**Abstract**

Problem Statement (Relevance): This article analyses the motion of a three-mass system of bridge crane mechanisms. The relevance of this work is in the application of Lagrange’s equations of second kind, which helped describe the system motion. Due to this approach, the authors could study the dynamic behaviour of the crane mechanisms since dynamic loads appear to be the prevailing cause of crane failures. Objectives: The objective of this research is to obtain the equations of motion for the bridge crane mechanisms, which can be used for mathematical and computer modelling of electrical drives for crane applications. Methods Applied: The analysis (of the bridge crane) conducted is based on combining separate mechanisms into a three-mass system in order to derive the dependencies of velocity and acceleration rates from other physical and process parameters. Originality: The mechanisms of a bridge crane are not examined as separate components but as an intergrated system where all the constituents influence each other. To describe the system motion without any simplifications, Lagrange’s equations of second kind were applied. This indicates that the obtained description offers high accuracy. Findings: The article defines the generalized coordinates of the lifting mechanism, the trolley and the bridge. The kinetic energy of the mechanical system was determined, and the generalized forces found that impact each mechanism. Lagrange’s equations of second kind were derived for each generalized coordinate. Equations of motion were derived for each component that prove the interrelated nature of the components. Practical Relevance: The authors indicate other possible applications for these equations for further study of bridge cranes. The derived equations can be used to build accurate mathematical and computer models, which can help learn more about the motion of bridge crane mechanisms.

**Keywords**

Bridge crane, Lagrange’s equation of second kind, generalized coordinate, kinetic energy of the system, generalized force, equation of the system motion.

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