The developed analytical approximate method for the calculation of small free and forced vibrations of elastic one-dimensional systems with dry friction is considered. The method is based on the reduction of systems to mechanical analogs. Mechanical analogue presented for each type of vibration in endless system of linear oscillators. In this postulated equality of frequencies of i-tone free vibration of each system to the i^th tone of the mechanical analogue vibrations. The model of dry friction, which has been applied to the calculation of the forced vibrations of the system with one degree of freedom in the classic textbook "Vibrations in engineering" by S.P. Tymoshenko. In this model, dry friction is not dependent on the sliding speed of the elements of vibration system. It is assumed that the amount of dry friction is small and that the forms of the natural oscillations do not change registered by friction. To account for the dry friction energy method is applied, which was suggested by S.P. Tymoshenko for the first time to be used to study the forced vibration system with one degree of freedom. This method allows to determine the equivalent viscous damping ratio for each i^th tone-free vibrations. Method is to equate the work of viscous friction operation of dry friction forces during the period of free oscillations for each i^th issue. In the construction method of calculating vibrations the method of reduced parameters (equivalent) was used. The examples of longitudinal, transverse and tension free and forced vibrations of elastic rods, shafts and beams with the dry friction are provides. Transcendental equations for a finding amplitudes and phase shifts of free oscillations of mechanical analogs with viscous resistance were written out. For the forced vibration of mechanical analogues are found private solutions, written out transcendental equations for the amplitudes and phase shifts and consider the equation of the transition process. The results can be used to study the dynamics of pipelines, for example, oil pipelines.