# Chapter 13: Governors¶

## Example 1, Page 459¶

In [2]:
#Variable declaration
#all lengths are in inches
W=120.#lb
w=15#lb
AB=12
BF=8
BC=12
BE=6.5
g=35230#inches rpm

#Calculations
#at Minimum radius
AF=(AB**2-BF**2)**(1./2)
CE=(BC**2-BE**2)**(1./2)
k2=(BE*AF)/(CE*BF)
N2=(((W/2)*(1+k2)+w)*g/(w*AF))**(1./2)
#At MAximum radius
BF1=10
BE1=8.5
AF1=(AB**2-BF1**2)**(1./2)
CE1=(BC**2-BE1**2)**(1./2)
k1=(BE1*AF1)/(CE1*BF1)
N1=(((W/2)*(1+k1)+w)*g/(w*AF1))**(1./2)

#Results
print "N1 (corresponding maximum radius) = %.1f rpm\nN2 (corresponding minimum radius) = %.1f rpm"%(N1,N2)

N1 (corresponding maximum radius) = 201.7 rpm
N2 (corresponding minimum radius) = 176.2 rpm


## Example 2, Page 462¶

In [4]:
import math

#Variable declaration
BG=4#in

#Calculations&Results
#solution a
w=15#lb
W=120.#lb
k=.720
BD=10.08#in
CE=BD
DG=BD+BG
#by equating quations 13.2 and 13.10 and reducing, we get
w1=(W/2*(1+k))/(((W/2*(1+k)+w)*DG/(BD*w))-1)
print "Weight of ball = %.3f lb"%w1
#solution b
CD=6.5#in
BC=12#in
BF=10.#in
AB=12#in
CG=(DG**2+CD**2)**(1./2)
gama=math.atan(CD/DG)
bita=math.asin(CD/BC)
alpha1=math.asin(BF/AB)
bita1=math.asin(8.5/BC)
gama1=gama+bita1-bita
F=((w1+W/2)*8.471*(math.tan(alpha1)+math.tan(bita1)))/(CG*math.cos(gama1))-(w1*math.tan(gama1))
print"F1= %.1f lb"%F
r1=CG*math.sin(gama1)+1.5#radius of rotation
N1=(30/math.pi)*(F*32.2*12/(w1*r1))**(1./2)
print "r1= %.2f in\nN1= %.1f rpm"%(r1,N1)

Weight of ball = 10.313 lb
F1= 113.1 lb
r1= 10.85 in
N1= 188.7 rpm


## Example 3, Page 466¶

In [6]:
import math

#Variable declaration
w=3#lb
g=32.2
N2=300

#Calculations
w2=(N2*math.pi/30)
r2=3./12#ft
N1=1.06*N2
r1=4.5/12#ft
a=4#in
b=2#in
ro=3.5/12#ft
F2=w*w2**2*r2/g
F1=F2*N1**2*r1/(N2**2*r2)
p=2*a**2*(F1-F2)/(b**2*(r1-r2))
Fc=F2+(F1-F2)*(.5/1.5)
N=(Fc*g/(ro*w))**(1./2)*30/math.pi
Ns=math.ceil(N)

#Result
print "N = %.1f rpm"%Ns

N = 308.0 rpm


## Example 4, Page 468¶

In [8]:
import math

#Variable declaration
w=5#lb
g=32.2
N2=240#rpm

#Calculations
w2=(N2*math.pi/30)
r2=5./12#ft
N1=1.05*N2
r1=7./12#ft
a=6.#in
b=4#in
pb=3./2
F2=w*w2**2*r2/g
F1=F2*N1**2*r1/(N2**2*r2)
p=2*(a/b)**2*((F1-F2)/(r1*12-r2*12)-4*pb)

#Result
print "Equivalent stiffness; p = %.f lb/in"%p

Equivalent stiffness; p = 23 lb/in


## Example 5, Page 470¶

In [9]:
import math

#Variable declaration
w=3.#lb
W=15.#lb
g=32.2
r2=2.5/12#ft
N2=240.#rpm

#Calculations
w2=N2*math.pi/30
F2=w*w2**2*r2/g
a=4.5#in
b=2#in
sleevelift=0.5
r1=r2*12+a*sleevelift/b#the increase of radius for a scleeve lift is 0.5 in
N1=1.05*N2
F1=(N1/N2)**2*(r1/(r2*12))*F2
#a) at minimum radius
S2=(F2*a/b-w)*2-W
#b) At maximum radius
DB=r1-r2*12
BI=1.936#in
AD=a
BI=b
S1=2*(F1*AD/BI-w*(DB+BI)/BI)-W
k=(S1-S2)/sleevelift

#Result
print "Stiffness of the spring is %.1f lb/in"%k

Stiffness of the spring is 59.3 lb/in


## Example 6, Page 475¶

In [10]:
#Variable declaration
c=0.01
W=120#lb
w=15#lb
k=.720
h=8.944#in

#Calculations
Q=c*(W+2*w/(1+k))
x=(2*c/(1+2*c))*(1+k)*h
P=Q*x

#Result
print "Governor power = Q*x = %.3f in lb"%P

Governor power = Q*x = 0.415 in lb


## Example 7, Page 475¶

In [12]:
#Variable declaration
r=6#in
a=6#in
b=4#in
#from example 4(using conditions and calculating constants A and B) we get F=11.1r-14.6
#when r=6 , F= 52
F=52#lb

#Calculations
inc=2*.01*52#increase neglecting very small values
F1=F+inc
F2=2*a*inc/b#Force required to prevent the sleeve from rising
F3=F2/2#Force is uniformly distributed
r2=-14.6/(F1/r-11.1)#from equation 1
x=r2-r#increase in radius of rotation
lift=b*x/a#sleeve lift
P=F3*lift#governor power

#Result
print "Governor power = %.3f in lb"%P

Governor power = 0.479 in lb


## Example 10, Page 483¶

In [13]:
#Variable declaration
fs=3.#lb
W=90#lb
w=15#lb

#Calculations
#fb=(fs/2)*(1+k)*(r/h) From equation 13.31
k=1#All the arms are of equal length
#fb=fs*(r/h)
#comparing the above result with the one obtained from example 8 , F=(W+w)*(r/h), we get coefficient of insensitiveness = k = (N1-N2)/N = fs/(W+w)
k=fs/(W+w)
K=k*100

#Result
print "Coefficient of insensitiveness = %.3f"%k

Coefficient of insensitiveness = 0.029


## Example 11, Page 484¶

In [14]:
#Variable declaration
a=4.5#in
b=2#in
r1=2.5#in
r2=4.5#in
F2=12.25#lb
F1=25.4#lb
fs=1.5#lb

#Calculations
fb=(fs/2)*(b/a)
#At minimum radii
k1=fb/F2
#At maximum radii
k2=fb/F1

#Results
print "Coefficient of insensitiveness\nAt minimum radii = %.4f\nAt maximum radii = %.4f"%(k1,k2)

Coefficient of insensitiveness
At minimum radii = 0.0272
At maximum radii = 0.0131