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The classical method of determination of velocities and accelerations of the closed kinematical chains consists in calculation of derivatives of the position vector in relation to time. In the paper, the methods of kinematical and dynamical analysis of the closed kinematical systems based upon the graph representation are presented. Planetary gears are the exemplary mechanical systems which have been considered. Two types of graphs are assigned to the gears. Both graphs allow for derivation of systems of equations which constitute the kinematical analysis. The results of classical and graph-based methods were compared. The graph based methods allow for further considerations like e.g. synthesis, enumeration of design solutions and even evolutionary design of mechanisms. Dynamical analysis is shown in graphical form, only.

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Graph-based modeling of planetary gears was applied in the presented research work for their synthesis. The tailored Hsu and contour graphs were used. Graphs encode the structure (layout) of a planetary gear and upon them - the systems of equations can be generated. These systems enable determination of rotational velocities of planetary gear elements. The method is algorithmic and simple. It allows for an easy comparison of different design solutions and selection of the most appropriate one. The list of the possible modes of operation for an exemplary planetary gear is also provided in the paper.

In the paper, graph-based models of planetary gears are presented. Especially, compound planetary gear boxes are analyzed. The rules of assignment of particular graphs are shortly described. Based upon these models, ratios for consecutive drives are calculated. Graph transformations are introduced aiming for presentation of every single drive separately. Kinematic equations are generated in algorithmic way based on the adequate subgraphs as well as their codes. Compatibility of results - obtained by means of different methods - was achieved. The presented method is simple and effective, moreover it can be used for other design tasks as e.g. optimization and further analyzes i.e. evaluation of effectiveness.