Increasing requirements for reducing the negative manifestations of vibrations in technical systems have created a fairly large number of different designs of passive relaxation shock absorbers with expanded functionality. The expansion of these devices’ functionality results from the introduction of mechanical and non-mechanical elements into their structures for controlling damping characteristics. Considering the approach to implementing the target damping characteristic common to all such devices, it should be noted that they all have some structural similarities and, at the same time, have specific differences in their structure. Identifying existing differences is an urgent task when determining specific priorities. The paper presents the research results on the analysis of the structures of existing types of passive relaxation shock absorbers made using modified kinematic graphs, making it possible to identify and classify the existing structural differences of such devices. When analyzing the structures of these mechanisms based on graph models, it is shown that comparison criteria are identified in the form of the number and integrity of cycles of the corresponding graph. Using the elimination method, the degree of a graph vertex is alternately reduced in graph models, and certain vertices are excluded. With parallel control of the integrity of cycles, the accuracy of the analysis results of the structures under consideration can significantly increase. The proposed structure analysis approach can be proposed for other passive mechanisms with expanded functionality.