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A theoretical vibration analysis of an elastically connected double-string system is presented. The double-string system is the simplest model of a complex continuous system, which is composed of two one-dimensional elastic solids attached by a Winkler elastic layer. The free and forced transverse vibrations of this system are considered. The present paper develops the free vibration theory, and a companion paper analyzes the forced vibrations. The motion of the system considered is described [1] by a non-homogeneous set of two partial di!erential equations, which are solved by using classical mathematical methods. The solutions of the free vibrations are derived from the Bernoulli}Fourier method. The boundary-value and initial-value problems are solved. The natural frequencies and natural mode shapes of vibration are determined. The free vibrations of an elastically connected double-string system are realized by synchronous and asynchronous de#ections.