## 5 Conclusions  
(1) Through the analysis of the motion of the input and output shafts, it can be seen that when installing the output shaft with self-aligning bearings on the three-way universal coupling, due to the small angle difference, especially when the deviation angle is very  
Figure 12: The relation curves of \(R_{X^{\prime}}-\varphi\) when the \(\beta\) angle is a different value.  
Figure 13: The relation curves of \(R_{Y^{\prime}}-\varphi\) when the \(\beta\) angle is a different value.  
small, the angle difference is extremely small. Therefore, the transmission of the three-way universal joint at this time can be called a quasi-constant angular velocity transmission.  
(2) Through the analysis of the forces on the output shaft, it is observed that the magnitude of the circumferential forces transmitted by the three link rods is equal. The output shaft is subjected only to axial forces, which exhibit periodic variation. The magnitude of the axial force increases with the increase of parameter \(\beta\), proportional to the square of the angular velocity. When \(\beta\leq 25^{\lx@math@degree}\) is large, the change in axial force is slow.  
(3) Through the analysis of the forces on the input shaft, it is evident that the reaction force varies periodically, increasing with the value of parameter \(\beta\) and proportional to the square of the angular velocity. When \(\beta\geq 25^{\lx@math@degree}\) is large, due to the sharp increase in the reaction force, stringent requirements are placed on the strength and rigidity of the input shaft.  
(4) The optimal range of application angle for a three-pronged universal joint coupling is \(\beta\leq 25^{\lx@math@degree}\). At this point, the system exhibits excellent transmission and mechanical performance. By reducing the angular velocity and adjusting the relevant dimensional parameters, the applicable angle range can be further expanded.  
The analysis methods used in this paper, such as the directional cosine matrix method and the single force pairing method, are effective analytical methods and play an important role in obtaining the correct analysis results. In addition, the consistency of torque and torque was maintained throughout the paper by means of checks.  
Applying the dynamics and kinematics models established in this study, our group has developed a number of three-pronged equal-angular velocity universal couplings, which have different new structures, but the kinematic principles and kinematic characteristics reflect the guiding significance of this research result.  
The new three-pronged sliding universal coupling as a new solution has significant advantages compared with the traditional solution. Especially in the application environment of high speed and heavy load, it reflects unique advantages, and is expected to replace traditional solutions and widely enter the application field.  
The study, kinematics, dynamics analysis has achieved the research objectives. The research results are mainly applied to the three-pronged universal coupling with constant angular velocity transmission. It is of guiding significance for the development of new structures and analytical mechanical properties in the future, and will be committed to systematic research in the future to reflect a wider range of application value.  
Software, L.Z.; Validation, Y.J.; Writing--original draft, D.C.; Funding acquisition, X.X. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
The data presented in this study are available in the article.  
The author declares that there is no conflict of interest.