Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling w...Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.展开更多
The problem of nonlinear aerothermoelasticity of a two-dimension thin plate in supersonic airflow is examined. The strain-displacement relation of the von Karman＇s large deflection theory is employed to describe the ...The problem of nonlinear aerothermoelasticity of a two-dimension thin plate in supersonic airflow is examined. The strain-displacement relation of the von Karman＇s large deflection theory is employed to describe the geometric non-linearity and the aerodynamic piston theory is employed to account for the effects of the aerodynamic force. A new method, the differential quadrature method （DQM）, is used to obtain the discrete form of the motion equations. Then the Runge-Kutta numerical method is applied to solve the nonlinear equations and the nonlinear response of the plate is obtained numerically. The results indicate that due to the aerodynamic heating, the plate stability is degenerated, and in a specific region of system parameters the chaos motion occurs, and the route to chaos motion is via doubling-period bifurcations.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10772129)
文摘Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.
基金the National Natural Science Foundation of China (10576024)
文摘The problem of nonlinear aerothermoelasticity of a two-dimension thin plate in supersonic airflow is examined. The strain-displacement relation of the von Karman＇s large deflection theory is employed to describe the geometric non-linearity and the aerodynamic piston theory is employed to account for the effects of the aerodynamic force. A new method, the differential quadrature method （DQM）, is used to obtain the discrete form of the motion equations. Then the Runge-Kutta numerical method is applied to solve the nonlinear equations and the nonlinear response of the plate is obtained numerically. The results indicate that due to the aerodynamic heating, the plate stability is degenerated, and in a specific region of system parameters the chaos motion occurs, and the route to chaos motion is via doubling-period bifurcations.
