## 4 Time-History Method for Impact Resistance Calculation
### Shock Spectrum Calculation  
In the time-domain impact of the propulsion shafting system, the shock input spectrum is typically represented by a trilinear shock spectrum for velocity, acceleration, and displacement. In the transformation between the frequency-domain and the time-domain, that is, the input of the impact load can be represented by the peak value, action time, and time-domain waveform. According to the Federal German Defense Navy BV043/85 ship specification, the trilinear shock input spectrum can be transformed into a combined triangular wave of impact acceleration time-domain variation curve, as shown in Figure 15.  
The combined triangular wave has the following formulaic relationship:  
\[a_{2}=0.6\times a_{0} \tag{11}\]  
\[V_{2}=\frac{3}{4}\times V_{0} \tag{12}\]  
\[t_{3}=2\times V_{2}/a_{2} \tag{13}\]  
\[(t_{5}-t_{3})=\frac{6\times 1.05\times d_{0}-1.6\times a_{2}t_{3}^{2}}{1.6 \times a_{2}t_{3}} \tag{14}\]In the aforementioned equations, \(t_{2}\), \(t_{3}\), \(t_{4}\), \(t_{5}\), represent various time points, \(s\); \(a_{2}\) denotes the peak value of the positive pulse acceleration, mm/s\({}^{2}\). \(a_{4}\) signifies the peak value of the negative acceleration pulse, mm/s\({}^{2}\). According to the specification requirements, when the mass of the equipment to be calculated exceeds 5 tons, it is necessary to consider the spectrum drop effect in the design spectrum. At this time, the impact load reduction coefficient (\(A_{a0}\), \(A_{v0}\)) is considered, and the impact response value is affected by the reduction coefficient to become \(A_{a0}\), \(A_{v0}\):  
Acceleration \(A_{a0}\):  
\[A_{a0}=(\frac{M_{i}}{M_{0}})^{-0.537} \tag{17}\]  
Velocity \(A_{v0}\):  
\[A_{v0}=(\frac{M_{i}}{M_{0}})^{-0.4} \tag{18}\]  
where \(M_{0}\) is the equipment mass, 5 t; \(M_{i}\) is the mass of the equipment subjected to impact, t.  
It is known from establishing the three-dimensional model of the propulsion shafting system that the total mass of the propulsion shafting system is approximately 31.946 tons. According to the impact environment specified in the specification and combined with the spectrum drop effect, the combined triangular wave time-history curve of the propulsion shafting system is calculated. This method is only for the vertical and lateral directions. The calculated acceleration time-history curve parameters are shown in Table 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Time-History Curve Parameter** & **Vertical** & **Transverse** \\ \hline a\({}_{2}\) (mm/s\({}^{2}\)) & 695,028.4 & 608,149.9 \\ t\({}_{2}\) (s) & 0.0029 & 0.0028 \\ t\({}_{3}\) (s) & 0.0072 & 0.0070 \\ a\({}_{4}\) (mm/s\({}^{2}\  
According to the acceleration time-history curve parameters in Table 6, five time steps are set for the simulation calculation. To see the triangular spectrum, the fifth step time is set to 0.1 s, and the impact of the acceleration time-history curves in the three directions are obtained. In the calculation, to analyze the state of the propulsion shafting system after being subjected to the impact load, the analysis time is selected to be 0.1 s. Due to the action of the system damping, the decay curve of the entire propulsion shafting system within 0.1 s after being subjected to the impact load can be obtained, which allows for a more intuitive analysis of the impact displacement response and stress response of the propulsion shafting system.