We studied the problem of bifurcation and chaos in a 4-side fixed rectangular thin plate in electromagnetic and me-chanical fields.Based on the basic nonlinear electro-magneto-elastic motion equations for a rectangula...We studied the problem of bifurcation and chaos in a 4-side fixed rectangular thin plate in electromagnetic and me-chanical fields.Based on the basic nonlinear electro-magneto-elastic motion equations for a rectangular thin plate and the ex-pressions of electromagnetic forces,the vibration equations are derived for the mechanical loading in a steady transverse magnetic field.Using the Melnikov function method,the criteria are obtained for chaos motion to exist as demonstrated by the Smale horseshoe mapping.The vibration equations are solved numerically by a fourth-order Runge-Kutta method.Its bifurcation dia-gram,Lyapunov exponent diagram,displacement wave diagram,phase diagram and Poincare section diagram are obtained.展开更多
The chaotic motion behavior of the rectangular conductive thin plate that is simply supported on four sides by airflow andmechanical external excitation in a magnetic field is studied.According to Kirchhoff’s thin pl...The chaotic motion behavior of the rectangular conductive thin plate that is simply supported on four sides by airflow andmechanical external excitation in a magnetic field is studied.According to Kirchhoff’s thin plate theory,considering geometric nonlinearity and using the principle of virtualwork,the nonlinearmotion partial differential equation of the rectangular conductive thin plate is deduced.Using the separate variable method and Galerkin’s method,the system motion partial differential equation is converted into the general equation of the Duffing equation;the Hamilton system is introduced,and the Melnikov function is used to analyze the Hamilton system,and obtain the critical surface for the existence of chaos.The bifurcation diagram,phase portrait,time history response and Poincarémap of the vibration system are obtained by numerical simulation,and the correctness is demonstrated.The results showthatwhen the ratio of external excitation amplitude to damping coefficient is higher than the critical surface,the system will enter chaotic state.The chaotic motion of the rectangular conductive thin plate is affected by different magnetic field distributions and airflow.展开更多
An analytical method for predicting chaos in perturbed planar non Hamiltonian integrable systems with slowly varying parameters was developed. Based on the analysis of the geometric structure of unperturbed systems, ...An analytical method for predicting chaos in perturbed planar non Hamiltonian integrable systems with slowly varying parameters was developed. Based on the analysis of the geometric structure of unperturbed systems, the condition of transversely homoclinic intersection was given. The generalized Melnikov function of the perturbed system was found by applying the theorem on the differentiability of ordinary differential equation solutions with respect to parameters.展开更多
将无扰闭轨道变量变换到作用-角变量,再将微扰变量在无扰闭轨道附近展开,获得了有微扰的作用-角变量一级近似表达式。以无扰闭轨道的周期为采样时间,用作用-角变量表达式建立了二维多频驱动的Poincar'e映射,由其中的作用变量映射定...将无扰闭轨道变量变换到作用-角变量,再将微扰变量在无扰闭轨道附近展开,获得了有微扰的作用-角变量一级近似表达式。以无扰闭轨道的周期为采样时间,用作用-角变量表达式建立了二维多频驱动的Poincar'e映射,由其中的作用变量映射定义了多频驱动的次谐Melnikov函数,并用该函数,给出了Hopf分岔条件。并应用到多频驱动的Duffing-Van der pol系统中,导出了该系统的Hopf分岔条件。按分岔条件取参数,对三频驱动的Duffing-Van der pol方程进行了数值模拟,无一例外地均出现了Hopf分岔。展开更多
基金Project(No. A2006000190)supported by the Natural Science Foundation of Hebei Province,China
文摘We studied the problem of bifurcation and chaos in a 4-side fixed rectangular thin plate in electromagnetic and me-chanical fields.Based on the basic nonlinear electro-magneto-elastic motion equations for a rectangular thin plate and the ex-pressions of electromagnetic forces,the vibration equations are derived for the mechanical loading in a steady transverse magnetic field.Using the Melnikov function method,the criteria are obtained for chaos motion to exist as demonstrated by the Smale horseshoe mapping.The vibration equations are solved numerically by a fourth-order Runge-Kutta method.Its bifurcation dia-gram,Lyapunov exponent diagram,displacement wave diagram,phase diagram and Poincare section diagram are obtained.
基金funded by the Anhui Provincial Natural Science Foundation(Grant No.2008085QE245)the Natural Science Research Project of Higher Education Institutions in Anhui Province(2022AH040045)the Project of Science and Technology Plan of Department of Housing and Urban-Rural Development of Anhui Province(2021-YF22).
文摘The chaotic motion behavior of the rectangular conductive thin plate that is simply supported on four sides by airflow andmechanical external excitation in a magnetic field is studied.According to Kirchhoff’s thin plate theory,considering geometric nonlinearity and using the principle of virtualwork,the nonlinearmotion partial differential equation of the rectangular conductive thin plate is deduced.Using the separate variable method and Galerkin’s method,the system motion partial differential equation is converted into the general equation of the Duffing equation;the Hamilton system is introduced,and the Melnikov function is used to analyze the Hamilton system,and obtain the critical surface for the existence of chaos.The bifurcation diagram,phase portrait,time history response and Poincarémap of the vibration system are obtained by numerical simulation,and the correctness is demonstrated.The results showthatwhen the ratio of external excitation amplitude to damping coefficient is higher than the critical surface,the system will enter chaotic state.The chaotic motion of the rectangular conductive thin plate is affected by different magnetic field distributions and airflow.
文摘An analytical method for predicting chaos in perturbed planar non Hamiltonian integrable systems with slowly varying parameters was developed. Based on the analysis of the geometric structure of unperturbed systems, the condition of transversely homoclinic intersection was given. The generalized Melnikov function of the perturbed system was found by applying the theorem on the differentiability of ordinary differential equation solutions with respect to parameters.
文摘将无扰闭轨道变量变换到作用-角变量,再将微扰变量在无扰闭轨道附近展开,获得了有微扰的作用-角变量一级近似表达式。以无扰闭轨道的周期为采样时间,用作用-角变量表达式建立了二维多频驱动的Poincar'e映射,由其中的作用变量映射定义了多频驱动的次谐Melnikov函数,并用该函数,给出了Hopf分岔条件。并应用到多频驱动的Duffing-Van der pol系统中,导出了该系统的Hopf分岔条件。按分岔条件取参数,对三频驱动的Duffing-Van der pol方程进行了数值模拟,无一例外地均出现了Hopf分岔。
