7. governors

7/29/2019 7. Governors

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7. GOVERNORS

Governors are used to regulate fluctuation of speed.

7.1. Classification of Governors

Governors broadly classified into two categories, depending on thecontrolling force acting on the governor.

Centrifugal governors, and

Inertia governors.

7.1.1 . Centrifugal Governors (Loaded Governors)

Are based on the balancing centrifugal force acting on rotatingballs by an equal and opposite radial force known as the controllingforce.

The centrifugal force acting on the rotating balls cause a sleeve torise or fall until equilibrium is maintained.

fuel supply in engines is adjusted by linkages connected to the sleeve.

Centrifugal governors are of two type:

Dead weight governors, and

Spring loaded governors. (see figure below)


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7.1.2. Governor Characteristics

To regulate speed, a governor must posses certain qualities.

A governor has to maintain the mean speedof rotating member byfloating in the mean position b/n extreme positions.

At the same time it must readily respond to changes of speed. The characteristics that are related to the proper functioning of a

governor are

Stability, and

Sensitivity, which affect the controlling force of the governor.


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7.1.2.1 Controlling Force

Is the equivalent inward radial force that controls ball movement

It is due to sleeve weight, spring force, etc,

Constraining force acting on the balls.

Depends on the horizontal radial position of the balls and it may beexpressed as

Assume the governor to be rotating at a constant angular velocityrad/s, and let rbe the radius of the circle in which the balls

rotate.

Neglecting the effect of friction and inertia forces of the links, the

centripetal force acting on the balls is

At constant speed , the centripetalforce is plotted against rasshown below.

)1()(rfF

)2(2mrFc


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The graph of Fccuts the graph of the controlling force at P, forwhich

for a constant radial distance r and steady angular velocity .

)3()(2mrrfF


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7.1.2.2 Stability

If a governor is stable and the balls are slightly displaced from theequilibrium position, being constant, the balls have the tendency

to return to the equilibrium position. Let the balls be displaced from the equilibrium position by an

amount r.

The corresponding increment in the centripetal force Fc is

the corresponding increment in the controlling force Fis

dF/dris the slop of the controlling force at point Pand m 2 is theslop of the centripetal force Fc.

)4(2 rmrrFF cc

)5(rdr

dFF


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The restoring force on each ball is given by

For stability of the governor, the restoring force has to satisfy thecondition Fr> 0, i.e.

)7(

,

)6(

22rm

dr

dFrmr

dr

dFF

or

FFF

r

cr

)10(

,

)9(

)8(0

2

2

r

F

dr

dF

or

m

dr

dF

rmdr

dF

c


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But, at a radial distance r, Fc=F as shown in figure above.

the condition for stability is obtained to be

i.e. slope of the controlling force must be greater than the slope ofthe centripetal force as can be observed from the figure below.

)11(r

F

dr

dF


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For complete stability of the governor, the above condition must besatisfied at all speeds within the operating range of the governor.

At equilibrium speed,

From which we obtain

A governor is stable if r increases as increases; i.e. F/r mustincrease for increasing r.

)12(2mrFF c

)13(mr

F


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7.1.2.3 Sensitivity

Sensitivity of a governor is defined as the change in level of therevolving balls corresponding to a change of speed.

For maximum and minimum governor speeds, 1 and 2,respectively, the coefficient of sensitivity of a governor is definedas

Similarly, coefficient of insensitivity, is defined as

where = range of governor speed in which the controlmember remains stationary due to friction inthe system

and

)14()(2 21

21

21

meang

)15(m

gc

)(2

1 m


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7.1.2.4 Isochronous Governor

A governor is said to be isochronouswhen it is infinitely sensitive.

This condition is obtained when

For this case, the controlling force curve becomes a straight linecoinciding with the centripetal force.

the governor becomes isochronous when

For this condition, the balls immediately fly outward for a slightincrease of speed and move inward for a slight decrease of speed.

It is worth noting that sensitivityis obtained at the expense ofstability.

21

)16(rF

drdF


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7.1.2.5 Power of a Governor

Power of a governor is work done at the sleeve for a given changein speed.

power = total energy capacity

power = mean effort x sleeve movement (18)

)17()(2

2

2

1

2

1

Nmdrrf

FdrPower

r

r

r

r


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7.1.2.6 Effort of a Governor

Effort of a governor is defined as theforce exerted at the sleeve for a given

fractional change in speed of thegovernor.

The effort is exerted on the sleeve toovercome the resistance at the sleevewhich opposes the sleeve motion.

7.1.3 The Porter Governor

The porter governor is a dead weightgovernor.

As the speed of the shaft increases,

the balls move outward raising thesleeve height, and vise versa.

Let m1 be the mass of the sleeve andm2be the mass of each ball.


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Considering the equilibrium of the forces acting at C, we have

Considering vertical equilibrium at the balls

Considering horizontal equilibrium at the balls

)19(2

cos 13gm

T

)20(

2

coscos

2

1

232

gmgm

gmTT

)21(tan2

sin

cos

sin

2

sin

sinsin

1

2

1

2

32

gmFT

Fgm

T

FTT


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Dividing equation (21) by (20)

Let

And using

)22(tantan

tan

22

2

tan2

tan

12

1

21

1

Fgmgmgm

or

gmgm

gmF

,tan

tanq

rmFandhr 2

2,tan

)23(22

2

21

2

1

h

r

rmq

gmgm

gm


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Solving equation (23) for h,

For q = 1, where the pendulum arms and suspension links areequal and axes of the joints at B and C intersect atthe main axes, or at equal distance from the axis,

The change in level of the revolving balls is the same as thechange in the height of the governor. From equation (25), it can be

written

Where, Cis a constant that depends on the type of governor and the variousweights, and

nis the speed of the governor in revolutions per second.

)24(1

)1(2

22

21

m

gmqgm

h

)25(122

21

mgmgmh

)26(2nCh


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Increasing the speed from nto x.n, for x > 1, the height of thegovernor for this speed is

the change in height of the governor is

The change in the level of the revolving parts is a measure of thesensitivity of the governor, thus the sensitivity is directlyproportional to the height h.

For porter governor

)27(2221

x

h

xn

Ch

)28(1

2

2

21

x

xhh

x

hhhhh

)29(4

1

2

2

2

21

2

2

21

n

g

m

mm

mgmgmh


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7.1.3.1 Controlling Force of Porter Governor

Consider a Porter governor with all four links of equal length andpivoted at the axis of the shaft rotating at some angular velocity

with the lower and upper limits of 1 and 2. From equation (23) above, the controlling force is given as

Substituting for h, Fis obtained to be

The curve of the controlling force is obtained from this equation

it cuts the r-axis at the origin as shown.

A vertical line is now drawn through any arbitrary radius rr greaterthan the radius corresponding to the maximum sleeve height ormaximum speed of the shaft.

On this line (speed ordinate) values of the centripetal forceFc=m rr

2for various values, including 1 and 2, are

marked.

)30(21h

rgmgmF

)31(22

21

rl

rgmgmF


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Lines are then drawn joining the origin Oand the various values ofFcfor the different values of s marked on the speed ordinate.

These lines are known as speed lines.

The radius of the balls for intermediate values of , where1


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If the effect of friction is considered in the above analysis, then thegovernor needs to overcome the frictional resistance.

Thus, the controlling force becomes

for rising speed

for falling speed

N.B.

The values of for rr is obtained from

,22

21

rl

rfgmgmF

22

21

rl

rfgmgmF

)32(22

21

2

2

rl

rfgmgmrmF

)33(1

)(22

2

212

rlm

fgmgm


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From equation (33), for rising speed of the governor, we obtain

And for falling speed

The curves corresponding to 2and 1are indicated by abandcdrespectively.

It follows that the governor speed will rise to a maximum 2 > 2and will fall to 1 < 1 for the extreme positions of the sleeve

travel.

)34(1

)(22

2

212

2

rlm

fgmgm

)35(1

)(22

2

212

1

rlm

fgmgm


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)36(2

212

h

g

m

mm

)38(22

,1

)37(

1

2222

21

2

2

1

2

12

2

h

rgmgmrmF

qforor

hqr

gm

rgm

gm

rmF

7.1.3.2 Effort of Porter Governor

is the force which it is capable of exerting at the sleeve for a givenpercentage or fractional change of speed.

From equation (25) we have

Let the speed be increased from to x and let a force V be

applied at the sleeve in a downward direction, V being justsufficient to prevent the sleeve from rising.

This will increase the centripetal load from m1g to m1g + V.

The centrifugal force on the two balls is


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At the increased speed x ,

Subtracting F from F1 we obtain

From which V can be evaluated as

If V is made to diminish gradually to zero, the sleeve will rise untilthe height his obtained to be

And the average value of the effort on the sleeve during the risewill be V/2, i.e.

)39(2221

22

21h

rgmVgmrxmF

)40(2)1(222

21h

rVxrmFF

)41()1(

22

2

xhmV

)42(

122

2

21

xm

gmmh

)43()1(2

1

2

1 222 xhmPV


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Substituting for 2,

P is the resistance of the sleeve which the governor is capable ofovercoming with the increase of speed from tox .

For decrease of speed from tox , forx


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7.1.3.3 Power of Porter Governor

Power of governoris the amount of work which the governor iscapable of doing at the sleeve for a given fractional change of

speed

Where

And

power of Porter governor is given by

)46()()(

sPUliftsleeveeffortmeanpower

2

2

21

1

2

1

x

xgmgmP

2

21

22x

xhhs

movementsleeves

)48(1

)47(1

21

2

1

2

2

2

21

2

2

2

2

21

hx

xgmgmU

hx

x

x

xgmgmU


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7.1.3.4 Effect of Friction on the Porter Governor

At a definite sleeve height, the speed may vary from an upper limit

2 to a lower limit 1 without any change of sleeve height due to

the presence of friction in the system. Let the friction force be f.

From this we obtain the difference in the square of the speeds as:

)50(1

)49(1

2

212

1

2

212

2

hm

fgmgm

and

hm

fgmgm

)51(12

2

2

1

2

2hm

f


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Nothing that

For any speed 1


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7.1.4. Hartnel Governor

A Hartnel governor is a spring loaded governor as shown below.

The balls B are mounted on the vertical arms of the bell-crank

levers which are pivoted at O. The pivots are fixed on a frame keyed to the shaft.

The helical spring in compression provides downward forces onthe rollers through the collar of the sleeve


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Let m1 be the mass of the sleeve and m2be the mass of each ball.

The maximum and minimum positions of the governorcorresponding to the maximum and minimum speeds are shown in

fig (a) and (b) respectively.


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For the maximum position

and for the minimum position

Adding equations (55) and (56) and simplifying

)55(11

x

rr

y

h

)56(22

x

rr

y

h

)58()(

)57(

21

21

2121

x

yrrh

hhhletting

x

rr

y

hh


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For maximum position and taking moments about O,

For minimum position, again taking moments about O,

Subtracting equation (60) from equation (59)

)59(2

2

1

1112

1

11

1112111

xFrrgmy

Pgm

xFrrgmyPgm

)60(2

21

2222

2

21

2222221

xFrrgmy

Pgm

xFrrgmyPgm

)61(22

2222

2

1112

1

21 xFrrgmy

xFrrgmy

PP


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But

where k = spring constant and

Thus the spring constant k is obtained from

Neglecting the effect of obliquity of the arms for small angles, wehave

)62(21 hkPP

x

yrrh )( 21

)63(21

21

y

x

rr

PPk

)65(2

)64(2

,

2

21

21

2121

2121

y

x

rr

FFk

y

xFFPP

yyyxxx


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7.1.4.1 Controlling Force of the Hartnel Governor

Assume the position of arm OR to be inclined at an angle of tothe horizontal as shown below corresponding to the speed .


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Taking moments about O,

Let kbe the stiffness of the spring and Pobe the values of Pfor=0,

Neglecting sleeve weight

For small angle , tansin, therefore,

)66(tan2

cos2

1sincos

21

12

gmx

ygmPF

ygmPxgmxF

)67(sinykPP o

)68(tansin22

2

2

gmx

ky

x

yPF o

)69(sin22

2

2

gm

x

ky

x

yPF o


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Noting that ,x

crsin

)70(

22

2

2

2

crx

gm

x

ky

x

yPF o


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The slope of the controlling force F = f(r) is, therefore,

and passes through the point for which

If points Pand Qrepresent the limits of action,

the increment in the controlling force is

x

gm

x

ky2

2

2

2

crwhenx

yPF

2

0

)72(22

)71(22

2

2

2

2

0

2

1

2

2

2

0

1

cr

x

gm

x

ky

x

yPF

crx

gm

x

ky

x

yPF

)73(2

12

2

2

2

12rr

x

gm

x

kyFF


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7.2 SHAFT GOVERNORS

Shaft governors are directly fitted on the main shaft of the engineand the masses revolve at right angles to the shaft axis which in

general is horizontal. Shaft governors are classified into two

Centrifugal flywheel governors, and

Inertia governors.

Both shaft governors operate by a combination of radial and

transverse accelerating forces.7.2.1. Centrifugal Shaft Governor (Flywheel Type)

The effect of transverse accelerating force is neglected.

Heavy links carrying masses are pivoted on the disc which iskeyed

to the engine shaft as shown in the figure below.

Let the tension in each control spring be Pand the resistance dueto the control mechanism be Rfor a constant angular speed .


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The moment about the pivot A of the external force on link AC isdefined as the controlling couple and is denoted by L.

For constant , the acceleration of the mass center G is directedtoward O and

)76(PaRbL

)77(2

OGaG


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The moment about A due to the inertia force is equal to thecontrolling couple; i.e.

Noting that

The controlling couple is obtained to be

At minimum speed 1 the controlling couple is

and at maximum speed 2,

N.B. that the component of the controlling couple due to theresistance of the control mechanism remains unchanged with thechange in speed.

)78(cos2 AGOGmL

)79(sinOAOHcosOG

)80(2

sinAGOAmRbPaL

)81(12

11sinAGOAmRbaP

)82(22

22sinAGOAmRbaP


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Hence, from the above two equations the following relation isobtained.

For small angular displacement 2-1, the increment in the springforce is

7.2.1.1. Condition of Isochronism

Differentiating equation (80) with respect to , we have

From equation (84), noting that dP = P2- P1 for small angulardisplacement 2-1,

)83(12 12

12

2

2 sinsinAGOAmaPP

)84(1212 kaPP

)85(cos2

AGOAm

d

dPa

)86(kad

dP


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Hence, from the above two equations the following relation isobtained.

Substituting OA cos= AH, the spring constant kis given by

For successful regulation, the spring constant in an actual governormust be greater than the stiffness k given in the above equation.

)87(22

cosAGOAmka

)88(2

2

a

AHAGmk


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7.2.1.2 Effect of Angular Acceleration

Suppose the governor is subjected to an angular acceleration inaddition to the angular speed as shown below.

Let be the angular velocity and be the angular acceleration oflink AC.

The acceleration of the mass center Gis

)90(

)89(

tAGnAGtAnAyGxG

AGAG

aaaaaa

or

aaa


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where

In their respective directions as shown in the figure below.

AGa

AGa

OAa

OAa

tAG

nAG

tA

nA

2

2


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From the acceleration polygon

The acceleration of G is, therefore, given by

External forces acting on link AC are

the tension in the control spring P,

the resistance due to the control mechanism R, and

the reaction force at A, Ax and Ay.

The free-body-diagram of link AC is shown including the inertia forceand inertia torque.

)92(

)91(

AGOAcosOAsina

AGOAsinOAcosa

2

yG

22

xG

)93(yGxGG

aaa


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From conditions of dynamic equilibrium:

where kG is the radius of gyration of link ACabout the mass centerG

Substituting for Ay= (maG)y R P, the moment equation can be

written as

From which we obtain

)96()()(

0)()(0

)95(

00

)94(

00

2

2

aAGPAGbRAGAmk

AGAaAGPmkAGbRM

Aam

AamF

APRam

APmaRF

yG

yGG

xxG

xxGx

yy

G

yyGy

)97()()()(2 aAGPAGbRAGPRammk yGG

)98()(2 GyG mkAGamRbPa


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Again, substituting for (aG)y, the controlling couple L becomes

where kA is the radius of gyration of link ACabout A and is given by

At a steady where both are zero, and the controlling couplebecomes

which is the same as equation (80) obtained for steady angular velocity .

If the angular acceleration is suddenly imparted to the shaft whilerotating at a steady angular velocity , equation (100) is used todetermine , the acceleration at which the arms begin to moveoutwards.

(100)mkcos)sinAG(mOARbPaLor

(99))km(AGcos)sinAG(mOARbPaL

2

A

2

2

G

22

)101(222

AGkk GA

and

)102(sinAGmOAL 2


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At this instant, both equation (80) and (100) are satisfiedsimultaneously. Therefore,

From which the following relation is obtained;

is positive in the direction of increasing.

From relative acceleration relations

)103(mkcos)sinAG(mOAsinAGOAmL 2A22

)105(

)104(

2cos

k

AGOA

and

mkcosAGmOA

A

2

A

)107(

)106(

or


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7.2.1.1 Action of flywheel governor under changing speed.

During speed change from 1 to 2, equation (100) holds. Whenthe speed 2 is reached,

But link AChas an angular velocity relative to the governorframe which keeps increasing beyond the equilibrium position 2.

The equation which governs this motion is

Assuming small movements and further assuming that 2 = at

the same instant when 2= , the term in brackets in equation(109) becomes zero; i.e.

)108(0,0

)109(0sin2222

2 AGmOALdt

dmkA

)110(0sin2

2 AGmOAL


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At an angle slightly greater than 2, the term in brackets is givenby

where is the small angular movement measured from the equilibrium

position 2 .

Thus equation (109) becomes

Equation (112) is a differential equation in and can be written as

)111()cos(22 AGOAmka

increamentcouplelcentripetatheforcegcontrollinthe

)112(0)cos( 22222

2

2

AGmOAka

dt

dmkA

)114(

cos

)113(0

2

2

2

2

2

2

2

2

2

Amk

AGmOAkap

where

pdt

d


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This equation represents a simple harmonic motion about theequilibrium position 2 of period

For the case where

)115(cos

2

2

21

2

2

2

2

2

AGOAmka

mkT

or

pT

A

)116(cos

cos

22

2

2

2

2

2

2

a

AGOAmk

or

AGOAmka


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A condition of resonance is obtained for which case the governorbecomes isochronous.

For stability of the governor, the spring must satisfy the condition

)117(cos2

2

2 a

AGOAmk


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7.2.2 Inertia Governors

Inertia governors are operated by the rate of speed instead of thefinite change of speed.

As shown in the figure below point A is a pivot near the shaft axis.

For some rate of change of the speed, the link carrying the twoballs will start to move causing the spring to deflect.

For convenience, lets consider the displacement of the swinginglink to be measured from the inertial 1 where the speed is 1.

Thus the link is just floating at the speed 1 and the angleOAG=90o is assumed.


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After time t from the start of the angular acceleration , referringto the figure below,

and the angular velocity and controlling couple are

where is regarded as a small angle and is constant.

)118(90 o

)120(

)119(

2

1

1

kaLL

t


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Equation (100) shows that the controlling couple is maximum when= 90oand it may be written as

The equation of motion can now be written as

For small angle , the equation of motion becomes

,

,

)121()cossin(

2

2

2

2

AGmOAJ

mkI

where

IJLdt

d

I

AA

AA

)122()2

cos()(2

1

2

1

AA IJtJkaLI

)123()(2

1

2

1 AA IJtJkaLI


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54

For the initial position at a steady speed 1, the control couple is

Hence, the equation of motion is

Equation (126) may be written as

)124(2

1

2

11 JAGOAmL

)126(2

)125(2][

1222

1

222

A

A

A

AA

I

ItJtJ

I

Jka

or

ItJtJJkaI

)127(22

qrtntp


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The general solution of the above equation is

The first term in the solution is the solution of the homogeneousequation and the second term is the particular solution.

The particular integral gives the position of the swinging link at

time t. To determine the constant B, we introduce the initial condition =0

when t=0 which yields

)128(2

cos42

2

p

n

p

qrtntptB

)129(2

0

02

42

42

p

n

p

qB

r

p

n

p

qB


56/56

56

qtwhen

q

I

J

r

I

Jn

where

p

npt

p

nq

prntpt

pn

pq

p

n

p

qrtntpt

p

n

p

q

A

A

,0

.

,2

,

)132(2

cos2

)131(2sin2

)130(2

cos2

1

2

22

23

42

2

42

7. governors

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