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The equation of motion and the fundamental mode of flexural vibration for a rod with different tensile and compressive properties

We present some results of a theoretical investigation on the free flexural vibration of a slender prismatic rod containing the materials for which the elastic moduli in tension and compression may differ. For simplicity the rod considered possesses a longitudinal plane of symmetry, and the initial condition adjustment and support are also symmetric with respect to this plane. The equation of motion that we have derived is nonlinear, no matter how low the amplitude of flexural vibration is. The nonlinearity is based on the fact that the flexural rigidity of the rod depends on the sign of a local curvature of the bent rod treated as an elastic line. We investigated the particular solution that corresponds to the fundamental mode of flexural vibration. In this mode, the rod performs periodic, though not simple harmonic, vibration. The frequency of this vibration is derived as a function of the material and geometric properties of the rod investigated.