# A General Framework for Crankshaft Balancing and Counterweight Design
## 1 Introduction  
The modern automotive market requires a continuous reduction of carbon emissions and fuel consumption and an ever increasing engine efficiency [1, 2]. These goals can be achieved by the use of a support electric motor for functions such as the Start/Stop or real hybrid engines. The engine compartment must then accommodate these new components; therefore, engine dimensions must be as small as possible, especially in the case of small/medium size cars. Furthermore, engine downsizing is recognised, in itself, as an effective method to reduce consumption and increase efficiency [3, 4] and many manufacturers are reducing engine displacement and the number of cylinders.  
From a design point of view, the combination of a reduced number of cylinders with weight and space limitations makes the task of balancing the crankshaft more difficult. Crankshaft unbalance translates into high vibrations that are transmitted to the entire car, thus compromising the user comfort level [5]. Crankshaft balancing is typically performed in two subsequent phases. The first is during the early design, where the configuration of the masses to be added to the crankshaft and to the additional balance shafts are defined [6]. The second involves the measurement of the residual unbalance [7] in machined crankshafts and the subsequent correction on a sample-to-sample basis [8, 9]. The contribution of the present work fits precisely into the first step of the balancing process and was developed to aid the designer in choosing, since the early stages, the weight and the configuration of masses to be added to the crankshaft and, when present, to the balance shafts.  
It must be noted that the goal of the present tool is to perform a \"global\" balancing of the system, in order to find the optimal counterweight configuration. The introduction of the counterweights produces a change in the reaction forces and in the loading diagram of each journal bearing. Therefore, once the counterweight configuration has been defined, it is of paramount importance to evaluate how this change may influence the design of thejournal bearings themselves [10; 11; 12; 13], an aspect which is, however, beyond the scope of the present work.  
The proposed tool, equipped with a Graphical User Interface, is based on a comprehensive formulation which allows the user to analyse a wide range of scenarios, i.e., crankshafts with any number of cylinders and counterweights, in-line and V configurations alike. The proposed framework needs as input only the main geometrical dimensions and key operating conditions (e.g., piston firing order), and offers, as outputs, the rotating vectors diagrams and the counterweights mass and configuration necessary to balance the centrifugal forces and the first and second order reciprocating forces and the respective moments.  
The basic concepts and formulation adopted by the tool are presented in Section 2.  
The matrix formulation adopted to express the balancing equations is presented and discussed to highlight its versatility in Section 3. Section 4 is devoted to the parametrization of the counterweight shape and its subsequent optimisation, a problem which is either tackled through an iterative procedure requiring a lengthy direct involvement from the designer [14] or through the use of computationally intensive CAD-based genetic algorithms [15].  
The numerical tool, whose user interface is described in Section 5, is then used in Section 6 to investigate different scenarios, demonstrating its effectiveness in exploring the counterweights' minimum weight-minimum dimension trade-off.