Constructively nonlinear problems with unilateral constraints are frequent in the calculation of various structures. At the same time, certain difficulties cause problems with the contact friction, as well as with the dynamic action of the load. In such cases, the contact problem becomes more complicated in terms of mathematics and its numerical solution becomes more complicated as well. This article is devoted to the construction of calculation models and methods for solving problems with non-ideal unilateral constraints under dynamic loading. As a result, a numerical algorithm has been developed based on the finite element model of contact and the step-by-step analysis method, which allows simultaneous integration of the motion equations and the realization of contact conditions with Coulomb friction. At the same time, to comply with the limitations under the conditions of ultimate friction-sliding, the method of compensating loads is applied. Using the proposed approach, numerical solutions of some problems of contact of a structure with a base have been obtained and analyzed. The reliability of the calculation results is confirmed by comparing them with the solution obtained by the alternative iteration algorithm. It can be concluded that the step-by-step analysis algorithm is more efficient in terms of computation time, showing satisfactory convergence, stability, and accuracy of the solution in a fairly wide range of time integration steps.