Chapter7-Governors

Ex1-pg196

In [2]:
##CHAPTER 7 ILLUSRTATION 1 PAGE NO 196
##TITLE:GOVERNORS
import math
##===========================================================================================
##INPUT DATA
L=.4##                     LENGTH OF UPPER ARM IN m
THETA=30.##                 INCLINATION TO THE VERTICAL IN degrees
K=.02##                    RISED LENGTH IN m
##============================================================================================
h2=L*math.cos(THETA/57.3)##         GOVERNOR HEIGHT IN m
N2=(895./h2)**.5##           SPEED AT h2 IN rpm
h1=h2-K##                  LENGTH WHEN IT IS RAISED BY 2 cm
N1=(895./h1)**.5##           SPEED AT h1 IN rpm
n=(N1-N2)/N2*100.##         PERCENTAGE CHANGE IN SPEED
##==========================================================================================
print'%s %.1f %s'%('PERCENTAGE CHANGE IN SPEED=',n,' PERCENTAGE')

PERCENTAGE CHANGE IN SPEED= 3.0  PERCENTAGE

Ex2-pg197

In [3]:
##CHAPTER 7 ILLUSRTATION 2 PAGE NO 197
##TITLE:GOVERNORS
##FIGURE 7.5(A),7.5(B)
import math
##===========================================================================================
##INPUT DATA
OA=.3##                          LENGTH OF UPPER ARM IN m
m=6.##                            MASS OF EACH BALL IN Kg
M=18.##                           MASS OF SLEEVE IN Kg
r2=.2##                          RADIUS OF ROTATION AT BEGINING IN m
r1=.25##                         RADIUS OF ROTATION AT MAX SPEED IN m
##===========================================================================================
h1=(OA**2.-r1**2.)**.5##             HIEGHT OF GOVERNOR AT MAX SPEED IN m
N1=(895.*(m+M)/(h1*m))**.5##      MAX SPEED IN rpm
h2=(OA**2.-r2**2.)**.5##             HEIGHT OF GONERNOR AT BEGINING IN m
N2=(895.*(m+M)/(h2*m))**.5##      MIN SPEED IN rpm
##===========================================================================================
print'%s %.1f %s %.1f %s %.1f %s'%('MAX SPEED = ',N1,' rpm'' MIN SPEED = ',N2,' rpm''RANGE OF SPEED = ',N1-N2,' rpm')

MAX SPEED =  146.9  rpm MIN SPEED =  126.5  rpmRANGE OF SPEED =  20.4  rpm

Ex3-pg197

In [4]:
##CHAPTER 7 ILLUSRTATION 3 PAGE NO 197
##TITLE:GOVERNORS
##FIGURE 7.6
import math
##===========================================================================================
##INPUT DATA
OA=.25##                                 LENGHT OF UPPER ARM IN m
CD=.03##                                 DISTANCE BETWEEN LEEVE AND LOWER ARM IN m
m=6.##                                    MASS OF BALL IN Kg
M=48.##                                   MASS OF SLEEVE IN Kg
AE=.17##                                  FROM FIGURE 7.6
AE1=.12##                                 FROM FIGURE 7.6
r1=.2##                                  RADIUS OF ROTATION AT MAX SPEED IN m
r2=.15##                                 RADIUS OF ROTATION AT MIN SPEED IN m
##============================================================================================
h1=(OA**2-r1**2)**.5##                     HIEGHT OF GOVERNOR AT MIN SPEED IN m
TANalpha=r1/h1
TANbeeta=AE/(OA**2-AE**2)**.5
k=TANbeeta/TANalpha
N1=(895.*(m+(M*(1.+k)/2.))/(h1*m))**.5##    MIN SPEED IN rpm
h2=(OA**2-r2**2)**.5##                    HIEGHT OF GOVERNOR AT MAX SPEED IN m
CE=(OA**2-AE1**2)**.5
TANalpha1=r2/h2
TANbeeta1=(r2-CD)/CE
k=TANbeeta1/TANalpha1
N2=(895.*(m+(M*(1.+k)/2.))/(h2*m))**.5##    MIN SPEED IN rpm
##========================================================================================================
print'%s %.1f %s %.1f %s %.1f %s'%('MAX SPEED = ',N1,' rpm'' MIN SPEED = ',N2,' rpm''RANGE OF SPEED = ',N1-N2,' rpm')

MAX SPEED =  215.5  rpm MIN SPEED =  188.2  rpmRANGE OF SPEED =  27.2  rpm

Ex4-pg199

In [5]:
##CHAPTER 7 ILLUSRTATION 4 PAGE NO 199
##TITLE:GOVERNORS
##FIGURE 7.7
import math
##===========================================================================================
##INPUT DATA
g=9.81##                   ACCELERATION DUE TO GRAVITY 
OA=.20##                  LENGHT OF UPPER ARM IN m
AC=.20##                  LENGTH OF LOWER ARM IN m
CD=.025##                 DISTANCE BETWEEN AXIS AND LOWER ARM IN m
AB=.1##                   RADIUS OF ROTATION OF BALLS IN m
N2=250##                  SPEED OF THE GOVERNOR IN rpm
X=.05##                   SLEEVE LIFT IN m
m=5.##                     MASS OF BALL IN Kg
M=20.##                    MASS OF SLEEVE IN Kg
##===========================================================
h2=(OA**2.-AB**2.)**.5##               OB DISTANCE IN m IN FIGURE
h21=(AC**2.-(AB-CD)**2.)**.5##         BD DISTANCE IN m IN FIGURE
TANbeeta=(AB-CD)/h21##            TAN OF ANGLE OF INCLINATION OF THE LINK TO THE VERTICAL
TANalpha=AB/h2##                  TAN OF ANGLE OF INCLINATION OF THE ARM TO THE VERTICAL
k=TANbeeta/TANalpha
c=X/(2.*(h2*(1.+k)-X))##            PERCENTAGE INCREASE IN SPEED 
n=c*N2##                          INCREASE IN SPEED IN rpm
N1=N2+n##                          SPEED AFTER LIFT OF SLEEVE
E=c*g*((2.*m/(1.+k))+M)##            GOVERNOR EFFORT IN N
P=E*X##                            GOVERNOR POWER IN N-m

print'%s %.1f %s  %.2f %s  %.1f %s '%('SPEED OF THE GOVERNOR WHEN SLEEVE IS LIFT BY 5 cm = ',N1,' rpm'' GOVERNOR EFFORT = ',E,' N' 'GOVERNOR POWER = ',P,' N-m')

SPEED OF THE GOVERNOR WHEN SLEEVE IS LIFT BY 5 cm =  275.6  rpm GOVERNOR EFFORT =   25.95  NGOVERNOR POWER =   1.3  N-m

Ex5-pg200

In [6]:
##CHAPTER 7 ILLUSRTATION 5 PAGE NO 200
##TITLE:GOVERNORS
##FIGURE 7.8
import math
##===========================================================================================
##INPUT DATA
g=9.81##                   ACCELERATION DUE TO GRAVITY 
OA=.30##                  LENGHT OF UPPER ARM IN m
AC=.30##                  LENGTH OF LOWER ARM IN m
m=10.##                     MASS OF BALL IN Kg
M=50.##                    MASS OF SLEEVE IN Kg
r=.2##                    RADIUS OF ROTATION IN m
CD=.04##                  DISTANCE BETWEEN AXIS AND LOWER ARM IN m
F=15.##                    FRICTIONAL LOAD ACTING IN N
##============================================================
h=(OA**2-r**2)**.5##           HIEGTH OF THE GOVERNOR IN m
AE=r-CD##                   AE VALUE IN m
CE=(AC**2-AE**2)**.5##         BD DISTANCE IN m
TANalpha=r/h##              TAN OF ANGLE OF INCLINATION OF THE ARM TO THE VERTICAL
TANbeeta=AE/CE##            TAN OF ANGLE OF INCLINATION OF THE LINK TO THE VERTICAL
k=TANbeeta/TANalpha
N=((895./h)*(m+(M*(1.+k)/2.))/m)**.5##      EQULIBRIUM SPEED IN rpm
N1=((895./h)*((m*g)+(M*g+F)/2.)*(1.+k)/(m*g))**.5##        MAX SPEED IN rpm
N2=((895./h)*((m*g)+(M*g-F)/2.)*(1.+k)/(m*g))**.5##        MIN SPEED IN rpm
R=N1-N2##                                   RANGE OF SPEED
print'%s %.1f %s %.1f %s '%('EQUILIBRIUM SPEED OF GOVERNOR = ',N,' rpm'' RANGE OF SPEED OF GOVERNOR= ',R,' rpm')

EQUILIBRIUM SPEED OF GOVERNOR =  145.1  rpm RANGE OF SPEED OF GOVERNOR=  3.4  rpm

Ex6-pg202

In [7]:
##CHAPTER 7 ILLUSRTATION 6 PAGE NO 202
##TITLE:GOVERNORS
##FIGURE 7.9
import math
##===========================================================================================
##INPUT DATA
g=9.81##                   ACCELERATION DUE TO GRAVITY 
OA=.30##                  LENGHT OF UPPER ARM IN m
AC=.30##                  LENGTH OF LOWER ARM IN m
m=5.##                     MASS OF BALL IN Kg
M=25.##                    MASS OF SLEEVE IN Kg
X=.05##                   LIFT OF THE SLEEVE
alpha=30.##                ANGLE OF INCLINATION OF THE ARM TO THE VERTICAL
##==============================================
h2=OA*math.cos(alpha/57.3)##        HEIGHT OF THE GOVERNOR AT LOWEST POSITION OF SLEEVE
h1=h2-X/2.##                HEIGHT OF THE GOVERNOR AT HEIGHT POSITION OF SLEEVE
F=((h2/h1)*(m*g+M*g)-(m*g+M*g))/(1.+h2/h1)##      FRICTION AT SLEEVE IN N
N1=((m*g+M*g+F)*895./(h1*m*g))**.5##          MAX SPEEED OF THE GOVVERNOR IN rpm
N2=((m*g+M*g-F)*895./(h2*m*g))**.5##          MIN SPEEED OF THE GOVVERNOR IN rpm
R=N1-N2##                                   RANGE OF SPEED IN rpm

print'%s %.1f %s %.1f %s'%('THE VALUE OF FRICTIONAL FORCE= ',F,' F'' RANGE OF SPEED OF THE GOVERNOR = ',R,' rpm')

THE VALUE OF FRICTIONAL FORCE=  14.9  F RANGE OF SPEED OF THE GOVERNOR =  14.9  rpm

Ex7-pg203

In [8]:
##CHAPTER 7 ILLUSRTATION 7 PAGE NO 203
##TITLE:GOVERNORS
import math
##===========================================================================================
##INPUT DATA
PI=3.147
m=3##                 MASS OF EACH BALL IN Kg
a=.12##               LENGTH OF VERTICAL ARM OF BELL CRANK LEVER IN m
b=.08##               LENGTH OF HORIZONTAL ARM OF BELL CRANK LEVER IN m
r2=.12##              RADIUS OF ROTATION OF THE BALL FOR LOWEST POSITION IN m
N2=320.##               SPEED OF GOVERNOR AT THE BEGINING IN rpm
S=20000.##                 STIFFNESS OF THE SPRING IN N/m
h=.015##                  SLEEVE LIFT IN m
##==================================================
Fc2=m*(2.*PI*N2/60.)**2*r2##               CENTRIFUGAL FORCE ACTING AT MIN SPEED OF ROTATION IN N
L=2*a*Fc2/b##                           INITIAL LOAD ON SPRING IN N
r1=a/b*h+r2##                           MAX RADIUS OF ROTATION IN m
Fc1=(S*(r1-r2)*(b/a)**2/2)+Fc2##         CENTRIFUGAL FORCE ACTING AT MAX SPEED OF ROTATION IN N
N1=(Fc1/(m*r1)*(60./2./PI)**2)**.5
print'%s %.1f %s %.1f %s '%('INITIAL LOAD ON SPRING =',L,' N'' EQUILIBRIUM SPEED CORRESPONDING TO LIFT OF 15 cm =',N1,' rpm')

INITIAL LOAD ON SPRING = 1217.0  N EQUILIBRIUM SPEED CORRESPONDING TO LIFT OF 15 cm = 327.9  rpm

Ex7-pg204

In [9]:
##CHAPTER 7 ILLUSRTATION 8 PAGE NO 204
##TITLE:GOVERNORS

##===========================================================================================
##INPUT DATA
PI=3.147
m=3##                           MASS OF BALL IN Kg
r2=.2##                         INITIAL RADIUS OF ROTATION IN m
a=.11##               LENGTH OF VERTICAL ARM OF BELL CRANK LEVER IN m
b=.15##               LENGTH OF HORIZONTAL ARM OF BELL CRANK LEVER IN m
h=.004##                  SLEEVE LIFT IN m
N2=240.##                INITIAL SPEED IN rpm
n=7.5##                    FLUCTUATION OF SPEED IN %
##===================================
w2=2.*PI*N2/60.##                  INITIAL ANGULAR SPEED IN rad/s
w1=(100.+n)*w2/100.##              FINAL ANGULAR SPEED IN rad/s
F=2.*a/b*m*w2**2.*r2##              INITIAL COMPRESSIVE FORCE IN N
r1=r2+a/b*h##                    MAX RDIUS OF ROTATION IN m
S=2.*((m*w1**2.*r1)-(m*w2**2.*r2))/(r1-r2)*(a/b)**2.
print'%s %.1f %s %.1f %s'%('INITIAL COMPRESSIVE FPRCE = ',F,' N'' STIFFNESS OF THE SPRING = ',S/1000,' N/m')

INITIAL COMPRESSIVE FPRCE =  557.8  N STIFFNESS OF THE SPRING =  24.1  N/m

Ex9-pg204

In [10]:
##CHAPTER 7 ILLUSRTATION 9 PAGE NO 204
##TITLE:GOVERNORS
##FIGURE 7.3(C)

##===========================================================================================
##INPUT DATA
g=9.81##                   ACCELERATION DUE TO GRAVITY 
PI=3.147
r=.14##                          DISTANCE BETWEEN THE CENTRE OF PIVOT OF BELL CRANK LEVER AND AXIS OF GOVERNOR SPINDLE IN m
r2=.11##                         INITIAL RADIUS OF ROTATION IN m
a=.12##                          LENGTH OF VERTICAL ARM OF BELL CRANK LEVER IN m
b=.10##                          LENGTH OF HORIZONTAL ARM OF BELL CRANK LEVER IN m
h=.05##                         SLEEVE LIFT IN m
N2=240##                         INITIAL SPEED IN rpm
F=30##                           FRICTIONAL FORCE ACTING IN N
m=5##                            MASS OF EACH BALL IN Kg
##==========================================
r1=r2+a/b*h##                    MAX RADIUS OF ROTATION IN m
N1=41.*N2/39.##                 MAX SPEED OF ROTATION IN rpm
N=(N1+N2)/2.##                 MEAN SPEED IN rpm
Fc1=m*(2.*PI*N1/60.)**2.*r1##     CENTRIFUGAL FORCE ACTING AT MAX SPEED OF ROTATION IN N
Fc2=m*(2.*PI*N2/60.)**2.*r2##     CENTRIFUGAL FORCE ACTING AT MIN SPEED OF ROTATION IN N
c1=r1-r##                     FROM FIGURE 7.3(C) IN m
a1=(a**2.-c1**2.)**.5##            FROM FIGURE 7.3(C) IN m
b1=(b**2.-(h/2.)**2.)**.5##             FROM FIGURE 7.3(C) IN m
c2=r-r2##                     FROM FIGURE 7.3(C) IN m
a2=a1##                       FROM FIGURE 7.3(C) IN m
b2=b1##                       FROM FIGURE 7.3(C) IN m
S1=2.*((Fc1*a1)-(m*g*c1))/b1##          SPRING FORCE EXERTED ON THE SLEEVE AT MAXIMUM SPEED IN NEWTONS
S2=2.*((Fc2*a2)-(m*g*c2))/b2##          SPRING FORCE EXERTED ON THE SLEEVE AT MAXIMUM SPEED IN NEWTONS
S=(S1-S2)/h##                   STIFFNESS OF THE SPRING IN N/m
Is=S2/S##                       INITIAL COMPRESSION OF SPRING IN m
P=S2+(h/2.*S)##                  SPRING FORCE OF MID PORTION IN N
n1=N*((P+F)/P)**.5##             SPEED,WHEN THE SLEEVE BEGINS TO MOVE UPWARDS FROM MID POSITION IN rpm
n2=N*((P-F)/P)**.5##             SPEED,WHEN THE SLEEVE BEGINS TO MOVE DOWNWARDS FROM MID POSITION IN rpm
A=n1-n2##                        ALTERATION IN SPEED IN rpm
print'%s %.1f %s %.1f %s '%('INTIAL COMPRESSION OF SPRING= ',Is*100,' cm''ALTERATION IN SPEED = ',A,' rpm')

INTIAL COMPRESSION OF SPRING=  6.8  cmALTERATION IN SPEED =  6.7  rpm

Ex10-pg206

In [1]:
##CHAPTER 7 ILLUSRTATION 10 PAGE NO 206
##TITLE:GOVERNORS
##FIGURE 7.10
import math
##===========================================================================================
##INPUT DATA
PI=3.147
AE=.25##                  LENGTH OF UPPER ARM IN m
CE=.25##                  LENGTH OF LOWER ARM IN m
EH=.1##                   LENGTH OF EXTENDED ARM IN m
EF=.15##                  RADIUS OF BALL PATH IN m
m=5.##                     MASS OF EACH BALL IN Kg
M=40.##                    MASS OF EACH BALL IN Kg
##===================================================================
h=(AE**2.-EF**2.)**.5##           HEIGHT OF THE GOVERNOR IN m
EM=h
HM=EH+EM##                   FROM FIGURE 7.10
N=((895./h)*(EM/HM)*((m+M)/m))**.5
print'%s %.1f %s'%('EQUILIBRIUM SPEED OF GOVERNOR =',N,' rpm')

EQUILIBRIUM SPEED OF GOVERNOR = 163.9  rpm

Ex11-pg207

In [2]:
##CHAPTER 7 ILLUSRTATION 11 PAGE NO 207
##TITLE:GOVERNORS
##FIGURE 7.11
import math
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                  ACCELERATION DUE TO GRAVITY IN N/mm**2
AE=.25##                  LENGTH OF UPPER ARM IN m
CE=.25##                  LENGTH OF LOWER ARM IN m
ER=.175##                 FROM FIGURE 7.11
AP=.025##                 FROM FIGURE 7.11
FR=AP##                   FROM FIGURE 7.11
CQ=FR##                   FROM FIGURE 7.11
m=3.2##                     MASS OF BALL IN Kg
M=25.##                    MASS OF SLEEVE IN Kg
h=.2##                    VERTICAL HEIGHT OF GOVERNOR IN m
EM=h##                    FROM FIGURE 7.11
AF=h##                    FROM FIGURE 7.11
N=160.##                   SPEED OF THE GOVERNOR IN rpm
HM=(895.*EM*(m+M)/(h*N**2.*m))
x=HM-EM##                LENGTH OF EXTENDED LINK IN m
T1=g*(m+M/2.)*AE/AF##     TENSION IN UPPER ARM IN N
print'%s %.3f %s %.1f %s'%('LENGTH OF EXTENDED LINK = ',x,' m''TENSION IN UPPER ARM =',T1,' N')

LENGTH OF EXTENDED LINK =  0.108  mTENSION IN UPPER ARM = 192.5  N

Ex12-pg208

In [3]:
##CHAPTER 7 ILLUSRTATION 12 PAGE NO 208
##TITLE:GOVERNORS
##FIGURE 7.12,7.13
import math
##===========================================================================================
##INPUT DATA
PI=3.147
EF=.20##               MINIMUM RADIUS OF ROTATION IN m
AE=.30##               LENGTH OF EACH ARM IN m
A1E1=AE##              COMPARING FIRUES 7.12&7.13
EC=.30##               LENGTH OF EACH ARM IN m
E1C1=EC##              LENGTH OF EACH ARM IN m
ED=.165##              FROM FIGURE 7.12 IN m
MC=ED##                FROM FIGURE 7.12
EH=.10##                FROM FIGURE 7.12 IN m
m=8.##                  MASS OF BALL IN Kg 
M=60.##                 MASS OF SLEEVE IN Kg
DF=.035##              SLEEVE DISTANCE FROM AXIS IN m
E1F1=.25##             MAX RADIUS OF ROTATION IN m
g=9.81
##=========================================================
alpha=math.asin((EF/AE))*57.3##     ANGLE OF INCLINATION OF THE ARM TO THE VERTICAL IN DEGREES
beeta=math.asin((ED/EC))*57.3##     ANGLE OF INCLINATION OF THE ARM TO THE HORIZONTAL IN DEGREES
k=math.tan(beeta/57.3)/math.tan(alpha/57.3)
h=(AE**2.-EF**2.)**.5##        HEIGHT OF GOVERNOR IN m
EM=(EC**2.-MC**2.)**.5##       FROM FIGURE 7.12 IN m
HM=EM+EH
N2=(895.*EM*(m+(M/2.*(1.+k)))/(h*HM*m))**.5##      EQUILIBRIUM SPEED AT MAX RADIUS
HC=(HM**2.+MC**2.)**.5##                      FROM FIGURE 7.13 IN m
H1C1=HC
gama=math.atan((MC/HM))*57.3
alpha1=math.asin((E1F1/A1E1))*57.3
E1D1=E1F1-DF##                             FROM FIGURE 7.13 IN m
beeta1=math.asin((E1D1/E1C1))*57.3
gama1=gama-beeta+beeta1
r=H1C1*math.sin(gama1/57.3)+DF##                      RADIUS OF ROTATION IN m
H1M1=H1C1*math.cos((gama1/57.3))
I1C1=E1C1*math.cos(beeta1/57.3)*(math.tan(alpha1/57.3)+math.tan(beeta1/57.3))## FROM FIGURE IN m
M1C1=H1C1*math.sin(gama1/57.3)
w1=(((m*g*(I1C1-M1C1))+(M*g*I1C1)/2.)/(m*r*H1M1))**.5##   ANGULAR SPEED IN rad/s
N1=w1*60./(2.*PI)##                         ##SPEED IN m/s
print'%s %.1f %s %.1f %s '%('MINIMUM SPEED OF ROTATION =',N2,' rpm'' MAXIMUM SPEED OF ROTATION = ',N1,' rpm')

MINIMUM SPEED OF ROTATION = 146.6  rpm MAXIMUM SPEED OF ROTATION =  156.3  rpm