Chapter 15: Moments of Inertia

Example 15.15-11, Page no 305

In [2]:
import math

#Initilization of variables
y1=1 #in
y2=4 #in
d1=2.2-1 #in
d2=4-2.2 #in
A1=12 #in**2
A2=8 #in**2
b1=6 #in
b2=2 #in
h1=2 #in
h2=4 #in

#Calculations
y_bar=(A1*y1+A2*y2)/(A1+A2) #in
I1=(12**-1)*(b1)*(h1**3) #in**4
I2=(12**-1)*(b2)*(h2**3) #in**4
#Using Parallel Axes Theorem
I=(I1+(A1*d1**2))+(I2+(A2*d2**2)) #in**4

#Result
print'The moment of inertia is',round(I,1),"in**4"

The moment of inertia is 57.9 in**4

Example 15.15-12, Page no 306

In [7]:
import math

#Initilization of variables
d=60 #mm diameter of the hole
#Areas
At=100*100 #mm**2
Ab=200*100 #mm**2
Ac=((pi/4)*d**2) #mm**2
bt=100 #mm
ht=100 #mm
bb=200 #mm
hb=100 #mm
#Distance of centroids of each area
yt=150 #mm
yb=50 #mm
yc=150 #mm

#Calculations
y_bar=((At*yt)+(Ab*yb)-(Ac*yc))/(At+Ab-Ac) #mm
#Distances
dt=yt-y_bar #mm
db=y_bar-yb #mm
dc=yc-y_bar #mm
#Values of Inertia
It=(12**-1)*(bt)*(ht**3) #mm**4
Ib=(12**-1)*(bb)*(hb**3) #mm**4
Ic=(4**-1)*(pi)*((d/2)**4) #mm**4
#Moment of inertia
I=((It+At*dt**2)+(Ib+Ab*db**2)-(Ic+Ac*dc**2)) #mm**4

#Result
print'The moment of inertia is',round(I,1),"mm**4"

The moment of inertia is 77156533.6 mm**4

Example 15.15-14, Page no 308

In [8]:
import math

#Initilization of variables
b1=2 #in
b2=4 #in
h1=8 #in
h2=2 #in
bo=8 #in
ho=8 #in
bi=4 #in
hi=4 #in

#Calculations
I1=(12**-1)*(b1)*(h1**3) #in**4
I2=(12**-1)*(b2)*(h2**3) #in**4
I=2*(I1+I2) #in**4
Io=(12**-1)*(bo)*(ho**3) #in**4
Ii=(12**-1)*(bi)*(hi**3) #in**4
I_bar=Io-Ii #in**4

#Result
print'The moment of inertia is',round(I_bar),"in**4"

The moment of inertia is 320.0 in**4

Example 15.15-15, Page no 308

In [4]:
import math

#Initilization of variables
b1=75 #mm
b2=12 #mm
h1=12 #mm
h2=162 #mm
d1=75 #mm

#Calculations
A=(h2*b2)+(2*b1*h1) #mm**2
I1=(12**-1)*(b1)*(h1**3)+(b1*h1*d1**2) #mm**4
I2=(12**-1)*(b2)*(h2**3) #mm**4
I_bar=2*I1+I2 #mm**4
k=sqrt(I_bar/A) #mm

#Result
print'The axial moment of inertia is',round(I_bar,1),"mm**4"
print'The radius of gyration is',round(k,1),"mm"

# Here value of k is off by 0.1 mm

The axial moment of inertia is 14398128.0 mm**4
The radius of gyration is 62.0 mm

Example 15.15-20, Page no 311

In [11]:
import math

#Initilization of variables
r=50 #mm

#Calculations
Ixy=(8**-1)*(50**4) #mm**4

#Result
print'The moment of inertia is',round(Ixy,2),"mm**4"

The moment of inertia is 781250.0 mm**4

Example 15.15-24, Page no 314

In [16]:
import math

#The notation has been changed for ease

#Calculations
x=(5*1*3.5+8*1*0.5)/(5*1+8*1) #in
y=(5*1*0.5+8*1*4)/13 #in
#Moment of inertia 
Ix=(12**-1)*(5)*(1**3)+(5*2.15*2.15)+(12**-1)*(1*8**3)+(8*1.35**2) #in**4
Iy=(12**-1)*(1)*(5**3)+(5*1.85*1.85)+(12**-1)*(8)*(1**3)+(8*1.15**2) #in**4
Ixy=(8*1*(-1.15)*1.35)+(5*1*1.85*(-2.15)) #in**4
#Mohr circle calculations
d=0.5*(Ix+Iy) #distance to center of the cirlce 
r=sqrt((21**2)+(32.3**2)) 
maxI=d+r #in**4
theta=arctan(32.3/21)*(180/pi) #degrees maxI occurs at this angle
minI=d-r #in**4

#Result
print'The moment of inertias are as follows:'
print'Ix=',round(Ix,1),"in**4"
print'Iy=',round(Iy,1),"in**4"
print'Ixy=',round(Ixy,1),"in**4"
print'maxI=',round(maxI,1),"in**4"
print'minI=',round(minI,1),"in**4"

The moment of inertias are as follows:
Ix= 80.8 in**4
Iy= 38.8 in**4
Ixy= -32.3 in**4
maxI= 98.3 in**4
minI= 21.2 in**4

Example 15.15-25, Page no 315

In [18]:
import math

#Notations have been changed

#Calculations
x=-(25*125*0.5*125+25*100*0.5*25)/(25*125+25*100) #mm
y=(25*125*0.5*25+25*100*75)/5625 #mm 
Iy=(12**-1)*25*125**3+25*125*(62.5-40.3)**2+(12**-1)*100*25**3+100*25*(40.3-12.5)**2 #mm**4
Ix=Iy #mm**4 for L-section
#The second computation checks the first
Ixy=(125*25*22.2*27.8)+(100*25*(-27.8)*(-34.7)) #mm**4
#Mohr Circle analysis
Imax=Ix+Ixy #mm**4
Imin=Ix-Ixy #mm**4

#Result
print'The moment of inertias are as follows:'
print'Ix=',round(Ix,2),"mm**4"
print'Iy=',round(Iy,2),"mm**4"
print'Ixy=',round(Ixy,2),"mm**4"
print'Imax=',round(Imax),"mm**4"
print'Imin=',round(Imin,2),"mm**4"

The moment of inertias are as follows:
Ix= 7671443.75 mm**4
Iy= 7671443.75 mm**4
Ixy= 4340275.0 mm**4
Imax= 12011719.0 mm**4
Imin= 3331168.75 mm**4

Example 15.15-30, Page no 320

In [18]:
import math

#Initilization of variables
rho=490 #lb/ft**3
t=0.02 #in
d=4 #in
r=d/2 #in
g=32.2 #ft/s**2

#Calculations
W=(pi*r**2*t*rho)*1728**-1 #lb
#Mass
m=W*g**-1 #slugs
#Momemt of inertia
I=(4**-1)*m*(r*12**-1)**2 #slug-ft**2

#Result
print'The moment of inertia is',round(I,6),"slug-ft**2"

The moment of inertia is 1.5e-05 slug-ft**2

Example 15.15-36, Page no 322

In [27]:
import math

#Initilization of variables
#The integration involves variables hence the direct formula is being used in this coding
m=500 #kg
R=0.25 #m
h=0.5 #m

#Calculations
Ix=(3*10**-1)*m*R**2 #kg.m**2
Iy=(3*5**-1)*m*((4**-1)*R**2+h**2) #kg.m**2

#Result
print'Hence proved that Ix=',round(Ix,2),"kg.m**2" 
print'and Iy=',round(Iy,1),"kg.m**2"

Hence proved that Ix= 9.38 kg.m**2
and Iy= 79.7 kg.m**2

Example 15.15-37, Page no 323

In [20]:
import math

#Initilization of variables
delta=450 #lb/ft**3
h1=9*12**-1 #ft
h2=10*12**-1 #ft
ro1=4*12**-1 #ft
ri1=2*12**-1 #ft
ro2=18*12**-1 #ft
ri2=16*12**-1 #ft
a=2.5*24**-1 #ft
b=3.5*24**-1 #ft
l=1 #ft
g=32.2 #ft/s**2

#Calculations
Whub=(pi*ro1**2-pi*ri1**2)*h1*delta #lb
Wrim=(pi*ro2**2-pi*ri2**2)*h2*delta #lb
#For one spoke
Wspoke=(pi*a*b*l*delta) #lb
#Moment of inertia calculations
Ihub=0.5*(Whub*g**-1)*(ro1**2+ri1**2) #lb-s**2-ft
Irim=0.5*(Wrim*g**-1)*(ro2**2+ri2**2) #lb-s**2-ft
Ispoke=6*((12**-1)*(Wspoke*g**-1)*l**2+(Wspoke*g**-1)*h2**2) #lb-s**2-ft
Iwheel=Ihub+Irim+Ispoke #lb-s**2-ft

#Result
print'The moment of inertia of the wheel is',round(Iwheel,1),"lb-s**s-ft"

The moment of inertia of the wheel is 38.1 lb-s**s-ft