10: Vibrations in Bars

Example number 1, Page number 348

In [3]:
#importing modules
import math
from __future__ import division

#Variable declaration
l=1;     #bar length(m)
R=0.01/2;    #radius(m)
V=3560;     #wave velocity(m/sec)
x=3.0112; 

#Calculation
k=R/2;    #geometric radius(m)
fL=V/(2*l);    #fundamental frequency of longitudinal vibrations(Hz)
fT=math.pi*V*k*x**2/(8*(l**2));         #fundamental frequency of transverse vibrations(Hz)

#Result
print "fundamental frequency of longitudinal vibrations is",fL,"Hz"
print "fundamental frequency of transverse vibrations is",round(fT,3),"Hz"
print "answer in the book varies due to rounding off errors"

fundamental frequency of longitudinal vibrations is 1780.0 Hz
fundamental frequency of transverse vibrations is 31.691 Hz
answer in the book varies due to rounding off errors

Example number 2, Page number 348

In [6]:
#importing modules
import math
from __future__ import division

#Variable declaration
C=5050;     #sound velocity(m/sec)
rho=7700;     #steel density(kg/m**3)

#Calculation
Y=C**2*rho;        #youngs modulus(N/m**2)

#Result
print "youngs modulus is",round(Y/10**11,2),"*10**11 N/m**2"
print "answer given in the book is wrong"

youngs modulus is 1.96 *10**11 N/m**2
answer given in the book is wrong

Example number 3, Page number 349

In [10]:
#importing modules
import math
from __future__ import division

#Variable declaration
l=1;     #bar length(m)
R=0.004;    #radius(m)
C=3560;     #wave velocity(m/sec)
x=3.0112; 

#Calculation
k=R/2;    #geometric radius(m)
f1=math.pi*C*k*x**2/(8*(l**2));     #fundamental frequency of transverse mode of vibration(Hz)
f2=math.pi*C*k*5**2/(8*(l**2));     #first overtone of transverse mode of vibration(Hz)

#Result
print "fundamental frequency of transverse vibrations is",round(f1,2),"Hz"
print "first overtone of transverse vibrations is",round(f2,1),"Hz"
print "answer for first overtone in the book varies due to rounding off errors"

fundamental frequency of transverse vibrations is 25.35 Hz
first overtone of transverse vibrations is 69.9 Hz
answer for first overtone in the book varies due to rounding off errors

Example number 4, Page number 349

In [20]:
#importing modules
import math
from __future__ import division

#Variable declaration
l=0.2;     #bar length(m)
C=4990;    #wave velocity(m/sec)
x=3.0112; 
f1=250;     #frequency(Hz)

#Calculation
a=f1*8*(l**2)*math.sqrt(12)/(math.pi*C*(x**2));     #value of a(m)

#Result
print "value of a is",round(a,5),"m"
print "answer given in the book is wrong"

value of a is 0.00195 m
answer given in the book is wrong

Example number 5, Page number 350

In [23]:
#importing modules
import math
from __future__ import division

#Variable declaration
Y=21*10**10;          #youngs modulus(N/m**2) 
rho=8800;             #nickel density(kg/m**3)
R=0.01;               #radius(m)

#Calculation
k=R/2;                     #geometric radius(m)
C=math.sqrt(Y/rho);        #sound velocity(m/sec)
f=C/(2*math.pi*k);         #frequency(Hz)

#Result
print "frequency is",round(f/10**6,3),"MHz"

frequency is 0.155 MHz