# Mechanical Performance Analysis and Experimental Study of Four-Star-Type Crank-Linkage Mechanism
## 2 Numerical Calculation of Crank Linkage  
### Kinematic Relation of Crank Linkage  
The rotating crankshaft compresses the gas by driving the piston in a reciprocating motion through the linkage. The kinematic relationship of the single-row crank linkage is shown in Figure 1. The expressions of piston displacement \(x\), velocity \(v\), and acceleration \(a\) are obtained from the geometric position relationship of the crank linkage as follows:  
\[\begin{array}{c}x=r\Big{[}(1-\cos\theta)+\frac{1}{\lambda}\Big{(}1-\sqrt{1- \lambda^{2}\sin^{2}\theta}\Big{)}\Big{]},\ v=\omega r\Big{[}\sin\theta+\frac{ \lambda\sin 2\theta}{\lambda\sqrt{1-\lambda^{2}\sin^{2}\theta}}\Big{]},\\ a=\omega^{2}r\Big{[}\cos\theta+\frac{\lambda\cos 2\theta}{\sqrt{1-\lambda^{2} \sin^{2}\theta}}+\frac{\lambda^{3}\sin^{2}2\theta}{4\big{(}1-\lambda^{2} \sin^{2}\theta\big{)}^{3/2}}\Big{]}\end{array} \tag{1}\]  
where \(r\) indicates the crank length, \(\lambda=r/l\) indicates the crank linkage length ratio, \(\omega\) indicates the crankshaft speed, and \(\theta\) indicates the crank rotation angle.  
Expanding Equation (1) by Taylor formula and removing higher-order terms above the third order, we obtain the following:  
\[\begin{array}{c}x=r\Big{[}(1-\cos\theta)+\frac{\lambda(1-\cos 2\theta)}{4} \Big{]},\ v=\omega r\Big{[}\sin\theta+\frac{\lambda\sin 2\theta}{2}\Big{]},\ a= \omega^{2}r(\cos\theta+\lambda\cos 2\theta)\end{array} \tag{2}\]  
From Equation (2), only second-order steps exist, and thus, the calculated inertial forces can only go to the second order. If higher-order inertia forces are to be calculated numerically, more higher-order terms should be retained. The linkage combines the rotational motion of the crank  
Figure 1: Kinematic relation of crank linkage.  
and the reciprocating motion of the piston and oscillates reciprocally in the plane, and the angular displacement \(\beta\), velocity \(\omega_{c}\), and acceleration \(\varepsilon_{c}\) of the barycenter can be expressed as follows:  
\[\beta=\arcsin(\lambda\sin\theta),\ \omega_{c}=\frac{\lambda\omega\cos\theta}{ \sqrt{1-\lambda^{2}\sin^{2}\theta}},\ \varepsilon_{c}=\frac{-\lambda(1-\lambda^{2})\omega^{2}\sin\theta}{\left(1- \lambda^{2}\sin^{2}\theta\right)^{3/2}} \tag{3}\]  
The rectangular coordinate schematic diagram of the crank-linkage mechanism is shown in Figure 2. In the diagram, \(\varphi\) indicates the angle between the center line of any column of the cylinder and the \(y\)-axis. The right side of the \(y\)-axis is specified as positive, and the left side of the \(y\)-axis is negative. \(\theta\) indicates the angle of crank relative to the \(y\)-axis, \(\omega\) indicates the angular speed of crank rotation, and \(a\) indicates the axial distance between the barycenter of two adjacent rows of linkage.  
Based on the kinematic relationship of the crank linkage, \(r\) = 35 mm, \(\lambda\) = 25/280, and \(\omega\) = 1480 r/min are substituted into Equations (2) and (3). The phases of one, two, three, and four cylinders differ by 180\({}^{\circ}\), 270\({}^{\circ}\), and 90\({}^{\circ}\), respectively, and the piston acceleration curve and linkage angular acceleration curve are obtained as shown in Figure 3. The acceleration change trend of the opposed two piston columns is the same, and there is some difference in the values. The acceleration curve of a single piston is different from a simple sinusoidal law curve. The acceleration maxima in positive and negative directions are not equal. In one rotation period of the crankshaft, there are two moments when the acceleration of all four pistons is equal and four moments when the acceleration of the opposed two pistons is equal. Moreover, the angular acceleration of the linkage is symmetrical about the equilibrium position, and the acceleration phase is consistent with the geometric phase relationship of each column of the cylinder.