Research Of Stability And Bifurcation On The Airfoil With Structural Nonlinearity

The flutter of an airfoil is a phenomenon resulting from the interaction of structural,inertial,and aerodynamic forces.As one of the typical self-excited systems,the airfoil can generate sustained stable or unstable vibrations by extracting energy from the air,which are called benign flutter or disaster flutter respectively.Benign flutter can be used reasonably to change our life.However,disaster flutter will cause great harm.Therefore,it is important to predict the aeroelastic characteristics.There are many nonlinear factors that can cause the flutter of the airfoil,such as structural nonlinearity,aerodynamic nonlinearity,damping nonlinearity and so on.These nonlinear factors permit aeroelastic system to avoid the direct aerodynamic instabilities and present different nonlinear dynamical behaviors,such as Hopf bifurcation,limit cycle and chaos.Different bifurcation characteristics will lead to benign limit cycle or malignant limit cycle.Therefore,it is important to analysis the bifurcation characteristic of the airfoil under the effects of various nonlinear factors.The master's thesis studies the following several aspects in smooth and non-smooth airfoil system.In the first part,the qualitative dynamical behaviors in two-degree-of-freedom smooth airfoil system are studied base on center manifold theory.First of all,the existence of Hopf bifurcation in airfoil system is analyzed by using the explicit critical criteria of Hopf bifurcation of continuous time dynamical system.The general term formulas of the first Lyapunov coefficient are derived to provide the basis of determining the stability of the Hopf bifurcation.Secondly,the bifurcation area of double parameter is obtained by analysis of the existence of degenerate co-dimensions two Hopf bifurcation.Then the general term formulas of the second Lyapunov coefficient are derived to analyze the stability of the co-dimension two Hopf bifurcation by combining the center manifold reduction principle and homogeneoustransformation.Finally,the thrid Lyapunov coefficient is derived to determine the stability of the co-dimension three Hopf bifurcation.In the second part,the quantitative dynamical behaviors in two-degree-of-freedom smooth airfoil system are studied by using the method of harmonic balance.First of all,the limit cycle amplitude and frequency are obtained by the first harmonic balance.Then the transfer matrix of the system is obtained to analysis the stability of periodic solutions of the system by using Floquet theory.Finally,for comparison,the limit cycle amplitude with the change of parameters obtained from the method of harmonic balance agrees well with that from direct numerical simulation,which further verifies the correctness of theoretical derivation.In the third part,the qualitative dynamical behaviors in two-degree-of-freedom non-smooth airfoil system are studied base on generalized center manifold theory.The generalized center manifold theory is different from the center manifold reduction theory of smooth high-dimensional system.It depends on the geometric properties of trajectory,rather than a smooth analytic properties.A space region that contains transversal periodic trajectory is obtained.An approximate invariant cone manifold in piecewise linear system is obtained on the basis of the existence region of transversal periodic solutions and the mapping fixed point derived.A generalized non-smooth invariant cone is obtained by analysis of the piecewise nonlinear system under the nonlinear disturbance.In the fourth part,the quantitative dynamical behaviors in three-degree-of-freedom non-smooth airfoil system are studied by using the average method.First of all,aiming at the freeplay nonlinearity on control surface,the relationship between the limit cycle amplitude and frequency in non-smooth system is obtained by using the first-order average method.Then the transfer matrix of the system is obtained by using Floquet theory.Finally,the dynamic response of the system is analyzed under the effects of freeplay on control surface.