Balancing of V-engines

Consider a symmetrical two cylinder V-Engine as shown in Fig., The common crank OC is driven by two connecting rods PC and QC. The lines of stroke OP and OQ are inclined to the vertical OY, at an angle α as shown in Fig.,

Let                          m= Mass of reciprocating parts per cylinder,

                                l = Length of connecting rod, 

                                r = Radius of crank,

                                n= Ratio of length of connecting rod to crank radius = l/r

                                θ = Inclination of crank to the vertical at any instant,

                                ω  = Angular velocity of crank.

We know that inertia force due to reciprocating parts of the cylinder 1, along the line of stroke, 

and the inertia force due to reciprocating parts of the cylinder 2, along the line of stroke,

The balancing of V-Engines is only considered for primary and secondary forces as discussed below:

Considering primary forces:

Considering secondary forces: