Stochastic P-Bifurcation In Multi-stable System And Its Application In High Dimensional Wing Flutter System

Stochastic P-bifurcation is an important topic in the field of stochastic nonlinear dynamics. For high dimensional systems with coexisted equilibriums and limit cycles, few theories and methodologies are available for their P-bifurcation studies.This thesis includes investigations on two levels:(1) the new stochastic P-bifurcations caused by the interaction between high order nonlinear damping and random excitation which exist the generalized van der Pol-Duffing equation,(2) the P-bifurcation analysis method suitable for high dimensional systems is developed and applied to determine the critical flutter velocity of a typical 2-DOF airfoil.This thesis consists of six major aspectslisted below.1) The Stochastic P-bifurcation of multi-stable van der Pol-Duffing equation excited by additive noise is studied. Critical parameter condition for stochastic P-bifurcation and all possible types of probability density function(PDF) are obtained. All theoretical results are verified by Monte Carlo simulations. The peak number on a PDF curve is the same as the number of coexisted attractors of the corresponding deterministic system. The multi-modal response region shaped like triangle(s) inclines to higher damping direction if the noise intensity increases. The inreasion of the noise corrleation time will decrease the influnece of the random excitation.2) The Stochastic P-bifurcation of van der Pol-Duffing equation excited by multiplicative noise is analyzed. The multi-modal response regions are in the shape like strips bounded by several parallel lines. Just like the additive noise, the increasion of its correlation time will decrease the influence of random excitation.3) The Stochastic P-bifurcations of the van der Pol-Duffing system are investigated for the case both additive and multiplicative noises coexisted. It is shown that the multiplicative noise causes the translation of the multi-modal response regions to higher damping level on the plane of the linear damping coeffcient and the intensity of the addtive noise.4) The Stochastic P-bifurcation of two degree-of-freedom airfoil under steady flow is studied respectively for the two cases, random perturbations added on the the airflow forces(additive noise) and the airflow speed(multiplicative noise). Results indicate additive noise may lower the critical velocity of airfoil flutter while the multiplicative noise will increase this value.5) Airfoil flutter in the unsteady air flow is studied. Longitudinal and transverse turbulence effects are taken into consideration by using Dryden model. Effects of the turbulence parameters and the incoming flow velocity on P-bifurcation and critical velocity are disscussed. Results show the turbulence decreases the stochastic flutter velocity, which lets the large amplitude vibration of the airfoil happens in advance.6) The method for Stochastic P-bifurcation is developed to analysis of multi-stable high dimension systems subjected to stochastic excitations. This methodology combines the dimension reduction, stochastic averaging and singularity theory to obtain the critical parametric condition of Stochastic P-bifurcation, which facilitates the parametric study. In addition, this method does not require the construction of the Hamiltonian function, which implies its potential for highly complicated systems.