Supercritical and subcritical Hopf bifurcation and limit cycle oscillations of an airfoil with cubic nonlinearity in supersonic\hypersonic flow

In this paper, the Hopf bifurcations and limit cycle oscillations (LCOs) of an airfoil with cubic nonlinearity in supersonic\hypersonic flow are investigated. The harmonic balance method and multivariable Floquet theory are applied to analyze the LCOs of the airfoil. Four distinct cases of the LCOs response are detected in this system: (I) supercritical Hopf bifurcation, (II) a single subcritical Hopf bifurcation, (III) two subcritical Hopf bifurcations, and (IV) no Hopf bifurcation. Furthermore, the parameter variations domains separating the supercritical and subcritical Hopf bifurcations are presented using singularity theory.     

Aeroelastic suppression of an airfoil with control surface using nonlinear energy sink

Aeroelasticity exists in airfoil with control surface freeplay, which may induce instability in an incompressible flow. In this paper, a nonlinear energy sink (NES) is used to suppress the aeroelasticity of an airfoil with a control surface. The freeplay and cubic nonlinearity in pitch are taken into account. The harmonic balance method is used to analytically determine the limit cycle oscillations (LCOs) amplitudes of the airfoil–NES system. Linear and nonlinear flutter speeds are detected from the airfoil with control surface freeplay. When NES is attached, both the linear flutter speed of airfoil without freeplay and the nonlinear flutter speed of airfoil with a freeplay are increased. Moreover, the LCO amplitude of airfoil is decreased due to NES. Then, the influences of NES parameters on the increase in flutter boundary of airfoil are carefully studied.     

Transonic limit cycle oscillation behavior of an aeroelastic airfoil with free-play

The limit cycle oscillation (LCO) behaviors of an aeroelastic airfoil with free-play for different Mach numbers are studied. Euler equations are adopted to obtain the unsteady aerodynamic forces. Aerodynamic and structural describing functions are employed to deal with aerodynamic and structural nonlinearities, respectively. Then the flutter speed and flutter frequency are obtained by V-g method. The LCO solutions for the aeroelastic airfoil obtained by using dynamically linear aerodynamics agree well with those obtained directly by using nonlinear aerodynamics. Subsequently, the dynamically linear aerodynamics is assumed, and results show that the LCOs behave variously in different Mach number ranges. A subcritical bifurcation, consisting of both stable and unstable branches, is firstly observed in subsonic and high subsonic regime. Then in a narrow Mach number range, the unstable LCOs with small amplitudes turn to be stable ones dominated by the single degree of freedom flutter. Meanwhile, these LCOs can persist down to very low flutter speeds. When the Mach number is increased further, the stable branch turns back to be unstable. To address the reason of the stability variation for different Mach numbers at small amplitude LCOs, we find that the Mach number freeze phenomenon provides a physics-based explanation and the phase reversal of the aerodynamic forces will trigger the single degree of freedom flutter in the narrow Mach number range between the low and high Mach numbers of the chimney region. The high Mach number can be predicted by the freeze Mach number, and the low one can be estimated by the Mach number at which the aerodynamic center of the airfoil lies near its elastic axis. Influence of angle of attack and viscous effects on the LCO behavior is also discussed.     

Limit cycle oscillations of rectangular cantilever wings containing cubic nonlinearity in an incompressible flow

Limit cycle oscillations (LCO) as well as nonlinear aeroelastic analysis of rectangular cantilever wings with a cubic nonlinearity are investigated. Aeroelastic equations of a rectangular cantilever wing with two degrees of freedom in an incompressible potential flow are presented in the time domain. The harmonic balance method is modified to calculate the LCO frequency and amplitude for rectangular wings. In order to verify the derived formulation, flutter boundaries are obtained via a linear analysis of the derived system of equations for five different cases and compared with experimental data. Satisfactory results are gained through this comparison. The problem of finding the LCO frequency and amplitude is solved via applying the two methods discussed for two different cases with hardening cubic nonlinearities. The results from first-, third- and fifth-order harmonic balance methods are compared with the results of an exact numerical solution. A close agreement is obtained between these harmonic balance methods and the exact numerical solution of the governing aeroelastic equations. Finally, the nonlinear aeroelastic analysis of a rectangular cantilever wing with a softening nonlinearity is studied.     

A high dimensional harmonic balance approach for an aeroelastic airfoil with cubic restoring forces

This is a study of a two-dimensional airfoil including a cubic spring stiffness placed in an incompressible flow. A new formulation of the harmonic balance method is employed for the aeroelastic airfoil to investigate the amplitude and frequency of the limit cycle oscillations. The results are compared with the results from the classical harmonic balance approach and from the conventional time marching integration method.     

On the nonlinear theory of the wing in a plane unsteady flow

The main aspects of the nonlinear theory of the wing in a plane unsteady fluid flow are generalized on the basis of the author’s previous results. An initial-boundary problem for complex velocity is formulated. A system of differential equations with conditions at points of vortex wake shedding is presented, which allows a large class of problems to be solved correctly. The Cauchy problem is solved by using a standard discretization procedure. The boundary-value problem is reduced at each time step to singular integral equations of the first and second kind. The accuracy of solving these equations by the method of discrete vortices and by the method of panels is compared. Specific features of pressure calculations in the case of a separated flow around the airfoil contour are discussed     

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.     

Localized Basis Function Method for Computing Limit Cycle Oscillations

An alternate approach to the standard harmonic balance method (based on Fourier transforms) is proposed. The proposed method begins with an idea similar to the harmonic balance method, i.e. to transform the initial set of differential equations of the dynamics to a set of discrete algebraic equations. However, as distinct from previous harmonic balance techniques, the proposed method uses a set of basis functions which are localized in time and are not necessarily sinusoidal. Also as distinct from previous harmonic balance methods, the algebraic equations obtained after the transformation of the differential equations of the dynamics are solved in the time domain rather than the frequency domain. Numerical examples are provided to demonstrate the performance of the method for autonomous and forced dynamics of a Van der Pol oscillator.     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Equivalent damping of aeroelastic system of an airfoil with cubic stiffness

This paper presents a mathematical study on the subsonic aerodynamics acting on an airfoil with a cubic stiffness. One portion of aerodynamics is assumed as and replaced by an equivalent damping. Using the harmonic balance method, an equivalent system is deduced and studied by a numerical integration method. Numerical examples show the validity and feasibility of the proposed mathematical treatment of the aerodynamics. It reveals that the unsteady aerodynamics acting on the airfoil can not only affect the pitch stiffness but also result in additional damping.     

一种单元谐波平衡法

基于有限元离散,对于工程中的非线性响应问题,提出一种单元谐波平衡法．与常规的谐波平衡法不同,本文将谐波平衡方程建立在有限元素上,从而兼顾了有限元素法和常规谐波平衡法两大优势．有限元技术的应用能使得求解问题的范围扩大到复杂工程结构,而谐波平衡概念的使用将使得含有复杂变形和复杂本构关系的动力学响应问题得到有效解决．所提方法能适用于工程结构中具有复杂非线性关系的动力学响应问题．由于谐波平衡法的实施依赖于谐波系数方程及其切线刚度矩阵的解析推导,尽管已经局限到有限元素上,但对于较为复杂一些的本构关系,推导仍非易事．为解决这些问题,放弃通常对于变形梯度和应变张量所作的向量假设,而是从连续介质力学中基本的几何关系入手,提出一种矩阵分解形式．通过利用张量的内蕴导数定义以及关于迹函数的有关性质,给出应力增量的一种新的表现形式．当它与变形梯度的矩阵分解相结合时,使得切线刚度矩阵的导出变得十分简单,而且所得计算形式也比通常紧凑和方便许多．     

Modeling the flow past an oscillating airfoil by the method of viscous vortex domains

The Lagrangian vortex method for solving the Navier-Stokes equations is applied for numerically modeling the unsteady flow past a wing airfoil executing angular oscillations in a viscous incompressible flow. Formulas relating the unsteady forces on the airfoil and the vorticity field are derived. The calculated results are compared with the experimental data for the NACA-0012 airfoil executing harmonic oscillations in an air flow at the Reynolds number Re = 4.4 × 10⁴.     

具任意脱层复合材料梁的非线性谐波响应

研究了具任意脱层复合材料梁的非线性谐波响应问题。基于弹性理论，建立了考虑剪切变形时的复合材料梁脱层的基本方程式。在空间上采用B样条函数和Galerkin积分法，在时间上采用增量谐波平衡法进行计算。通过实例计算，得出了简谐力作用下的非线性动力响应曲线。认为基谐波振动仍是非线性振动的主要部分。     

Some Results of the Finite Deflection Analysis of Clamped Skew Sandwich Plates

The suitability of Galerkin's method for the solution of the problem of the finite deflection analysis of clamped skew sandwich plates is studied. The five coupled nonlinear governing differential equations for sandwich plates are transformed into nonlinear algebraic equations by using Galerkin's method of error minimization. These equations are then solved using an iterative algorithm suggested by Brown. Comparisons of the results of the present analysis with available solutions show good agreement. Numerical results are presented for skew sandwich plates for a wide range of values of the core modulus for different skew angles and aspect ratios. Simplicity in formulation and computation is the advantage of the method as compared with other methods of nonlinear analysis. Computing time and memory requirements in a digital computer are relatively very small, which makes the method attractive.     

BIFURCATIONS OF AIRFOIL IN INCOMPRESSIBLE FLOW

Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.     

A new method for predicting the maximum vibration amplitude of periodic solution of non-linear system

An original method based on the proposed framework for calculating the maximum vibration amplitude of periodic solution of non-linear system is presented. The problem of determining the worst maximum vibration is transformed into a non-linear optimization problem. The harmonic balance method and the Hill method are selected to construct the general non-linear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness of the proposed approach is illustrated through two numerical examples. Numerical examples show that the proposed method can, at much lower cost, give results with higher accuracy as compared with numerical results obtained by a parameter continuation method.     

Amplification and amplitude limitation of heave/pitch limit-cycle oscillations close to the transonic dip

Recent results from flutter experiments of the supercritical airfoil NLR 7301 at flow conditions close to the transonic dip are presented. The airfoil was mounted with two degrees-of-freedom in an adaptive solid-wall wind tunnel, and boundary-layer transition was tripped. Flutter boundaries exhibiting a transonic dip were determined and limit-cycle oscillations (LCOs) were measured. The local energy exchange between the fluid and the structure during LCOs is examined and leads to the following findings: at supercritical Mach numbers below that of the transonic-dip minimum the presence of a shock-wave and its dynamics destabilizes the aeroelastic system such that the decreasing branch of the transonic dip develops. At higher Mach numbers the shock-wave motion has a stabilizing effect such that the flutter boundary increases to higher flutter-speed indices with increasing Mach number. Amplified oscillations near this branch of the flutter boundary obtain energy from the flow mainly due to the dynamics of a trailing-edge flow separation. A slight nonlinear amplitude dependency of the shock motion and a possibly occurring boundary-layer separation cause the amplitude limitation of the observed LCOs. The impact of the findings on the numerical simulation of these phenomena is discussed.     

On accurate boundary conditions for a shape sensitivity equation method

This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier–Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least‐squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite‐element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of dynamic stall for a NACA 0012 airfoil using a vortex method

The unsteady, incompressible, viscous laminar flow over a NACA 0012 airfoil is simulated, and the effects of several parameters investigated. A vortex method is used to solve the two-dimensional Navier–Stokes equations in the vorticity/stream-function form. By applying an operator-splitting method, the “convection” and “diffusion” equations are solved sequentially at each time step. The convection equation is solved using the vortex-in-cell method, and the diffusion equation using a second-order ADI finite difference scheme. The airfoil profile is obtained by mapping a circle in the computational domain into the physical domain through a Joukowski transformation. The effects of several parameters are investigated, such as the reduced frequency, mean angle of attack, location of pitch axis, and the Reynolds number. It is observed that the reduced frequency has the most influence on the flow field.     

Implicit Calculations of an Aeroelasticity Problem

The purpose of this work is to show that a linearized implicit scheme for the flow resolution can be an efficient and accurate method for solving fluid-structure interaction. The fluid is modeled by the Euler equations in two dimensions and the structure by a one (free piston) or a two (NACA0012 airfoil) degrees of freedom system. The schemes are developed using a finite volume/finite element formulation and, stating the moving boundary problem in the space-time domain, the Riemann solver is generalized in a suitable manner. Assuming a modal decomposition for the structure's response, an analytical solution to the equation of motion is obtained.

The effects of the linearized implicit scheme on the aeroelastic response are demonstrated on the free piston and the NACA 0012 airfoil problems. In the latter case, we focus on the capability of the linearized implicit scheme to accurately predict the stability limit of the coupled response (wing flutter analysis). Although the above analysis is performed using a rigid transformation, a robust moving mesh strategy is presented for more general 2-D and 3-D deformations.