## 2 Methodology
### True Dynamic Bending Moment Establishment  
Direct reading of the strain gauge results will be presented in the rotating coordinate system. Two of the above methods can only obtain the nominal bending moment. This affects the position of the shaft in the bearing as well as the pressure distribution of the lubrication film. However, the vibration component of the propeller forces leads to a continuously changing pressure distribution in the bearing, which repeats itself for every rotation of shaft. To identify the propeller-induced forces of the vertical and horizontal position for every rotating angle, it is necessary to evaluate the true dynamic bending  
Figure 3: Local bending moment measured at every rotating angle position of strain gauge. moment. Typically, this requires two bridges of strain gauge (S1 and S2) positioned 90 degrees to each other. When the bridge S1 is located at the angle \(\theta\) related to the upper vertical axis, then the bridge S2 is located at the angle of \((\theta-90^{\circ})\). The local bending moments measured by these bridges are denoted by \(M_{s1}\) and \(M_{s2}\), respectively, as described in Figure 4.  
The true dynamic bending moment of shaft, \(M_{b}^{*}\), is obtained by:  
\[\overset{\rightarrow}{M_{b}^{*}}=\overset{\rightarrow}{M_{s1}}+\overset{ \rightarrow}{M_{s2}} \tag{6}\]  
We have:  
\[\overset{\rightarrow}{M_{b}^{*}}=\overset{\rightarrow}{M_{v}}+\overset{ \rightarrow}{M_{h}} \tag{7}\]  
\[\overset{\rightarrow}{M_{s1}}=\overset{\rightarrow}{M_{v1}}+\overset{ \rightarrow}{M_{h1}} \tag{8}\]  
\[\overset{\rightarrow}{M_{s2}}=\overset{\rightarrow}{M_{v2}}+\overset{ \rightarrow}{M_{h2}} \tag{9}\]  
Substituting the Equations (7)-(9) into Equation (6), the vertical and horizontal bending moments are determined as follows:  
\[\overset{\rightarrow}{M_{h}}=\overset{\rightarrow}{M_{h1}}+\overset{ \rightarrow}{M_{h2}} \tag{10}\]  
\[\overset{\rightarrow}{M_{v}}=\overset{\rightarrow}{M_{v1}}+\overset{ \rightarrow}{M_{v2}} \tag{11}\]  
or:  
\[ \begin{array}{rl}M_{h}&=M_{s1}\sin\theta+M_{s2}\sin\bigl{(}\theta-90^{\circ} \bigr{)}\\ &=M_{s1}\sin\theta-M_{s2}\cos\theta\end{array}  \tag{12}\]  
\[ \begin{array}{rl}M_{v}&=M_{s1}\cos\theta+M_{s2}\cos\bigl{(}\theta-90^{\circ} \bigr{)}\\ &=M_{s1}\cos\theta+M_{s2}\sin\theta\end{array}  \tag{13}\]  
The true dynamic bending angle is determined by  
\[\lambda=tan^{-1}\Bigl{(}\frac{M_{h}}{M_{v}}\Bigr{)}\quad\text{if }M_{v}\geq 0 \tag{14}\]  
or:  
\[\lambda=tan^{-1}\Bigl{(}\frac{M_{h}}{M_{v}}\Bigr{)}+180^{\circ}\quad\text{if }M_{v}<0 \tag{15}\]  
Figure 4: True dynamic bending moment measurement using two bridges of strain gauge. In the case that only one bridge of strain gauge, S1, is installed, a new measurement technique is required to obtain the true dynamic bending moment. The propeller has four blades, the true dynamic bending moment includes the nominal bending (static) and the fourth-order bending moment fluctuation (dynamic) due to the propeller force. The nominal bending moment does not change during a shaft rotation. Assuming that the propeller force distribution by all blades are even, the main fluctuation of the fourth-order repeats its period after every \(90^{\lx@math@degree}\). Here is the key point of this research. Consider the state where the bridge S1 was at angle (\(\theta-90^{\lx@math@degree}\)) and the local bending moment was measured as \(M_{\text{s1}}{}^{\prime}\). In the current state, S1 is at angle \(\theta\), and the local bending moment now is \(M_{\text{s1}}\). The bridge S2 (if available) now is at angle (\(\theta-90^{\lx@math@degree}\)) and obtains \(M_{\text{s2}}\). The local bending moment repeats after shaft rotation every \(90^{\lx@math@degree}\), meaning that \(M_{\text{s2}}=M_{\text{s1}}{}^{\prime}\) (both are measured at the same position of angle but 90 degrees out of phase).