Variation of Tractive Force

The tractive force is the reasultant unbalanced force due to the twi cylinders along the line of stroke. The unbalanced portion of the primary force which along the line of stroke causes the variation in the tractive force or effect. Let C is the friction of reciprocating masses that is balanced. Let the crank for the first cylinder be inclined at an angle θ with the line of stroke, as shown in Fig., Since the crank for the second cylinder is at right angle to the first crank, therefore the angle of inclination for the second crank will be (90° + θ).

Let        m  = Mass of the reciprocating parts per cylinder, and 

              C   = Fraction of the reciprocating parts to be balanced.

        We know that unbalanced force along the line of stroke for cylinder 1, 

         Similarly, unbalanced force along the line of stroke for cylinder 2,

As per defination the tractive force,

                         FT = Resultant unbalanced force along the line of stroke, 

The tractive force is maximum or minimum when (cos θ - sin θ)  is maximum or minimum.

For (cos θ - sin θ) to be maximum or minimum.

Maximum or minimum value of tractive force or the variation of tractive force is

Swaying couple

2. Swaying couple:

        The unbalanced portions of forces acting along the line of stroke are distance 'l' apart. These forces constitutes a couple about YY axis, which tends to make the leading wheels sway from side to side. This couple is known as swaying couple.

Note: In order to reduce the magnitude of the swaying couple, revolving balancing masses are introduced. But, as discussed in the previous article, the revolving balancing masses cause unbalanced forces to act at right angles to the line of stroke. These forces vary the downward pressure of the wheels on the rails and cause oscillation of the locomotive in a vertical plane about a horizontal axis. Since a swaying couple is more harmful than an oscillating couple, therefore a value of 'C' from 2/3 to 3/4, in two-cylinder locomotive with two pairs of coupled wheels, is usually used. But in large four cylinder locomotives with three or more pairs of coupled wheels, the value of 'C' is taken as 2/5.

Hammer Blow

3. Hammer Blow:

          The maximum magnitude of unbalanced force acting perpendicular to the line of stroke is called Hammer Blow. At high speed, this unbalanced force tries to lift the wheels from the rail. 

Let W be the dead weight of the wheel. We know that the unbalanced force along the perpendicular to the line of stroke due to the balancing mass 'm', at a radios 'r' in order to balance reciprocating parts only is mω²r sin θ.  This force is maximum when the value of the unbalanced force is equal to mω²r (when sin θ is unity, i.e. θ = 90° or 270° ). 
Thus the threshold condition for lifting of wheel (Hammer blow) can be written as 

                             Hammer Blow  ( W )   =  mω²r 

The effect of hammer blow is to cause the variation in pressure between the wheel and the rail. This variation is shown in Fig., for one revolution of the wheel. 
Let P be the downward pressure on the rail (or static wheel load)

∴ Net pressure between the wheel and the rail 

= P ± mω²r 

If  (P - mω²r )  is negative, then the wheel will be lifted from the rails. Therefore the limiting condition in order that the wheel does not lift from the rails is given by 
                                               P = mω²r    
and permissible value of angular speed  (or) lifting angular velocity of the crank :

Balancing of Coupled Locomotives

          In a coupled locomotive, the driving wheels are connected to the leading and trailing wheels by an outside coupling rod. By such an arrangement, a greater portion of the engine mass is utilised by tractive purposes. In coupled locomotives, the coupling rod cranks are placed diametrically opposite to the adjacent main cranks (i.e. Driving cranks). The coupling rods together with cranks and pins may be treated as rotating masses and completely balanced by masses in the respective wheels. Thus in a coupled engine, the rotating and reciprocating masses must be treated separately and the balanced masses for the two systems are suitably combined in the wheel.

          It may be noted that the variation of pressure between the wheel and the rail (i.e. Hammer blow) may be reduced by equal distribution of balanced mass between the driving , leading and trailing wheels respectively.