# A Novel Coaxial Balance Mechanism for Reciprocating Piston Engines
## 3 Dynamics Simulations  
In this section, Adams, a commercial software program used for dynamic analysis, was applied to the dynamic analysis of a DK32 single-cylinder engine in which a balance mechanism was installed. The vibration reduction effect of this coaxial second-order balance mechanism was then verified by numerical simulation. The complete engine model was established in the MSC Adams software (MSC Corp., Irvine, CA, USA), version 2020, as shown in Figure 6. The engine body, crankshaft, and the support structure supporting the single-cylinder engine were modeled as flexible bodies. The component modal parameters of the DK32 engine are shown in Tables 1 and 2. The engine was connected with the plate via two flexible supports, the front support was near the cylinder head, the rear support was near the balance mechanism, and the plate was fixed to the ground. Figure 5: The layout of compact coaxial balance mechanism on a single-cylinder engine. To verify the vibration reduction effect, we select six test points on the engine body, the location of which are shown in Figure 7. Using dynamic simulation, we compared the first-order balance mechanism with the novel coaxial balance mechanism based on the closed-form method under four different crankshaft speeds (800, 1500, 1800, and 2350 r/min) and one based on the novel closed-form method. Additionally, the average root mean square velocity (denoted as the velocity (RMS)) of the six test points in the direction of piston motion (Y direction) was compared under two cases in which the coaxial balance mechanism based on the novel closed-form method was installed or the coaxial \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component 1** & **Mass (kg)** & **Moment of Inertia (kg-mm2) (I\({}_{\text{xx}}\), I\({}_{\text{yy}}\), I\({}_{\text{zz}}\))** & **Simulation Model** \\ \hline Engine body & 41.060 & \(1.768\times 10^{6}\), \(4.950\times 10^{6}\), \(1.566\times 10^{6}\) & Flexible body \\ Crankshaft & 37.959 & \(1.124\times 10^{6}\), \(9.673\times 10^{5}\), \(9.800\times 10^{5}\) & Flexible body \\ Piston set & 2.712 & \(3.587\times 10^{3}\), \(3.252\times 10^{3}\), \(3.162\times 10^{3}\) & Rigid body \\ Connecting rod & 2.433 & \(1.667\times 10^{4}\), \(1.567\times 10^{4}\), \(1.589\times 10^{3}\) & Rigid body \\ Front support & 2.832 & \(3.775\times 10^{5}\), \(2.027\times 10^{5}\), \(2.064\times 10^{5}\) & Flexible body \\ Rear support & 2.843 & \(2.329\times 10^{5}\), \(2.088\times 10^{5}\), \(5.578\times 10^{4}\) & Flexible body \\ First-order balance weight & 4.868 & \(2.006\times 10^{4}\), \(1.882\times 10^{4}\), \(7.634\times 10^{3}\) & Rigid body \\ Second-order balance weight & 0.971 & \(1.839\times 10^{3}\), \(1.836\times 10^{3}\), \(7.682\times 10^{1}\) & Rigid body \\ Second-order balance weight & 1.000 & \(1.896\times 10^{3}\), \(1.893\times 10^{3}\), \(7.920\times 10^{1}\) & Rigid body \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Component mass and simulation model information of the DK32 engine. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component 1** & **Component 2** & **Constraint Relationship** \\ \hline Crankshaft & Connecting rod & Revolute \\ Crankshaft & Engine body & Revolute \\ Piston set & Connecting rod & Revolute \\ Piston set & Engine body & Translational \\ First-order balance weight & Engine body & Revolute \\ Second-order balance weight & Engine body & Revolute \\ Engine body & Front support & Fixed \\ Engine body & Rear support & Fixed \\ Front support & Plate & Fixed \\ Rear support & Plate & Fixed \\ Plate & Ground & Fixed \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constraint relationship between components of the DK32 engine model. Figure 6: Complete engine model of the DK32 single-cylinder engine. balance mechanism using the second-order balance mass of the conventional design was installed. When the velocity (RMS) was smaller in this direction, the vibration reduction effect was improved.