LIMIT CYCLE FLUTTER OF AIRFOILS IN STEADY AND UNSTEADY FLOWS

The limit cycle flutter of a two-dimensional wing with non-linear pitching stiffness is investigated. For modelling the aerodynamic forces of the wing steady linear and non-linear models as well as an unsteady model were used. The flutter speed was calculated using the harmonic balance method and by predicting Hopf bifurcation. Analytical solutions based on the centre manifold theory and normal forms were obtained as were results given by the harmonic balance method. The analytical solutions were compared with those obtained by numerical integration. The results show that the harmonic balance method can forecast flutter speed with a good accuracy while analytical solutions based on centre manifold theorem are accurate only in a small neighbourhood of the bifurcation point. The oscillation of the airfoil after flutter for two different models, linear and non-linear pitching stiffness were compared with each other and the flutter speeds for two linear steady and an unsteady aerodynamic model calculated. The obtained results show that flutter analysis based on the linear steady model is conservative only for the ratios of plunge frequency to pitch frequency lower than 1.     

带有结构非线性的跨音速翼型颤振特性研究

以非定常N-S方程为主管方程,采用时间推进的方法,计算翼型振荡的瞬态非定常气动力,并与带有结构非线性的颤振方程耦合求解,计算了带有结构刚度非线性(间隙型,三次型刚度非线性)和结构阻尼非线性(三次型阻尼非线性)的结构响应特性和颤振特性.计算研究表明,由于同时具有结构和气动非线性,振荡极限环和气动力极为复杂.     

Limit cycle oscillation of missile control fin with structural non-linearity

Non-linear aeroelastic characteristics of a deployable missile control fin with structural non-linearity are investigated. A deployable missile control fin is modelled as a two-dimensional typical section model. Doublet-point method is used for the calculation of supersonic unsteady aerodynamic forces, and aerodynamic forces are approximated by using the minimum-state approximation. For non-linear flutter analysis structural non-linearity is represented by an asymmetric bilinear spring and is linearized by using the describing function method. The linear and non-linear flutter analyses indicate that the flutter characteristics are significantly dependent on the frequency ratio. From the non-linear flutter analysis, various types of limit cycle oscillations are observed in a wide range of air speeds below or above the linear divergent flutter boundary. The non-linear flutter characteristics and the non-linear aeroelastic responses are investigated.     

A high-efficiency aerothermoelastic analysis method

In this paper,a high-efficiency aerothermoelastic analysis method based on unified hypersonic lifting surface theory is established.The method adopts a two-way coupling form that couples the structure,aerodynamic force,and aerodynamic thermo and heat conduction.The aerodynamic force is first calculated based on unified hypersonic lifting surface theory,and then the Eckert reference temperature method is used to solve the temperature field,where the transient heat conduction is solved using Fourier’s law,and the modal method is used for the aeroelastic correction.Finally,flutter is analyzed based on the p-k method.The aerothermoelastic behavior of a typical hypersonic low-aspect ratio wing is then analyzed,and the results indicate the following:(1)the combined effects of the aerodynamic load and thermal load both deform the wing,which would increase if the flexibility,size,and flight time of the hypersonic aircraft increase;(2)the effect of heat accumulation should be noted,and therefore,the trajectory parameters should be considered in the design of hypersonic flight vehicles to avoid hazardous conditions,such as flutter.     

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.     

Investigation on transonic flutter active auppression with CFD-Based ROMs

The calculation of accurate unsteady aerodynamic forces is critical in the analysis of aeroelastic problems,however the efficiency is low because of high computational costs of the computational fluid dynamics(CFD)portion.Additionally,direct integrated CFD and computational structural dynamics(CSD)technique is unsuitable for the analysis of ASE and the flutter active suppression in state-space form.A reduced-order model(ROM)based on Volterra series was developed using CFD calculation and used to predict the flutter coupled with the structure.The closed-loop control systems designed by the sliding mode control(SMC)and linear quadratic Gaussian(LQG)control were constructed with ROM/CSD to suppress the AGARD 445.6wing flutter.The detailed implementation of the two control approaches is presented,and the flutter suppression effectiveness is discussed and compared.The results indicate that SMC method can make the controlled object response decay to the stable equilibrium more rapidly and has better control effects than the LQG control.     

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...     

A theoretical investigation of flow-induced instabilities in compliant coatings

The flow-induced instabilities of a fairly general class of compliant coatings are investigated theoretically. The coatings are of finite length and consist of elastic plates or membranes stretched over a fluid substrate having a density which may be different from the main flow. Provision is also made for the plate to be backed by an elastic foundation of arbitrary spring stiffness. Fairly standard aeroelastic methods are followed. The aerodynamic forces generated by the main flow are calculated by using thin aerofoil theory with a correction factor to allow for the presence of a boundary layer. The pressure induced in the substrate fluid is calculated by assuming potential flow and applying the method of images. A single-mode analysis shows that coatings with laminar boundary layers suffer a divergence-type instability in contrast to turbulent boundary layers which always give rise to a flutter-type instability with a higher critical velocity. The order of the most dangerous mode is calculated and found to rise with an increase in equivalent spring stiffness for fixed tension or flexural rigidity. Results are presented for plates and membrane coatings with an air stream over air and water substrates. Taking account of the substrate fluid dynamics reduces the growth rate of the instability by an order of magnitude and completely suppresses flutter with water substrates. Single-, double- and triple-mode analyses are carried out and the results compared. The critical velocity is adequately predicted by single-mode analysis but a coupling of odd and even modes can lead to flutter even with a laminar boundary layer.     

Free vibration analysis of continuous orthotropic bridge decks

While studies of the free vibration problem of single span bridge slabs have been undertaken by a number of authors, literature on continuous span orthotropic bridge slabs is rather scarce. Furthermore, general continuous bridge deck problems have been dealt with by approximate methods only for specific types of boundary conditions. In this paper an attempt is made to formulate a general analytical solution which would be applicable to all types of boundary conditions. The solution developed is discussed here with special reference to “bridge type” boundary conditions. The analysis is based on the ordinary theory of thin plates and is formulated for linearly elastic materials with isotropic or orthotropic properties. A Levy-type series solution is employed and the problem of free vibration analysis of continuous isotropic and orthotropic bridge slabs is solved by using the principle of superposition. The solution is tested for convergence by varying the number of terms in the solution and the convergence is found to be excellent. Results obtained for continuous isotropic bridge decks are compared with published solutions and close agreements are found. For orthotropic bridge decks a similar comparison was not possible because of a lack of published results in the technical literature.     

三维机翼跨音速颤振的分布式并行计算

以多台微机组成分布式计算系统,利用无限插值方法(TFI)生成三维多块贴体运动网格,以Navier Stokes方程为控制方程,求解三维机翼的跨音速非定常气动力,并与颤振方程耦合迭代计算,求解机翼广义位移响应的时间历程,根据广义位移的时间历程的衰减、等幅和发散振荡等情况确定机翼跨音速颤振临界条件.通过算例验证,计算结果与实验结果和理论分析相吻合.     

Effects of hysteretic and aerodynamic damping on supersonic panel flutter of composite plates

The governing equation for the finite element analysis of the panel flutter of composite plates including structural damping is derived from Hamilton's principle. The first order shear deformable plate theory has been applied to structural modelling so as to obtain the finite element eigenvalue equation. The unsteady aerodynamic load in a supersonic flow is computed by using the linear piston theory. The critical dynamic pressures for composite plates have been calculated to investigate the effects of structural damping on flutter boundaries. The effects are dependent on fiber orientation because flutter mode can be weak or strong in the fiber orientation of composite plates. Structural damping plays an important role in flutter stability with low aerodynamic damping but would not affect the flutter boundary with high aerodynamic damping.     

Uncertainty analysis near bifurcation of an aeroelastic system

Variations in structural and aerodynamic nonlinearities on the dynamic behavior of an aeroelastic system are investigated. The aeroelastic system consists of a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. We follow two approaches to determine the effects of variations in the linear and nonlinear plunge and pitch stiffness coefficients of this aeroelastic system on its stability near the bifurcation. The first approach is based on implementation of intrusive polynomial chaos expansion (PCE) on the governing equations, yielding a set of nonlinear coupled ordinary differential equations that are numerically solved. The results show that this approach is capable of determining sensitivity of the flutter speed to variations in the linear pitch stiffness coefficient. On the other hand, it fails to predict changes in the type of the instability associated with randomness in the cubic stiffness coefficient. In the second approach, the normal form is used to investigate the flutter (Hopf bifurcation) boundary that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing LCO. The results show that this mathematical approach provides detailed aspects of the effects of the different system nonlinearities on its dynamic behavior. Furthermore, this approach could be effectively used to perform sensitivity analysis of the system's response to variations in its parameters.     

Lie symmetry analysis,conservation laws and analytical solutions for a generalized time-fractional modified KdV equation

In this paper, a generalized time fractional modified KdV equation is investigated, which is used for representing physical models in various physical phenomena. By Lie group analysis method, the invariance properties and the vector fields of the equation are presented. Then the symmetry reductions are provided. Moreover, we construct the explicit solutions of the equation by using sub-equation method. Based on the power series theory, the approximate analytical solution for the equation are also constructed. Finally, the new conservation theorem is applied to constructed conservation laws for the equation.     

Integrated analysis on static/dynamic aeroelasticity of curved panels based on a modified local piston theory

A flow field modified local piston theory, which is applied to the integrated analysis on static/dynamic aeroelastic behaviors of curved panels, is proposed in this paper. The local flow field parameters used in the modification are obtained by CFD technique which has the advantage to simulate the steady flow field accurately. This flow field modified local piston theory for aerodynamic loading is applied to the analysis of static aeroelastic deformation and flutter stabilities of curved panels in hypersonic flow. In addition, comparisons are made between results obtained by using the present method and curvature modified method. It shows that when the curvature of the curved panel is relatively small, the static aeroelastic deformations and flutter stability boundaries obtained by these two methods have little difference, while for curved panels with larger curvatures, the static aeroelastic deformation obtained by the present method is larger and the flutter stability boundary is smaller compared with those obtained by the curvature modified method, and the discrepancy increases with the increasing of curvature of panels. Therefore, the existing curvature modified method is non-conservative compared to the proposed flow field modified method based on the consideration of hypersonic flight vehicle safety, and the proposed flow field modified local piston theory for curved panels enlarges the application range of piston theory.     

DYNAMICS BEHAVIOR OF AXISYMMETRIC AND BEAM-LIKE ANISOTROPIC CYLINDRICAL SHELLS CONVEYING FLUID

The flow-induced vibration characteristics of anisotropic laminated cylindrical shells partially or completely filled with liquid or subjected to a flowing fluid are studied in this work for two cases of circumferential wave number, the axisymmetric, where n=0 and the beam-like, where n=1. The shear deformation effects are taken into account in this theory; therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The present method is a combination of finite element analysis and refined shell theory in which the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear co-ordinates. Mass and stiffness matrices are determined by precise analytical integration. A finite element is defined for the liquid in cases of potential flow that yields three forces (inertial, centrifugal and Coriolis) of moving fluid. The mass, stiffness and damping matrices due to the fluid effect are obtained by an analytical integration of the fluid pressure over the liquid element. The available solution based on Sanders' theory can also be obtained from the present theory in the limiting case of infinite stiffness in transverse shear. The natural frequencies of isotropic and anisotropic cylindrical shells that are empty, partially or completely filled with liquid as well as subjected to a flowing fluid, are given. When these results are compared with corresponding results obtained using existing theories, very good agreement is obtained.     

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.     

Effect of phase angle on multibladed rotor flutter

A theoretical technique for predicting the flutter characteristics of a helicopter rotor is presented. The effect of phase angle on flutter speed of a two-bladed rotor in hovering and axial flight is determined. For this purpose, a uniform and untwisted rotor blade with coupled flapwise bending and torsional degrees of freedom is considered. The transmission matrix method is used to obtain the natural vibration characteristics of the system. An unsteady aerodynamic theory is used to obtain the aerodynamic loading in compressible flow.     

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.     

An analysis of binary flutter of bridge deck sections

This paper is concerned with an analytical and experimental study of binary flutter of bridge deck sections. A set of analytical formulas giving the frequency and rate of growth of oscillation, the position of the equivalent center of rotation and the phase difference between bending and torsion near the critical flutter point is presented. The formulas provide an analytical basis for the previously proposed method of classification of binary flutter of bluff structures. The results of wind tunnel experiments on models with simple geometrical shapes confirm that the present formulas are applicable to a variety of structures ranging from a flat plate to much more bluff bridge deck sections.     

Galloping of bundled power line conductors

This paper describes an experimental and analytical study of the galloping of a two-dimensional section model of a two-conductor bundle in which ice-accreted conductors are replaced by two identical square prisms, with both vertical and torsional movements allowed but the horizontal one blocked, in a uniform wind tunnel flow. Emphasis is placed on elucidating the vital role played by the aerodynamic coupling in the stability of bundled conductors. It is shown that, apart from galloping type flutter, two other types of instability, namely, torsional and classical type flutter, can also occur for bundled conductors. In particular, it is shown that the aerodynamic coupling can cause violent classical type flutter to occur when the resonant condition is approached.