The equivalent linearization method （ELM） is modified to investigate the nonlinear flut- ter system of an airfoil with a cubic damping. After obtaining the linearization quantity of the cubic nonlinearity by the ELM...The equivalent linearization method （ELM） is modified to investigate the nonlinear flut- ter system of an airfoil with a cubic damping. After obtaining the linearization quantity of the cubic nonlinearity by the ELM, an equivalent system can be deduced and then investigated by linear flut- ter analysis methods. Different from the routine procedures of the ELM, the frequency rather than the amplitude of limit cycle oscillation （LCO） is chosen as an active increment to produce bifurca- tion charts. Numerical examples show that this modification makes the ELM much more efficient. Meanwhile, the LCOs obtained by the ELM are in good agreement with numerical solutions. The nonlinear damping can delay the occurrence of secondary bifurcation. On the other hand, it has marginal influence on bifurcation characteristics or LCOs.展开更多
The wash-out filter （WF） technique is used to control the flutter of a two dimensional airfoil with cubic non-linearity in incompressible flow. Firstly, Hopf bifurcation theory is used to determine the point at whic...The wash-out filter （WF） technique is used to control the flutter of a two dimensional airfoil with cubic non-linearity in incompressible flow. Firstly, Hopf bifurcation theory is used to determine the point at which the nonlinear controller is introduced. The system is then transformed into Jordan canonical form, based on analysis of linearized eigenvalues of the system. Secondly, for the introduced WF controller, the linear control gain is determined according to Hopf bifurcation condition. The sym- bolic computing program of normal form direct method （NFDM） is also used to obtain the normal form of the controlled system. The non-linear control gain can be determined based on the relation of the type of bifurcation and the parameters of the normal form, to transform sub-critical Hopf bifurcation to be su- per-critical one. Lastly, numerical simulations are used to certify the validity of theoretical analysis, in which the amplitude of flutter or limit cycle of the controlled system is reduced greatly, comparing to the original system.展开更多
Based on modified Leishman-Beddoes （L-B） state space model at low Mach number （lower than 0.3）, the airfoil aeroelastic system is presented in this paper. The main modifications for L-B model include a new dynamic...Based on modified Leishman-Beddoes （L-B） state space model at low Mach number （lower than 0.3）, the airfoil aeroelastic system is presented in this paper. The main modifications for L-B model include a new dynamic stall criterion and revisions of normal force and pitching moment coefficient. The bifurcation diagrams, the limit cycle oscillation （LCO） phase plane plots and the time domain response figures are applied to investigating the stall flutter bifurcation behavior of airfoil aeroelastic systems with symmetry or asymmetry. It is shown that the symmetric periodical oscillation happens after subcritical bifurcation caused by dynamic stall, and the asymmetric periodical oscillation, which is caused by the interaction of dynamic stall and static divergence, only happens in the airfoil aeroelastic system with asymmetry. Validations of the modified L-B model and the airfoil aeroelastic system are presented with the experimental airload data of NACA0012 and OA207 and experimental stall flutter data of NACA0012 respectively. Results demonstrate that the airfoil aeroelastic system presented in this paper is effective and accurate, which can be applied to the investigation of airfoil stall flutter at low Mach number.展开更多
The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attenti...The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincare mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincare section also approves the previous conclusion.展开更多
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 a...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.展开更多
This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） air...This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.展开更多
This study explores the fluid mechanics and force generation capabilities of an inverted heaving airfoil placed close to a moving ground using a URANS solver with the Spalart-Allmaras turbulence model. By varying the ...This study explores the fluid mechanics and force generation capabilities of an inverted heaving airfoil placed close to a moving ground using a URANS solver with the Spalart-Allmaras turbulence model. By varying the mean ground clearance and motion frequency of the airfoil, it was possible to construct a frequency-height diagram of the various forces acting on the airfoil. The ground was found to enhance the downforce and reduce the drag with respect to freestream. The unsteady motion induces hysteresis in the forces’ behaviour. At moderate ground clearance, the hysteresis increases with frequency and the airfoil loses energy to the flow, resulting in a stabilizing motion. By analogy with a pitching motion, the airfoil stalls in close proximity to the ground. At low frequencies, the motion is unstable and could lead to stall flutter. A stall flutter analysis was undertaken. At higher frequencies, inviscid effects overcome the large separation and the motion becomes stable. Forced trailing edge vortex shedding appears at high frequencies. The shedding mechanism seems to be independent of ground proximity. However, the wake is altered at low heights as a result of an interaction between the vortices and the ground.展开更多
研究了在不可压缩流中具有立方非线性俯仰刚度的二元机翼的颤振问题。首先,对系统平衡点进行了特征值分析,得到在Hopf分岔点附近的颤振参数方程。然后,应用规范形直接法的M ap le推导程序,计算得到Hopf分岔的规范形,并数值模拟验证了超...研究了在不可压缩流中具有立方非线性俯仰刚度的二元机翼的颤振问题。首先,对系统平衡点进行了特征值分析,得到在Hopf分岔点附近的颤振参数方程。然后,应用规范形直接法的M ap le推导程序,计算得到Hopf分岔的规范形,并数值模拟验证了超、亚临界类型分岔情形下响应对初值的依赖性。最后,分析了线性和非线性刚度系数对颤振响应拓扑结构的影响。展开更多
基金supported by the National Natural Science Foundation of China (Nos.:11002088,11172333,11272361)
文摘The equivalent linearization method （ELM） is modified to investigate the nonlinear flut- ter system of an airfoil with a cubic damping. After obtaining the linearization quantity of the cubic nonlinearity by the ELM, an equivalent system can be deduced and then investigated by linear flut- ter analysis methods. Different from the routine procedures of the ELM, the frequency rather than the amplitude of limit cycle oscillation （LCO） is chosen as an active increment to produce bifurca- tion charts. Numerical examples show that this modification makes the ELM much more efficient. Meanwhile, the LCOs obtained by the ELM are in good agreement with numerical solutions. The nonlinear damping can delay the occurrence of secondary bifurcation. On the other hand, it has marginal influence on bifurcation characteristics or LCOs.
文摘The wash-out filter （WF） technique is used to control the flutter of a two dimensional airfoil with cubic non-linearity in incompressible flow. Firstly, Hopf bifurcation theory is used to determine the point at which the nonlinear controller is introduced. The system is then transformed into Jordan canonical form, based on analysis of linearized eigenvalues of the system. Secondly, for the introduced WF controller, the linear control gain is determined according to Hopf bifurcation condition. The sym- bolic computing program of normal form direct method （NFDM） is also used to obtain the normal form of the controlled system. The non-linear control gain can be determined based on the relation of the type of bifurcation and the parameters of the normal form, to transform sub-critical Hopf bifurcation to be su- per-critical one. Lastly, numerical simulations are used to certify the validity of theoretical analysis, in which the amplitude of flutter or limit cycle of the controlled system is reduced greatly, comparing to the original system.
基金Aeronautical Science Foundation of China (08A52003)Science and Technology Foundation of Rotorcraft Aeromechanics Laboratory (9140C4001010901)
文摘Based on modified Leishman-Beddoes （L-B） state space model at low Mach number （lower than 0.3）, the airfoil aeroelastic system is presented in this paper. The main modifications for L-B model include a new dynamic stall criterion and revisions of normal force and pitching moment coefficient. The bifurcation diagrams, the limit cycle oscillation （LCO） phase plane plots and the time domain response figures are applied to investigating the stall flutter bifurcation behavior of airfoil aeroelastic systems with symmetry or asymmetry. It is shown that the symmetric periodical oscillation happens after subcritical bifurcation caused by dynamic stall, and the asymmetric periodical oscillation, which is caused by the interaction of dynamic stall and static divergence, only happens in the airfoil aeroelastic system with asymmetry. Validations of the modified L-B model and the airfoil aeroelastic system are presented with the experimental airload data of NACA0012 and OA207 and experimental stall flutter data of NACA0012 respectively. Results demonstrate that the airfoil aeroelastic system presented in this paper is effective and accurate, which can be applied to the investigation of airfoil stall flutter at low Mach number.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10872141)the Research Fund for the Doctoral Program of Higher Education (Grant No. 20060056005)the National Basic Research Program of China (GrantNo. 007CB714000)
文摘The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincare mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincare section also approves the previous conclusion.
基金Project supported by the National Natural Science Foundation of China (No. 10272092) Science Foundation of Southwest Jiaotong University (No.2003B09).
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.11172199)the KeyProgram of Tianjin Natural Science Foundation of China(No.11JCZDJC25400)
文摘This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.
文摘This study explores the fluid mechanics and force generation capabilities of an inverted heaving airfoil placed close to a moving ground using a URANS solver with the Spalart-Allmaras turbulence model. By varying the mean ground clearance and motion frequency of the airfoil, it was possible to construct a frequency-height diagram of the various forces acting on the airfoil. The ground was found to enhance the downforce and reduce the drag with respect to freestream. The unsteady motion induces hysteresis in the forces’ behaviour. At moderate ground clearance, the hysteresis increases with frequency and the airfoil loses energy to the flow, resulting in a stabilizing motion. By analogy with a pitching motion, the airfoil stalls in close proximity to the ground. At low frequencies, the motion is unstable and could lead to stall flutter. A stall flutter analysis was undertaken. At higher frequencies, inviscid effects overcome the large separation and the motion becomes stable. Forced trailing edge vortex shedding appears at high frequencies. The shedding mechanism seems to be independent of ground proximity. However, the wake is altered at low heights as a result of an interaction between the vortices and the ground.
文摘研究了在不可压缩流中具有立方非线性俯仰刚度的二元机翼的颤振问题。首先,对系统平衡点进行了特征值分析,得到在Hopf分岔点附近的颤振参数方程。然后,应用规范形直接法的M ap le推导程序,计算得到Hopf分岔的规范形,并数值模拟验证了超、亚临界类型分岔情形下响应对初值的依赖性。最后,分析了线性和非线性刚度系数对颤振响应拓扑结构的影响。
