# A Novel Coaxial Balance Mechanism for Reciprocating Piston Engines
## 2 Design of a Novel Coaxial Balance Mechanism
### Closed-Form Method for the Calculation of Inertia Forces  
A schematic of a single-cylinder piston engine crankshaft-connecting rod mechanism is shown in Figure 1. Herein, \(m\) is the reciprocating mass of the piston crankshaft-connecting rod mechanism, \(R\) is the crankshaft radius, \(L\) is the length of the connecting rod, and \(\lambda\) is the ratio of crankshaft radius to connecting rod length (i.e., \(\lambda=R/L\), generally taking the value of \(1/3\)). Further, \(\alpha=\omega t\) is the crank angle and \(\omega\) is the angular velocity of crankshaft rotation.  
Based on the kinematic analysis of the crankshaft-connecting rod mechanism, the composition of the inertia force produced by the reciprocating motion of the single-cylinder engine piston obtained by the traditional method [23] is expressed as follows:  
\[F_{TM}=b_{1-TM}\cos\alpha+b_{2-TM}\cos 2\alpha+\sum_{m=2}^{\infty}b_{2n-TM}\cos 2n\alpha \tag{1}\]  
where \(b_{1-TM}=m\omega^{2}R,b_{2-TM}=m\omega^{2}R\lambda\).  
Figure 1: Schematic of a single-cylinder piston engine crankshaft-connecting rod mechanism.  
Using the novel closed-form method [24], the composition of single-cylinder engine inertia force can be obtained as:  
\[F_{CF}=b_{1-CF}\cos\alpha+b_{2-CF}\cos 2\alpha+\sum_{n=2}^{\infty}b_{2n-CF}\cos 2n\alpha \tag{2}\]  
where \(b_{1-CF}=m\omega^{2}R\) and \(b_{2-CF}=16m\omega^{2}L[(2-\lambda^{2}) Elliptic(\lambda^{2})-2(1-\lambda^{2}) EllipticK\)\((\lambda^{2})]/3\pi\lambda^{2}\); _Elliptic_E and _EllipticK_ are the complete elliptic integrals of the first and second kind, respectively.  
In Equation (2), the inertial force can be divided into first-order inertial force, second-order inertial force, fourth-order inertial force, etc., by multiples of the angular velocity of the engine rotation; that is, \(\omega,2\omega,4\omega\ldots\) For commercial piston single-cylinder engines, its balance mechanism generally only balances the first-order inertia force. In multi-cylinder engines, the first-order inertia force can be self-balanced by designing the spatial configuration of the crankshaft, and the second-order inertia force needs to be balanced by setting a second-order balance mechanism.  
It can be observed that the first-order inertia force obtained by the two methods is the same. Therefore, if only the first-order inertia force of the single-cylinder engine is considered to be balanced, either method can be used. However, although the first-order inertia force can be balanced using these methods, the remaining second-order and higher-order inertia force is still not negligible, which is the main cause of engine vibration. A more accurate expression of second-order inertia force is beneficial to eliminate the effect of the second-order inertia force. In fact, from the comparative analysis in [24], we know that when the \(\lambda\) variation range is 0.25-0.4, the calculation accuracy of second-order inertia force can be improved by 1.5-4% by adopting a novel closed-form method. Hence, using a novel closed-form method to calculate second-order inertia force is an important means of reducing single-cylinder engine vibration. This method can also be adapted to improve the high-order balance performance of the multiple-cylinder engine because it still provides accurate results if the effect of fourth-order inertia force is further considered.