# A Novel Coaxial Balance Mechanism for Reciprocating Piston Engines
## 3 Dynamics Simulations
### Vibration Reduction Effect of the Novel Coaxial Balance Mechanism  
The duration of the simulation was 2 s, and the number of steps was 20,000. The integrator was GSTIFF. The vibration reduction effects from the installations of a first-order balance mechanism and a novel coaxial balance mechanisms based on the closed-form method were then compared. Figure 11 shows the average velocity (RMS) in the direction of piston motion (Y direction) of six test points under four different crankshaft speeds.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multicolumn{3}{c}{**Frequency (Hz)**} \\ \cline{2-4}  & **Experiment** & **Computation** & **Error** \\ \hline No. 1 & 23.75 & 23.45 & 1.26\% \\ No. 2 & 30.88 & 30.68 & 0.65\% \\ No. 3 & 51.99 & 49.89 & 4.04\% \\ No. 4 & 61.05 & 62.24 & 1.95\% \\ No. 5 & 74.71 & 68.87 & 7.82\% \\ No. 6 & 87.3 & 81.55 & 6.59\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of experimental frequency and computational frequency.  
Figure 7: Test point positions on the engine body.  
Figure 8: Experimental DK32 single-cylinder engine.  
Figure 10: Acceleration time-domain curve in the direction of piston motion (Y direction) of test point 6 at a crankshaft speed of 1500 r/min.  
Figure 9: Acceleration time-domain curve in the direction of piston motion (Y direction) of test point 6 at a crankshaft speed of 800 r/min.  
According to Figure 11, compared with a conventional first-order balance device, the average velocity (RMS) of the six test points on the cylinder of the single-cylinder engine with the novel coaxial balance mechanism was significantly reduced. At 800, 1500, 1800, and 2350 r/min, the percentage decrease in the average velocity (RMS) in the direction of piston motion reached 86.25%, 94.81%, 96.19%, and 97.31%, respectively.  
The support reaction force between the plate and the ground, which is another important indicator of the vibration reduction effect, was also calculated when the single-cylinder engine was operating. The results show that, when the support reaction force was smaller, the vibration reduction effect was better. Figure 12 indicates the support reaction force from the installations of a conventional first-order balance mechanism and a novel coaxial balance mechanism under different crankshaft speeds. At 800, 1500, 1800, and 2350 r/min, compared with the first-order balance mechanism, the support reaction force of the novel coaxial balance mechanism was reduced by 90.62%, 94.98%, 96.12%, and 96.54%, respectively.  
Figure 11: The average velocity (RMS) in the direction of piston motion of the six test points from the installations of a first-order balance mechanism and a novel coaxial balance mechanism based on the closed-form method.  
Figure 12: Support reaction force in the direction of piston motion from the installations of a first-order balance mechanism and a novel coaxial balance mechanism based on the closed-form method.  
At a crankshaft speed of 2350 r/min, according to Figures 11 and 12, with the first-balance mechanism installed, the average velocity (RMS) of the six test points on the cylinder of the single-cylinder engine was 155.45 mm/s and the support reaction force amplitude was 11,825 N. With the novel coaxial balance mechanism installed, the average velocity (RMS) was reduced to 4.18 mm/s and the support reaction force amplitude was reduced to 409 N. Figure 13 shows the time-domain curve of the support reaction force in the direction of piston motion at a crankshaft speed of 2350 r/min.