## 3 Transient Response Simulation
### Simulation Result Analysis  
The classic lumped-parameter equivalent system model and the Newmark-\(\beta\) method for step-by-step integration were utilized to conduct a time domain transient simulation of the propulsion shaft system, which includes a shaft-driven generator. The external excitation loads are defined by Equations (9) - (12). From Equation (1), the synthesized amplitude vector for each harmonic of the propulsion shaft system can be obtained. The product of this amplitude vector with the stiffness matrix yields the desired vibration torque. The vibration stress can be calculated using Equation (14). \[F_{t}=\frac{T_{t}}{\frac{\pi}{16}D^{3}\left(1-\left(\frac{d}{D}\right)^{4} \right)} \tag{14}\]  
Figure 3: Propeller damping curve diagram (Prodam damping model). where \(T_{t}\) represents the vibration torque, kNm; \(D\) represents the outer diameter of the shaft system, mm; and \(d\) represents the inner diameter of the shaft system, mm. Through the calculation of vibration torque and vibration stress under full-speed operating conditions for different unit equipment, the torsional vibration stress curves at each cylinder were obtained, as shown in Figure 4. It can be observed from Figure 4 that the torsional vibration stress at the 8 th harmonic cylinder reaches its maximum, and there is a distinct resonance speed point at a propeller shaft system rotational speed of 23.0 r/min. The calculated vibration stresses for the first five cylinders are all below the allowable continuous stress; the vibration stress at the sixth cylinder approaches the allowable continuous stress, but the vibration stresses at the seventh and eighth cylinders exceed the allowable continuous stress without surpassing the allowable instantaneous stress curve. Therefore, during actual navigation, when the rotational speed is near 23 r/min, a strict speed limit should be set, and the response characteristics of the shaft system and the state of cylinders seven and eight should be closely monitored. The propulsion shaft system of this type of low-speed diesel engine is significantly affected by the 8 th and 16 th harmonic excitation torques. The article employs the system matrix method for numerical analysis, and the simulation calculation results for the torsional vibration stress and vibration torque at the intermediate shaft under normal firing conditions, as shown in Figure 5. Analysis of the curves reveals that the maximum torsional vibration stress occurs at the 8 th harmonic of the intermediate shaft, with a resonance point present. It is noteworthy that the vibration stress near the resonance point and its vicinity has exceeded the allowable continuous stress but has not surpassed the allowable instantaneous stress curve; the vibration torque reached \(10^{7}\) Nm, which exceeds the allowable continuous torque. Thus, in actual navigation, it is recommended to strictly limit the rotational speed range to near 23 r/min and to closely monitor the response characteristics of the shaft system near this speed to avoid adverse effects caused by resonance and to ensure the safe and reliable operation of the system. Using the system matrix method for numerical analysis and through MATLAB R2022b programming under normal firing conditions, the vibration stress and vibration torque curves of the propeller shaft were obtained, as shown in Figure 6. Analysis of Figure 6 indicates that at the 8 th harmonic of the propeller shaft, the torsional vibration stress reaches its maximum value, up to 65 MPa, which is far beyond the allowable continuous stress. At this point, the vibration torque reaches \(10^{7}\) Nm, which is far beyond the allowable continuous stress but does not exceed the allowable instantaneous stress curve.