# Equivalent Simulation Study of Delta-Rotor Engine
## 2 Equivalence of Movement Modes
\begin{split} S_{1}=2F+BL+BL^{\prime}+S_{r1}\\ F=\Big{[}\frac{\pi}{3}+2\sqrt{K^{2}-9}-\frac{3\sqrt{3}}{2}Ksin \big{(}\frac{2}{3}\alpha+\frac{\pi}{6}\big{)}+\big{(}\frac{2}{3}K^{2}+4\big{)} sin^{-1}\frac{3}{K}\Big{]}e^{2}\\ L=\frac{\sqrt{3}[(R+a)^{2}-2(R+a)+4\epsilon^{2}](R+a)(R+a-4\epsilon) }{2(R+a-4\epsilon)[(R+a)^{2}+4\epsilon^{2}-2(R+a-4\epsilon)]}\\ L^{\prime}=e(K+3){\int_{0}^{\frac{\pi}{3}}}\left(\sqrt{1-k^{2} sin^{2}\frac{\pi}{3}}\right)d\frac{\pi}{3}+\frac{3\alpha}{2}{\int_{0}^{ \frac{\pi}{3}}}\frac{\pi^{2}}{\left(\frac{2}{3}K^{2}+1+\frac{4}{3}Kcos^{2} \frac{\pi}{3}\right)}d\frac{2\alpha}{3}\\ k=\frac{2\sqrt{K}}{\sqrt{3}\big{(}\frac{1}{3}K+1\big{)}},\end{split}  \tag{7}\]  
where \(S_{r1}\) is the increased in-cylinder surface area after the addition of the combustion chamber and \(a\) is the translation distance. Equation (8) shows the curve of the change in the cylinder surface area \(S_{2}\) of a four-stroke engine with the crankshaft angle. \[S_{2}=2\pi\sigma^{2}+2\pi\sigma\Big{[}(1-cos\alpha)+\frac{r}{4l_{1}}(1-cos2 \alpha)\Big{]}+2\pi\sigma l_{2}+S_{r2}, \tag{8}\]  
where \(S_{r1}\) is the increased in-cylinder surface area after the addition of the combustion chamber. To simplify the calculation, the equivalent four-stroke engine combustion chamber is assumed to be a standard cylinder:  
\[ \begin{split} V_{r2}&=l_{3}\pi\rho^{2}\\ S_{r2}&=2l_{3}\pi\rho,\end{split}  \tag{9}\]  
where \(l_{3}\) is the height of the combustion chamber and \(\rho\) is the radius of the combustion chamber. To ensure that the inner surface areas of the two cylinders are equal at all times, Equation (7) must be equal to Equation (8); that is, \(S_{1}=S_{2}\). This equation is transcendental. Figure 2: Cross section of triangular rotor engine. In Equations (6)-(9), there are five unknown quantities (crank radius \(r\), piston radius \(\sigma\), distance \(l_{2}\) from the cylinder head when the piston moves to top dead centre, combustion chamber height \(l_{3}\), and combustion chamber radius \(\rho\)). Taking Equation (6) as the constraint condition and Equation (10) as the fitness function \(Y\), we use the NSGA-II algorithm to solve \(r\), \(\sigma\), and \(l_{2}\). The NSGA-II is an optimisation method based on linear weighting and Pareto ranking that can effectively solve various multi-objective optimal solution problems ([PERSON] et al., 2024; [PERSON] et al., 2023; [PERSON] et al., 2023) [21; 22; 23]. \[ \begin{split} Y=0.5\times\left(S_{1}^{0}-S_{2}^{0}\right)+0.5 \times\frac{\frac{3\pi}{1^{2}}}{S_{1}^{0}}\times\left(S_{1}^{\frac{3\pi}{2}}-S_ {2}^{\pi}\right)\\ +0.1\times\frac{S_{1}^{\frac{3\pi}{2}}}{2S_{1}^{0}}\times\frac{ \sum_{1=1}^{179}\left(S_{1}^{\frac{1\pi}{2}}-S_{2}^{\frac{1\pi}{200}}\right)} {178},\end{split}  \tag{10}\]  
where \(S_{1}^{n}\) and \(S_{2}^{n}\), respectively, represent the in-cylinder area of the triangular rotor engine and the four-stroke engine at the equivalent working time when the crankshaft angle is \(n\), respectively. By using the fitness function of Equation (10), the changes in the cylinder area at other angles can be made similar to those of the triangular rotor engine while ensuring that the maximum and minimum cylinder areas of the equivalent four-stroke engine are as equal as possible to those of the triangular rotor engine. Through the above calculation method, the equivalent basic structural parameters of the four-stroke engine can be obtained.