# A General Framework for Crankshaft Balancing and Counterweight Design
## 2 Methodology  
### Balancing Concepts  
The creation of the tool starts from the definition of a suitable model of the crankshaft with the relative forces that need balancing. A generic crankshaft can be divided into multiple elementary elements, called cranks, identified and oriented based on the firing order of the engine. Each crank is made of three parts:  
* two main journals, which are supported by the engine block;
* one crank-pin, on which the big-eye of the connecting rod is placed;
* two crank webs, which connect the main journals and the crank-pin. These components are modelled as rigid beams, made of the same material. Such model can be seen in Figure 1. In this figure the dimensions relevant to the balancing process can be identified, alongside the reference system adopted throughout the analysis. It is possible to identify three forces in need of balancing in the engine:  
* Centrifugal force, \(F_{\omega}=m_{rot}\omega^{2}r\);
* Primary reciprocating force, \(F_{a}^{\prime}=-m_{alt}\omega^{2}r(\cos\theta)\);
* Secondary reciprocating force, \(F_{a}^{\prime}=-m_{alt}\omega^{2}r(\Lambda\cos 2\theta)\)  
Figure 1: Model of the crankshaft for a 3 cylinder in-line engine, with the relative polar plot on the right. where \(m_{rot}\) is the rotational mass of the crank mechanism (including crank webs, crankpin and conrod's big-eye), \(m_{alt}\) is the crank mechanism reciprocating mass (including piston, wristpin, piston rings and conrod's small-eye), \(\omega\) is the crankshaft rotational speed, \(\theta\) is the crank-angle with respect to the \(z\)-axis and \(\Lambda\) is the elongation ratio, equal to the ratio between the crank radius \(r\) and the conrod length. The minus sign for the reciprocating forces is representative of the fact that the inertia forces have opposite direction with respect to the direction of motion of the pistons. The centrifugal force \(F_{\omega}\) has constant magnitude but its direction continuously changes coherently with the crankshaft rotation angle, meanwhile the reciprocating forces \(F_{\omega}^{\prime}\), \(F_{a}^{\prime\prime}\) are always aligned with their respective cylinder axis and the magnitude changes continuously, with the secondary one having double the frequency of the primary reciprocating force. In any case, the behaviour is clearly oscillating; therefore, these forces, and the possible moments they may form, can generate deformations of the crankshaft which must be minimised. The balancing of the forces can be performed by adding to the engine properly sized masses, called counterweights. These masses will rotate during the operation of the engine, generating centrifugal forces that counter those of the crankshaft. It is possible to place the counterweights through two main methods: either directly on the crankshaft or on additional components called balancing shafts. For what concerns the centrifugal force balancing, it depends on the rotational masses of the crankshaft \(m_{rot}\), composed of the masses of the crank-pin, the two crank-webs and the big-eye of the connecting rod, and, due to its constant magnitude, it is sufficient to place the counterweights on the crankshaft, as shown in Figure 2, where the following quantities can be identified: \(F_{c}\) are the centrifugal forces generated by the counterweights \(m_{c}\), \(M_{\omega,1}\) and \(M_{\omega,3}\) are the moments by the cranks (1 and 3 respectively), \(M_{\omega,res}\) is the resultant of the previously defined moments and \(M_{c,res}\) is the moment generated by the counterweight centrifugal forces. The numerical subscripts refer to the cylinder number and in the following discussion are removed for simplicity, since the elements share the same magnitude (e.g., \(m_{c,1}\) and \(m_{c,3}\) will be defined as \(m_{c}\)). The reciprocating forces, instead, due to their direction always aligned with the cylinders axis, cannot be treated as the centrifugal force.