## 2 Crankshaft System Substructure Modelling  
The six-cylinder diesel engine crankshaft is selected in this study due to its representative structural complexity and its widespread application in both marine and automotive diesel engines. The six-cylinder configuration provides a structurally balanced layout and exhibits typical vibration characteristics, making it an ideal platform for validating the effectiveness and applicability of the proposed modelling approach. The selection of master nodes and their degrees of freedom in the crankshaft substructure model must accurately represent crankshaft deformation, apply loads and constraints, and capture dynamic responses [25]. Using the crankshaft system of a six-cylinder diesel engine as an example, bearing constraints need to be defined at each main journal, assigning \(m\) master nodes; loads need to be applied at the crankpins, also defining \(n\) master nodes; an output torque need to be applied at the flywheel end, defining one master node; and the free end need to be connected to a damper, with its properties specified, also defining one master node. Typically, these numbers \(m\) and \(n\) are odd. Each master node contains six degrees of freedom, including three translational and three rotational components. A schematic diagram of the main nodes of the crankshaft substructure of the six-cylinder diesel engine is shown in Figure 1.  
To reduce the degrees of freedom, the degrees of freedom of the finite element nodes must be condensed based on the established relationship between master nodes and finite element nodes. This process employs the Craig-Bampton fixed-interface modal synthesis method, combined with the dynamic substructuring approach based on double modal analysis proposed by [PERSON] and [PERSON], to ensure the preservation of the structural physical characteristics during the degree-of-freedom condensation process [7]. Assuming that appropriate node numbering is applied, the stiffness matrix of the substructure, along with the corresponding nodal displacement and load vectors, can be expressed in a partitioned form as follows [26]:  
\[ \begin{bmatrix}K_{bb}&K_{bi}\\ K_{ib}&K_{ii}\end{bmatrix}\begin{pmatrix}a_{b}\\ a_{i}\end{pmatrix} = \begin{pmatrix}P_{b}\\ P_{i}\end{pmatrix}  \tag{1}\]  
where \(a_{b}\) and \(a_{i}\) represent the vectors of the crankshaft system's master and finite element node degrees of freedom, respectively. The degrees of freedom of the internal nodes to be condensed within each substructure can be expressed as follows:  
\[a_{i}=K_{ii}^{-1}(P_{i}-K_{ib}a_{b}) \tag{2}\]  
Thus, the condensed equation is  
\[K_{bb}^{*}a_{b}=P_{b}^{*} \tag{3}\]  
where,  
\[K_{bb}^{*}=K_{bb}-K_{bi}K_{ii}^{-1}K_{ib} \tag{4}\]  
\[P_{b}^{*}=P_{b}-K_{bi}K_{ii}^{-1}P_{i} \tag{5}\]  
Define \(K_{ib}^{*}\) and \(P_{i}^{*}\) as the relevant matrices that convert the substructure interface degrees of freedom to internal degrees of freedom. After eliminating and modifying the corresponding original matrices, the resulting equation is  
\[P_{i}^{*}=K_{ii}^{-1}P_{i} \tag{6}\]  
\[K_{ib}^{*}=K_{ii}^{-1}K_{ib} \tag{7}\]  
The Equation (1) can be transformed into the following form using Gaussian elimination:  
\[ \begin{bmatrix}K_{bb}^{*}&0\\ K_{ib}^{*}&I\end{bmatrix}\begin{pmatrix}a_{b}\\ a_{i}\end{pmatrix} = \begin{pmatrix}P_{b}^{*}\\ P_{i}^{*}\end{pmatrix}  \tag{8}\]  
Figure 1: Schematic diagram of the main nodes of the crankshaft substructure. By the minimum half-bandwidth requirement, nodes within the substructure are numbered consecutively to ensure that the interface nodes and internal nodes of the substructure are not concentrated in the same sequence. Typically, they may be grouped into several segments. Taking \(a_{i}\) and \(a_{b}\), each divided into two segments, the system equation of the substructure is as follows:  
\[