## 3 Effects of Variable Speed on the Structure Vibration  
### Materials and Methods  
The experiment diagram and sensors installation are shown in Figures 5 and 6 following the requirement of ISO 8528-9. All vibration channels were recorded simultaneously with a sampling rate of 8192 samples/second using the Vibration Monitoring software developed by Green Ship Technology Laboratory of Korea Maritime and Ocean University. Sensors in both types of acceleration and velocity were used to measure the vibrations of the structure. The nonlinearity of sensors is \(\pm\)1%. Vibration data from accelerometers were converted to velocity using integration. The diesel engine excites vibrations with frequencies mainly between 2 and 300 Hz. However, when considering the overall genset structure and components, a range of 2 to 1000 Hz is required to evaluate the vibration [18]. In the general case of any vibration in the time-domain, the vibration level is given by the root mean square (RMS) value as known as effective value of vibration. It is defined as square root of the arithmetic mean of the squares of the values. In the case of a set of \(N\) velocity values [\(v_{1}\), \(v_{2}\),..., \(v_{\text{N}}\)], the RMS value is defined by:  
\[v_{RMS}=\sqrt{\frac{1}{N}(v_{1}{}^{2}+v_{1}{}^{2}+\ldots+v_{N}{}^{2})} \tag{2}\]  
RMS value calculation using the general definition equation takes a long time. It is more complicated but less precise when using velocity integrated from acceleration data in time-domain, especially with a band-pass filter. The calculation method using frequency-domain can reduce work. In mathematics, [PERSON]'s theorem [19,20] states that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. In this application, the Fourier transform is referred to. The theorem becomes [21]:  
\[\sum_{i=1}^{N}\lvert v_{i}\rvert^{2}=\frac{1}{n}\sum_{j=1}^{N}\lvert V_{j} \rvert^{2} \tag{3}\]  
Here, \(V_{j}\) is the discrete Fourier transform of \(v_{i}\), both of length \(N\). Substituting Equation (2) into Equation (3) results in:  
\[v_{rms^{2}}=\frac{1}{n^{2}}\sum_{j=1}^{N}\lvert V_{j}\rvert^{2} \tag{4}\]  
Figure 5: Sensors installation for structure vibration measurement based on ISO 8528-9. (**a**) Generator main bearing housing; (**b**) Generator base; (**c**) Engine top girder; (**d**) Engine base.  
Here, \(\hat{v_{k}}\) is amplitude spectrum [22], also known as magnitude of velocity at each frequency amplitude in frequency-domain.  
Time-domain signals are first analyzed using the fast Fourier transform (FFT), and then converted to a frequency-domain representation. From there, the magnitudes of all frequency components of the vibration are expressed. The interval of frequency in Hz is equal to the inverse of the recording time in seconds. In case of acceleration, it can be converted to velocity by:  
\[\hat{v}=\frac{\hat{a}}{2\pi f} \tag{6}\]  
Based on Equation (5), the RMS value can be obtained from the frequency-domain:  
\[v_{RMS}=\sqrt{\frac{{{\hat{v}_{1}}^{2}+{\hat{v}_{2}}^{2}+\ldots+{{\hat{v}_{k}} }^{2}}}{2}} \tag{7}\]  
where, \(\hat{v_{1}}\), \(\hat{v_{2}}\),..., \(\hat{v_{k}}\) are the peak velocity values of all frequencies in the range of 2 to 1000 Hz. This method is more precise and simpler than using the band-pass filter in the time-domain. The measurements were obtained in both constant maximum speed mode and variable speed operation mode with varying load.