A mechanical system, where the load in the form of material point is suspended on inextensible thread screwed on the rotating cylindrical drum, but the drum is connected to the boom rotating around fixed horizontal axis is considered. Using the Lagrange equation of the second kind, a mathematical model of the motion of the mechanical system is obtained. The system has three degrees of freedom, two of which are cylindrical. The investigation of the system motion is carried out using computer technology. As a result, the dependences of linear and angular coordinates and velocities in time at different values of the output data for two main modes of the system operation, namely – under the conditions of lifting and lowering the load are obtained. Appropriate graphs are constructed, including the trajectories of the cargo motion. The mathematical model takes into account nonlinearities of the system and allows you to find the amount of tension of the hoisting rope at any time. The analysis showed that vertical oscillations occur twice as fast as horizontal ones. The phase portrait of the generalized coordinate (angle of the rope with the vertical axis) is the focus, which is untwisted when lifting due to nonlinearity in the system, and when the load moves down, the focus, which twists and approaches the mathematical pendulum is obtained. The obtained results can be used in modeling of controlled pendulum motions for different mechanical systems. The methodology and program are recommended to the students and graduate students in terms of learning the principles of construction and analysis of complex nonlinear dynamical systems.

The article considers the spatial motion of a mechanical system where a heavy beam of a given mass and dimensions is suspended at one end by a weightless inextensible cable to a trolley, which can move along horizontal guides without resistance. The system has five degrees of freedom. Based on the apparatus of analytical mechanics and Lagrange equations, a mathematical model of the considered mechanical system in the form of a system of five nonlinear differential equations of the second order is obtained. The mathematical model is implemented in the form of a computer program that allows you to determine the coordinates (positions) of the beam at any time, build the trajectory of the center of mass, determine the kinematic characteristics of the movement, calculate the cable tension and determine its extreme value. Based on the numerical experiment, graphs and phase trajectories of these parameters are constructed, including the 3D trajectory of the center of mass of the beam. The system can show quite complex dynamics depending on the initial conditions, as evidenced by the results of numerical calculations. Under certain conditions, chaotic behavior of the system is possible. Having a mathematical model and a calculation program, it is possible to conduct further studies of the system under consideration, revealing the positions of stable and unstable equilibrium, modes of self-oscillations, revealing areas of periodic and chaotic modes, bifurcations, and so on.