Abstract: Lie groups and Lie algebras are introduced to represent a beam with spatial compliance, that is the beam under the forces of bending, twisting and extending. Based on the material theory and fundamental kinematic assumptions, the general spatial compliance matrix of the beam can be obtained by congregating. Based on the analysis of potential energy, the potential energy density and compliance density of the beam are obtained. The frequencies of such a beam are analyzed using Rayleigh-Ritz method, and its eigenfrequencies are compared with the beam under the condition of each individual deformation case, pure bending, extension or torsion. The theory of Lie groups and Lie algebras is a facilitated mathematical tool for the representation and analysis of the mechanism with spatial compliance links is convinced.

Key words: Eigen frequency, Elastic beam, Lie algebras, Lie groups, Spatial compliance