# A General Framework for Crankshaft Balancing and Counterweight Design
## 4 Counterweight Characterisation  
Another crucial step in performing a complete crankshaft balancing procedure is the definition of the counterweight final geometry. In fact, most codes are limited to computing the total mass and mass centre position, leaving to the designer the actual design of the counterweight, often through a cumbersome trial and error procedure. The present tool aims at overcoming these limitations through a careful parametrization of the most common counterweight configurations. For balancing purposes, the counterweights can be considered as concentrated masses, as in the previous section. In fact, the actual counterweights added to engines have complex shapes which must be designed in order to cope with both the balancing needs of the system and the weight and geometrical constraints of the system. Specifically, the balancing components that need more careful design are the counterweights added directly onto the crankshaft, since the clearances for the crankshaft are tighter and the biggest balancing masses are involved. Therefore, it is very important to size and shape properly these components in order to maximise their balancing potential and at the same time reduce the volume and masses used. To this end, a model for the counterweights has been defined, with the necessary dimensional parameters, and it is shown in Figure 5.  
It can be noticed that the surfaces of the counterweight have been divided into different zones, each made of a simple geometrical element:  
* In red, the circle representing the base crank web (no counterweights), defined by \(r_{cw}\);
* A rectangle in orange, defined by \(h_{R}\) and \(2r_{cw}\);
* A trapezoid in blue, defined by \(2r_{cw}\), \(\delta\) and \(l_{c}\);
* A circular segment in green, defined by \(R_{cw}\), \(r_{cw}\) and \(l_{c}\). With the variation of these parameters, multiple shapes and sizes for the counterweights can be obtained, closely representing the actual counterweights used in internal combustion engines. A few examples of these elements can be visualised in Figure 6.  
It is noticeable that the variety of shapes that can be obtained through this modelling involves the possible intersection of the red area (which is constant for a certain radius \(r_{cw}\)) with the other ones, with consequent subtraction of varying areas. These quantities are defined, apart from the parameters introduced earlier, by the angles \(\gamma\) and \(\lambda\). The angle \(\lambda\) can be defined as follows:  
\[ \begin{cases}\lambda=2 cos^{-1}\Big{(}\frac{h_{R}+l_{c}tan\delta}{r_{cw}}\Big{)} &\text{if }h_{R}+l_{c}tan\delta\leq r_{cw}\\ \lambda=0&\text{if }h_{R}+l_{c}tan\delta>r_{cw}\end{cases}  \tag{24}\]  
While the angle \(\gamma\) can be defined as:  
\[ \begin{cases}\gamma=2 cos^{-1}\Big{(}\frac{h_{R}}{r_{cw}}\Big{)}&\text{if }h_{R} \leq r_{cw}\\ \gamma=0&\text{if }h_{R}>r_{cw}\end{cases}  \tag{25}\]  
It is then possible to derive the equations for the computation of the total lateral surface area of the counterweights:  
\[ \begin{cases}A_{green}=R_{cw}^{2}(\frac{\pi}{2}-sin\frac{\pi}{2}cos\frac{\pi} {2})-r_{cw}^{2}\Big{(}\frac{\lambda}{2}-sin\frac{\lambda}{2}cos\frac{\lambda} {2}\Big{)}\\ A_{blue}=(2r_{cw}+l_{c})l_{c}tan\delta-r_{cw}^{2}(\frac{\gamma}{2}-sin\frac{ \gamma}{2}cos\frac{\gamma}{2})+r_{cw}^{2}\Big{(}\frac{\lambda}{2}-sin\frac{ \lambda}{2}cos\frac{\lambda}{2}\Big{)}\\ A_{orange}=2r_{cw}h_{R}-r_{cw}^{2}\frac{\pi}{2}+r_{cw}^{2}\Big{(}\frac{\gamma}{2} -sin\frac{\gamma}{2}cos\frac{\gamma}{2}\Big{)}\\ l_{c}=\frac{2R_{cw}sin\frac{\pi}{2}-2r_{cw}}{2}\end{cases}  \tag{26}\]  
Figure 5: Model and quotas of the crankshaft counterweight. Figure 6: Examples of characterisation of a crankshaft counterweight. where \(A_{green}\), \(A_{blue}\) and \(A_{orange}\) are, respectively, the green, blue and orange areas in Figure 5. It can be noticed that the red area has not been included in the system, since it is not part of the balancing components and, being symmetrical with respect to the crankshaft rotational axis, it is naturally centrifugally balanced.