Nonlinear Dynamics Of Rotating Blade With Variable Cross-section And Rotating Speed

Blade is an important component of aero-engine,which has an important impact on the overall performance of the engine.During the flight of an aircraft,the engine blades in the high-speed rotating state are subjected to external forces such as aerodynamic force and centrifugal force.The special working conditions make the blades present complex non-linear behavior,which can easily lead to fatigue,fracture and other failure problems of the blades,and even lead to the failure of the aircraft.Therefore,the analysis of non-linear dynamic problems of blades under high-speed rotating conditions has practical application value,and can provide reliable theoretical basis for avoiding dangerous working conditions of engines.In this paper,the nonlinear dynamics of rotating blades with variable cross-section is studied.Taking aero-engine compressor blade as the main research object,the theoretical model of variable cross-section rotating blade is established firstly,then the partial differential equation of the system is obtained by Hamilton principle,and the ordinary differential equation of the system is obtained by Galerkin discretization method.Finally,the influence of the parameters such as aerodynamic,disturbance amplitude of rotating speed and damping on the nonlinear dynamic response of variable cross-section blade is analyzed.The research contents of this paper are mainly divided into the following parts:(1)The blade is simplified as a cantilever plate model with variable cross-section fixed on rigid hub.Considering the effects of variable cross-section,variable speed,centrifugal force,cross-section warping,pre-installation angle and pre-torsion angle,the partial differential equations of the system are obtained according to Reddy's high-order shear deformation theory,von Karman's large deformation theory and Hamilton's principle.After dimensionless treatment of partial differential equation,Galerkin method is used to discretize it into two-degree-of-freedom ordinary differential equation.Compressor blades of different grades have different length-width ratios.The first two natural frequencies corresponding to different aspect ratios of compressor blades are calculated by Ansys.The curves of the first two natural frequencies varying with aspect ratio are plotted.Through frequency analysis,it is found that there may be 1:2,1:3 and 1:4 internal resonance relations in the system.(2)Considering the 1:2 internal resonance of blade,the perturbation analysis is carried out by using multiple scales method,and the average equations in polar and rectangular coordinates are obtained.Based on the average equation in polar coordinate form,the amplitude-frequency response characteristics and force-amplitude response characteristics of rotating blades are studied,and the effects of disturbance amplitude of rotating speed,aerodynamic force and non-linear term coefficient on the amplitude-frequency response characteristics and force-amplitude response characteristics of the system are analyzed.Based on the average equations in polar coordinate form,numerical simulation is carried out with aerodynamic force,disturbance amplitude of rotating speed and damping as control parameters,respectively.The bifurcation and chaotic dynamics of the system are studied by using the bifurcation diagram,maximum Lyapunov exponential diagram,phase diagram,spectrum diagram and waveform diagram.The results show that the periodic motion and chaotic motion alternate when the parameters change.(3)Considering the 1:3 internal resonance of the blade,the perturbation analysis is carried out by using multiple scales method,and the average equations in polar and rectangular coordinates are obtained.The influence of parameters such as disturbance amplitude of rotating speed,aerodynamic force and coefficient of nonlinear term on amplitude-frequency response of the system is studied by numerical simulation of polar coordinate form average equation.Based on the average equation in polar coordinate form,the velocity disturbance amplitude and aerodynamic force are taken as the research objects,and the bifurcation diagram,phase diagram,spectrum diagram and waveform diagram of the system are obtained by numerical simulation.The results show that there are periodic and chaotic motions in the system with the change of parameters.(4)Considering the 1:4 internal resonance of the blade,the perturbation analysis is carried out by using the asymptotic perturbation method,and the average equations in polar and rectangular coordinates are obtained.By means of numerical simulation of polar average equation,the effects of aerodynamic and disturbance amplitude of rotating speed on the amplitude frequency response of the system are studied.Based on the numerical simulation of the average equation in rectangular coordinates,the influence of aerodynamic force on bifurcation and chaotic kinematics of high-speed rotating blades is analyzed with aerodynamic force as control parameter.The bifurcation diagram,phase diagram,waveform diagram and spectrum diagram of the system are obtained.The results show that with the increase of aerodynamic force,the system changes from periodic motion to chaotic motion and then to periodic motion.