# Equivalent Simulation Study of Delta-Rotor Engine
## 2 Equivalence of Movement Modes  
Unlike conventional engines that convert the reciprocating motion of the piston into the rotational motion of the crankshaft, the triangular rotor engine uses the expansion work generated by the combustion of the mixture in the combustion chamber to rotate the triangular rotor and then acts on the eccentric journal of the eccentric shaft to produce tangential force. With the support of the front and rear primary bearings, the main shaft rotates and performs work, thereby generating mechanical energy. The working process of a triangular rotor engine can be divided into four strokes: intake, compression, power, and exhaust. A comparison with a conventional piston engine is shown in Figure 1. The relationship between the internal and external gears of the triangular rotor engine makes its working cycle \(1080^{\circ}\) CA, and it also has the characteristics of a two-stroke engine. Based on the above analysis, this study treats the delta-rotor engine as an equivalent study to a four-stroke engine. The movement of the engine is primarily reflected in the change in the cylinder volume with the crankshaft angle. Figure 2 shows a cross-sectional view of a triangular rotor engine; its single cylinder volume \(V_{1}\) can be expressed as  
\[V_{1}=V_{f}+V_{r1}, \tag{1}\]where \(V_{r1}\) is the combustion-chamber volume with a fixed value. \(V_{f}\) is the cylinder surface volume, which changes with the crankshaft angle and can be expressed as  
\[V_{f}=\left[\frac{\pi}{3}+2\sqrt{K^{2}-9}-\frac{3\sqrt{3}}{2}Ksin\left(\frac{2} {3}\alpha+\frac{\pi}{6}\right)+\left(\frac{2}{9}K^{2}+4\right)sin^{-1}\frac{3} {K}\right]e^{2}B, \tag{2}\]  
where \(B\) is the cylinder thickness, \(e\) is the eccentricity, and \(\alpha\) is the crankshaft angle. The shape parameter, \(K\), can be expressed as  
\[K=\frac{R}{e}, \tag{3}\]  
where \(R\) denotes the radius of creation. Then, we combined Equations (1)-(3) to discover  
\[V_{1}=\left[\left(\frac{\pi}{3}+2\sqrt{K^{2}-9}+\left(\frac{2}{9}K^{2}+4 \right)sin^{-1}\frac{3}{K}\right)e^{2}B+V_{r}\right] \tag{4}\]  
where the first term is a constant, and the second term is a sine function of \(\frac{2}{3}\alpha\). Equation (5) shows the curve of the cylinder volume \(V_{2}\) of a four-stroke engine that changes with the crankshaft angle. \[V_{2}=\left[\left(1-cos\alpha\right)+\frac{r}{4l_{1}}\left(1- cos2\alpha\right)\right]\pi r\sigma^{2} \tag{5}\] \[+l_{2}\pi\sigma^{2}+V_{r2},\]  
where \(r\) is the crank radius, \(l_{1}\) is the connecting rod length, \(\sigma\) is the piston radius, \(V_{r2}\) is the combustion chamber volume, and \(l_{2}\) is the distance between the piston and the cylinder head when the piston moves to the top dead centre. We combined Equations (4) and (5)  
Figure 1: Comparison of working processes between reciprocating piston engine and triangular rotor engine. and converted the two motion periods to 720\({}^{\circ}\) CA to ensure the inner volume of the two cylinders were equal at all times:  
\[ \begin{split}\Big{(}\frac{\pi}{3}+2\sqrt{K^{2}-9}+\big{(}\frac{2}{ 3}K^{2}+4\big{)}sin^{-1}\frac{3}{K}\Big{)}e^{2}B+V_{r1}\\ =\pi r\sigma^{2}+l_{2}\pi\sigma^{2}+V_{r2}\\ \frac{3\sqrt{3}}{2}Ke^{2}B=\pi r\sigma^{2}\\ l_{1}\rightarrow+\infty.\end{split}  \tag{6}\]  
In addition to the cylinder volume of the triangular rotor engine, the cylinder surface area is an important parameter that has a significant influence on the performance of the engine, such as heat dissipation loss and stress. The change in cylinder surface area \(S_{1}\) with the crankshaft angle can be expressed as  
\[