| High-precision membrane materials are widely used in printing and packaging,biological medicine,information,aerospace and other fields.The membrane products are transmitted at a high speed in a continuously tensioned state during the manufacturing process.The tensioned membranes are prone to intense flutter,wrinkles and drift that seriously affect the preparation precision of membrane products due to the influence of material characteristics,moving speed,tension non-uniformity,air accompanying,external excitation force of the drying and curing process.At the same time,it also restricts the further improvement of printing speed.Therefore,the research of dynamic characteristics of high-precision membrane has important engineering application value for improving overprint precision and production efficiency.In this paper,the printing membrane production equipments Shaanxi Beiren FR400ELS shaftless transmission high-speed gravure printing machine and Shaanxi Beiren B624 web printing machine are taken as examples,the dynamic characteristics of the moving flexible membrane under multiple working conditions are systematically studied.The specific research work is as follows:(1)The vibration characteristics of the variable density membrane subjected to follower force and the nonuniform tension are investigated.The transverse vibration differential equations of a moving membrane with variable density are established by the D'Alembert principle and solved by using the differential quadrature method(DQM).The effects of tension ratio,follower force,aspect ratio,nonuniform tension coefficient and density coe:fficient on the vibration characteristics of the membrane are analyzed.The stable working regions of the membrane and the critical speed of the divergence instability are obtained,and the main factors affecting the vibration characteristics of the membrane are also obtained.(2)The parametric vibration of the variable speed membrane subjected to tangential uniformity follower force and nonuniform tension is analyzed.The vibration differential equation of the membrane with variable speed of periodic disturbance is established by the D'Alembert principle.The differential quadrature method is used to discretize the differential equation,and a second-order periodic coefficient differential equation only containing time variables is obtained.The Floquet theory is used to analyze the influence of system parameters such as follower force,tension ratio,aspect ratio,average speed and nonuniform tension coefficient on the unstable region of the membrane.This research is important to improve the stability of printing membranes on variable speed motion conditions.(3)The nonlinear vibration and stability of a moving membrane with variable density are studied.Based on the D'Alembert principle and von Karman large deflection theory,the vibration equations of the large deflection of the moving membrane are deduced.The partial differential equations are discretized by the Bubnov-Galerkin method,then the ordinary differential equations of the system are obtained,and the ordinary differential equations are solved by elliptic integral method.The influence of dimensionless speed,aspect ratio and density coefficient on the complex frequency of nonlinear vibration of the moving membrane.For the nonlinear forced vibration of a moving membrane with variable density,the ordinary differential equations are numerically solved by 4-th order Runge-Kutta method.The bifurcation diagrams,time histories,phase-plane portraits,Poincare maps and power spectrum are used to analyze the effects of the dimensionless speed,the amplitude of external excitation,aspect ratio and density coefficient on the nonlinear dynamics behavior of the moving membrane.The regions of chaotic motion of the membrane are obtained,and the main factors affecting the stability of the membrane are also obtained.(4)The nonlinear dynamics behavior of a moving membrane with variable speed has been examined.The nonlinear vibration partial differential equations of the moving membrane with variable speed are established.The Galerkin truncation is used to discretize the partial differential equations containing spatial variables and time variables,stress functions and displacement functions,and the ordinary differential equations of the moving membrane with variable speed are obtained.The bifurcation diagrams,phase-plane portraits,time histories,Poincare maps and power spectrum are used to study the influence of different initial conditions,mean speed,speed variation amplitude and aspect ratio on the nonlinear vibration of the moving membrane with variable speed.The periodic motion,quasi-periodic motion and chaotic motion of the system with different parameters are obtained.(5)Considering the material properties of the membrane,the geometric nonlinear vibration of large deflection of the moving orthotropic membrane is studied,and a nonlinear vibration mathematical model of the moving orthotropic membrane is established.The nonlinear dynamics behavior of the system is identified by time histories,phase-plane portraits,Poincare maps,power spectrum and bifurcation diagrams.The effects of speed,aspect ratio and orthotropic coefficient on the laws of nonlinear vibration of the moving membrane are analyzed,and the chaotic motion regions of orthotropic membranes are obtained.In summary,the research results of this paper provide a theoretical basis for the stability of membrane transmission and the structural optimization,design,manufacture and acceleration of the printing press.At the same time,the research results enrich the stability theory of membrane transmission,which is of great significance to improve the overprinting precision and printing quality of membrane products.This achievement can also be extended to the manufacturing and research of flexible electronic products. |
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