# Study on the Propeller Rope Cutter Concerning Transient Torsional Vibration Due to Cutting Action
## 3 Permissible Torsional Stresses Determination  
### Methodology  
In this section, we will find out the critical bolt diameter for a rope cutter installed on the same propulsion system as Section 2, with the following specifications:  
- Intermedia shaft diameter, \(d=120\) mm,  
- Tensile strength of intermediate shaft, \(\sigma_{b}=600\) MPa,  
- Tensile strength of connecting bolt, \(\sigma_{ult}=800\) MPa. More details of intermediate shaft and connecting bolt materials are described in Table 1.  
- Shear strength of bolt, \(\tau_{ult}=0.75\times\sigma_{ult}=600\) MPa,  
- Distance from center of bolt to center of moment, \(D_{1}=20\) mm, see Figure 2,  
- Vertical distance from rope cutting point to shear section of bolt, \(D_{2}=20\) mm,  
- Distance from rope cutting point to center of propeller shaft, \(D_{3}=250\) mm.  
Figure 1: Relationship between the torsional vibration stress amplitude, the number of reversals to failure, and the fractional damage.  
During operation, the connecting bolts are subjected to tensile and shear force, as shown in Figure 3. The body part has the cross section equal to 1.161 times the area of the threaded part. Shear failure occurs only in the cutting section. The cutting forces, \(F_{2}=F_{3}\), are generated when trying to cut a rope. Force, \(F_{3}\), creates a torque on the shaft:  
\[T=F_{3}D_{3} \tag{13}\]  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Intermediate Shaft** & **Connecting Bolt** \\ \hline Grade (Name) & EN-GJS-600-10 & 34 CrMo4 \\ EN steel number & 5.3110 & 1.7220 \\ Classification & Spheroidal graphite cast iron & Alloy steel \\ Standard & EN 1563 & EN 10263-4 \\ Tensile strength & 600 MPa & 800 MPa \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of the intermediate shaft and connecting bolt materials.  
Figure 3: Tensile and shear forces applied to bolts.  
Figure 2: Illustration of rope cutter system and cutting forces.  
The additional torsional stress applied on the intermediate shaft is calculated by:  
\[\tau_{IS}=\frac{16T}{\pi d^{3}} \tag{14}\]  
The opposite force, \(F_{2}\), creates tension and shear forces on bolts. The forces producing tension are denoted by \(F_{1t}\) and \(F_{2t}\); the others producing shear are denoted by \(F_{1s}\) and \(F_{2s}\). Applying the equivalent condition, we get:  
\[F_{2}=F_{2t}+F_{2s} \tag{15}\]  
\[F_{1t}=\frac{F_{2t}\cdot D_{2}}{D_{1}} \tag{16}\]  
and  
\[F_{1s}=F_{2s} \tag{17}\]  
Assuming that the forces distribute equally on all \(n\) bolts, the forces applied on each bolt in tension and shear are determined by:  
\[F_{bt}=\frac{F_{1t}}{n} \tag{18}\]  
and  
\[F_{bs}=\frac{F_{1s}}{n} \tag{19}\]  
The tensile and shear stresses created on each bolt are:  
\[\sigma=\frac{F_{bt}}{A} \tag{20}\]  
and  
\[\tau=\frac{F_{bs}}{A} \tag{21}\]  
In this case of combined shear and uniaxial tension loading, to predict the ultimate failure, the maximum tensile and shear stresses are given as follows [37]:  
\[\sigma_{max}=\frac{\sigma}{2}+\sqrt{\left(\frac{\sigma}{2}\right)^{2}+\tau^{2}} \tag{22}\]  
and  
\[\tau_{max}=\sqrt{\left(\frac{\sigma}{2}\right)^{2}+\tau^{2}} \tag{23}\]  
The destruction is defined when the maximum shear stress is \(\tau_{ult}\) or the maximum tensile stress is \(\sigma_{ult}\); the above equations can then be written as the following failure criteria:  
\[\sigma_{ult}=\frac{F_{bt}}{2A}+\sqrt{\left(\frac{F_{bt}}{2A}\right)^{2}+ \left(\frac{F_{bs}}{A}\right)^{2}} \tag{24}\]  
and  
\[\tau_{ult}=\sqrt{\left(\frac{F_{bt}}{2A}\right)^{2}+\left(\frac{F_{bs}}{A} \right)^{2}} \tag{25}\]  
Then the criteria cross section area of the connecting bolt is obtained by the larger of followings:  
\[A=\frac{\frac{F_{bt}}{2}+\sqrt{\left(\frac{F_{bt}}{2}\right)^{2}+F_{bs}{}^{2 }}}{\sigma_{ult}} \tag{26}\]or  
\[A=\frac{\sqrt{\left(\frac{F_{\text{th}}}{2}\right)^{2}+{F_{bs}}^{2}}}{\tau_{ult}} \tag{27}\]  
The criteria diameter is determined corresponding to specified failure conditions as shown in Figure 4.