# Changes in Propeller Shaft Behavior by Fluctuating Propeller Forces during Ship Turning
## 2 Approaches Adopted
### Introduction to the Strain Gauge Method  
The Wheatstone bridge circuit method is a common method that is applied for measurement using a strain gauge. There are 3 types of strain gauge configuration based on the number of active gauges: quarter-bridge (1 active gauge); half-bridge (2 active gauges); and full-bridge (4 active gauges). In general, the half-bridge configuration for bending moment measurement has all the advantages of the full-bridge configuration. The full-bridge configuration creates an output voltage that is double that of the half-bridge configuration, but it was not necessary for the measurements in this study. Therefore, as shown in Figure 3, a half-bridge strain gauge circuit was applied in consideration of the high accuracy compared with the installation time. \begin{table} \begin{tabular}{c c c} \hline \hline
**Vessel type** & \multicolumn{2}{c}{50,000 DWT medium-range oil/chemical tanker} \\ \hline L \(\times\) B \(\times\) D (**m**) & 174.0 \(\times\) 32.2 \(\times\) 19.1 & Four blades at a fixed pitch \\ \hline \multirow{3}{*}{**Main engine**} & Type: MAN B\&\&W 650 ME-B \\ \cline{2-3}  & MCR: 7700 kW \(\approx\) 93.4 rpm & **Propeller** \\ \cline{2-3}  & NCR: 5344 kW \(\approx\) 82.7 rpm & Mass: 18,200 kg \\ \hline \multicolumn{2}{c}{**Flywheel**} & Mass: 11,207 kg & Cap and nut mass: 1538 kg \\ \hline \end{tabular} \begin{tabular}{c c} \hline \hline
**L**, length; B, breadth; D, depth; MCR, maximum continuous rate; NCR, nominal continuous rate. \\ \hline \end{tabular} 
\begin{tabular}{c c} \hline \hline
**RIAL  
The resistance of the strain gauge varies in proportion to the strain at the gauge position when the shaft rotates. It can be expressed as shown in Figure 4.  
The strain curve has a sine wave since the strain depends on the bending moment and the angle position where the strain is measured. The strain \(\varepsilon\), as measured by strain gauges at any point, can be expressed as Equation (1) [35]. \[\varepsilon=\varepsilon_{a}\cos(\theta-\lambda)+\overline{\varepsilon} \tag{1}\]  
where  
\(\varepsilon_{a}\) is the strain amplitude, \(\varepsilon_{a}=(\varepsilon_{max}-\varepsilon_{min})/2\),  
\(\theta\) is the angle between the vertical direction and the gauge,  
\(\lambda\) is the angle between the normal to neutral axis and the vertical direction or bending angle, and  
\(\overline{\varepsilon}\) is the average strain. The average strain value does not matter because, to determine the moment, only the strain amplitude is needed. The strain is at its maximum when \(\theta=\lambda\)  
\[\varepsilon=\varepsilon_{a}\cos(0)+\overline{\varepsilon}=\varepsilon_{a}+ \overline{\varepsilon} \tag{2}\]  
The uniaxial stress can be expressed as Equation (3) by applying the beam relation used in the calculation of uniaxial stress on the shaft surface. \[\sigma=E\varepsilon_{a}=\frac{M_{b}c}{I} \tag{3}\]  
Figure 4: Typical angle-domain shaft strain curve. Figure 3: Typical half-bridge strain gauge circuit configuration. where  
\(\varepsilon_{a}\) is the strain amplitude,  
\(M_{b}\) is the bending moment of the shaft,  
\(E\) is [PERSON]'s modulus,  
\(I\) is the second moment of the shaft's cross-section, and  
\(c\) is the radius of the shaft's cross-section. In a hollow axis or a solid axis, \(I\) is to be \(I=\frac{\pi(d_{o}^{4}-d_{i}^{4})}{64}\) or \(I=\frac{\pi d_{o}^{4}}{64}\), respectively, and \(c=\frac{d_{o}}{2}\) in common. Thereby, Equation (3) can be rearranged as Equation (4). \[M_{b}=E\frac{\pi}{32}\frac{\left(d_{o}^{4}-d_{i}^{4}\right)}{d_{o}}\varepsilon _{a},\ M_{b}=E\frac{\pi d_{o}^{3}}{32}\varepsilon_{a},\ M_{b}=\sigma Z \tag{4}\]  
The bending moment measured at the position of the strain gauge is obtained by:  
\[M_{s}=E\frac{\pi d_{o}^{3}}{32}\varepsilon \tag{5}\]  
Therefore, Equation (4) becomes:  
\[M_{b}=\frac{M_{s\ max}-M_{s\ min}}{2} \tag{6}\]  
where  
\(Z\) is the section modulus of the shaft. Then, the vertical and horizontal moments can be expressed by Equations (7) and (8), respectively.