The article deals with the problem of a dynamic model of the truck crane considering the elasticity of the rope and the ground. The dynamic model of a truck crane in the vertical plane has been established. When considering the oscillation of a truck crane, it is a dynamic system with a large number of degrees of freedom. It is crucial to develop a dynamic model for the truck crane. This model will allow us to evaluate the dynamic response of the machine, analyze the vibration of the heavy object being lifted and lowered, and subsequently study stability and control problems. In this paper, the dynamic models of the cranes in the working plane have been established, built by Lagrange's multiplier equations, which are systems of differential-algebraic equations with the generalized coordinates of the machine's motions and the elastic coordinates. The Lagrange equation provides a simple method for solving dynamic problems. The advantage of this equation is that the form and number of equations do not depend on the number of objects in the investigated system, nor do they depend on the way the objects move. The number of Lagrange equations depends only on the number of degrees of freedom of the system.