## 2 Problem Description
### Engine Balancing Criteria  
The balancing of an internal combustion engine is closely related to the kinematics of the crank mechanism [20]. The movement of the components generates forces that are transmitted to the main journals and, subsequently, to the engine mounts. Improper balancing can lead to undesirable phenomena such as excessive wear, unexpected failures, vibrations, and increased noise [20]. Therefore, balancing is essential to ensure optimal operation of the system. Vibrations in an internal combustion engine are not only caused by crankshaft imbalance but also by other factors, such as clearances or wear of moving components. This aspect has not been analyzed in detail, as the focus has been on reducing crankshaft imbalance, which is the first step in minimizing vibrations. The inertial forces generated by the crank mechanism motion can be split into two main components: centrifugal and reciprocating forces. Centrifugal inertial forces are associated with rotating masses: the crankpin, the crank webs, the big end bearing, and the rotating portion of the connecting rod (big end). For a single crank mechanism, the magnitude of this force is defined by the equation:  
\[F_{rot}=\omega^{2}r(m_{c,rot}+m_{cp}+m_{cw1}\frac{r_{cw1}}{r}+m_{cw2}\frac{r_ {cw2}}{r}), \tag{1}\]  
where \(\omega\) is the angular velocity of the engine, \(r\) is the crank radius, \(m_{c,rot}\) is the mass of the rotating portion of the connecting rod (big end and bearing), \(m_{cp}\) is the crankpin mass, \(m_{cw}\) is the mass of the crank web, and \(r_{cw}\) is the crank web center of mass radius. Balancing in terms of centrifugal forces is achieved when the following conditions are met:  
* the center of mass of the system lies on the crankshaft axis of rotation which for multi-cylinder architectures, with a generic number of cylinders \(n_{cm}\), means that the vector sum of the centrifugal forces, \(\mathbf{F}_{rot}\), generated by each crank mechanism is zero: \[\mathbf{F}_{rot}=\sum_{i=1}^{n_{cm}}\mathbf{F}_{rot,i}\] (2)
* the vector sum of the moments generated by the centrifugal forces of each crank mechanism composing the multicylinder architecture, \(\mathbf{M}_{rot}\), is zero. Specifically, given any point along the crankshaft axis, each moment is obtained multiplying the centrifugal force by its respective lever arm with respect to the considered point, \(\mathbf{b}_{i}\): \[\mathbf{M}_{rot}=\sum_{i=1}^{n_{cm}}\mathbf{F}_{rot,i}\times\mathbf{b}_{i}\] (3)  
Reciprocating inertial forces arise from reciprocating masses: the piston, the pin, the piston rings, and the reciprocating portion of the connecting rod (small end). For a single crank mechanism, the magnitude of this force is defined by the equation:  
\[F_{rec}=m_{rec}\omega^{2}r(cos\theta+\lambda cos2\theta) \tag{4}\]  
where \(m_{rec}\) is the sum of the reciprocating masses, \(\omega\) is the angular velocity of the engine, \(r\) is the crank radius, \(\theta\) is the crank angle, and \(\lambda\) is the connecting rod ratio defined as \(\lambda=\frac{r}{L}\) where \(l_{c}\) is the connecting rod length. Equation (4) consists of the sum of twoterms and, for the purpose of studying engine balancing, it is convenient to analyze their contributions separately:  
\[F^{\prime}_{rec}=m_{rec}\omega^{2}rcos\theta, \tag{5}\]  
which represents the first-order reciprocating force and:  
\[F^{\prime\prime}_{rec}=m_{rec}\omega^{2}r\lambda cos2\theta, \tag{6}\]  
which represents the second-order reciprocating force. These forces are represented by vectors aligned with the cylinder axis with a variable amplitude function of the crank angle. However, this representation is not optimal for balancing analysis, since it requires complex calculations to define the level of imbalance for a specific engine configuration. A more effective representation exploits the harmonic nature of Equations (5) and (6).