The displacement graphs (d-graphs) are the final form of planetary gear trains (PGTs) synthesis. The existing configuration synthesis methods for the d-graphs of PGTs need rotation graphs (r-graphs) as the intermediate step and produce a large amount of isomorphic KCs. This paper proposes a method to obtain d-graphs directly from the geared graph. Meanwhile, a uniform isomorphism identification method suitable for parent graphs, geared graphs, and d-graphs is presented. Finally, a fully automatic method based on parent graphs is tested on synthesizing 1-DOF PGTs with up to nine links. Our synthesis results and those in the literature are comparatively analyzed, and the reasons for contradictory synthesis results are analyzed.

An important step in the structural synthesis of kinematic chains (KCs) or mechanisms is the detection of isomorphic structures. Although many detection methods have been proposed, most of them require complex computations and have poor versatility. In this study, a simple detection method is proposed based on a compound topological invariant (CTI), which comprises the fourth power of adjacency matrix and eigenvalues. Besides two complex 15- and 28-link planar simple-joint KCs (PSKCs), the method is tested on the complete atlas of contracted graphs with up to six independent loops, PSKCs with up to 13 links, 8-link 1-degree-of-freedom (DOF) planar multiple-joint KCs (PMKCs), and 6-link 1-DOF planetary gear trains (PGTs). All the results are in agreement with the reported results in the literature. Our method possesses good versatility and has been verified as being reliable and efficient.

The structural synthesis of planar kinematic chains (KCs) with prismatic pairs (P-pairs) is the basis of innovating mechanisms containing P-pairs. In the literature, only a little research has been carried out to synthesize planar KCs with P-pairs. Moreover, these synthesis methods for KCs with P-pairs involve all possible combinations of edges, resulting in a large number of isomorphic KCs and a low synthesis efficiency. In this study, our previous similarity recognition algorithm is improved and applied to synthesize planar KCs with P-pairs. Only a small number of isomorphic KCs are generated in the synthesis process and the synthesis efficiency is greatly enhanced. Our method is applied to synthesize 9-link 2-DOF, 10-link 1-DOF, and 11-link 2-DOF KCs with one and two P-pairs. Our synthesis results are consistent with those of the existing literature. The present work is helpful to design mechanisms with P-pairs and can be extended to mechanisms with other types of kinematic pairs.

Isomorphism identification is an essential step in mechanism configuration synthesis. Although various detection methods have been proposed, some of them can only effectively identify kinematic chains (KCs) within 10 links or complex programs that are needed to identify multilink KCs. In this study, a new isomorphism identification method is proposed based on the distance concept of graphs and the graph theory definition of isomorphism. In addition to two complex 21- and 28-link planar simple-joint KCs (PSKCs), the proposed algorithm is tested on the complete atlas of 8-link 1-DOF, 9-link 2-DOF, 10-link 1-DOF, 12-link 1-DOF, and 13-link 2-DOF PSKCs. The algorithm is also tested on 6-link 1-DOF and 7-link 1-DOF planetary gear trains (PGTs) to detect isomorphism. All results are in agreement with those of the existing literature. The method is fully automated via a computer program and has been verified to be reliable and efficient.

A plane kinematic chain inversion refers to a plane kinematic chain with one link fixed (assigned as the ground link). In the creative design of mechanisms, it is important to select proper ground links. The structural synthesis of plane kinematic chain inversions is helpful for improving the efficiency of mechanism design. However, the existing structural synthesis methods involve isomorphism detection, which is cumbersome. This paper proposes a simple and efficient structural synthesis method for plane kinematic chain inversions without detecting isomorphism. The fifth power of the adjacency matrix is applied to recognize similar vertices, and non-isomorphic kinematic chain inversions are directly derived according to non-similar vertices. This method is used to automatically synthesize 6-link 1-degree-of-freedom (DOF), 8-link 1-DOF, 8-link 3-DOF, 9-link 2-DOF, 9-link 4-DOF, 10-link 1-DOF, 10-link 3-DOF and 10-link 5-DOF plane kinematic chain inversions. All the synthesis results are consistent with those reported in literature. Our method is also suitable for other kinds of kinematic chains.

The structural design is a prerequisite for the creation of mechanical equipment. This study focuses on the structural synthesis of epicyclic chains (ECs), and proposes a new topological model and synthesis method of ECs to address the existing research problems (i.e. synthesis dependence on multiple graphic transformations and the generation of pseudo-isomorphism). First, the structural characteristics of ECs are analyzed to explore the relation between their axles, gears, and planet carriers, and the corresponding topological model and the adjacency matrix expression of ECs are provided. Then, the topological elements corresponding to the axles, gears, and gear pairs are added sequentially to the topological graph corresponding to the planet carriers, and the constraints are established based on similarity to inhibit the generation of isomorphism. Finally, all the databases of feasible ECs are obtained through gear loop discrimination. All processes are carried out with computer assistance. The synthesis results of the ECs with 1–3 degrees of freedom and up to 10 components are presented. The logic of this paper is to simplify (or optimize) the synthesis process to improve efficiency of structural synthesis, which will break through the threshold of computing power and provide more possibilities for the design of mechanical equipment.