This paper presents an analysis of mechanical systems. It considers matrix methods for determining the positions of links and the transformation of the simplest movements of the output links of motors into the movements of the working bodies of the machine, which are carried out by a mechanical system consisting of transmission mechanisms. The object of study is a mechanical system including an executive organ, a transmission mechanism and a motor. The technological processes mechanised with the help of this system are used in a wide variety of industries. The practice of engineering calculations demonstrates that in many cases the links of transmission mechanisms are the most pliable, directly transmitting dynamic loads. Consequently, the modelling of transmission mechanisms in mechanical systems, taking into account both kinematic and dynamic characteristics, is a highly relevant area of study. This research presents a set of methods and techniques for determining the kinematic and dynamic characteristics of mechanical systems. It is shown that geometric characteristics significantly affect the dynamics of the mechanical system as a whole. The Lagrangian equation of the second kind is employed in the formulation of the equation of motion of mechanical systems. A crucial phase in the modelling of mechanical systems is the assessment of the stress-strain state of the supporting metallic structure. Autodesk Inventor software was utilised to ascertain the stress-strain state and safety factor of the mechanical system.