## 3 Numerical Analysis
### Diagnostic Analysis  
To achieve the diagnostic quantitative analysis for different operating conditions, the RMS results of four local responses, respectively marked as Impact Co and Impact Cs, the blue rectangle and red rectangle in Figure 14, have been calculated and drawn as shown in Figures 15 and 16. Figure 14: The Short-time Fourier transform (STFT) of dynamic responses to piston slaps of biodiesel; (**a**) STFT result of predicted displacement response; (**b**) Piston side-thrust force, predicted displacement and acceleration responses. Figure 15: The variation trends of combustion shock-induced events; (**a**) Root mean square (RMS) results of pressure oscillations and Impact Co event; (**b**) Root mean square (RMS) results of Impact Cs under different loads. Figure 15a presents the RMS results of local features with regard to the oscillations in combustion pressure and the simulated responses to combustion oscillations Impact Co. It can be clearly seen that the changing trends of simulated dynamic responses Impact Co are highly proportional to those of combustion oscillations. This consistency provides clear evidence that the event associated with Impact Co is primarily the forced response caused by the high frequency oscillations of in-cylinder pressure. The RMS results of modal responses of Impact Cs induced by different fuels show more abundant information. As can be seen in the bottom plot of Figure 15b, under high load conditions, with the fuel injection quantity increasing, the atomisation of biodiesel during the premixed combustion phase deteriorated significantly, thereby resulting in a reduction of the shock strength and the consequent dynamic responses. To analyze the variation trends of dynamic responses to piston slaps, the RMS results of local responses around two TDCs, marked with blue and red rectangles in Figure 14, have been calculated and drawn, as shown in Figure 16.  
As shown in Figure 16a, under low load conditions, almost all of the RMSs of dynamic responses to slap events after the exhaust TDC increase steadily with the running speed rising. The upward trend of RMSs fueled with biodiesel shows the most rapid rise, which may be due to the prolonged combustion duration as a result of the larger amount of biodiesel injection (biodiesel possesses a lower calorific value). The delayed mixing-controlled combustion phase can cause a relative increase in the kinetic energy of piston side slaps in the exhaust stroke, thereby resulting in the rise of RMSs around the exhaust TDC. The RMS results of two fuels after combustion TDC show more complex variation tendencies, as shown in Figure 16b. Under low-load conditions, with the increase in the operating speed, both fuels show similar trends to the forced responses Impact Co of combustion oscillations, indicating that the slap-induced responses around the combustion TDC are basically consistent with the in-cylinder combustion behaviors under low-load conditions. Under high-load conditions, the RMS result of biodiesel shows a clearly decreasing tendency after an increase, which cannot be well interpreted by the combustion characteristics of biodiesel itself. To reveal the reasons for this trend, it is necessary to have an in-depth understanding of the formation mechanism of the piston side-thrust force, and especially the effects of combustion pressure on the peak value of piston thrust force around the combustion TDC. Figure 17 shows a typical characteristic of the piston side-thrust force, combined with the longitudinally acting combustion force \(P_{c}\) and axial inertia force of piston assembly \(P_{i}\), obtained based on the engine in this study. Figure 16: The variation trends of combustion piston slap-induced events; (**a**) Root mean square (RMS) results of predicted displacement response after exhaust top dead center (TDC); (**b**) RMS results of predicted displacement response after combustion TDC.