Nonlinear Flutter Characteristics Research Articles

A Study on Post-Flutter Characteristics of a Large-Span Double-Deck Steel Truss Main Girder Suspension Bridge

To investigate the nonlinear flutter characteristics of long-span suspension bridges under different deck ancillary structures and configurations, including those with and without a central wind-permeable zone, as well as to analyze the hysteresis phenomenon of a subcritical flutter and elucidate the mechanisms leading to the occurrence of nonlinear flutter, this paper studies first the post-flutter characteristics of the torsion single-degree-of-freedom (SDOF) test systems and vertical bending–torsion two-degree-of-freedom (2DOF) test systems under different aerodynamic shape conditions are further analyzed, and the role of the vertical vibration in coupled nonlinear flutter is discussed. The results indicate that better flutter performance is achieved in the absence of bridge deck auxiliary structures with a central wind-permeable zone. The participation of vertical vibrations in the post-flutter vibration increases with the increase in wind speed, reducing the flutter performance of the main girder. Furthermore, the hysteresis phenomenon in the subcritical flutter state is observed in the wind tunnel experiment, and its evolution law and mechanism are discussed from the perspective of amplitude-dependent damping. Finally, the vibration-generating mechanism of the limit oscillation ring is elaborated in terms of the evolution law of the post-flutter vibration damping.

Investigation on nonlinear flutter modes competition of a long-span suspension bridge based on full-bridge aeroelastic model tests

Nonlinear flutter has attracted wide attention due to the bottleneck caused by linear flutter theory on flutter-resistant design of super-long-span bridges. The longer the span, the more closely spaced the natural modes, which may cause competition among flutter-modes in a nonlinear flutter, but it is rarely reported so far. To this end, this study experimentally investigates the 3D nonlinear flutter characteristics of a long-span suspension bridge with closely spaced natural modes based on full-bridge aeroelastic model wind tunnel tests. Since there are two unstable flutter-modes within the interested post-critical regime and thus various complex but interesting flutter-modes competition processes were observed, such as continuous modes competition where the vibration amplitude evolution exhibits various different types of “sawtooth” shapes during stable vibration stages. Meanwhile, the complex nonlinear bifurcation behaviors caused by modes competition were also observed. The evolutionary characteristics of flutter-modes competition are analyzed in detail and relevant mechanisms are discussed. As the wind speed increases, the flutter of the studied bridge mainly undergoes a transition from being dominated by the symmetric mode to being dominated by the antisymmetric mode, accompanied by a continuous modes competition zone in between as a transitional zone. The results show that the continuous modes competition will result in a decrease in the maximum amplitude RMS of the full-span, which is beneficial for the structure. But it may also lead to a transient increase in the maximum amplitude of the full-span due to the coupling vibration shape formed by the two significant flutter-modes, which may be bad for the structure.

Study on Nonlinear Flutter Characteristics of Single-Box Section and Twin-Box Section as a Three-Degree-of-Freedom System

The accurate prediction of the nonlinear flutter response is one of the important prerequisites for accurately evaluating the post-flutter performance of long-span bridges. In fact, as the vibration amplitude increases, the participation of lateral degree-of-freedom (DOF) will become more and more obvious and thus ignoring self-excited forces related to lateral vibration may no longer be applicable. To this end, based on 18 amplitude-dependent flutter derivatives (FDs), a 3-DOFs coupled nonlinear flutter analysis method was proposed in this study. Then, 18 amplitude-dependent FDs of two typical bridge decks were identified, and they were used to calculate the 3-DOF coupled nonlinear flutter responses based on the proposed flutter analysis method. Meanwhile, the correctness of the above flutter analysis method was also verified by compared with the numerical simulation results. Finally, the influence and mechanism of the lateral DOF, structural modal parameters and amplitude-dependent FDs on the nonlinear flutter of these typical bridge decks were investigated. The results show that when the torsional amplitude of the nonlinear flutter is large, the participation of lateral DOF will significantly increase the steady-state amplitude of the nonlinear flutter, and the larger the torsional amplitude, the more significant this increase is. The main reason is the participation of the lateral DOF introduces the negative coupled aerodynamic damping to the system, which then reduces the stability of the system. Besides, for the twin-box section, the influence of the lateral DOF on the nonlinear flutter is relatively smaller compared with the single-box section since the proportion of negative aerodynamic damping by the lateral DOF is smaller. Compared with the single-box section, the mass of the twin-box section has a smaller effect on the critical wind speed, and the torsional structural damping ratio of the twin-box section has a greater effect on the nonlinear flutter.

Nonlinear thermal flutter of variable angle tow composite laminates in supersonic airflow

Compared with traditional composite materials, variable stiffness composite materials with curvilinear fiber paths have higher design flexibility and the potential to improve structural efficiency. In view of this, the nonlinear flutter characteristics of Variable Angle Tow (VAT) composite laminates with thermal effect in supersonic airflow are investigated in this paper. The proposed panel flutter model is developed based on the von Kármán large deformation plate theory. The thermal stress is approximated according to the quasi-steady thermal stress theory, whilst the aerodynamic pressure is calculated based on the first-order piston theory. The aerothermoelastic equations of motion are first established using Hamilton’s variational principle, and the Rayleigh-Ritz method is then applied to solve the governing equations in the space domain. The time-domain nonlinear equations are solved through the fourth-order Runge-Kutta method. The accuracy and convergence of the proposed method are verified through the comparisons with the results available in the literature. The effects of several parameters such as fiber orientation angle, thermal load, and aerodynamic pressure on the nonlinear flutter responses of VAT laminates are comprehensively discussed. Several dynamic behaviors of VAT laminates, including stable flat, buckled, Limit Cycle Oscillation (LCO), multi-periodic motion, quasi-periodic motion, and chaotic motion, are revealed by using the time history, phase portrait, Poincaré map, and bifurcation diagram. The results presented herein may be beneficial for the design of VAT laminates in supersonic airflow.

Numerical study on the nonlinear characteristics of shock induced two-dimensional panel flutter in inviscid flow

For an aeroelastic system of a two-dimensional elastic panel subjected to an impinging inviscid oblique shockwave, the nonlinear flutter characteristics are affected by many factors such as shock impingement location, cavity pressure and initial perturbation. The effects of the above factors on the variation of system bifurcation type and dynamic behaviors are investigated numerically. A low-fidelity computational method coupled with local piston theory and van Karman plate model, and a high-fidelity computational method coupled with Euler equations and finite element model are used for fluid-structure interaction simulations. Two sets of new findings are unveiled. First, either the variation of shock impingement location or cavity pressure can induce the aeroelastic system to transition between a subcritical bifurcation and a supercritical bifurcation. For some cases, the system bifurcation characteristics exhibit strong sensitivity to these two factors. Second, it is found that in addition to the limit cycle oscillation in the form of a combination of the second and third structural modes, multiple stable limit cycle oscillations due to the coupling of higher-order modes can be triggered by suitable initial perturbations. These limit cycle oscillations have high frequencies and amplitudes, which suggest a higher risk of structural fatigue failure.

Influence of Single External Store on Geometrical Nonlinear Flutter of High-aspect-ratio Wings

High-altitude and long-endurance UAVs, distributed electric propulsion aircraft and large passenger aircraft often use high-aspect-ratio wings, which have obvious geometric nonlinear large deformation effect, and are usually arranged with a certain number of external stores, which have an impact on their geometric nonlinear aeroelastic characteristics. Based on the “quasi-linear” hypothesis, the updated Lagrange incremental finite element method and doublet-lattice method are used to perform the geometrically nonlinear static aeroelastic analysis, and the mass, stiffness and aerodynamic matrices of large deformation are updated, and the geometrically nonlinear flutter characteristics of high-aspect-ratio wings are calculated. The influence of the position and mass of the external store on geometrical nonlinear flutter characteristics of high-aspect-ratio wing structure is investigated. Results show that the change of the position and mass of the external store causes the relative position between the elastic axis and the center of gravity of the wing, and the wing’s large deformation equilibrium state to change, resulting in differences in the stiffness characteristics of the wing and the natural vibration frequency of the equilibrium state, thus affecting the wing flutter characteristics. When a single external store is installed near the wing root, changing the mass and direction of the store has no significant effect on the geometric nonlinear flutter characteristics of the wing. The improvement of wing flutter speed is most obvious when the hanging point is moved to the leading edge of the wing near 0.65 times the wingspan.

Nonlinear Aeroelastic Analysis of Heated Curved Panels in Supersonic Air Flow

A nonlinear aeroelastic model for a two-dimensional heated curved panel in supersonic air flow is established by using Galerkin method. The von Karman large deflection theory and the third-order piston theory appended with the static aerodynamic loading are used in the governing equations. The static aero-thermal deflection of the curved panel is firstly obtained by using Newton iterative approach. Then the stability boundary under different temperature elevations is achieved by using Lyapunov’s indirect method. Lastly, the motion equations of a heated curved panel in supersonic air flow are solved by a fourth-order Runge–Kutta numerical scheme. Time history responses, phase plots, Poincare maps and bifurcation diagrams are used for better understanding of the pre/post-flutter responses of a curved panel by varying parameters, including curvatures, dynamic pressures and temperature elevations. The results demonstrate that the flutter boundary drops significantly with increasing temperature elevation for panels with small curvatures. However, the flutter boundary almost keeps the same for panels with large curvatures. The nonlinear flutter characteristics of the curved panel differs from those of flat panels significantly, the nonlinear flutter response of heated curved panels changes from static equilibrium point or LCO to chaos with the increase of temperature elevation or dynamic pressure and the static, LCO motions also exist in the chaotic area. For panels with large curvatures, the chaotic motions cannot occur, and the amplitude of LCO is limited in a certain range.

Characterization of vibration amplitude of nonlinear bridge flutter from section model test to full bridge estimation

This paper presents a comprehensive study of nonlinear flutter characteristics of a bridge deck section through free vibration wind tunnel testing and theoretical analysis. The section model is spring-supported in a single degree of freedom (SDOF) in torsion and 2DOFs in both vertical direction and torsion, respectively. The amplitude-dependent aerodynamic damping and other response characteristics are determined through use of Hilbert Transform of response time histories at different wind speed of smooth flow. An approach is proposed to extract flutter derivatives as nonlinear functions of amplitude of torsional motion at various reduced wind speeds. The flutter derivatives are then used to estimate the nonlinear flutter response of bridge section and a prototype suspension bridge with a main span length of 1700 ​m. The results showed that the magnitude of negative aerodynamic damping increases with increasing wind speed but decreases with vibration amplitude. The flutter of this bridge example is initialed from torsion but the coupled vertical motion further generates negative damping, thus reduces the flutter onset wind speed and increase the vibration amplitude. The estimation using three-dimensional bridge model with coupled vertical and torsional modal responses leads to increase in flutter onset wind speed and decrease in flutter amplitude as compared to section model estimation.

Sensitivity of Aeroservoelastic Stability Characteristics Using Parametric Flutter Margins

The recently developed parametric flutter margin method is expanded to perform efficient sensitivity studies in which linear and nonlinear flutter characteristics are calculated for changes in selected parameters. Two approaches are suggested: one for structural, aerodynamic, or control parameters that can be formulated as single-input/single-output feedback loops and one for parameters that require multiple-input/multiple-output realization. The single-input/single-output analysis is based on frequency response functions, with the formulation aiming to track the important flutter mechanisms, and is arranged such that the sensitivity to different parameters, over large-variation ranges, may be based on the same set of response functions. The multiple-input/multiple-output analysis is similarly effective but is based on eigensolutions that indicate the parametric changes needed to obtain flutter. Three numerical examples using a generic transport aircraft model, in which sensitivity with respect to stiffness and mass variations were studied, are presented. The single-input/single-output and multiple-input/multiple-output approaches are also applied for calculating damping and frequency variations with velocity that may be used as traditional plots in aeroelastic design and certification tasks. The results demonstrate excellent agreement with those obtained by separate solutions of the MSC/NASTRAN code and nonlinear time marching simulations.

Nonlinear flutter analysis of composite panels

Nonlinear panel flutter is an interesting subject of fluid–structure interaction. In this paper, nonlinear flutter characteristics of curved composite panels are studied in very low supersonic flow. The composite panel with geometric nonlinearity is modeled by a nonlinear finite element method; and the responses are computed by the nonlinear Newmark algorithm. An unsteady aerodynamic solver, which contains a flux splitting scheme and dual time marching technology, is employed in calculating the unsteady pressure of the motion of the panel. Based on a half-step staggered coupled solution, the aeroelastic responses of two composite panels with different radius of R = 5 and R = 2.5 are computed and compared with each other at different dynamic pressure for Ma = 1.05. The nonlinear flutter characteristics comprising limited cycle oscillations and chaos are analyzed and discussed.

Aero-thermoelastic stability and nonlinear flutter analysis of functionally graded panels

Nonlinear flutter characteristics under the thermal load as well as aerodynamic heating on functionally graded material panels are investigated in the hypersonic flow. Mathematical models are defined such as power-law, sigmoid and exponential functionally graded materials by the volume fractions of the constituents. Further, third-order piston theory of aerodynamics and von Karman strain-displacement relationship is adopted in order to perform the aero-thermoelastic analysis. Applying the principle of virtual work, equations of motion are derived in the matrix form, and then finite element method is used to obtain the discretized equations. Then, the time domains analysis is performed using Newmark’s time integration method. To check the validity of the present formulation, numerical results are compared with the previous data in the literatures.

Structural Nonlinear Flutter Characteristics Analysis for an Actuator-fin System with Dynamic Stiffness

The flutter characteristics of an actuator-fin system are investigated with structural nonlinearity and dynamic stiffness of the electric motor. The component mode substitution method is used to establish the nonlinear governing equations in time domain and frequency domain based on the fundamental dynamic equations of the electric motor and decelerator. The existing describing function method and a proposed iterative method are used to obtain the flutter characteristics containing preload freeplay nonlinearity when the control command is zero. A comparison between the results of frequency domain and those of time domain is studied. Simulations are carried out when the control command is not zero and further analysis is conducted when the freeplay angle is changed. The results show that structural nonlinearity and dynamic stiffness have a significant influence on the flutter characteristics. Limit cycle oscillations (LCOs) are observed within linear flutter boundary. The response of the actuator-fin system is related to the initial disturbance. In the nonlinear condition, the amplitude of the control command has an influence on the flutter characteristics.

A finite element study on the large amplitude flexural vibration characteristics of FGM plates under aerodynamic load

The large amplitude flexural vibration characteristics of functionally graded material (FGM) plates are investigated here using a shear flexible finite element approach. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The effective material properties are then evaluated based on the rule of mixture. The FGM plate is modeled using the first-order shear deformation theory based on exact neutral surface position and von Kármán’s assumptions for large displacement. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations of motion are solved by harmonic balance method to study the vibration amplitude of FGM plates under supersonic air flow. Thereafter, the non-linear equations of motion are solved using Newmark’s time integration technique to understand the flexural vibration behavior of FGM plates in time domain (simple harmonic or periodic or quasi-periodic). This work is new in the sense that it deals with the non-linear flutter characteristics of FGM plates under high supersonic airflow accounting for both the geometric and aerodynamic non-linearities. Some parametric study is conducted to understand the influence of these non-linearities on the flutter characteristics of FGM plates.

Nonlinear flutter analysis of stiffened composite panels in supersonic 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 stiffening scheme on the flutter critical dynamic pressure is demonstrated through the numerical example, and the nonlinear flutter characteristics of stiffened composite panels are also analyzed in the time domain. This will lay the foundation for design of panel structures employed in aerospace vehicles.

구조 비선형을 고려한 이차원 단면 날개 모델의 이중 제한 주기 운동

이선형 플런지 스프링을 가지는 2차원 단면 익형 모델에 대하여 초음속 비선형 공탄성 해석을 수행하였다. 초음속 비정상 공기력 계산을 위해 DPM을 사용하였고 최소상태접근법을 사용하여 근사하였다. 비선형 플러터 해석을 위해 구조 비선형성을 비대칭 이선형 스프링으로 모델링하고 기술 함수 법을 사용하여 선형화하였다. 선형 및 비선형 플러터 해석 결과들은 공력탄성학적 특성들이 주파수 비에 중요한 영향을 받는다는 것을 보여준다. 비선형 플러터 해석으로부터 다양한 제한 주기 운동이 선형플러터 속도 이하 또는 이상에서 관측되었다. 또한 플러터 특성과 응답을 시간영역에서도 조사하였다. Nonlinear aeroelastic characteristics of a two dimensional typical section model with bilinear plunge spring are investigated. Doublet-point method(DPM) is used for the calculation of supersonic unsteady aerodynamic forces which are approximated by using the minimum-state approximation. For nonlinear flutter analysis structural nonlinearity is represented by an asymmetric bilinear spring and is linearized by using the describing function method. The linear and nonlinear flutter analyses indicate that the flutter characteristics are significantly dependent on the frequency ratio. From the nonlinear flutter analysis, various types of limit cycle oscillations are observed in a wide range of air speeds below or above the linear flutter boundary. The nonlinear flutter characteristics and the nonlinear aeroelastic responses are investigated.

구동장치의 동강성을 고려한 미사일 조종날개의 비선형 플러터 해석

구동기의 백래쉬와 동강성을 고려한 미사일 조종날개의 비선형 공탄성 해석이 수행되었다. 아음속 비정상 공기력 계산을 위해 DHM을 사용하였고 최소상태접근법을 사용하여 근사하였다. 비선형 플러터 해석을 위해 백래쉬는 유격으로 모델하고 기술 함수법을 사용하여 선형화하였다. 또한, 동강성은 주파수의 함수로 모터의 운동방정식으로부터 계산하였다. 선형 및 비선형 플러터 해석 결과들은 공력탄성학적 특성들이 백래쉬와 동강성에 중요한 영향을 받는다는 것을 보여준다. 비선형 플러터 해석에서 다양한 제한 주기 운동이 선형플러터 속도 이하에서 관측되었다. 또한 플러터 특성과 응답을 시간영역에서도 조사하였다. Nonlinear aeroelastic analyses of a missile control fin are performed considering backlash and dynamic stiffness of actuator. Doublet-Hybrid method is used for the calculation of subsonic unsteady aerodynamic forces, and aerodynamic forces are approximated by the minimum-state approximation. For nonlinear flutter analysis backlash is represented by a free-play and is linearized by using the describing function method. Also, dynamic stiffness is function of frequency and is calculated by solving equation of motion for actuator. The linear and nonlinear flutter analyses show that the aeroelastic characteristics are significantly dependent on the backlash and dynamic stiffness. From the nonlinear flutter analysis, various types of limit cycle oscillations are observed in a range of air speeds below the linear divergent flutter boundary. The nonlinear flutter characteristics and the nonlinear aeroelastic responses are also investigated in the time domain.

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.

Nonlinear Flutter Characteristics of Composite Missile Wing in Transonic/Low-Supersonic Flows

This work was supported through funding from the Korea Ministry of Science and Technology. The authors acknowledge the support (Subject 2000-N-NL-01-C-250) of the National Research Laboratory Program.

Finite element time domain - Modal formulation for nonlinear flutter of composite panels

A finite element time domain modal approach is presented for determining the nonlinear flutter characteristics of composite panels at elevated temperatures. The von Karman large-deflection strain-displacement relations, quasisteady first-order piston theory aerodynamics, and quasisteady thermal stress theory are used to formulate the nonlinear panel flutter finite element equations of motion in nodal displacements. A set of nonlinear modal equations of motion of much smaller degrees of freedom for the facilitation in time numerical integration is then obtained through a modal transformation and reduction. All five types of panel behavior-flat, buckled, limit-cycle, periodic, and chaotic motions-can be determined

Vector unsymmetric eigenequation solver for nonlinear flutter analysis on high-performance computers

An efficient, vector-version generalized unsymmetric eigenequation solver is developed for large-amplitude vibration and nonlinear panel flutter analyses by finite-element methods on high-performance computers. The proposed eigensolver is designed to extract extreme eigenpairs which are all that are required in solving panel flutter problems. A loop-unrolling technique is employed to enhance the vector speed of the eigensolver. A finiteelement approach is presented for determining the nonlinear flutter characteristics of isotropic and composite panels using unsteady, third-order piston theory aerodynamics. Both nonlinear structural (large-amplitude, and nonlinear aerodynamics terms are considered in the finite-element formulation. Solution procedures are presented to solve the nonlinear panel flutter and large amplitude-free vibration finite-element equations. Nonlinear flutter analyses are performed for different dynamic pressures A; and maximum deflection-to-thickness ratios clh. Large-amplitude vibration finite-element results are compared with classical analytical solutions. Nonlinear aerodynamic and linear structural finite-element flutter results for composite panels are also presented.