Partial Balancing of Unbalanced Primary Force in a reciprocating Engine

          The primary unbalanced force may be considered as the component of the centrifugal force produced by a rotating mass 'm' placed at the crank radius 'r' as shown in below Fig.,

          The primary force act from O to P along the line of stroke. Hence, the balancing of primary force is considered as equivalent to the balancing of mass 'm' rotating at the crank radius 'r'. This is balanced by having a mass B at radius b, placed diametrically opposite to the crank pin C. 

We know that centrifugal force due to mass B, 

                              = B. w^2. b.
and horizantal component of this force acting in opposite direction of primary force
                              = B. w^2. b cos θ.
The primary force is balnced if 
                    B. w^2. b cos θ = m w^2. r cos θ 
                                                 or
                     B.b = m. r
          A little consideration will show, that the primare force is completely balanced if B.b = m. r , but the centrifugal force produced due to the revolving mass B, has also a vertical  component perpendicular to the line of stroke having magnitude B. w^2. b sin θ. This force remains unbalnced. The maximum of this force occurs at value of θ = 90° or 270°. 
          From the above discussion we know that there are two unbalced force, one along the line of stroke where as the second unbalnced force acts along the perpendicular to the line of stroke. The maximum valve of force remains same  in both the cases.It is thus obvious, that the effect of the above method of balancing is to change the direction of the maximum unbalnced force from the line of stroke to the perpendicular of line of stroke. A compromise let a fraction 'c' of the reciprocationg masses is balnced such that

Partial Balancing of Locomotives

          The locomotives, usually, have two cylinders with cranks located at right angles (90°) to each other in order to have uniformly turning moment diagram and also the engine can be started easily after stopping in any position. Balance masses are placed on the wheels in both types. 

        In coupled locomotive, wheels are coupled by connecting their crank pins with coupling rods. As the coupling rod revolves with the crank pin, its proportionate mass can be considered as a revolving mass which can be completely balanced. 

Locomotive engines depending upon the location of cylinders they, are classified in two types they are:

           1. Inside cylinder engine and 2. Outside cylinder engine. 

          In the Inside cylinder locomotives, the two cylinders are placed in between the planes of two driving wheels as shown in Fig., Whereas in the Outside cylinder locomotives, the two cylinders are placed outside the driving wheels one on each side of the driving wheel, as shown in Fig., 

          They are further classified as Coupled and Uncoupled. If two or more pairs of wheels are coupled together to increase the adhesive force between the wheels and the track, it is called as coupled locomotive. otherwise, it is called as uncoupled locomotive.
          A single or uncoupled locomotive is one, in which the effort is transmitted to one pair of the wheels only; Whereas in coupled locomotives, the driving wheels are connected to the leading and trailing wheel by an outside connecting rod.
          Thus, whereas in uncoupled locomotive, there are four planes for consideration, two of the cylinders and two of the driving wheels, In coupled locomotives there are six planes, two of cylinders, two of coupling rods and two of wheels. The planes which contain the coupling rod masses lie outside the planes that contain the balance (counter) masses. Also, in case of coupled locomotives, the mass required to balance the reciprocating parts is distributed among all the wheels which are coupled. Thus, results in a reduced Hammer Blow.