The task of determining the vibration stress at the suspension point of the jack pump drill rods, which drives the plunger pump of an oil-well pumping unit, is associated with the need to solve a one-dimensional wave differential equation. The accuracy of determining the magnitude of the vibration stress depends both on the speed of movement of the rods’ suspension point and on the elastic displacements and their speeds of the rod string cross sections during the initial deformation of the column, and at the moment the pump plunger begins to move upward. All this, taken together, forms the initial conditions of the problem. In this regard, the purpose of this article is to determine the vibration stress, taking into account the actual nonlinear speed of the rods suspension point and its replacement by the linear velocity and the successively found values of the elastic displacement velocities of the rod string sections at the moment the pump plunger starts to move upward. In this case, the elastic movement of the lower end of the rod string upwards during their initial deformation is not considered. The elastic displacement of the rod sections at the moment the pump plunger begins to move is assumed to be zero. First, the speed of the rods’ suspension point during their initial deformation was determined using the kinematics of the crank-rocker mechanism and its replacement by the linear speed. After that, the elastic displacement velocities of the rod sections were obtained at the moment the pump plunger began to move upwards as a result of solving additional problems (a round rod, one end of which is pinched, and the other moves at the speeds indicated above; as a result of solving these problems, the elastic displacement velocities of the rod sections are determined). These problems were solved by the Laplace integral transform method. Finally, knowing the rates of elastic displacements of the rod string sections at the initial moment of the pump plunger moving upwards and assuming the elastic displacements of the rods’ sections at this moment to be equal to zero, boundary-value problems were set to determine the elastic displacements of the rod string sections when the pump plunger moved upwards. These problems were solved by the Fourier method. The solutions obtained made it possible to get vibration stress at the point of suspension of the rods. It has been established that taking into account the nonlinearity of the speed of the suspension point of the rods has little effect on the magnitude of the vibration stress. However, the obtained values of vibration stresses are approximate since their determination does not take into account the elastic displacement of the lower end of the rod string during their initial deformation, and the elastic displacements of the rod sections at the initial moment of the plunger movement are taken equal to zero. Therefore, a mathematical formulation of another auxiliary problem has been additionally developed, the solution of which will make it possible to obtain a more accurate value for the vibration stress in the future