Bifurcation And Chaos Of Nonlinear Flutter Or Chatter System

Nonlinear flutter or chatter often occurs in large engineering equipments. It not only leads to the equipments or their accessory breakage but also results in catastrophic accident. The paper investigates the nonlinear dynamics character of flutter or chatter system and analyses detailed the bifurcation and chaos in two-dimensional airfoil and milling machine separately. The main innovative contributions achieved are as follows:(1) The two degree-of-freedom (2-DOF) airfoil system with freeplay nonlinearity in pitch is investigated numerically. The effect of parameters of the freeplay nonlinearity on the system responses is obtained. The two probability density function (PDF) including Marginal PDF,Bi-dimension PDF and random bifurcation are all used in investigation of the random system. The results show the two PDFs have different shapes in low level turbulence at pre- and post-flutter speeds, but they keep similar shape in high level turbulence. The random bifurcation analysis indicates that the P-bifurcation can happen at both pre- and post-flutter speeds but the D-bifurcation never occurs.(2) Investigation on the control of the airfoil system excited by the random turbulence is important and critical for designing of the airfoil. The feedback control strategy of the airfoil system is studied in the stochastic domain instead of in the time domain. The differential equation of the two-degree-of-freedom (2-DOF) airfoil excited by random turbulence is found, and then the first or second order moment equation are derived from the differential equation in the stochastic domain. The optimal control theory applies to the second order moment equation. Especially, the numerically simulation shows that the mean airspeed in unstable regime, the optimal control input feedback on the airfoil system can make the system convergent to zero in short time. The results confirm that the optimal control can suppress random flutter of the airfoil system effectively.(3) The dynamics character of a two degree-of-freedom aeroelastic airfoil with freeplay and cubic stiffness nonlinearities combined in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincarémapping method and Floquet theory are adopted to analyze 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 Poincarésection also approves previous conclusion.(4) Chatter in machine leads to poor surface finish, promotes wear of the tool and hamper productivity. The shifted Chebyshev polynomials and Floquet theory are adopted for the prediction chatter stability and bifurcation in milling. The stability lobes diagram is obtained. The stability in milling can well be predicted by the lobes diagram. The muliti-periodic and Hopf bifurcations are detected by the Eigen-values analysis. The result shows that the stability solution of the system transform from the stable equilibrium point to the limit cycle oscillatory after multiple cycle bifurcation, and it transforms to the quasi-periodic oscillation after Hopf bifurcation. The numerical result of the Poincarésection proves that the occurrence of the quasi-periodic oscillation. The paper presents more accurately calculation methods for analysis of milling stability, which offers new methods to solve the critical questions for the design of 5-axis machine.