# Mechanical Performance Analysis and Experimental Study of Four-Star-Type Crank-Linkage Mechanism
## 2 Numerical Calculation of Crank Linkage
### Force Analysis of Crank Linkage
#### 2.2.1 Reciprocating Inertia Forces and Moments  
The reciprocating inertia force on the crank-linkage mechanism is the combined force of the reciprocating inertia forces generated by the four piston groups and the linkage group [13]. The inertia force is decomposed in the plane in the x-direction and y-direction. The formula for calculating the reciprocating inertia force and moment for a multi-column reciprocating air compressor is shown in Table 1, where \(F_{si}^{min}\) is the \(m\)-order reciprocating inertia force in the \(n\) direction for any column \(i\). \(m_{si}\) indicates the mass of reciprocating motion in any column \(i\). \(M_{si}^{min}\) denotes the \(m\)-order reciprocating moment of inertia in the direction of \(n\) for any column \(i\). \(b_{i}\) denotes the distance of the inertia force to the center of symmetry. In the process of crank-linkage design, different cylinder bore requirements are met by choosing different materials while keeping the crank and linkage masses equal. Therefore, there is \(m_{s1}=m_{s2}=m_{s3}=m_{s4}=m_{s}\), and the combined force of the first-order reciprocating inertia forces in each column on the \(x\)-\(x\) and \(y\)-\(y\) axes can be expressed as follows:  
\[\left\{ \begin{array}{l}F_{s1}^{1x}=m_{s}r\omega^{2}\cos\theta\sin 0\\ F_{s2}^{1x}=m_{s}r\omega^{2}\cos(\theta-\pi)\sin\pi\\ F_{s3}^{1x}=m_{s}r\omega^{2}\cos(\theta+\frac{\pi}{2})\sin(-\frac{\pi}{2})\\ F_{s4}^{1x}=m_{s}r\omega^{2}\cos(\theta-\frac{\pi}{2})\sin\left(\frac{\pi}{2} \right)\\ \end{array} \right. \tag{4}\]  
\[F_{s}^{1x}=\sum_{i=1}^{4}F_{si}^{1x}=2m_{s}r\omega^{2}\cos\theta \tag{5}\]  
Similarly, according to the calculation formula in Table 2, we can obtain the following:  
\[ \begin{array}{l}F_{s}^{1y}=\sum_{i=1}^{4}F_{si}^{1y}=2m_{s}r\omega^{2}\sin \theta,\;F_{s}^{1}=\sqrt{\left(F_{s}^{1x}\right)^{2}+\left(F_{s}^{1y}\right)^ {2}}=2m_{s}r\omega^{2},\\ M_{s}^{1}=\sqrt{\left(M_{s}^{1x}\right)^{2}+\left(M_{s}^{1y}\right)^{2}}=0,\;F_ {s}^{2}=\sqrt{\left(F_{s}^{2x}\right)^{2}+\left(F_{s}^{2y}\right)^{2}}=0,\\ M_{s}^{2}=\sqrt{\left(M_{s}^{2x}\right)^{2}+\left(M_{s}^{2y}\right)^{2}}=\sqrt {10}b\lambda m_{s}r\omega^{2}\cos 2\theta\end{array}  \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **The Component Force of Any** & **The Component Force of Any** \\  & **Column \(i\) in the X-Axis Direction** & **Column \(i\) in the Y-Axis Direction** \\ \hline First-order reciprocating inertial force & \(F_{si}^{1x}=m_{si}r\omega^{2}\cos(\theta-\varphi_{i})\sin\varphi_{i}\) & \(F_{si}^{1y}=m_{si}r\omega^{2}\cos(\theta-\varphi_{i})\cos\varphi_{i}\) \\ \hline First-order reciprocating moment of inertia & \(M_{si}^{1x}=F_{si}^{1x}b_{i}\) & \(M_{si}^{1x}=F_{si}^{1y}b_{i}\) \\ \hline Second-order reciprocating inertia force & \(F_{si}^{2x}=\lambda m_{si}r\omega^{2}\cos 2(\theta-\varphi_{i})\sin\varphi_{i}\) & \(F_{si}^{2y}=\lambda m_{si}r\omega^{2}\cos 2(\theta-\varphi_{i})\cos\varphi_{i}\) \\ \hline Second-order reciprocating moment of inertia & \(M_{si}^{2x}=F_{si}^{2x}b_{i}\) & \(M_{si}^{2x}=F_{si}^{2x}b_{i}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Formula for calculating reciprocating inertia force and reciprocating moment of inertia of multi-row air compressors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Materials** & **Ultimate** & **Maximum** & **Maximum Stress Area** \\  & & **Strength (MPa)** & **Stress (MPa)** & **Maximum Stress Area** \\ \hline First-stage piston & ZL108 & 195 & 46.6 & Location of the piston pin hole \\  & & & & connecting to the support gusset \\ \hline First-stage piston pin & 20 CrMn & 735 & 83.6 & Both ends in contact with needle \\  & & & & roller bearings \\ \hline Two-stage linkage & 42 CrMo & 930 & 29.95 &
\begin{tabular}{c} Transition position of the linkage \\ small end to the rod body \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Strength calibration of main components.