# Chapter 19: Mechanical Vibrations¶

## Example 19.19-2, Page No:465¶

In [4]:
import math
#Initilization of variables
k=18 #lb/in
g=386 #in/s**2
W=35 #lb

#Calculations
f=(1/(2*pi))*sqrt((k*g/W)) #cps
period=1/f #s

#Result
print'The period of vibration is',round(period,2),"s"

The period of vibration is 0.45 s


## Example 19.19-11, Page No:471¶

In [6]:
import math

#Initilization of variables
ds=0.2 #m
ts=0.05 #m
rhos=7850 #kg/m**3 density of  steel
dw=0.002 #m
lw=0.9 #m
G=80*10**9 #Pa

#Calculations
#Torsional Constant
#Mass Calculations
m=(4**-1)*pi*(ds**2)*ts*rhos #kg
#Moment of Inertia
Io=(0.5)*m*(ds*2**-1)**2 #kg.m**2
#Frequency
f=(1*(2*pi)**-1)*(sqrt(K*Io**-1)) #Hz

#Result
print'The natural frequency of the system is',round(f,2),"Hz"

The natural frequency of the system is 0.24 Hz


## Example 19.19-13, Page No 472¶

In [2]:
import math

#Initilization of variables
m=120 #kg
k=0.3 #m
ls=0.6 #m
ds=0.05 #m
G=80*10**9 #Pa

#Calculations
#Polar Moment of Inertia
J1=m*k**2 #kg.m**2
J2=J1 #kg.m**2
J=(32**-1)*pi*(ds**4) #m**4
#Frequency
f=(1/(2*pi))*(sqrt((J*G*(J1+J2))/(ls*J1*J2))) #Hz

#Result
print'The natural frequency of the torsional oscillation is',round(f,1),"Hz"

The natural frequency of the torsional oscillation is 19.6 Hz


## Example 19.19-14, Page No: 473¶

In [14]:
import math

#Initilization of variables
ds=2 #in
L=15 #in
Wf1=300 #lb
k1=6 #in
Wf2=100 #lb
k2=4 #in
G=12*10**6 #Pa
g=386 #in/s**2

#Calculations
#Moment of inertia of flywheel
Jf=(Wf1*g**-1)*k1**2 #lb-s**2-in
#Moment of inertia of the rotor
Jr=(Wf2*g**-1)*k2**2 #lb-s**2-in
#Moment of inertia of the shaft cross section
J=(32**-1)*pi*ds**4 #in**4
#Frequency
f=((pi*2)**-1)*(sqrt((J*G*(Jf+Jr))*(L*Jf*Jr)**-1)) #cps

#Result
print'The natural frequency of the system is',round(f,1),"cps"

#The answer may wary due to decimal point descrepancy

The natural frequency of the system is 93.9 cps


## Example 19.19-15, Page No: 473¶

In [15]:
import math

#Initilization of variables
W=10 #lb
A=2 #in**2
#Calculations

#Result

The frequency of oscillation is 2.89 rad/s


## Example 19.19-16, Page No:474¶

In [16]:
import math

#Initilization of variables
f=50 #cps
g=386 #in/s**2
E=30*10**6 #lb/in**2
l=4 #in
I=2.08*10**-6 #in**4

#Calculations
W=(3*E*I*g)/(((f*2*pi)**2)*l**3) #lb

#Result
print'The value of W is',round(W,3),"lb"

The value of W is 0.011 lb


## Example 19.19-19, Page No:478¶

In [18]:
import math

#Initilization of variables
F=10 #lb
v=20 #in/s
g=386 #in/s
W=12 #lb
k=20 #lb/in

#Calculations
#Coefficient of damping
c=F*(v**-1) #lb-s/in
#Natural Frequency
#Critical Damping coefficient
cr=(2*W*(g**-1))*wn #lb-s/in
#Damping Coefficient
d=c*(cr**-1)
#Frequency of damped vibrations

#Result
print'The frequency of damped vibrations is',round(wd,1),"rad/s"

# The answer is off by 0.1 units

The frequency of damped vibrations is 24.0 rad/s


## Example 19.19-20, Page No 478¶

In [19]:
import math

#Initilization of variables
t=0.261 #s
d=0.316

#Calculations
delta=d*t*wn #logarithmic decay

#Result
print'The rate of decay is',round(delta,3)

The rate of decay is 2.095


## Example 19.19-24, Page No 483¶

In [27]:
import math

#Initilization of variables
F=9 #N
m=5 #kg
k=6000 #N/m
f1=1 #Hz
f2=5.4 #Hz
f3=50 #Hz

#Calculations
#Natural Frequency
fn=((pi*2)**-1)*(sqrt(k/m)) #Hz
deltaf=F*(k/1000)**-1 #mm
#Part(a)
r1=f1*fn**-1
amp1=deltaf*(1-r1**2)**-1 #mm amplitude
#Part (b)
r2=f2*fn**-1
amp2=deltaf/(1-r2**2) #mm amplitude
#Part (c)
r3=f3*fn**-1
amp3=deltaf/(1-r3**2) #mm amplitude

#Result
print'The amplitudes in part (a),(b) and (c) respectively are',round(amp1,3),"mm ,",round(amp2,1),"mm and",round(amp3,3),"mm"

# The answer for amp2 is incorrect in textbook

The amplitudes in part (a),(b) and (c) respectively are 1.551 mm , 36.9 mm and -0.018 mm


## Example 19.19-25, Page No 483¶

In [28]:
import math

#Initilization of vraiables
g=386 #in/s**2
W=20 #lb
w=600 #rpm
ratio=12**-1

#Calculations
r=sqrt((1*ratio**-1)+1)
fn=((w/60)/r) #cps
k=((fn*2*pi)**2*W)/(g) #lb/in

#Result
print'The value of k is',round(k,1),"lb/in"

The value of k is 15.7 lb/in


## Example 19.19-28, Page No 487¶

In [29]:
import math

#Initilization of variables
X=12 #mm
me_M=1.3 #mm

#Calculations
d=(me_M)/(2*X)

#Result
print'The damping ratio is',round(d,3)

The damping ratio is 0.054