On the aeroelastic stability and bifurcation structure of subsonic nonlinear thin panels subjected to external excitation

Dynamic behavior of panels exposed to subsonic flow subjected to external excitation is investigated in this paper. The von Karman’s large deflection equations of motion for a flexible panel and Kelvin’s model of structural damping is considered to derive the governing equation. The panel under study is two-dimensional and simply supported. A Galerkin-type solution is introduced to derive the unsteady aerodynamic pressure from the linearized potential equation of uniform incompressible flow. The governing partial differential equation is transformed to a series of ordinary differential equations by using Galerkin method. The aeroelastic stability of the linear panel system is presented in a qualitative analysis and numerical study. The fourth-order Runge-Kutta numerical algorithm is used to conduct the numerical simulations to investigate the bifurcation structure of the nonlinear panel system and the distributions of chaotic regions are shown in the different parameter spaces. The results shows that the panel loses its stability by divergence not flutter in subsonic flow; the number of the fixed points and their stabilities change after the dynamic pressure exceeds the critical value; the chaotic regions and periodic regions appear alternately in parameter spaces; the single period motion trajectories change rhythmically in different periodic regions; the route from periodic motion to chaos is via doubling-period bifurcation.     

亚音速气流中复合材料悬臂板的非线性振动响应研究

随着材料科学的发展,越来越多的新型材料应用到了工程实践中.在气流激励的作用下,对于以航空航天工程为背景、采用复合材料的板壳结构的非线性动力学问题仍是动力学领域的研究热点.本文研究了复合材料悬臂板在亚音速气流条件下的非线性振动和响应.根据理想不可压缩流体的流动条件和 Kutta--Joukowski升力定理,基于升力面理论,利用涡格法计算了三维有限长平板机翼上的亚音速气动升力.将亚音速气动力施加到复合材料悬臂板上,利用Hamilton原理,考虑Reddy三阶剪切变形理论并引入冯$\cdot$卡门非线性应变位移关系,建立了有限长平板的非线性动力学微分方程.利用有限元方法考察了不同几何参数下层合板悬臂板的固有特性,通过比较不同材料和几何参数的线性系统的固有频率,得到不同比例的内共振关系.利用Galerkin方法将偏微分方程截断为两自由度非线性常微分方程,在这里考虑了1:2的内部共振关系并利用多尺度法进行了摄动分析.对应多个选取参数,得到了频率响应曲线.结果展示了硬化弹簧型行为和跳跃现象.      

Nonlinear dynamics analysis of a two-dimensional thin panel with an external forcing in incompressible subsonic flow

Based on the potential theory of incompressible flow and the energy method, a two-dimensional simply supported thin panel subjected to external forcing and uniform incompressible subsonic flow is theoretically modeled. The nonlinear cubic stiffness and viscous damper in the middle of the panel is considered. Transformation of the governing partial differential equation to a set of ordinary differential equations is performed through the Galerkin method. The stability of the fixed points of the panel system is analyzed. The regions of different motion types of the panel system are investigated in different parameter spaces. The rich dynamic behaviors are presented as bifurcation diagrams, phase-plane portraits, Poincaré maps and maximum Lyapunov exponents based on carefully numerical simulations.     

轮对非线性动力学系统蛇行运动的解析解

在对铁路车辆系统的极限环幅值和非线性临界速度进行分析时通常采用数值方法, 不便于研究其随系统参数的变化规律. 轮对系统保留了影响车辆系统动力学性能的几个关键要素: 如轮轨几何非线性约束、轮轨接触蠕滑关系和悬挂系统等, 可以反映铁路车辆系统蛇行运动的本质特性. 轮对系统自由度少、参数少, 可以采用解析方法进行分析. 本文选取合适的特征量把轮对非线性动力学方程无量纲化, 得到了带有小参数的两自由度微分方程; 采用多尺度方法对该方程进行了解析求解; 给出了轮对系统极限环幅值的解析表达式并对其稳定性进行了判定; 给出了轮对系统的分岔速度解析表达式, 并进而获得系统的非线性临界速度的解析表达式. 在对得到的解析解用数值结果进行验证后, 用得到的解析解进行了系统参数影响分析. 传统的分岔图计算方法(如降速法、路径跟踪法等)需对微分方程进行大量数值积分计算方可求解系统的非线性临界速度值, 而通过本文获得的解析表达式可直接给出系统的非线性临界速度值和极限环幅值, 便于研究轮对系统动力学特性随参数的变化规律,进行快速方案比对和筛选, 为转向架结构优化设计提供参考.      

DEVELOPMENT OF A THREE-DIMENSIONAL VISCOUS AEROELASTIC SOLVER FOR NONLINEAR PANEL FLUTTER

A new three-dimensional (3-D) viscous aeroelastic solver for nonlinear panel flutter is developed in this paper. A well-validated full Navier–Stokes code is coupled with a finite-difference procedure for the von Karman plate equations. A subiteration strategy is employed to eliminate lagging errors between the fluid and structural solvers. This approach eliminates the need for the development of a specialized, tightly coupled algorithm for the fluid/structure interaction problem. The new computational scheme is applied to the solution of inviscid two-dimensional panel flutter problems for subsonic and supersonic Mach numbers. Supersonic results are shown to be consistent with the work of previous researchers. Multiple solutions at subsonic Mach numbers are discussed. Viscous effects are shown to raise the flutter dynamic pressure for the supersonic case. For the subsonic viscous case, a different type of flutter behavior occurs for the downward deflected solution with oscillations occurring about a mean deflected position of the panel. This flutter phenomenon results from a true fluid/structure interaction between the flexible panel and the viscous flow above the surface. Initial computations have also been performed for inviscid, 3-D panel flutter for both supersonic and subsonic Mach numbers.     

Reduced order nonlinear aeroelasticity of swept composite wings using compressible indicial unsteady aerodynamics

Nonlinear dynamic aeroelasticity of composite wings in compressible flows is investigated. To provide a reasonable model for the problem, the composite wing is modeled as a thin walled beam (TWB) with circumferentially asymmetric stiffness layup configuration. The structural model considers nonlinear strain displacement relations and a number of non-classical effects, such as transverse shear and warping inhibition. Geometrically nonlinear terms of up to third order are retained in the formulation. Unsteady aerodynamic loads are calculated according to a compressible model, described by indicial function approximations in the time domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to derive the final explicit form of the coupled fluid-structure equations of motion. The final nonlinear governing aeroelastic system of equations is solved using the eigenvectors of the linear structural equations of motion to approximate the spatial variation of the corresponding degrees of freedom in the Ritz solution method. Direct time integrations of the nonlinear equations of motion representing the full aeroelastic system are conducted using the well-known Runge–Kutta method. A comprehensive insight is provided over the effect of parameters such as the lamination fiber angle and the sweep angle on the stability margins and the limit cycle oscillation behavior of the system. Integration of the interpolation method employed for the evaluation of compressible indicial functions at any Mach number in the subsonic compressible range to the derivation process of the third order nonlinear aeroelastic system of equations based on TWB theory is done for the first time. Results show that flutter speeds obtained by the incompressible unsteady aerodynamics are not conservative and as the backward sweep angle of the wing is increased, post-flutter aeroelastic response of the wing becomes more well-behaved.     

Subsonic axisymmetric wake in a viscous gas

The problem of the subsonic axisymmetric flow of a compressible viscous perfect gas in the wake behind a cylindrical body with a flat base section is considered under the condition that the stream parameters are given at infinity and at some distance x_w upstream of the base section. (Let us note that the possibility of the existence of an axisymmetric wake at moderate Reynolds numbers has been shown experimentally [1].) The problem is solved by the numerical build-up method in a cylindrical x, y coordinate system on the basis of the Navier-Stokes equations, by a method elucidated in [2, 3]. Equations obtained from the fundamental system by a passage to the limit while taking into account the symmetry conditions on the axis y=0 are used on the axis of symmetry.     

激波主导流动下壁板的热气动弹性稳定性理论分析

针对激波主导流动下弹性壁板的热气动弹性稳定性分析问题,建立了基于当地活塞流理论的分析模型,并用数值仿真方法来验证其正确性. 首先基于Hamilton原理和Von-Karman大变形理论,建立壁板的热气动弹性运动方程,其中假设壁板受热后温度均匀分布,激波前后区域的气动力模型采用当地一阶活塞流理论;利用Galerkin方法将具有连续参数系统的偏微分颤振方程离散为有限个自由度的常微分方程;基于李雅普诺夫间接法将非线性颤振方程组在平衡位置处进行线化,再用Routh-Hurwits判据来判断线性系统的稳定性,从而来推论出非线性颤振系统的气动弹性稳定性. 在时域中采用龙格--库塔法对非线性颤振方程进行数值积分,得到壁板非线性颤振响应的时间历程,与理论分析结果进行对比. 研究结果表明,壁板受到斜激波冲击时,更容易发生颤振失稳,并且激波强度越大,极限环幅值和频率越大;激波主导流场中的壁板失稳边界不同于传统单纯超声速气流中壁板颤振的失稳边界;只有在斜激波前后不同的动压值都满足颤振稳定性边界的条件下,壁板才可能保持其气动弹性稳定性.      

A Mach‐uniform preconditioner for incompressible and subsonic flows

In this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.     

Global stability of separated flows: subsonic flow past corners

The triple-deck equations for the steady subsonic flow past a convex corner are solved numerically using a novel technique based on Chebychev collocation in the direction normal to the body combined with finite differences in the direction along the flow. The resulting set of nonlinear algebraic equations are solved with Newton linearization and using the GMRES method for the solution of the linear system of equations. The stability of the computed steady flows is then examined using global stability analysis. It is found that for small corner angles, the Tollmien?CSchlichting modes are globally unstable and these persist to larger corner angles. Multiple steady state solutions also exist beyond a critical corner angle but these are globally unstable because of the presence of the Tollmien?CSchlichting modes.     

Nonlinear flutter of viscoelastic rectangular plates and cylindrical panels of a composite with a concentrated masses

The problem of flutter of viscoelastic rectangular plates and cylindrical panels with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate and panel, the effect of concentrated masses is accounted for using the δ-Dirac function. The problem is reduced to a system of nonlinear ordinary integrodifferential equations by using the Bubnov-Galerkin method. The resulting system with a weakly singular Koltunov-Rzhanitsyn kernel is solved by employing a numerical method based on quadrature formulas. The behavior of viscoelastic rectangular plates and cylindrical panels is studied and the critical flow velocities are determined for real composite materials over wide ranges of physicomechanical and geometrical parameters.     

外激励作用下亚音速二维壁板的复杂响应研究

研究了亚音速流中二维壁板在外激励作用下的复杂响应问题。采用迦辽金方法将非线性运动控制方程离散为常微分方程组,采用数值方法进行计算,研究了壁板系统的复杂响应。应用最大李亚普诺夫指数和庞加莱截面方法对系统的运动性质进行了判定。结果表明,系统随着参数的变化呈现出复杂的响应,系统的周期运动与混沌运动会相间出现;系统由周期运动进...     

Numerical Investigation of Multi-Jet Channel Flows

The results of a numerical simulation of the three-dimensional outflow of a system of circular supersonic turbulent jets into a cocurrent supersonic (or subsonic) air flow in a partially bounded region are given. Solutions are obtained by the splitting method using a matrix sweep of the parabolized Navier-Stokes equations. Assuming that the flow is nonseparated in the boundary layer, features of the three-dimensional structure of the jet system are investigated as functions of the pressure ratio number and the jet and cocurrent flow Mach numbers.     

Stability of membrane in low subsonic flow

Stability of an isolated membrane lying in a uniform two-dimensional low subsonic flow is studied theoretically and experimentally. The problem is formulated in a form of a boundary integral equation and differential equations. The boundary integral equation is solved by the boundary element method and the finite difference method is used to solve the differential equations. An effect of a membrane wake is used in the analysis. The theoretical critical divergence velocity is compared with the experimental value.     

Dynamics of a system comprising an orthotropic layer and orthotropic half-plane under the action of an oscillating moving load

This paper investigates the dynamic response to a time-harmonic oscillating moving load of a system comprising a covering layer and half-plane, within the scope of the piecewise-homogeneous body model utilizing of the exact equations of the linear theory of elastodynamics. It is assumed that the materials of the layer and half-plane are anisotropic (orthotropic), and that the velocity of the line-located time-harmonic oscillating moving load is constant as it acts on the free face of the covering layer. Our investigations were carried out for a two-dimensional problem (plane-strain state) under subsonic velocity for a moving load in complete and incomplete contact conditions. The corresponding numerical results were obtained for the stiffer layer and soft half-plane system in which the modulus of elasticity of the covering layer material (for the moving direction of the load) is greater than that of the half-plane material. Numerical results are presented and discussed for the critical velocity, displacement and stress distribution for various values of the problem parameters. In particular, it is established that the critical velocity of the moving load is controlled mainly with a Rayleigh wave speed of a half-plane material and the existence of the oscillation of the moving load causes two types of critical velocity to appear: one of which is less, but the other one is greater than that attained for the case where the mentioned oscillation is absent.     

热环境下壁板非线性颤振分析

基于一阶活塞气动力理论,采用Von Karman大变形应变-位移关系建立了无限展长壁板热环境下颤振方程,采用伽辽金方法对方程进行离散处理.取温度为分叉参数,研究壁板颤振时的分叉及混沌等复杂动力学特性.结果表明:温度载荷降低了系统的颤振临界动压,改变了颤振特性.在整个分岔参数范围内,系统呈现出较为复杂的变化,包括衰减振动、极限环振动、拟周期振动和混沌型振动.当考虑材料热效应时,系统的颤振动压将进一步降低,其响应也表现出更为丰富的非线性动态力学行为.     

Elasticity solution for free vibration analysis of four-parameter functionally graded fiber orientation cylindrical panels using differential quadrature method

In this paper, three-dimensional free vibrations analysis of a four-parameter functionally graded fiber orientation cylindrical panel is presented. The panel is simply supported at the edges and assumed to have an arbitrary variation of fiber orientation in the radial direction. A generalization of the power-law distribution presented in literature is proposed. Symmetric and asymmetric fiber orientation profiles are studied in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method to obtain the natural frequency. The main contribution of this work is to illustrate the influence of the power-law exponent, of the power-law distribution choice and of the choice of the four parameters on the natural frequencies of continuous grading fiber orientation cylindrical panels. Numerical results are presented for a cylindrical panel with arbitrary variation of fiber orientation in the shell’s thickness and compared with discrete laminates composite panels. It is shown maximum natural frequencies will be obtained by using symmetric fiber orientation profiles.     

Electric Charging of Soot Particles in Aircraft Engine Exhaust Plumes

A physico-mathematical model is developed and the variation of the charge on soot particles interacting with positive and negative ions in the exhaust of modern subsonic aircraft is simulated numerically. The calculations are based the gas dynamic system of equations for an axisymmetric turbulent isobaric jet. The system is supplemented with the thermodynamic relations, kinetic equations, and equations for the turbulent viscosity. Ion and neutral and charged soot particle concentration distributions over the exhaust jet on the ground and under cruising flight conditions are obtained.     

THE DYNAMIC BUCKLING OF AN ORTHOTROPIC CYLINDRICAL THIN SHELL UNDER TORSION VARYING AS A POWER FUNCTION OF TIME

The subject of this investigation is to study the buckling of orthotropic cylindricalthin shells under torsion,which is a power function of time.The dynamic stability and compati-bility equations are obtained first.These equations are subsequently reduced to a time dependentdifferential equation with variable coefficient by using Galerkin's method.Finally,the critical dy-namic and static loading,the corresponding wave numbers,the dynamic factors,critical time andcritical impulse are found analytically by applying the Ritz type variational method.Using thoseresults,the effects of the variations of the power of time in the torsion load expression,of theloading parameter,the ratio of the Young's moduli and the ratio of the radius to thickness onthe critical parameters are studied numerically.It is observed that these factors have appreciableeffects on the critical parameters of the problem in the heading.     

Upwind discretization of the steady Navier–Stokes equations

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.