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In This Chapter We Cover the Following Topics

Art. Content Page

5.1 Types of Governors 2

5.2 Centrifugal Governors 3

5.3 Characteristics of Centrifugal Governors 3

5.4 Gravity-Controlled Centrifugal Governors 7

5.5 Spring-Controlled Centrifugal Governors 15

5.6 Pickering Governor 20

5.7 Governor Effort and Power 21

5.8 Hunting of Centrifugal Governor 22

5.9 Centrifugal Effect of the Revolving 24

5.10 Inertia Governors 25

References:

1. Bevan, T., The Theory of Machines, CBS Publishers and Distributors, 1984.

2. Shigley, J.E., Uicker (Jr.), J.J. and Pennock, G.R. Theory of Machine and Mechanism,

Oxford University Press, New York, 2003.

3. Mallik, A. K., Ghosh, A., Theory of Mechanism and Machines, Affiliated East-West

Press (P) Ltd., New Delhi, 2004.

4. Martin, G.H., Kinematics and Dynamics of Machines, MaGraw-Hill, New York, 1982.

5. Rao, J.S., Dukkipati, R.V., Mechanism and Machine Theory, New Age International

Publishers, New Delhi, 2006.

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].




2 Chapter 5: Governor Mechanisms

The function of a governor is to automatically regulate the power input to the engine as

demanded by the variation of load so that the engine speed is maintained at or near the

optimum value.

In otherworld’s, the function of a governor is to regulate the mean speed of an engine,

when there are variations in the load. This means that, when the load on an engine

increases, its speed decreases, therefore it becomes necessary to increase the supply of

working fluid. On the other hand, when the load on the engine decreases, its speed

increases and thus less working fluid is required. The governor automatically controls

the supply of working fluid to the engine with the varying load conditions and keeps the

mean speed within certain limits.

If the efficiency of an engine is plotted against the speed of the engine, a curve similar to

that shown in Diagram 6.1 is obtained. As can be observed, there exists an optimum

speed Nopt for which the efficiency of the engine is maximum. So, for an efficient

operation, it is desirable that, irrespective of the load, the speed of the engine should

remain close to Nopt. To achieve this, engines are provided with a regulatory control or

governor.

Diagram 5.1 Diagram 5.2 Function of Governor Vs flywheel

We know that, the function of governor is to maintain the speed of an engine with in

prescribed limits for the various altering load conditions. It maintains speed within set

limits right from no load on the engine to full rated load on the engine. The function of

governor is distinct from that of a flywheel. As we have seen in Diagram 5.2, a flywheel

smoothens the cyclic fluctuations of speed which are inevitable because of variations in

the turning moment on the crank shaft. Flywheel does not controlled speed variations

caused by a varying load.

5.1 TYPES OF GOVERNORS

Governors may be classified on the basis of their operating principles. The types most

commonly used are (i) centrifugal governors, and (ii) inertia and flywheel governors.

In the centrifugal governor, the change in the centrifugal forces of the rotating masses

due to a change in the speed of the engine is utilized for the movement of the governor

sleeve.

Governor

Centrifugal

governor

Inertia

governor

Watt governor

Pendulum

type governor

Loaded type

governor

Dead weight

governor

Spring control

governor

Wilson governor

Hartung governor

Hartnell governor

Pickering governor

Porell governor

Porter governor

N Cyclic speed variation

controlled by flywheel

Governor control

Load

Variation Time




3 Theory of Mechanism and Machines By Brij Bhooshan

In a gravity-controlled governor, the movement of the governor balls is regulated by the

force of gravity, whereas in a spring-controlled governor, this regulation is provided by

means of springs.

In the inertia governor, the inertia forces caused by the angular acceleration of the

engine shaft (or of the flywheel) by the change in speed are utilized for the movement of

the governor balls. Thus, the movement of the balls is decided by the rate of change of

speed (rather than the change in speed itself, as in a centrifugal governor), with the

result that such a governor is more sensitive than a centrifugal governor. Nevertheless,

a centrifugal governor is more commonly used because of the simplicity of its operation.

5.2 CENTRIFUGAL GOVERNORS

The centrifugal governors are based on the balancing of centrifugal force on the rotating

balls by an equal and opposite radial force, known as the controlling force. There are two

balls of equal masses, are attached the upper link as termed as governor balls or fly

balls as shown in Diagram 5.3. The balls are revolving with a spindle, which is driven by

the engine through bevel gears. The sleeve regarded with the spindle, and free to slides

up and down. The balls and the sleeve rises when the spindle speed increases, and falls

when the speed decreases. In order to limit the travel of the sleeve in upward and

downward directions are provided on the spindle. The sleeve is connected by a bell crank

lever to a throttle valve. The supply of the working fluid decreases when the sleeve rises

and increases when it falls.

Diagram 5.3 Centrifugal governor

When the load on the engine decreases, the mean speed of engine is increases, hence

rotation of flywheel increases and it will swing upward so sleeve moves upwards. Due to

the upward movement of the sleeve its operates a throttle valve at the other end bell

crank leaver to reduces the supply of the working fluid and thus the engine speed is

decreased and vice-versa.

5.3 CHARACTERISTICS OF CENTRIFUGAL GOVERNORS

A centrifugal governor should have the following qualities for satisfactory performance:

1. When its sleeve reaches its lowest position, the engine should develop maximum

power.

2. Its sleeve should at once reach the topmost position when the load on the engine

is suddenly removed.

3. Its sleeve should float at some intermediate position under normal operating

conditions.

4. Its response to a change in speed should be fast.

Throttle valve

Spindle Upper link

Lower link

Sleeve

(free to move)

Base

Fly ball

Bell crank lever To engine





5. It should have sufficient power so that it can exert the required force at the

sleeve to operate the control mechanism.

The definitions of some terms used for describing these qualities of a governor follow.

Controlling Force

In a centrifugal governor, the resultant of all the external forces which control the

movement of the ball can be regarded as a single inward radial force acting at the centre

of the ball. The variation of this force F with the radius of rotation of the ball can be

studied under static conditions by measuring the outward radial force on the ball which

is necessary to keep the ball in equilibrium at various configurations (i.e., for different

values of r). The force F is known as the controlling force and is a function of a single

variable r. Thus,

F = F(r) [5.1]

Diagram 5.4

Diagram 5.4 shows a typical plot of the controlling-force characteristic (curve AB). The

controlling force is derived from purely statical considerations without reference to the

speed of rotation.

Now, let us suppose that the governor ball rotates at a speed ω. The centripetal force

needed for maintaining the radius of rotation r is given by mω2r, where m is the mass of

each ball. The plot of this force against r for a given speed ω will obviously be a straight

line passing through the origin as shown by the line OC in Diagram 5.4. So, the

equilibrium radius for this speed ω will be determined by the intersection of the curve

AB with the line AC (at the point D). For this value of r, controlling force will be equal to

the centripetal force. Mathematically, we can express this equilibrium condition as

F(r) = mω2r [5.2]

Stability

If the governor ball is displaced from its equilibrium position for a particular speed

without any change in the speed of rotation, and thereafter if it tends to return to its

original equilibrium position, then the governor is said to be stable.

A governor is said to be stable when for each speed within the working range there is

only one radius of rotation of the fly balls at which the governor is in equilibrium. For a

stable governor, if the equilibrium speed increases, the radius of governor balls must

also increase. A governor is said to be unstable, if the radius of rotation decreases as the

speed increases.

To determine the condition necessary for stability, suppose the speed of the governor be

ω at the equilibrium position given by the point D (Diagram 5.4). If the speed remains

the same, and if the radius changes to r + δr, the increment in the controlling force EF

E

r

G

F C

D

F B

r

A





will be (dF/dr) δr, and the corresponding increment in the centripetal force EG will be

EG = mω2δr. The restoring force FG on the ball is

FG = EF EG = (dF/dr mω2)δr

This should be greater than zero for the equilibrium position to be regained. Thus, for

stable operation, we get dF/dr > mω2. Using (5.2), we get

In other words, the condition for the stability of a governor is that the slope of the curve

for the controlling force should be more than that of the line representing the centripetal

force at the speed considered.

Sensitiveness

If a governor operates between the speed limits ω1 and ω2, then sensitiveness is defined

as the ratio of the mean speed to the difference between the maximum and minimum

speeds. Thus,

Isochronism

If a governor is at equilibrium only for a particular speed, it is called an isochronous

governor, for which ω1 = ω2 = ω. Thus, from (5.4), we can say that an isochronous

governor is infinitely sensitive. The controlling-force curve for an isochronous governor

coincides with the centripetal-force line corresponding to isochronous speed. Therefore,

in this case, we have

Comparing (5.5) with (5.3), we see that isochronism (or sensitiveness) can be achieved

only at the expense of stability.

Let us suppose the movement of balls in a parabolic track as shown in Diagram

5.5(a). One of the geometrical properties of a parabola is that the subnormal is of

constant length at any point (x1, y1) on the curve y = mx2. Slope (dy/dx) = 2mx1.

Diagram 5.5

So the slope of normal of –(1/2 mx1), but the slope of normal is –h/x1, h is the subnormal

on the y-axis which is the height of the governor.

Thus for any such parabola

(a)

y = mx2

h

x,y

y

x

(b)

θ

b

h

a





Since parabolic governor has a constant height, so it has only one speed for all positions

of the balls, hence it is isochronous.

For crossed arm governor as shown in Diagram 5.5(b). The parabola the radius of

curvature at the point (x1, y1).

R = h sec θ

For crossed arm governor b = R.

h = b cos θ a cot θ

a = b sin θ h tan θ {b = h sec2 θ}

a = h tan2 θ h tan θ = h tan3 θ

An isochronous governor will be infinitely sensitive and hence even a small speed

variation when the governor is at lowest position will shift the sleeve to the other

extreme.

In case of spring controlled governors, the controlling force curve will be straight line, as

shown in Diagram 5.6. i.e. they are isochronous. Then

Diagram 5.6

There are following points, for the stability of spring-controlled governors, may be noted:

For the governor to be stable, the controlling force (F) must increase as the radius of

rotation (r) increases, i.e. F/r must increase as r increases. Hence the controlling force

line C1D1 when produced must intersect the –ve rotation axis, as shown in Diagram 5.6.

The relation between the controlling force and the radius of rotation for the stability

of spring controlled governors is given by

F = Ar B [a]

where A and B are constants.

The value of B in equation (a) may be made either zero or positive by increasing the

initial tension of the spring. If B is zero, the controlling force line CD passes through the

origin and the governor becomes isochronous because F/r will remain constant for all

radii of rotation.

The relation between the controlling force and the radius of rotation, for an isochronous

governor is, therefore,

F = Ar [b]

If B is greater than zero or positive (CC2 – B) > 0, then F/r decreases as r increases, so

that the equilibrium speed of the governor decreases with an increase of the radius of

rotation of balls, which is not possible. So for governor is unstable, then

C2D2 Unstable F = Ar + B

CD Isochronous F = Ar

C1D1 Stable F = Ar B

r

F

D

D1

D2

C2

C1 C





F = Ar + B [c]

Hunting

If the frequency of fluctuations in engine speed happens to coincide with the natural

frequency of oscillations of the governor, then, due to resonance, the amplitude of

oscillations becomes very high, with the result that the governor tends to intensify the

speed variation instead of controlling it. Such a situation is known as hunting. The

problem of hunting becomes more acute when the sensitiveness of a governor is high,

i.e., when a change in speed causes a large sleeve movement. For example, an

isochronous governor (i.e., one that is infinitely sensitive) will oscillate between the

highest and the lowest positions if the speed deviates from the isochronous speed.

Capacity

The magnitude of the controlling force is a measure of the force that can be exerted at

the sleeve to operate the control mechanism. The area under the controlling-force curve

(i.e., between the curve and the r-axis, see Diagram 5.4) for the limits of operation rmin

and rmax represents the work done by the governor against all external forces (since the

controlling force is the resultant of all external forces). This also represents the energy

released by the governor ball when the speed falls from its maximum to its minimum

value, i.e., as r varies from rmin and rmax. Thus, the total energy capacity of the governor

is

as there are two rotating balls.

5.4 GRAVITY-CONTROLLED CENTRIFUGAL GOVERNORS

Three commonly-used gravity-controlled centrifugal governors. The simplest (but

obsolete) type is Watt's pendulum governor shown in Diagram 5.7. The Porter governor

as shown in Diagram 5.8 differs from Watt's governor only in the extra sleeve weight,

which gives it more sensitiveness than Watt's governor. The poor sensitiveness of Watt's

governor, particularly at a high speed, limits its field of application. In the Proell

governor (Diagram 5.11), the balls, instead of being placed at the junction of the arms,

are carried on extensions rigidly fixed to the lower arms. This increases the

sensitiveness of the governor (of course, with the consequent loss in stability). We shall

first consider the characteristics of a Porter governor. The characteristics of Watt's

governor can then easily be derived by taking the sleeve mass M to be zero.

Watt Governor

The simplest form of a centrifugal governor is a Watt governor, as shown in Diagram

5.7. It is basically a conical pendulum with links attached to a sleeve of negligible mass.

Suppose m = Mass of each ball (kg),

T = Tension in the arm in (Newtons),

ω = Angular velocity of the arm and ball about the spindle axis in (rad/s),

r = Radius of rotation of the ball i.e. horizontal distance

h = Height of the governor in (m).

Let us suppose the links to be mass less and neglecting the friction of the sleeve.





Diagram 5.7 Watt governor

Now, equation of equilibrium

Force balance in horizontal direction

T cos θ = mg (i)

Force balance in vertical direction

T sin θ = F (ii)

From equation (i) and (ii), we have

F = mg tan θ

Now from Diagram 5.7 tan θ = r/h, then

we know that ω = 2N/60, then Eq. (5.7a) will be

Height h of the governor is the vertical distance between the centre of the ball to the

intersection position of the arms and spindle.

Porter Governor

It is differs from Watt's governor only in the extra sleeve weight, which gives it more

sensitiveness than Watt's governor. The poor sensitiveness of Watt's governor,

particularly at a high speed, limits its field of application.

Suppose the both arms of the governor be of length L and hinged at a distance e from

the axis of rotation of the governor. One-half of the governor is shown in the Diagram

5.8.

Diagram 5.8 Porter governor

l

r

F

l

I C

B

A

θ h

A

C

e

B Fc

mg

r





Suppose

m = mass of the rotating ball,

M = mass on the sleeve,

θ = inclination of the arms with the vertical when the radius of rotation of the governor

ball is r,

Tu,Tl = tension in the upper, lower arms, respectively, and

F = controlling force (i.e., the resultant of Tu, Tl, and mg).

The friction force at the sleeve is neglected for the time being.

Taking moments about I (the instantaneous centre of rotation of the lower link at this

position) of all the forces acting on the ball, we get

Considering the vertical equilibrium of forces on the sleeve, we have

Substituting Tl from (6.10) in (6.9), we obtain

For equilibrium, we also have F = mrω2. Thus, the equilibrium radius r at a speed ω is

given by

Diagram 5.9

The controlling-force curve given by (5.12) is shown as AB in Diagram 5.9 (F = 0 at r = e,

F at r = l + e). From the nature of this curve, it is obvious that the governor is

stable throughout its range of operation. This is because the slope of this curve will be

more than that of any intersecting line passing through the origin (the slope of the curve

being measured at the point of intersection).

Let

rmin = minimum radius of operation when the equilibrium speed is ωmin,

rmax = maximum radius of operation when the equilibrium speed is ωmax.

The energy capacity of the governor, from (5.6), is

B

D

E

F

C

A

F





(Since dr = l dθ cos θ). Thus,

Now, the total vertical movement of the ball is l (cos θmin – cos θmax) and the total

movement of the sleeve is 2 l (cos θmin – cos θmax). So,

E = (m + M)g (movement of the sleeve) [5.15]

Let us now consider the effect of frictional resistance. The friction force at the sleeve fr is

assumed to be constant and to always oppose the motion of the sleeve. Thus, in all the

preceding equations, the sleeve weight Mg should be replaced by (Mg + fr) for rising

speed and by (Mg – fr) for falling speed (as the friction force will act upwards).

The controlling-force curve with friction is represented by the loop CDEF in Diagram

5.9; the curve for rising speed is CD and that for falling speed, EF. Now, the minimum

speed of operation is given by ω1 and the maximum by (Diagram 5.9). As ω1 < ωmin

(without friction) and ω2 > ωmax (without friction), the sensitiveness of the governor

decreases with friction (note that ωmean remains almost the same with and without

friction). At a given value of r, the lower speed limit ωl [from (5.13)] is given by

and the upper speed limit ωu is given by

From (5.13), (5.16a), and (5.16b), we get

where ω refers to the speed at the same value of r without considering friction. With the

approximation (ωl + ωu)/2 ≈ ω, from (5.17), we get

The ratio given by (5.18) is known as the coefficient of insensitiveness or detention by

friction.

Another arrangement of a Porter governor is shown in Fig. 6.7, where the upper arm

is hinged on the axis of rotation and the lower arm is connected to the sleeve at a

distance e from the axis of rotation.

Diagram 5.10

A

C

B





Suppose θ be the inclination of upper arm, β be the inclination of lower arm.

Now from equilibrium conditions

Now using Eqns. (ii) and (iii), we have

Now using Eqns. (i) and (iv), we have

Now using Eqns. (v) and (iv), we have

Suppose tan β/tan θ = k, and tan θ = r/h, then

If frictional force fr to be considered, then Eq. (5.19) will be

Application 5.1: The arms of a Porter governor are 17.8 cm long and are hinged at a

distance of 3.8 cm from the axis of rotation. The mass of the balls is 1.15 kg each and the

mass on the sleeve is 20 kg. The governor sleeve begins to rise at 280 rpm when the

links are at an angle of 30° to the vertical. Assuming the friction force to be constant,

determine the higher and lower speeds when the angle of inclination of the arms to the

vertical is 45°. Find also the detention by friction at the second position.

Solution: Referring to Diagram 5.8, we have e = 0.038 m, l = 0.178 m, m = 1.15 kg, and

M = 20 kg. Let fr (in N) be the constant friction force at the sleeve.

For θ = 30°,

From (5.15b), we have

After solving we get fr = 10.29 N.

For θ = 45°,

From (5.16a), we have





Similarly, from (5.16b), we get

Again, from (5.13), we get

Then ω = 33.2 rad/s.

Thus, the detention by friction is

Proell Governor

In the Proell governor (Diagram 5.11), the balls, instead of being placed at the junction

of the arms, are carried on extensions rigidly fixed to the lower arms. This increases the

sensitiveness of the governor (of course, with the consequent loss in stability).

A typical Proell governor is shown in Diagram 5.11. Here, the upper arm is hinged on

the axis of rotation and the lower arm is connected to the sleeve at a distance e from the

axis of rotation. The governor ball is not placed at the joint of the upper and lower links.

Instead, it is carried on a rigid extension of the lower link.

Diagram 5.11

Taking moments about I (the instantaneous centre of rotation of the lower link) of all

forces on the ball, it can be shown that

Suppose tan β/tan θ = k, and tan θ = r/h, then

B

C

D I

E

O





From (5.21a), we can obtain F for different positions of the governor ball (i.e., for

different values of r). This controlling-force curve will show smaller values of dF/dr

than those for a Porter governor of the same dimensions (with the ball placed at B').

Thus, this governor has more sensitiveness as compared to a Porter governor.

Consequently, there may be chances of instability in a Proell governor which were

absent in a Porter governor. This is illustrated by the example which follows.

Application 5.2: Let the upper and lower arms of a Proell governor be of equal length

L, and assume that the arms are pivoted on the axis of rotation as shown in Diagram

5.12. The extensions of the lower arms, to which the governor balls of mass m are

attached, are of a length a. The central mass on the sleeve is M. At the minimum radius

of rotation r0, the extensions are parallel to the governor axis (Diagram 5.12(a)), and the

arms are inclined at an angle θ0 to the axis of the governor. Determine the minimum

value of r0 to make the governor stable throughout its range of operation. Given L = 15

cm, a = 5 cm, m = 1 kg, M =10 kg, and θ0 = 30°.

Diagram 5.12

Solution: Diagram 5.12(b) shows the configuration of this governor for any radius r

when the arms are inclined to the governor axis at an angle θ, the inclination of the

extensions to the vertical being δ. Let us now obtain the controlling-force curve for this

governor.

From Diagram 5.12(a), B0B0C0 = – θ0, and Diagram 5.12(b), BBC = – θ + δ. As

the extension is rigidly connected to the lower link, we have

B0B0C0 =BBC,

– θ0 = – θ + δ or δ = θ – θ0

Thus, from Diagram 5.12(b), we get

Taking moments about I, we get

From (ii), we get r0 = 7.5 cm, and from (i) and (iii), we get the values of F and r.

The tangent to the curve is drawn from the origin. The radius at the point of

tangency is r1 = 9.5 cm (from Diagram 5.13). So, for r < r1, dF/dr < F/r, and the

(b)

(a)

A

D I

B

C

A





governor is unstable, whereas, for r > r1, dF/dr > F/r, and the governor is stable. As r1 >

r0, we see that the governor is unstable for the zone from the minimum radius r0 to a

radius r1.

Diagram 5.13

Let us now find out the minimum inclination θ0 to make the governor stable throughout

its range of operation, for which it is necessary that the governor be stable at the

minimum radius. In this case, the condition for stability can be stated as

Now, from (i), we get

and from (iii), we get

Since dF/dr = (dF/dθ)/(dr/dθ), the foregoing equations give

Now, from (i) and (c), we have

Hence, the condition for stability is

On simplification, we get

Let = a/L, β = (M + 2m)/(M + m). Then

Let

With reasonable values of and β, the only real root of (v) is

Thus, for stability, from (iv) and (v), we get

F

r1 r





For this example, = 1/3 and β = 12/11. So, θ0 > 39.5°. Therefore, r0 ≥ 10.38 cm or

(r0)min = 10.38 cm.

5.5 SPRING-CONTROLLED CENTRIFUGAL GOVERNORS

There are two commonly-used spring-controlled centrifugal governors. The simpler one

is called the Hartung governor, in which the controlling springs are directly connected to

the rotating balls. For the Hartnell governor, the controlling spring is connected to the

sleeve. In both cases, the movement of the balls is transmitted to the sleeve by means of

a bell-crank lever (with arms at right angles).

Hartung Governor

The free-body diagram of a bell-crank lever are shown in Diagram 5.14, when the radius

of rotation of the ball centre is r.

Let

k = stiffness of the spring,

Fs = spring force at radius r,

M = mass at the sleeve,

m = mass of the ball,

F = controlling force (at the centre of the ball),

c = radius at which spring force is zero, i.e., when the spring attains its free length,

b = length of the arm to which the ball is connected,

a = length of the arm connected to the sleeve, and

= inclination of the arms as shown in Diagram 5.14.

Diagram 5.14 Hartung governor Diagram 5.15

We shall neglect the friction and the moment of the weight mg about the pivot point O.

Taking moments of all the external forces about O, we have

Further, we see that Fs = k(r – c) as (r – c) is the compression of the spring. Thus,

So, the controlling-force curve is a straight line with a slope k and an intercept [c –

(Mga)/(2bk)] on the r-axis (Diagram 5.15). From this curve, the governor is seen to be

stable everywhere.

b

F r

c

O

r

F





The equilibrium speed ω at radius r is given by

By adjusting the initial compression of the spring, if c is made equal to M.ga/(2bk), the

controlling-force curve (the straight line) passes through the origin and the governor

becomes isochronous. To determine the isochronous speed, we have dF/dr = F/r and k =

mω2. Thus, the isochronous speed

and the condition for isochronism is

If a Hartung governor is made isochronous without considering friction as in the

foregoing analysis, it can be readily shown that with friction such a governor is stable

only for falling speed.

Hartnell Governor

The free-body diagram of the bell-crank lever are shown in Diagram 5.16, when the

radius of rotation of the ball is r.

Let


F1 = total force at the sleeve due to the weight of the sleeve and the spring,

m = mass of the ball,

F = controlling force of the ball,

p = distance of the pivot O from the axis of rotation,

b = length of the arm to which the ball is connected,

a = length of the arm connected to the sleeve, and

= inclination of the arms as indicated in Diagram 5.16.

Diagram 5.16 Hartnell governor

For the present, we shall neglect the friction at the sleeve.

Taking moments about the pivot O, we have

Further, we see that F1 = F0 + ka sin , where F0 is the value of F1 with = 0, and

p

O





Using this value of F1 in (5.26), we get,

The equilibrium speed ω at any radius r is given by

where is found from (5.27) and F0 depends on the initial compression of the spring (by

initial compression, we do not necessarily mean the compression for = 0; it is the

compression of the spring at the lowest position of the sleeve).

To start with, let us neglect the moment of the weight of the ball, i.e., the last term in

(5.28), so that

So, the controlling-force curve is a straight line CD with a slope of ka2/(2b2), passing

through the points A [p, F0a/(2b)] as shown in Diagram 5.17.

Diagram 5.17 Diagram 5.18 Diagram 5.19

As a second approximation, we can use sin = tan which is valid for small values of .

Then, from (5.27) and (5.28), we get

This curve is also a straight line EF passing through the point A, with a slope of

[ka2/(2b2) –mg/b], as shown in Diagram 5.17. The true curve given by (5.28) also passes

through A and is tangential to the line EF at the point A (because, for = 0, the values

of F and dF/dr are the same in both cases). The nature of the true curve is shown in

Diagram 5.18. If the tangent to this curve, drawn from the origin, meets the curve at B,

then the radius rB corresponding to the point B gives the limit for the stability of

operation. For r > rB, the governor becomes unstable as seen from Diagram 5.18 (dF/dr

becomes less than F/r).

If, by the proper choice of parameters, the straight line EF is made to pass through

the origin O, then the governor becomes locally isochronous at A (Diagram 5.19) and is

unstable anywhere else.

The condition for isochronism at A is

D

E C

F

A

r

F

True curve

r

F

F

B

A

O,E

F

r

True curve F

A





where ω0 is the isochronous speed. For the analysis with friction, the force at the sleeve

should be replaced by (F1 ± fr), where fr the friction force. The plus sign applies to rising

speed and the minus sign to falling speed.

Analysis for minimum and maximum position

Consider the forces acting at one bell crank lever. The minimum and maximum position

is shown in Diagram 5.20. The subscript 1 for minimum position and 2 for maximum

position.

Let h be the compression of the spring when the radius of rotation changes from r1 to r2.

Diagram 5.20

For the minimum position i.e. when the radius of rotation changes from r to r1, as shown

in Diagram 5.20(a), the compression of the spring or the lift of sleeve h1 is given by

Similarly, for the maximum position i.e. when the radius of rotation changes from r to

r2, as shown in Diagram 5.20(b), the compression of the spring or lift of sleeve h2 is given

by

Now

Taking moments about point A1, we get

Again, taking moments about point A2, we get

Now

where

(b) Maximum position

(a) Minimum position

A





for small angle θ, the obliquity effect of the arms can be neglected (i.e. x1 = x2 = b, and y1

= y2 = a) and the moment due to weight of the balls (i.e. mg), we have

Spring Controlled Governor with Auxiliary Spring (Wilson-Hartnell Governor)

As shown in Diagram 5.21 in Wilson-Hartnell governor, an auxiliary spring ka is

attached by the lever ABC, besides the fly balls connected by a primary spring kb. Let F

be the centrifugal force, Fs the spring force and S the auxiliary spring force.

Diagram 5.21

The total downward force on the sleeve is evidently,

Taking moments about O the fulcrum of the bell crank lever and neglecting the gravity

on the ball, we have

At minimum equilibrium speed

At maximum equilibrium speed

From the above, we get

But if the radius increases from r1 to r2, the ball springs extend by the amour 2(r2 – r1)

and the auxiliary spring extends by the amount

Therefore

x y

F

F





Hence,

It is clear from the above equation that either ka of kb may be fixed arbitrarily and the

value of the other stiffness can be calculated. If no auxiliary spring is used i.e ka = 0 then

Eq. (5.34) reduces to

5.6 PICKERING GOVERNOR

Another important and widely used type is the Pickering governor. This is widely used

for driving a gramophone. It consists of three straight leaf springs arranged at equal

angular intervals (120 deg) round the spindle. Refer Diagram 5.22(a) and (b). Each

spring carries a mass at the centre. The masses move outwards and the springs bend as

they rotate about the spindle axis with increasing speed.

Diagram 5.22a shows the governor at rest When the governor rotates, the springs

together with the masses are deflected as shown in Diagram 5.22b. The upper end of the

spring is attached by a screw to a hexagonal nut fixed to the governor spindle. The lower

end of the spring is attached to a sleeve which is free to slide on the spindle. The spindle

runs in a bearing at each end and is driven through gearing by the motor. The sleeve

can rise until it reaches a stop whose position is adjustable.

Diagram 5.22 Pickering Governor

Let r be the distance of mass M attached to the leaf spring when the governor is at rest,

ω be the angular speed in rad/sec and δ be the deflection of the centre of the leaf spring

at angular speed . Then r + δ is the distance from the spindle axis to the centre of

gravity of the mass when the governor is rotating. Let h be the lift of the sleeve

corresponding to the deflection δ. We know that the maximum deflection a leaf spring

with both ends fixed with a load at the centre is

L - The distance between the fixed ends of the spring.

E - Youngs modulus of the spring material.

I - Second moment of area of its cross section about the neutral axis.

In the above the central load W is the centrifugal force of rotating mass M. Hence

Mass

Leaf spring

Axis

r

r + δ

h





As an approximation we can use

L = La – h where La = the actual length of leaf spring and h = sleeve movement.

An empirical relation for the sleeve motion h corresponding to the deflection δ can be

used

5.7 GOVERNOR EFFORT AND POWER

Effort

A governor can exert force at the sleeve of the mechanism which controls the supply of

energy to the engine. This force is referring to effort of governor. When the speed of

rotation is constant then effort is zero as the sleeve does not move at all but if a sudden

variation of speed takes place then sleeve tends to move toils new equilibrium position

and a force is exerted on the sleeve mechanism. This force gradually diminishes to zero

as the sleeve moves to the equilibrium position corresponding to the new speed. The

mean force exerted during the given change of speed is termed the effort. For

convenience to compare different governor is defined as the effort or force that can be

exerted by a 1% change of speed.

Power

The power of a governor is defined as the work done at the sleeve for a given percentage

change of speed. It is the product of the governor effort and the displacement of the

sleeve. The power required depends on the controlling mechanism which the governor

operates. Where large power is required usually compressed air or hydraulic pressure is

used in order to change the position of the valve that controls the supply of energy to the

engine.

Analysis

Evaluation of governor effort and power is illustrated with the help of Porter governor.

The principle can be applied to any other governor in a similar manner.

Diagram 5.23 Governor Effort and Power

Suppose N = Equilibrium speed corresponding to the configuration as shown in Diagram

5.23, and k = Percentage increase in speed.

Increase in speed = kN

and increased speed = N + kN = N (1 + k)





The magnitude of this force may be determined by finding what increase is requited in

the gravitational force on the sleeve in order to cause the governor to revolve in

equilibrium in the full line position at the increased speed.

Let the angles θ and β be equal so that k = 1 and speed be N rpm. Then the height h is

given by Eq. 5.19

If the speed increases to (1 + k) N r.p.m. and the height of the governor remains the

same, the load on the sleeve increases to M1g. Therefore

using equations (i) and (ii), we have

Now mean force Fm,

Then effort is

Suppose x = Lift of the sleeve.

Then governor power (Pg) = Fm × x

If the height of the governor at speed N is h and at an increased speed (1 + k) N is h1,

then

Now we know that

then we have

Then x will be

Then power will be

5.8 HUNTING OF CENTRIFUGAL GOVERNOR

Whenever there is a change in the mean speed, centrifugal governors develop a tendency

to oscillate around the desired new mean position. This is because of the fact that when

there is a change in the load on the engine, with a consequent change in engine speed,

the governor balls and the sleeve seek a new position to restore the original speed.

However, due to inertia, they overshoot the mark and thereafter again move towards





the desired position in the opposite direction with the same result. The process is them

repeated, and oscillations are set up. If the frequency of fluctuations in engine speed

happens to coincide with the natural frequency of oscillations of the governor, then, due

to resonance, the amplitude of oscillations becomes very high, with the result that the

governor tends to intensify the speed variation instead of controlling it. Such a situation

is known as hunting.

The problem of hunting becomes more acute when the sensitiveness of a governor is

high, i.e., when a change in speed causes a large sleeve movement. For example, an

isochronous governor (i.e., one that is infinitely sensitive) will oscillate between the

highest and the lowest positions if the speed deviates from the isochronous speed.

Let us calculate the natural period of oscillation of the governor ball considering a

Hartnell governor. Let

r = radius of the governor ball at a steady speed, and

δr = change in r at the same speed.

Considering the oscillation of the governor ball in the radial direction, δr = x is the

amount of displacement from the equilibrium position. So, the equation of motion for the

oscillation of the governor ball can be written as

where

[(dF/dr)δr – mω2δr] = change in the controlling force – change in the centripetal force dr

= restoring force,

m = mass of the governor ball, and

meq = equivalent mass of the governor ball for acceleration in the radial direction.

From (5.40), we have

So, the natural period of oscillation is

For an isochronous governor, dF/dr = mω2 [from (5.5)] and the time period becomes

infinite.

For a Hartnell governor, to find meq, we use the fact that the kinetic energy of the ball

remains the same. If V is the actual velocity of the ball, then ẋ = V cos (see Diagram

5.16). Now,

From (5.27), we get

From (5.28), we get

Using (5.43), we find this equation becomes





Substituting the values obtained from (5.42) and (5.44) in (5.41), for a Hartnell governor,

we get

5.9 CENTRIFUGAL EFFECT OF THE REVOLVING ARMS

The effect of distributed mass of the arms can be accounted as follows.

In the case of Porter governor, let the mass of the upper arm AB be m1 and that of the

lower arm CD be m2.

Diagram 5.24 The Effect of Distributed Man of Arms

(i) Centrifugal moment M1, of the upper arms:

Let the pivot A be at a radius a from the centre line of rotation as shown in Diagram

5.24(a). Let (x + a) be the radius of any element of mass m vertically below A at a

distance y.

Centrifugal force acting on the element is

The moment of the sum of all these elemental force about point A is

AG is the distance of C.G of mass from pivot A. If the rod can be regarded uniform rod of

small cross-section then we have

where L is the length of the rod.

(ii) Gravity moment M2 of the upper arms:

The effect of the gravity forces of the distributed particles comprising the body is the

same as if the mass of the rod were concentrated at its center of gravity G.

(iii) Centrifugal moment M3 about A due to lower links:

(b) (a)

L/2

L B

G

A

D

C

A





The moment about D of the centrifugal forces due to the mass m2 of the lower arms is

calculated as follows

If the radius of point D is a, then moment about D is given by

The reaction F' at the pin C is,

For a case where AC = CD and β = θ.

(iv) Gravity moment M4 about A due to lower links:

If there is no load L of the sleeve, the force at D must be horizontal. The downward force

m2g at G' is transmitted as an equal downward pull at C hence

Application 5.3: Let us consider a uniform rod of length L acting as a simple conical

pendulum. Neglecting the masses of lower links, establish a relation for moment with

speed and radius of gyration of the rod.

Solution: We know that from Diagram 5.24, centrifugal moment

Since offset of pivot is assumed to be zero

Gravity moment is

Hence

If the total centrifugal force of the rod F is assumed to be acting at the center of

Percussion E (with respect to point A), this will give the same centrifugal moment M1.

where IA is the moment of inertia of the rod about A.

5.10 INERTIA GOVERNORS

Diagram 5.25a shows a schematic representation of an inertia governor. As depicted, the

relative movement of the governor balls is controlled by the action of a spring. The arm

connecting the balls is hinged at A on the flywheel connected to the shaft. The relative

position of the ball arm with respect to the flywheel is represented by θ which depends

on the angular velocity ω as well as on the instantaneous angular acceleration of the

shaft. The relative movement of the ball arm is used to control the power input to an





engine. The dependence of the relative movement (Diagram 5.25b) on the angular

acceleration can be obtained as follows.

Diagram 5.25 Inertia governor

The inertia forces on the governor balls, each of mass m, are obtained from

Now, taking moments of all the forces acting on the rigid rod XY about the hinged point

A, the equilibrium equation, we have

where

Fs = Spring force when is equal to zero (considered to be the starting position),


β (angular velocity of the rod XY) = – ω, and

θ = θ0 + (θ0 is the starting equilibrium position when the shaft is rotating at a

constant speed ω), being small.

Eq. (5.46) can be written with θ0 = /2 (assumed) as

where

Thus the differential equation we get for is

where

where ω0 is the initial design speed and is constant. Solving (5.48), we get

The adjustments are such that when the shaft rotates at the design speed ω0, θ = θ0 =

/2. So,

Thus

Using the initial conditions

(b) (a)

O

G

A

B

X Y

A

O





We have

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