The stability and vibration of a thin elastic helical rod with circular cross section in a viscous medium are discussed. The dynamical equations of the rod in the viscous medium are established in the Frenet coordinat...The stability and vibration of a thin elastic helical rod with circular cross section in a viscous medium are discussed. The dynamical equations of the rod in the viscous medium are established in the Frenet coordinates of the centreline with the Euler angles describing the attitudes of the cross section as variables. We have proved that the Lyapunov and Euler conditions of stability of a helical rod in the space domain are the necessary conditions for the asymptotic stability of the rod in the time domain. The free frequencies and damping coefficients of torsional and flexural vibrations of the helical rod in the viscous medium are calculated.展开更多
The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler＇s ...The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler＇s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov＇s and Euler＇s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form.展开更多
For designing the thick equal-cross-section ultrasonic flexural vibration rod, the Timoshenko Theory works satisfactorily. For the vari-cross-section ultrasonic flexural vibration rod, however, there has not been an a...For designing the thick equal-cross-section ultrasonic flexural vibration rod, the Timoshenko Theory works satisfactorily. For the vari-cross-section ultrasonic flexural vibration rod, however, there has not been an accurate and convenient design theory to date. That is, no frequency equation is available. In the present study, we extend the Timoshenko Theory to the vari-cross-section rod. For equal-thickness vari-cross-section exponent-mode rods, the frequency equation was derived. Experiments were carried out to verify the equation. The theoretical calculation shows good agreement with the experi-展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No 10472067）.
文摘The stability and vibration of a thin elastic helical rod with circular cross section in a viscous medium are discussed. The dynamical equations of the rod in the viscous medium are established in the Frenet coordinates of the centreline with the Euler angles describing the attitudes of the cross section as variables. We have proved that the Lyapunov and Euler conditions of stability of a helical rod in the space domain are the necessary conditions for the asymptotic stability of the rod in the time domain. The free frequencies and damping coefficients of torsional and flexural vibrations of the helical rod in the viscous medium are calculated.
基金the National Natural Science Foundation of China (10472067)
文摘The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler＇s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov＇s and Euler＇s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form.
基金Project supported by the National Natural Science Foundation of China.
文摘For designing the thick equal-cross-section ultrasonic flexural vibration rod, the Timoshenko Theory works satisfactorily. For the vari-cross-section ultrasonic flexural vibration rod, however, there has not been an accurate and convenient design theory to date. That is, no frequency equation is available. In the present study, we extend the Timoshenko Theory to the vari-cross-section rod. For equal-thickness vari-cross-section exponent-mode rods, the frequency equation was derived. Experiments were carried out to verify the equation. The theoretical calculation shows good agreement with the experi-
