PLANAR VIBRATIONS OF A THIN ELASTIC ROD WITH CIRCULAR CROSS SECTION
LIU Yanzhu (Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, China)
Based on the Kirchhoff 's theory the planar vibrations of a thin elastic rod with circular cross section are studied. The dynamical equations of the rod are established in the Frenet coordinates of the centerline as the reference frame. In the case of planar motion the torsional vibration is decoupled from the flexural vibration. The torsional vibration of a rod with arbitrary planar shape and the flexural vibration of a straight rod without torsion under axial compression are discussed. It is proved that conditions of Lyapunov's and Euler's stability of equilibrium of a straight rod in static analysis are necessary conditions of its dynamic stability. The influence of the axial force and the inertial effect of the cross section on the natural frequency of flexural vibration is considered.
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