# A Novel Coaxial Balance Mechanism for Reciprocating Piston Engines
## 2 Design of a Novel Coaxial Balance Mechanism
### Application Cases  
The novel coaxial balance mechanism was applied to the DK32 single-cylinder diesel engine developed in the laboratory [20,21].  
A simplified balance shaft mechanism is shown in Figure 4. The light blue semicircle in the picture represents the counter-rotating mass, in which \(M_{i}\) is the mass of balance weight of a single balance shaft, and \(\omega_{i}\) is the rotational angular velocity of balance shaft. The rotational angular velocity of the first-order balance shaft is the same as that of the crankshaft. The rotational angular velocity of the second-order balance shaft is twice that of the crankshaft; that is, \(\omega_{1}=\omega\) and \(\omega_{2}=2\omega\). \(\alpha_{i}=\omega_{i}t\), \(r\), denotes the distance from the center of mass of balance weight to the axle center, and \(F_{i1}\) and \(F_{i2}\) denote the resultant forces of the two balance weight inertia forces. When the balance shaft rotates, the resultant inertia force of each balance weight acts on the center of mass, and the action line of the force passes through the balance shaft axle center, which has a size of:  
\[|F_{i1}|=|F_{i2}|=M_{i}\omega_{i}^{2}r \tag{3}\]  
Because the speed of the two balance shafts is the same and the crank angle is also the same, the components of \(F_{i1}\) and \(F_{i2}\) perpendicular to the direction of piston motion are offset. Then, the resultant balance force of the components in the direction of piston motion can be expressed as:  
\[F_{bi}=2M_{i}\omega_{i}^{2}r\cos\alpha_{i} \tag{4}\]  
where \(F_{bi}\) is balanced with the inertia force generated by the piston.  
For the first-order balance shaft:  
\[2M_{1}\omega^{2}r=m\omega^{2}R \tag{5}\]  
Figure 4: Simplified diagram of counter-rotating balance shafts.  
Therefore, the balance weight mass \(M_{1}\) of the first-order balance shaft is:  
\[M_{1}=\frac{mR}{2r} \tag{6}\]  
Considering the second-order balance shaft:  
\[8M_{2}\omega^{2}r=\frac{16m\omega^{2}L\big{[}\big{(}2-\lambda^{2}\big{)}EllipticE \big{(}\lambda^{2}\big{)}-2\big{(}1-\lambda^{2}\big{)}EllipticK\big{(} \lambda^{2}\big{)}\big{]}}{3\pi\lambda^{2}} \tag{7}\]  
The balance weight mass \(M_{2}\) of the second-order balance shaft is:  
\[M_{2}=\frac{2 mL\big{[}\big{(}2-\lambda^{2}\big{)}EllipticE\big{(}\lambda^{2} \big{)}-2\big{(}1-\lambda^{2}\big{)}EllipticK\big{(}\lambda^{2}\big{)}\big{]}} {3\pi r\lambda^{2}} \tag{8}\]  
The required balance weight mass was thus designed, and the size of the balance shaft was then able to be determined to counteract the first- and second-order inertia forces produced by the reciprocating motion of the piston set and the small end of the connecting rod.  
In the proposed balance mechanism, both first- and second-order balance shafts were installed in the shell, and only a rotating input gear was needed to connect to the crankshaft. The unit-body design could be used to arrange the parts and components of the engine more flexibly without increasing the size of the engine. In addition, the structure improved the assembly form of a conventional engine and adopted an integral installation mode, which is quick and straightforward to install and disassemble. This structure could be widely used in various types of piston engines to reduce the vibration and noise of the engine and improve its service life. The layout of the compact coaxial balance mechanism installed on the single-cylinder engine is shown in Figure 5.