In this study, the nonlinear equation of motion of the beam on the elastic foundation is obtained via the Newton's second law of motion, and its free vibration nature is investigated. Considering the inextensional condition, the planar model of the beam accounting for the effects of the rotary inertia is derived. Then, the linear vibration and nonlinear vibration of the beam on the elastic foundation are examined. It is shown that the cut-off frequency can be observed in the frequency spectrum of the beam response. The effects of the rotary inertia on the natural frequencies are systematically investigated. Finally, the frequency differences, due to the different foundation models, and the possible modal interaction of the beam are discussed.

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