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Chin. Phys. B, 2009, Vol. 18(1): 01    DOI: 10.1088/1674-1056/18/1/001
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On dynamics of elastic rod based on exact Cosserat model

Liu Yan-Zhu
Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, China
Abstract  The analysis of kinematics and dynamics of an elastic rod with circular cross section is studied on the basis of exact Cosserat model under consideration of the tension and shear deformation of the rod. The dynamical equations of a rod with arbitrary initial shape are established in general form. The dynamics of a straight rod under axial tension and torsion is discussed as an example. In discussion of static stability in the space domain the Greenhill criteria of stability and the Euler load are corrected by the influence of tension and shear strain. In analysis of dynamical stability in the time domain it is shown that the Lyapunov and Euler stability conditions of the rod in space domain are the necessary conditions of Lyapunov's stability in the time domain. The longitudinal, torsional and lateral vibrations of a straight rod based on exact model are discussed, and an exact formula of free frequency of lateral vibration is obtained. The free frequency formulas of various simplified models, such as the Rayleigh beam, the Kirchhoff rod, and the Timoshenko beam, can be seen as special cases of the exact formula under different conditions of simplification.
Keywords:  dynamics of elastic rod      exact Cosserat model      lateral vibration of rod  
Received:  24 March 2008      Revised:  14 August 2008      Published:  20 January 2009
PACS:  46.70.Hg (Membranes, rods, and strings)  
  46.25.-y (Static elasticity)  
  46.40.Ff (Resonance, damping, and dynamic stability)  
  89.20.Kk (Engineering)  
Fund: Project supported by the National Natural Science Foundation of China (Grant No 10472067).

Cite this article: 

Liu Yan-Zhu On dynamics of elastic rod based on exact Cosserat model 2009 Chin. Phys. B 18 01

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