# Mechanical Performance Analysis and Experimental Study of Four-Star-Type Crank-Linkage Mechanism
## 3 Simulation of Mechanical Characteristics of Four-Star-Type Crank Linkage
### Ideal Four-Star-Type Crank Linkage
#### 3.1.2 Simulation Calculation of Crank Linkage with Friction for Kinematic Pair  
As the analysis in the previous section shows, the crank linkage flexibility does not affect the trend of the main shaft bearing force. Therefore, a rigid-body model is used in this section to investigate the effect of friction on the main shaft bearing excitation force. The ADAMS built-in IMPACT collision function is used to define the kinematic pair contact between the big end of the first-stage linkage and the crankshaft. The other kinematic pairs are kept in ideal condition. Meanwhile, the parameters of the Coulomb friction force model are selected. The expression of the contact force is calculated as follows:  
\[F_{n}=\left\{\begin{aligned} K\delta^{n}+\text{step}(\delta,0,0, \delta_{\text{max}},C_{\text{max}})\frac{d\delta}{dt}&\delta>0 \\ 0&\delta\leq 0\end{aligned}\right. \tag{19}\]  
\[\text{step}(\delta,0,0,\delta_{\text{max}},C_{\text{max}})=\left\{\begin{aligned} 0& \delta\leq 0\\ C_{\text{max}}\Big{(}\frac{\delta}{\delta_{\text{max}}}\Big{)}& 0<\delta<\delta_{\text{max}}\\ C_{\text{max}}&\delta\geq\delta_{\text{max}}\end{aligned}\right. \tag{20}\]  
where \(K\) is the collision stiffness, \(\delta\) is the penetration depth of the kinematic pair contact surface, \(\delta_{\text{max}}\) is the maximum penetration depth of the kinematic pair contact surface, \(d\delta/dt\) is the relative velocity of the collision, \(n\) is the force exponent, and \(C\) is the damping coefficient. The selection of each parameter is shown in Table 4.  
By reasonably setting the relevant parameters of the contact model, the barycenter of the piston is loaded with the gas force applied to the piston in Figure 4 according to the phase relationship, and by default, the pistons of all levels are subjected to equal forces, and there is only a difference in phase. The multi-body dynamics simulation is carried out to obtain the force on the main shaft with and without friction at the big end of the first-stage linkage, as shown in Figures 19 and 20, respectively.  
As can be seen from the figure, the presence of friction has basically no effect on the crankshaft fundamental frequency excitation force. However, it will greatly broaden the excitation frequency band. The friction of the kinematic pair brings the excitation energy in the middle and high frequencies. When the friction is not considered, the peak excitation force is obvious near 225 Hz. When the friction is considered, the excitation peak at this location is suppressed. However, the amplitude of the excitation force in the nearby frequency band is increased. The line spectrum of air compressor vibration energy has more components, and the moderate friction can control the line spectrum energy and reduce the air compressor vibration to some extent.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name of Parameter** & **Value of Parameter** & **Name of Parameter** & **Value of Parameter** \\ \hline Collision stiffness \(K(\text{N}/\text{m})\) & \(5\times 10^{5}\) & The coefficient of kinetic friction is \(f_{d}\) & \(8\times 10^{-2}\) \\ Contact force index e & 1.5 & The coefficient of static friction is \(f_{s}\) & \(5\times 10^{-2}\) \\ Damping coefficient \(C\) & \(3\times 10^{-1}\) & The speed of dynamic friction is \(v_{d}/(\text{m}/\text{s})\) & \(1\times 10^{-2}\) \\ Penetration depth \(\delta_{\text{max}}/m\) & \(1\times 10^{-4}\) & The speed of static friction is \(v_{s}/(\text{m}/\text{s})\) & \(1\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Setting of simulation parameters.  
Figure 18: Main shaft bearing force of flexible linkage.