Masticatory robots have a broad application prospect in the field of denture material tests and mandible rehabilitation. Mechanism type of temporomandibular joint structure is an important factor influencing the performance of the masticatory robot. In view of the wide application of elastic components in the field of the biomimetic robot, an elastic component was adopted to simulate the buffering characteristics of the temporomandibular joint disc and formed the elastic temporomandibular joint structure on the basis of point-contact high pair. Secondly, the influences of the elastic temporomandibular joint structure (on mechanism degree, kinematics, dynamics, etc.) were discussed. The position and velocity of the temporomandibular joint were analyzed based on geometric constraints of the joint surface, and the dynamic analysis based on the Lagrange equation was carried out. Finally, the influence of the preload and stiffness of the elastic component was analyzed by the response surface method. The results showed that the elastic temporomandibular joint structure could effectively guarantee the flexible movement and stable force of the joint. The elastic joint structure proposed in this paper further improves the biomimetic behavior of masticatory robots. It provides new ideas for the biomimetic design of viscoelastic joint discs.