7490 Woodridge Lane, Bremerton, Washington 98310

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.

The equations pertinent to the coil spring governor are as

follows:

Resistive force in the spring; (1)

1 WrN^{2}

P= -(n) (-)

4x

2933

Where:

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

(Fig. 1)*l* = Distance between hinge joints

on fly weight assembly (in.) (Fig. 1)*x* = Radial distance of center of the

fly weight to the axis of the hinges (in.) (Fig. 1)*n* = Number of fly weights on the

governor*W* = Weight of a single fly weight

(Lbs.)*r* = Radius of the center of the

fly weight from the shaft centerline (in.) (Fig. 1)*N * = Rotation of the fly weight assembly in

revolutions per minute (Fig. 1)

Spring rate; (2)

*P*

K= –

L-d

Where:

*K* = Spring rate of deflection (Lbs./in.)*P* = Resistive force (Lbs.) (Fig. 1)*L* = Free (no load) length of the spring (in.)*d* = 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. 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)

*
P*

_{2}

P

P

_{1}

K= – = –

*–*

L

L

*d*

_{2}L-

*d*

_{1}

Solving for ‘*L*‘: (4)

*
P*

_{2}

*d*

_{1 –}

P

P

_{1}

*d*

_{2}

*L*= —–

P

P

_{2}–

*P*

_{1}

3. 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’; (5)

1 * FL*^{3}

Delta = – ( – )

48 * 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; (6)

* WrN*^{2}

F= —

2933

Where:

F = Centrifugal force (Lbs.)

W = Weight of a fly weight (Lbs.)

r = 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; (7)

1

I = – wt^{3} 12

Where:

I = Area moment of inertia (in. )

W = Width of the spring material (in.)

t = Thickness of the spring material (in.)

Approximate relationship of spring deflected position to valve

rod position;

1

x = – Sqrt L2 – *l*2

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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

4Pi

1^{2}+4x^{2 }

8x^{2}

L= — ( — ) arcsin (1- — )

360

8x

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.