Assessing the effects that nonstationary dynamic processes have on the durability of structural elements belongs to an important trend in modern dynamics and technical diagnostics of machines. Normally, fatigue strength calculations are performed taking into account only periodically variable stresses, as steady operating modes of machines are much longer in comparison with transient modes. However, a significant role in fatigue failure in machines and engineering structures is also played by nonstationary loads. This is explained by emerging intensive oscillations in the mechanical system during accelerating, braking, or changing the operation mode of a machine unit, which often lead to the accumulation of fatigue damages in the materials of parts in heavy loaded assemblies. The combination of stationary and nonstationary dynamic loads manifests itself, particularly in drilling rigs, where technological cycles include steady motion modes, starts, and stops. This paper represents a generalized mathematical model describing nonstationary processes in the lift system of a drill rig, which considers the relationship between electromagnetic processes in asynchronous motors and mechanical oscillatory phenomena, with the purpose of determining dynamic loads and stresses in structural elements of the rigs. Nonlinear physical systems include mechanical members with both concentrated and clearly expressed distributed parameters. The durability of structural elements is evaluated by means of a computer algorithm for analysis of crack growth rates using the NASGRO equation obtained with the presence of plastic deformation zones. An example of the crown block axis illustrates the influence of nonstationary dynamic processes in drill rigs on the durability of structural elements.