There are two common types of spring balance governors used on
traction engines. They are distinguished by the manner in which the
spring balances the centrifugal force of the rotating weights. The
first, and most easily resolved, is the coil spring governor, which
uses a coil spring, generally around the throttle rod, to resist
the force induced by the fly weights' tending to move outward
as they rotate. Figure (1) is a line diagram of one weight assembly
of a coil spring governor. The other prominent type of governor
utilizes a flat (sometimes called a single leaf) spring to resist
the outward movement of the fly weights. Figure (2) is a line
diagram of one weight assembly of a flat spring governor.

First let's investigate the simpler of the two for
evaluation, the coil spring governor. This system operates by
movement of a vertical throttle valve rod up and down in response
to the radial movement of the fly weights. The position of the fly
weights is, in turn, governed by the centrifugal force caused by
their rotation balanced with the force in the compressed
spring.

Where:

*P*Â =Â Resistive force in the spring (Lbs.)
(Fig. 1)

*l*Â Â Â =Â Distance between hinge joints
on fly weight assembly (in.) (Fig. 1)

*x*Â Â =Â Radial distance of center of the
fly weight to the axis of the hinges (in.) (Fig. 1)

*n*Â Â =Â Number of fly weights on the
governor

*W*Â =Â Weight of a single fly weight
(Lbs.)

*r*Â Â Â =Â Radius of the center of the
fly weight from the shaft centerline (in.) (Fig. 1)

*NÂ * =Â Rotation of the fly weight assembly in
revolutions per minute (Fig. 1)

Spring rate; (2)

Â Â Â Â Â Â *P*

K= -

Â Â Â Â Â L-d

Where:

*K* = Spring rate of deflection (Lbs./in.)

*P*Â = Resistive force (Lbs.) (Fig. 1)

*L*Â = Free (no load) length of the spring (in.)

*d*Â = Installed length of the spring (in.)
(Fig.1)

Note: the variables 'x,' 'r,' 'd,' and
'*l*' are dependent upon, and vary with, the
position of the valve rod. The equation for the outward force,
'F,' of the fly weight is incorporated into the above
equation for 'P.' It is, however, shown as an independent
equation in the discussion of flat spring governors.

These equations can be used to help determine the dimensions and
specifications for missing parts. For instance, if the spring for a
governor were to be missing, the specifications for a new spring
could be determined by the following method.

1. First determine the values for the dimensions in Figure 1 for
two conditions. First (subscript 1), the position at which the
throttle valve is just at fully open. Then secondly (subscript 2),
when the valve is just closed. These positions correspond with, (1)
full power, and (2) no power or idle. The values for
'*N*' in these instances would be derived from (1)
the minimum allowable engine speed at full power, and (2) the
minimum, no load engine speed. The minimums are used here to allow
for speed adjustment by the controller on the governor.

2.Â Calculate the resistive force required in the spring
for each condition. Realizing that the spring constant would remain
the same for each of these conditions, an equation for the free
length of the spring can be derived as follows: (3)

*Â Â Â Â Â Â
P*_{2}*Â Â Â Â
P*_{1}

K= - =Â -

*Â Â Â Â Â
L*-*d*_{2}Â L-*d*_{1}

Solving for '*L*':Â (4)

*Â Â Â Â Â Â Â
P*_{2}*d*_{1Â Â -}*Â
P*_{1}*d*_{2}

*L* = -----

*Â Â Â Â Â Â Â Â Â
P*_{2} - *P*_{1}

3.Â By substituting the proper quantities in these
equations, the specifications for overall spring rate and free
length of the balancing spring can be determined.

By making careful measurements of the governor's physical
characteristics, the dimensions and specifications for missing or
damaged items may be calculated. For example, the weight of the fly
weights can be determined by experimentally measuring the spring
rate of deflection 'K' and, using that value, determine a
force 'P' from equation (3) for condition 2 above
(generally the easiest to measure since the throttle is just
closed). Then substitute the values into equation (1) and solve for
'W'.

The methodology given here is reasonably accurate and will
suffice for all shop work. The equations given do not take into
account friction in the assembly and the fact that not all the
weight (mass) of the fly weights is concentrated at their centers
(the weight of the arms causes a small error).

The above equations can be used to show that for a close, or
narrow speed range (*N*_{1} -
*N*_{2}), the resulting spring will be longer and
with lower spring rate than a spring resulting from a wide speed
range. The narrow speed range, while being more sensitive in
regulation, will also have a greater tendency toward speed
fluctuation ('hunting'). For an operating engine, the
problem of excessive speed fluctuation may be mitigated by
installing a shorter, stiffer spring meeting the above equations
for a broader speed range.

Now on to the flat, or leaf spring governor. This system works
by the fly weights' directly deflecting the flat spring and
thereby causing the valve rod to be positioned up or down. The
interdependence of the bending of the spring, its curved surface,
positioning of the valve rod, and centrifugal force cause the exact
equations to be a bit complicated, so some approximations will need
to be made.

The equations pertinent to the flat spring governor are as
follows:

Deflection of the fly weights due to the centrifugal force
'F';Â (5)

Â Â Â Â Â Â Â Â Â Â Â Â
1 *Â Â FL*^{3}

Delta = - ( - )

Â Â Â Â Â Â Â Â Â Â Â
48 *Â EI*

Where:

Delta = Deflection of the spring and fly weight from the at
rest, no load position (in.) (Fig 2)

F = Centrifugal force deflecting the spring (Lbs.) (Equation
(6))

L = Length of the flat spring (in.)

E = Modulus of elasticity of the spring material - normal value is
28,500,000 Lbs./in.^{2} for most spring material l=Area
moment of inertia for the cross-section of the spring
(in.^{4} ) (Equation (7))

Note: This equation is for a spring with hinged, or movable
ends. If the ends of the spring are fixed or clamped, a value of
'192' should be substituted for '48' in the above
equation and any equation derived from it.

Centrifugal force;Â (6)

*Â Â Â Â Â WrN*^{2}

F= --

Â Â Â Â Â 2933

Where:

FÂ Â = Centrifugal force (Lbs.)

W = Weight of a fly weight (Lbs.)

rÂ Â Â = Radius of the center of the fly weight from
the shaft centerline (in.) (Fig 2)

N=Rotation of the flyweight assembly in revolutions per minute
(Fig 2)

Note: The constant,'2933', has dimensions of inches.

Area moment of inertia;Â (7)

Â Â Â Â Â 1

I = -Â wt^{3
Â Â Â Â Â Â } 12

Where:

IÂ Â Â = Area moment of inertia (in. )

W = Width of the spring material (in.)

tÂ Â Â = Thickness of the spring material (in.)

Approximate relationship of spring deflected position to valve
rod position;

Â Â Â Â Â Â 1

x = -Â Sqrt L2 - *l*2

Â Â Â Â Â Â 2

Where:

x = Spring deflected position (in.) (Fig 2)

L = Length of flat spring (in.)

*l* = Distance between spring end attach points (in.) (Fig
2)

Note: The deflected spring position, 'x,' will equal
'5' only if there is no set, or curvature, of the spring
with no load. Figure 2 shows the spring with a set in it.

These equations may be used in a manner similar to that shown
above, for the coil spring, to determine the specifications and
dimensions of missing or damaged parts of a governor. Where direct
resolution of the equations for the coil spring governor is the
norm, here, due to the flat spring's functioning both to resist
centrifugal force and to provide 'linkage' to operate the
valve rod, many cases will, unfortunately, require the
pre-selection of some values in order to determine others
('Trial and Error').

First, we shall address, as an example, a problem with a direct
solution. This is the case in which only the fly weights for a
governor are missing. The method is as follows.

1.Â A value Delta for should be determined experimentally
for condition 2 above (throttle just closed). This will require
manipulating the valve and at least one of the flat spring
assemblies into the position where the throttle valve is just
closed and then measuring for Delta as shown in Figure 2. Also
measure, or otherwise determine a value for 'r' at this
same time; it will be needed later.

2.Â Next, determine the value of 'I' from equation
(7) utilizing the physical dimensions of the spring. Substitute
these values plus the given normal value of 'E' into
equation (5) and solve for a value of 'F'.

3.Â Select a value for 'N' corresponding to the
just closed throttle position (minimum idle speed) and, with the
other values previously determined, utilize equation (6) to
determine a value for the weight of the fly weight. Since here each
fly weight and spring form an independent assembly (unlike the coil
spring governor), only one weight needs to be considered for the
solution.

Next we will consider a more ominous task, that of designing a
complete fly ball assembly for a governor when only the valve
assembly remains. This is, unfortunately, more often the case than
not. The method is as follows.

1.Â In order to determine the amount of movement required
to operate the valve, the vertical movement of the valve rod from
just closed to fully open needs to be determined from the valve
assembly. Utilizing subscripts as done in the coil spring example,
this then corresponds to the value of
'*l*_{1}' minus
'*l*_{2}.' Let's call this value
'Z,' since we will need to address it again.

2.Â We now need to make the first decision. That is, for the
value of 'L,' the length of the flat spring. This may be
done as scientifically as measuring other similar governors, or as
simply as picking a number because it 'looks' right. For
the purposes of this example, we will design the springs with no
set, so that the length of '*l*' (see Figure 2) when
the governor is at rest (stopped) will be the same as
'L.'

3.Â Next we can determine the deflection of the spring at
minimum idle speed by determining a corresponding value for
'*l*_{2}.' This value is equal to the value
of '*l*' at rest (i.e. 'L') minus
'Z' (see above) and minus the movement required in the rod
from the fully open position to the at rest position. This last
dimension is normally the value 'Z' or less due to valve
body constraints. Therefore, for this example, the value for
'*l*_{2}' is given by the following
equation. (9)

*l*2 = *L* - 2Z

4.Â Now utilizing the equation (8) and substituting values
for 'l' and 'L', the value of the spring deflection
at minimum idle speed can be determined.

5.Â The solution from this point depends upon substituting
values for the variables in the deflection and centrifugal force
equations (5) and (6). We have determined values for Delta
(actually 'x,' since we have no set) and
'*L*.' The value for 'r' is the radial
distance from the centerline of the rod to the hinge point of the
spring plus 'x.' The minimum idle speed 'N' can be
decided upon, and the value of '*I*' will be
determined by the availability of spring steel stock. Then,
utilizing the normal value for 'E,' a value for the weight
of the fly weight can be derived.

6.Â If the resulting value of 'W' is too great for
the governor design, it can be reduced by selecting a spring stock
with a lower value of 'I.' If the speed range (idle to full
power) is too wide, it can be reduced by selecting a longer spring
(larger '*L*'). Longer springs will be more
sensitive, however, as with the coil springs, they will also have a
greater tendency for 'hunting.' If the spring has, or
requires due to design, a set to it, the amount of set must be
taken into account in determining the values for 'r,'
'x,' and Delta as shown in Figure 2.

The methodology given here is reasonably accurate for most, if
not all, shop work. The principal inaccuracies occur in the
equation for relating deflection to valve rod position since it
uses a straight line to approximate the curved spring surface. The
equation produces an error of from 0 - 2.5% for the value of
'*l*,' given a range of 'x' being 0 - 20% of
'L.' The calculated value is more than actual value.
Another source of error is the equation for the deflection which is
based upon a simple beam equation and does not account for the
effect a large deflection will have upon the actual beam length.
This error would not adversely effect the design of replacement
components.

For those interested, the following is a more accurate equation
for the relationship between deflection and valve rod position. It
approximates the spring as a segment of a curve. The arcsin is to
be solved for degrees vice radians.

Â Â Â Â Â Â Â 4PiÂ Â Â
1^{2}+4x^{2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â }
8x^{2}

L=Â Â Â -- ( --- ) arcsin (1- --- )

Â Â Â Â Â Â Â
360Â Â Â Â Â
8xÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
1^{2} + 4x^{2}

As a final issue, the speed control on governors is commonly a
simple coil spring, which through the mechanics of the system, adds
an additional balancing force to the movement of the fly weights.
It actually adds to the force 'P' shown on Figure 1. The
equations for the coil spring governor, with proper consideration
for force variation caused by the mechanics of the system (levers,
gears, etc.), may be used to determine values for the speed control
springs.

The speed control also comes into consideration when determining
the speed range for the governor design. Since the speed control
force is additive to the force 'P,' the actual speeds
selected for the range should be the lower expected or allowable
speeds for idle and full power.

I have provided in this article some analysis for the two most
common spring balance governors (at least from my experience) in
use on engines. If there are any other specific governors that you
are interested in, I would be happy to answer questions concerning
them, or any other interesting machine or equipment problem you may
come up with.