## 1 Introduction  
The stability and reliability of marine diesel engines, serving as the primary power source for maritime vessels, hold significant importance. During diesel engine operation, the crankshaft system generates cyclic torsional vibrations due to fluctuating torque. Due to the longer length of the crankshaft in marine high-power diesel engines and the larger moment of inertia within the crankshaft system, there is smaller torsional stiffness and lower natural frequency, making it highly susceptible to resonance within its operational speed [1; 2; 3; 4; 5]. Hence, preventive measures must be incorporated into diesel engine design to avoid severe noise and potential crankshaft fractures. The leaf spring torsional vibration damper, a pivotal component of the diesel engine, is positioned at the crankshaft's free end. When appropriately selected, it efficiently absorbs vibration energy, diminishing vibration and noise at the engine's limit speed [6; 7; 8; 9]. The occurrence of torsional vibration in diesel engines prompts the inner and outer rings of the leaf spring torsional vibration damper to exhibit relative movement, resulting in bending and deformation of the damper's leaf spring. Simultaneously, the combined effect of friction from the red copper gasket between the leaf springs and the flow of cooling oil absorbs vibration energy, effectively mitigatingstructural vibrations in the crankshaft system and the hull [10; 11; 12]. For the damper, the torsional stiffness parameter plays a crucial role in calculating its natural frequency, thereby influencing the damper's design and ensuring its compatibility with the engine. Typically, the natural frequency of the damper is selected to be close to a specific vibration frequency. The selected natural frequency of the damper is closely tied to the torsional vibration frequency of the diesel engine shaft system, which, in turn, depends on the speed and structure of the engine. During the operation of the diesel engine, the torsional vibration frequency typically falls within the range of 50-300 Hz. The torsional stiffness of the damper is dictated by the stiffness and force exerted by the leaf spring group. The leaf spring group within the torsional vibration damper is structured as a double-laminated beam, attracting continuous scholarly attention to its mechanical model and calculations. The Newmark model [13], a classical calculation model, simplifies the upper and lower sections of the composite beam as Euler-Bernoulli beams derived from their elastic interaction. This classic model remains extensively utilized in current research. [PERSON] et al. [14] utilized the Hamilton principle to derive partial differential equations, general solutions for deflection and internal forces, and corresponding consistent boundary conditions for the Euler-Bernoulli composite beam structure with interlayer slip subjected to general dynamic loads. Building upon [PERSON]'s high-order beam theory, [PERSON] et al. [15] introduced the implicit kinematic hypothesis for small deformation double-laminated beams. The Lagrange multiplier method was utilized to incorporate this assumption as a constraint within the minimum potential energy variational principle of high-order composite beams, resulting in a nonlinear displacement method for high-order composite beams that can be converted into nonlinear Euler-Bernoulli composite beams. [PERSON] et al. [16] utilized the differential quadrature element method to derive the complete Lagrange formulation for analyzing quadrature elements in composite beams, considering interface slip. This method is grounded in the small-strain Euler-Bernoulli beam theory and incorporates assumptions of large displacement and finite rotation. Furthermore, [PERSON] et al. [17] introduced the direct stiffness method to address partial interaction behavior within double-laminated beams and performed linear static analyses on such beams. The thickness of the damper leaf spring is uniformly varied, which is typical of a variable cross-section leaf spring.