Abstract:

The theory of equilibrium and stability of a thin elastic rod under action of external force has a practical background such of sub-ocean cable and oil-dill string. The recent development of this theory is promoted by its application in the molecular biology as macroscopic model of DNA and other biological fibers. According to the Kirchhoffs kinetic analogy of the elastic rod, all the approaches and results of dynamics, including the Lyapunovs stability theory can be applied to discuss the equilibrium and stability of thin elastic rod by changing the time t to arc-coordinate s as independent variable. It is shown that the Lyapunovs stability conditions are always satisfied for the helical rod, including the straight rod and the annular rod as special cases. Nevertheless, from the view point of Eulers stability concept the rod buckles when the axial load reaches a critical value. We discussed the relationship between two different concepts of stability, and proved that the Eulers critical load is a special value of the external force on the ends of the rod when both the Lyapunovs stability condition and the ends geometrical viewpoint are satisfied.

The discussion of stability within the scope of statics is based on the comparison of perturbed and unperturbed elastica of the rod from pure geometrical point of view. In order to analyze the stability problem on more stringent basis, the time factor should be introduced and the Lyapunov stability concepts have to be extended to a dynamical system with arc-coordinate s and time t as two arguments. The dynamical equations of a rod with circular cross section described by the Eulers angles are established in the Frenet coordinates of the centerline. When the conditions of Lyapunov stability are satisfied in statics, the eigenvalues corresponding to the arc-coordinate can be determined by the boundary conditions. Then the dynamic stability of the helical equilibrium depends on the eigenvalues in the time domain. We proved that the Eulers stability condition of the rod in statics is the necessary condition of Lyapunovs stability in dynamics. When the dynamical stability conditions are satisfied the flexural/torsional vibration of the helical rod about its equilibrium state is studied, and its free frequency can be derived in analytical form.

Biography:

Prof. Liu obtained BSc from Tsinghua University in 1959. He worked as visiting scholar in Moscow State University during 1960-62 in Moscow State University before took the academic post in Dept. Engineering Mechanics, Tsinghua University in 1962. He was appointed as Professor of Dept. Engineering Mechanics and Head of Institute of Engineering Mechanics, Shanghai Jiao Tong University since 1973. His main research areas include: Dynamics of rigid body, Gyro dynamics, Multi-body dynamics, Nonlinear dynamics, Dynamics of system containing fluid. He is vice-chief editor of Chinese Quarterly of Mechanics, editorial member of J. of Applied Mechanics, Mechanics in Engineering(China), Technische Mechanik(Germany). He also serves as member of executive committee of academic associations worldwide. He has published 8 books and many papers in international and national referred journals and conferences.