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.     

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.      

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

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.     

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.     

Subsonic flapping flutter

A mathematical model of two-dimensional flow through a flexible channel is analyzed for its stability characteristics. Linear theory shows that fluid viscosity, modelled by a Darcy friction factor, induces flutter instability when the dimensionsless fluid speed, S, attains a critical flutter speed, S₀. This is in qualitative agreement with experimental results, and it is at variance with previous analytical studies where fluid viscosity was neglected and divergence instability was predicted. The critical flutter speed and the associated critical flutter frequency depend on three other dimensionless parameters: the ratio of fluid to wall damping; the ratio of wall to fluid mass; and the ratio of wall bending resistance to elastance. Non-linear theory predicts stable, finite amplitude flutter for S>S₀, which increases in frequency and amplitude as S increases. Both symmetric and antisymmetric modes of deformation are discussed.     

Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkin's technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v₁. For v>v₁, there is a value of the internal viscous damping coefficient, C_iv, which depends on the rotational speed and torque, above which destabilization occurs.The value of C_iv (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of C_iv critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%.For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using C_iv critical, parametric instabilities are still found in the practical range of operation.     

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.     

Estimation of structural system parameters from stationary and non-stationary ambient vibrations: An exploratory-confirmatory analysis

The natural frequency and damping parameters of a building structure are estimated from a long ambient vibration record that shows considerable non-stationarity. The long record is segmented into 57 approximately independent one minute duration stationary time series segments. Each segment is low pass filtered to reject unwanted higher frequency modes and is analyzed by a 2SLS (two stage least squares) time domain parametric model procedure. The scatter diagrams of the estimates of the natural frequency and damping parameters exhibit considerable variability. Estimates of the natural frequency and damping parameters and coefficient of variation expressions of their reliability are obtained by an exploratory-confirmatory data analysis of those 57 vibration time series. A procedure that can obtain the structural parameter estimates with the reliability that is available from stationary analysis from a long and not necessarily stationary record is indicated.     

Power oscillation suppression by robust SMES in power system with large wind power penetration

The large penetration of wind farm into interconnected power systems may cause the severe problem of tie-line power oscillations. To suppress power oscillations, the superconducting magnetic energy storage (SMES) which is able to control active and reactive powers simultaneously, can be applied. On the other hand, several generating and loading conditions, variation of system parameters, etc., cause uncertainties in the system. The SMES controller designed without considering system uncertainties may fail to suppress power oscillations. To enhance the robustness of SMES controller against system uncertainties, this paper proposes a robust control design of SMES by taking system uncertainties into account. The inverse additive perturbation is applied to represent the unstructured system uncertainties and included in power system modeling. The configuration of active and reactive power controllers is the first-order lead–lag compensator with single input feedback. To tune the controller parameters, the optimization problem is formulated based on the enhancement of robust stability margin. The particle swarm optimization is used to solve the problem and achieve the controller parameters. Simulation studies in the six-area interconnected power system with wind farms confirm the robustness of the proposed SMES under various operating conditions.     

Aeroelastic flutter of a disk rotating in an unbounded acoustic medium

The linear aeroelastic stability of an unbaffled flexible disk rotating in an unbounded fluid is investigated by modeling the disk-fluid system as a rotating Kirchhoff plate coupled to the irrotational motions of a compressible inviscid fluid. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. Both a semi-analytical and a numerical method are employed to solve the fluid boundary value problem. The semi-analytical approach involves a perturbation series solution of the dual integral equations arising from the fluid boundary value problem. The numerical approach is a boundary element method based on the Hadamard finite part. Unlike previous works, it is found that a disk with zero material damping destabilizes immediately beyond its lowest critical speed. Upon the inclusion of small disk material damping, the flutter speeds become supercritical and increase with decreasing fluid density. The competing effects of radiation damping into the surrounding fluid and disk material damping control the onset of flutter at supercritical speed. The results are expected to be relevant for the design of rotating disk systems in data storage, turbomachinery and manufacturing applications.     

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.     

Estimation of fatigue life of railway bridges under traffic loads

Random modelling of railway bridge loading enables fatigue damage to be calculated on the basis of the cumulative damage theory of Palmgren-Miner and the classification of the stress-time history by means of the “rain-flow” counting method. The results of calculations are the mean value of the damage and the standard deviation of the stresses, and thus an estimation of the bridge fatigue life. Accordingly the bridge life is dependent on the number of stress cycles and their distribution, the standard deviation of stresses, and on the shape of the Wöhler curve. Bridge life increasing span and decreases with an increasing traffic load. Results are presented as obtained in a detailed study of the effects on the bridge life of different parameters (vehicle speed, damping of bridge vibrations, variability in length and time of the moving load and its magnitude, number of stress cycles and their distribution). The equivalent damage method (the λ_T-method) in the integral form enables one to compare the effects of the traffic loads with those of the standard loading.     

Chordwise spacing aerodynamic detuning for unstalled supersonic flutter stability enhancement

Unstalled supersonic flutter is a significant problem in the development of advanced gas turbines because it restricts the high speed operating range of the engine. A new approach to passive control of unstalled supersonic flutter is aerodynamic detuning, defined as designed passage-to-passage differences in the unsteady aerodynamics of a blade row. In this paper, a mathematical model is developed to predict the unstalled torsion mode stability of an aerodynamically detuned turbomachine rotor operating in a supersonic inlet flow field with a subsonic axial component, with the aerodynamic detuning accomplished by alternate chordwise spacing of adjacent rotor blades. The unsteady aerodynamic moments acting on the blading are calculated in terms of influence coefficients. The stability enhancement associated with this alternate chordwise aerodynamic detuning is demonstrated utilizing an unstable twelve bladed rotor based on Verdon's Cascade B flow geometry. This model and unstable baseline rotor configuration are then used to show that axial spacing detuning leads to greater flutter stability enhancement than does circumferential spacing aerodynamic detuning. Finally, the trade-offs between structural damping, alternate chordwise aerodynamic detuning, and alternate circumferential aerodynamic detuning are considered.     

Unfolding the conical zones of the dissipation-induced subcritical flutter for the rotationally symmetrical gyroscopic systems

Flutter of an elastic body of revolution spinning about its axis of symmetry is prohibited in the subcritical spinning speed range by the Krein theorem for the Hamiltonian perturbations. Indefinite damping creates conical domains of the subcritical flutter (subcritical parametric resonance) bifurcating into the pockets of two Whitney's umbrellas when non-conservative positional forces are additionally taken into account. This explains why in contrast to the common intuition, but in agreement with experience, symmetry-breaking stiffness variation can promote subcritical friction-induced oscillations of the rotor rather than inhibit them.     

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.     

Snap loads and torsional oscillations of the original Tacoma Narrows Bridge

In 1940, the original Tacoma Narrows Bridge was completed on June 10 and opened to traffic on July 1. On November 7, the deck collapsed. Before that day, significant vertical oscillations had occurred, but no torsion. The bridge as built was stable with respect to torsional motion under the winds of November 7 and previous winds with higher speeds. However, snap loads in the diagonal ties attached to the north midspan cable band helped to loosen the band, and the frictional resistance between the band and the north suspension cable passing through it was overcome. The cable began to slip through the band. For this new structural system, with longitudinal motion of the north cable, the wind speed was higher than the critical speed for torsional flutter, and torsional motion was initiated. Approximately 700 cycles of torsional oscillations occurred during the hour prior to the collapse. In the present study, the snap loads on the cable band are discussed first. Then a continuum model of the central span (deck, cables, and hangers) is formulated. The longitudinal motions of the cables are included, so that the slippage can be incorporated. Known information from the observed steady-state torsional motion is utilized with assumed forms of the vertical cable displacements, and the governing equations provide the horizontal cable displacements, the dynamic tensions in the cables, the vertical and torsional motions of the deck, and the resultant lift force and pitching moment (including damping) acting on the deck during its final hour.     

Nonlinear dynamics of a rub-impact micro-rotor system with scale-dependent friction model

Scale effects in dry friction at microscale and the coefficients of friction due to adhesion force and two- and three-body deformations are considered. A rub-impact micro-rotor model with scaling nonlinear rub-impact force is presented and the nonlinear dynamic characteristics of the system in micro-electro-mechanical systems (MEMS) are investigated when the rotating speed, imbalance, damping coefficient, scale length and fractal dimension are regarded as the control parameters. Effects of scale length, fractal dimension, velocity-dependent impact factor and contact form on the coefficients of dry friction are investigated and discussed, and used to study the nonlinear behavior of rub-related vibrations with a large number of numerical simulations. The effects of rotating speed, imbalance, damping coefficient, and friction coefficient on the micro-rotor system responses are studied. It is indicated that the rub-impact micro-rotor system with the scale effects in friction alternates among the periodic, quasi-periodic and chaotic motions as the system parameters change. The results can be effectively used to diagnose the rub-impact fault, reduce the failure and improve the characteristics of a micro-rotor system, and optimize the design of micro-rotating machinery in MEMS.     

The damping function of the van der Waals attraction in the potential between rare gas atoms and metal surfaces

The physisorption potentials of the rare gases on noble metals recently reported by Chizmeshya and Zaremba [Surf. Sci. 268 (1992) 432] are recalculated with new damping parameters for the van der Waals attraction. The new damping parameters, which are applicable when the repulsive potential is not purely exponential, are consistent with the original analysis of Tang and Toennies. The uncertainties resulting from the two approximate sets of damping parameters used by Chizmeshya and Zaremba are removed, moreover a potential minimum is now predicted for Xe on the Au surfaces for which the previous approximations failed.     

The influence of a wieghardt type elastic foundation on the stability of some beams subjected to distributed tangential forces

In this investigation, the influence of a Wieghardt type elastic foundation on the stability of cantilever and clamped-hinged beams subjected to either a uniformly or a linearly distributed tangential force is considered. In addition to the usual transverse foundation modulus, the Wieghardt model includes the effects of inertia and shear deformation in the foundation. Approximate solutions of the Ritz type are obtained for the pertinent eigenvalue problems, and numerical calculations are reported for various combinations of the internal damping, inertia, transverse foundation modulus and shear foundation modulus parameters. The numerical results reveal that, in general, for a fixed value of the transverse foundation modulus parameter κ, an increase in the shear foundation modulus increases the critical load, whereas an increase in the foundation inertia parameter tends to decrease the critical load. The system consisting of a clamped-hinged beam subjected to a uniformly distributed tangential force loses stability through divergence, provided that the value of κ is sufficiently small. However, when κ becomes large enough, stability will be lost through flutter. In this case, the critical load considered as a function of κ possesses a discontinuity at the transition between divergence and flutter, and its value will either increase or decrease, depending upon the degree of damping in the system.