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
Thermodynamic hypothesis and kinetic stabil- ity are currently used to understand protein folding. The former assumes that free energy minimum is the exclusive dominant mechanism in most cases, while the latter shows ...Thermodynamic hypothesis and kinetic stabil- ity are currently used to understand protein folding. The former assumes that free energy minimum is the exclusive dominant mechanism in most cases, while the latter shows that some proteins have even lower free energy in inter- mediate states and their native states are kinetically trapped in the higher free energy region. This article explores the stability condition of protein structures on the basis of our study of complex chemical systems. We believe that sep- arating one from another is not reasonable since they should be coupled, and protein structures should be dom- inated by at least two mechanisms resulting in different characteristic states. It is concluded that： （1） Structures of proteins are dynamic, showing multiple characteristic states emerging alternately and each dominated by a respective mechanism. （2） Compromise in competition of multiple dominant mechanisms might be the key to understanding the stability of protein structures. （3） The dynamic process of protein folding should be depicted through the time series of both its energetic and structural changes, which is much meaningful and applicable than the free energy landscape.展开更多
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supported by the National Natural Science Foundation of China(21103195)the Knowledge Innovation Program of Chinese Academy of Sciences(KGCX2-YW-124)
文摘Thermodynamic hypothesis and kinetic stabil- ity are currently used to understand protein folding. The former assumes that free energy minimum is the exclusive dominant mechanism in most cases, while the latter shows that some proteins have even lower free energy in inter- mediate states and their native states are kinetically trapped in the higher free energy region. This article explores the stability condition of protein structures on the basis of our study of complex chemical systems. We believe that sep- arating one from another is not reasonable since they should be coupled, and protein structures should be dom- inated by at least two mechanisms resulting in different characteristic states. It is concluded that： （1） Structures of proteins are dynamic, showing multiple characteristic states emerging alternately and each dominated by a respective mechanism. （2） Compromise in competition of multiple dominant mechanisms might be the key to understanding the stability of protein structures. （3） The dynamic process of protein folding should be depicted through the time series of both its energetic and structural changes, which is much meaningful and applicable than the free energy landscape.
