomenological similarities that exist between the results of the harmonium reed analysis and the observed behavior of a flat-plate wing undergoing torsional stall flutter.     

A simplified method for the free vibration and flutter analysis of bridge decks

A simplified method for the free vibration and flutter analysis of bridge decks is presented. Bending-torsion coupled beam theory with warping stiffness included is used in the structural idealization of bridge decks in order to derive explicit formulae for natural frequencies and mode shapes. These are used to perform the flutter analysis. The time-dependent aerodynamic forces are modelled using Theodorsen type flat plate theory. Expressions for generalized mass, generalized stiffness and generalized aerodynamic force terms are derived in compact explicit form. The flutter problem is then formulated by summing algebraically the analytical expressions for generalized mass, generalized stiffness and generalized aerodynamic forces, and the associated flutter determinant is expanded in analytical form. Finally, the flutter speed and flutter frequency are thereby determined by using a standard root finding procedure. The method is demonstrated by numerical results. This is followed by some concluding remarks.