## 6 Conclusions  
This study systematically presents the dynamic modelling and corresponding numerical solution methods based on the substructure model for the complex vibration problem of torsional, bending, and longitudinal coupling in the diesel engine crankshaft system. It also explores the impact of various modelling factors on the vibration of the diesel engine crankshaft system. The research provides valuable support and references for the diesel engine crankshaft system's design, analysis, and application. The main conclusions drawn from the study are as follows:  
1. The crankshaft substructure model was effectively established by setting predefined thresholds to construct coupling constraints between the main and auxiliary nodes and performing degree-of-freedom reduction. This method significantly reduces the computational degrees of freedom while retaining the primary dynamic characteristics of the system.
2. The Cholesky decomposition and Jacobi iterative methods were found to accurately and efficiently solve the free vibration of the crankshaft system, demonstrating that the substructure model can effectively replace the finite element model. Additionally, increasing the number of main nodes helps improve model accuracy.
3. Under the framework of the implicit Newmark-\(\beta\) method, the combination of a prediction-correction mechanism and dynamic relaxation factors to adjust the iterative convergence path, along with an adaptive time step strategy driven by energy gradients, can effectively balance numerical stability and the capture of high-frequency response details.
4. In dynamic response calculations, appropriately increasing the number of main nodes, reducing the time step size, and narrowing the coupling region enhance the description of high-frequency local vibration details. Increasing the coupling region can more accurately reflect the dynamic properties of the structure.
5. Using a smaller initial step size combined with the energy gradient criterion for adaptive step size can optimize computational accuracy and efficiency. An increase in support stiffness will cause an upward shift in the local response results.  
With the development of the substructure dynamic response calculation programme, it becomes feasible to conduct many simulations through various case studies. This enables a deeper investigation into the characteristics of coupled vibrations and the interaction mechanisms among torsional, bending, and axial vibration modes. By identifying underly  
Figure 14: Calculation results of angle difference between two sides of the crankshaft with different bearing support stiffnesses.  
ing patterns and dynamic behaviour, the method provides valuable technical support for fundamental diesel engine design and shaft system development.  
Conceptualization, Z.T. and Z.S.; methodology, Z.T.; software, Z.S.; validation, Z.S. and Y.Z. ([PERSON]); formal analysis, Y.Z. ([PERSON]); investigation, Z.T.; resources, Z.T.; data curation, Z.S.; writing--original draft preparation, Z.S.; writing--review and editing, Z.T.; visualization, Z.S.; supervision, Z.T.; project administration, Z.T.; funding acquisition, Z.T. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program of China (14 th Five-Year Plan), grant number 2023 YFD2400800.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.