# A General Framework for Crankshaft Balancing and Counterweight Design
## 6 Practical Examples
### 3 Cylinder in-Line Engine  
Regarding the preliminary balancing, once the inputs for the configuration, which can be seen in Table 1, are inserted in the relative tabs (1 and 2), the computation can be started. The numerical results, i.e., the counterweights mass, are reported in Table 2.  
The highest value obtained is the one relative to the crankshaft counterweights: this fact is expected, since, looking at the input data, the rotating masses are larger than the reciprocating ones. Moreover, the mass for the secondary reciprocating forces is quite a small value relative to the others. Therefore, avoiding the secondary balancing shafts may be an option, which would reduce the overall complexity of the engine. The geometry of the defined configuration is shown in Figure 11.  
The geometry corresponds to the most used configuration of a 3 cylinder engine. It is also noticeable that the crankshaft counterweights are not symmetrical to their relative crank. It should be noted that, by convention, the geometry is shown with the first crank exhibiting a crank angle equal to zero (\(\theta=0\)).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline Engine & \multirow{2}{*}{\(N_{cyl}\)} & \multirow{2}{*}{\(\mathbf{\phi}\)**[deg]**} & \multirow{2}{*}{\(\mathbf{a}\)**[mm]**} & \multirow{2}{*}{\(N_{cw}\)} & \multirow{2}{*}{\(N_{cw,1}\)} & \multirow{2}{*}{\(N_{cw,2}\)} & \multirow{2}{*}{\(N_{sh,1}\)} & \multirow{2}{*}{\(N_{sh,2}\)} & \multirow{2}  
The plots for the equilibrium of the centrifugal forces and moments, with their \(x\) and \(z\) components, are reported in Figure 12.  
It is possible to see that the application of the counterweights allows one to cancel the resultants and, therefore, to balance the crankshaft. A similar argument can be made for the reciprocating moments, both primary and secondary. Their plots are shown in Figure 13.  
Figure 11: Geometry of crankshaft and balancing shafts for a 3 cylinder in-line engine.  
Figure 12: Centrifugal forces (**left**) and moments (**right**) for a 3 cylinder in-line engine.  
Figure 13: Primary (**left**) and secondary (**right**) reciprocating moments for a 3 cylinder in-line engine.  
It is noticeable that the curves in each plot are symmetrical around the \(x\)-axis, allowing for the cancellation of the resultants. This phenomenon can be further confirmed by the geometry plot: in fact, all the balancing shafts have an opposite counterweights configuration, which allows one to counter the moments.  
Finally, an optimisation, through an evolutionary algorithm, of the counterweights to be added on the crankshaft, in order to balance it centrifugally, can be performed through the third tab, presented in Section 5. The input values used are those reported in Figure 10, while the results obtained are shown in Table 3. For what concerns the input data relative to the crankshaft, the set shown in Figure 8 is used.  
The optimisation reported is obtained by imposing a perfect balance of the crankshaft (100% Balance), so that the code will find all the solutions abiding by this constraint. From these solutions, the most relevant ones are highlighted: the one with the minimum mass (CW mass) and the one with the maximum possible radial size (\(R_{min}\)), reported in Table 1 as \"MIN Mass\" and \"MIN Clearance\", respectively. Alongside these two parameters, it is possible to see all the other geometrical dimensions, computed by the code, which define the geometry of the counterweight, and, therefore, complete its characterisation.