Chapter11-Vibrations¶

Ex1-pg290¶

In [1]:
##CHAPTER 11 ILLUSRTATION 1 PAGE NO 290
##TITLE:VIBRATIONS
import math
#calculate frequency of longitudinal vibration and transversve vibaration
##===========================================================================================
##INPUT DATA
PI=3.147
D=.1##                           DIAMETER OF SHAFT IN m
L=1.10##                          LENGTH OF SHAFT IN m
W=450##                          WEIGHT ON THE OTHER END OF SHAFT IN NEWTONS
E=200*10**9##                     YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
##   =========================================================================================
A=PI*D**2./4.##                    AREA OF SHAFT IN mm**2
I=PI*D**4./64.##                   MOMENT OF INERTIA
delta=W*L/(A*E)##               STATIC DEFLECTION IN LONGITUDINAL VIBRATION OF SHAFT IN m
Fn=0.4985/(delta)**.5##         FREQUENCY OF LONGITUDINAL VIBRATION IN Hz
delta1=W*L**3./(3.*E*I)##           STATIC DEFLECTION IN TRANSVERSE VIBRATION IN m
Fn1=0.4985/(delta1)**.5##        FREQUENCY OF TRANSVERSE VIBRATION  IN Hz
##============================================================================================
##OUTPUT
print'%s %.2f %s %.2f %s '%('FREQUENCY OF LONGITUDINAL VIBRATION =',Fn,' Hz' 'FREQUENCY OF TRANSVERSE VIBRATION =',Fn1,'Hz')

FREQUENCY OF LONGITUDINAL VIBRATION = 888.78  HzFREQUENCY OF TRANSVERSE VIBRATION = 34.99 Hz


Ex2-pg290¶

In [2]:
##CHAPTER 11 ILLUSRTATION 2 PAGE NO 290
##TITLE:VIBRATIONS
##FIGURE 11.10
#calculate natural frequency of transverse vibration
#import math
##===========================================================================================
##INPUT DATA
PI=3.147
L=.9##                          LENGTH OF THE SHAFT IN m
m=100##                         MASS OF THE BODY IN Kg
L2=.3##                         LENGTH WHERE THE WEIGHT IS ACTING IN m
L1=L-L2##                       DISTANCE FROM THE OTHER END
D=.06##                         DIAMETER OF SHAFT IN m
W=9.81*m##                      WEGHT IN NEWTON
E=200.*10**9.##                     YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
##==========================================================================================
##CALCULATION
I=PI*D**4./64.##                   MOMENT OF INERTIA IN m**4
delta=W*L1**2*L2**2./(3.*E*I*L)##  STATIC DEFLECTION
Fn=.4985/(delta)**.5##           NATURAL FREQUENCY OF TRANSVERSE VIBRATION
##=========================================================================================
##OUTPUT
print'%s %.1f %s'%('NATURAL FREQUENCY OF TRANSVERSE VIBRATION=',Fn,' Hz')

NATURAL FREQUENCY OF TRANSVERSE VIBRATION= 51.9  Hz


Ex3-pg291¶

In [3]:
##CHAPTER 11 ILLUSRTATION 3 PAGE NO 291 ##TITLE:VIBRATIONS
##FIGURE 11.11
import math
#calculate frequency of longitudnial vibration and frequency of transverse vibration and torisional vibration
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                                      ACCELERATION DUE TO GRAVITY IN N /m**2
D=.050##                                       DIAMETER OF SHAFT IN m
m=450##                                      WEIGHT OF FLY WHEEL IN IN Kg
K=.5##                                       RADIUS OF GYRATION IN m
L2=.6##                                     FROM FIGURE IN m
L1=.9##                                     FROM FIGURE IN m
L=L1+L2
E=200.*10**9##                     YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
C=84.*10**9##                      MODUKUS OF RIDITY OF SHAFT MATERIAL IN Pascals
##=========================================================================================
A=PI*D**2./4.##                               AREA OF SHAFT IN mm**2
I=PI*D**4./64.##
m1=m*L2/(L1+L2)##                           MASS OF THE FLYWHEEL CARRIED BY THE LENGTH L1 IN Kg
DELTA=m1*g*L1/(A*E)##                       EXTENSION OF LENGTH L1 IN m
Fn=0.4985/(DELTA)**.5##                      FREQUENCY OF LONGITUDINAL VIBRATION IN Hz
DELTA1=(m*g*L1**3*L2**3)/(3*E*I*L**3)##        STATIC DEFLECTION IN TRANSVERSE VIBRATION IN m
Fn1=0.4985/(DELTA1)**.5##                    FREQUENCY OF TRANSVERSE VIBRATION  IN Hz
J=PI*D**4./32.##                               POLAR MOMENT OF INERTIA IN m**4
Q1=C*J/L1##                                 TORSIONAL STIFFNESS OF SHAFT DUE TO L1 IN N-m
Q2=C*J/L2##                                  TORSIONAL STIFFNESS OF SHAFT DUE TO L2 IN N-m
Q=Q1+Q2##                                    TORSIONAL STIFFNESS OF SHAFT IN Nm
Fn2=(Q/(m*K**2))**.5/(2.*PI)##                  FREQUENCY OF TORSIONAL VIBRATION  IN Hz
##=======================================================================================
print'%s %.3f %s %.3f %s %.3f %s '%('FREQUENCY OF LONGITUDINAL VIBRATION = ',Fn,' Hz''FREQUENCY OF TRANSVERSE VIBRATION = ',Fn1,' Hz'' FREQUENCY OF TORSIONAL VIBRATION = ',Fn2,' Hz')

FREQUENCY OF LONGITUDINAL VIBRATION =  248.014  HzFREQUENCY OF TRANSVERSE VIBRATION =  14.916  Hz FREQUENCY OF TORSIONAL VIBRATION =  5.673  Hz


Ex6-pg294¶

In [4]:
##CHAPTER 11 ILLUSRTATION 6 PAGE NO 294
##TITLE:VIBRATIONS
##FIGURE 11.14
import math
#calculate frequency of transverse vibration
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                                      ACCELERATION DUE TO GRAVITY IN N /m**2
D=.06##                                       DIAMETER OF SHAFT IN m
L=3.##                                         LENGTH OF SHAFT IN m
W1=1500.##                                     WEIGHT ACTING AT C IN N
W2=2000.##                                     WEIGHT ACTING AT D IN N
W3=1000.##                                     WEIGHT ACTING AT E IN N
L1=1.##                                        LENGTH FROM A TO C IN m
L2=2.##                                        LENGTH FROM A TO D IN m
L3=2.5##                                        LENGTH FROM A TO E IN m
I=PI*D**4./64.
E=200.*10**9.##                     YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
##===========================================================================================
DELTA1=W1*L1**2.*(L-L1)**2./(3.*E*I*L)##           STATIC DEFLECTION DUE TO W1
DELTA2=W2*L2**2.*(L-L2)**2./(3.*E*I*L)##           STATIC DEFLECTION DUE TO W2
DELTA3=W2*L3**2.*(L-L3)**2./(3.*E*I*L)##           STATIC DEFLECTION DUE TO W2
Fn=.4985/(DELTA1+DELTA2+DELTA3)**.5##          FREQUENCY OF TRANSVERSE VIBRATION  IN Hz
##==========================================================================================
print'%s %.3f %s'%('FREQUENCY OF TRANSVERSE VIBRATION = ',Fn,' Hz')

FREQUENCY OF TRANSVERSE VIBRATION =  4.080  Hz


Ex10-pg296¶

In [5]:
##CHAPTER 11 ILLUSRTATION 10 PAGE NO 296
##TITLE:VIBRATIONS
##FIGURE 11.18
import math
#calculate FREQUENCY OF TRANSVERSE VIBRATION
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                                      ACCELERATION DUE TO GRAVITY IN N /m**2
E=200.*10**9##                                  YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
D=.03##                                       DIAMETER OF SHAFT IN m
L=.8##                                        LENGTH OF SHAFT IN m
r=40000.##                                     DENSITY OF SHAFT MATERIAL IN Kg/m**3
W=10.##                                        WEIGHT ACTING AT CENTRE IN N
##===========================================================================================
I=PI*D**4./64.##                                 MOMENT OF INERTIA OF SHAFT IN m**4
m=PI*D**2./4.*r##                                MASS PER UNIT LENGTH IN Kg/m
w=m*g
DELTA=W*L**3./(48.*E*I)##                       STATIC DEFLECTION DUE TO W
DELTA1=5.*w*L**4./(384.*E*I)##                   STATIC DEFLECTION DUE TO WEIGHT OF SHAFT
Fn=.4985/(DELTA+DELTA1/1.27)**.5
##==========================================================================================
print'%s %.3f %s'%('FREQUENCY OF TRANSVERSE VIBRATION = ',Fn,' Hz')

FREQUENCY OF TRANSVERSE VIBRATION =  39.426  Hz


Ex11-pg297¶

In [6]:
##CHAPTER 11 ILLUSRTATION 11 PAGE NO 297
##TITLE:VIBRATIONS
##FIGURE 11.19
import math
#evaluvate CRITICAL SPEED OF SHAFT
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                                      ACCELERATION DUE TO GRAVITY IN N /m**2
E=210.*10**9.##                                  YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
D=.18##                                       DIAMETER OF SHAFT IN m
L=2.5##                                       LENGTH OF SHAFT IN m
M1=25.##                                       MASS ACTING AT E IN Kg
M2=50.##                                       MASS ACTING AT D IN Kg
M3=20.##                                       MASS ACTING AT C IN Kg
W1=M1*g
W2=M2*g
W3=M3*g
L1=.6##                                       LENGTH FROM A TO E IN m
L2=1.5##                                      LENGTH FROM A TO D IN m
L3=2.##                                        LENGTH FROM A TO C IN m
w=1962.##                                      SELF WEIGHT OF SHAFT IN N
##==========================================================================================
I=PI*D**4./64.##                                 MOMENT OF INERTIA OF SHAFT IN m**4
DELTA1=W1*L1**2.*(L-L1)**2./(3.*E*I*L)##           STATIC DEFLECTION DUE TO W1
DELTA2=W2*L2**2.*(L-L2)**2./(3.*E*I*L)##           STATIC DEFLECTION DUE TO W2
DELTA3=W3*L3**2.*(L-L3)**2./(3.*E*I*L)##           STATIC DEFLECTION DUE TO W3
DELTA4=5.*w*L**4./(384.*E*I)##                    STATIC DEFLECTION DUE TO w
Fn=.4985/(DELTA1+DELTA2+DELTA3+DELTA4/1.27)**.5
Nc=Fn*60##                                   CRITICAL SPEED OF SHAFT IN rpm
##========================================================================================
print'%s %.3f %s'%('CRITICAL SPEED OF SHAFT = ',Nc,' rpm')

CRITICAL SPEED OF SHAFT =  3111.629  rpm


Ex12-pg298¶

In [7]:
##CHAPTER 11 ILLUSRTATION 12 PAGE NO 298
##TITLE:VIBRATIONS
##FIGURE 11.20
import math
#calculate  FREQUENCY OF FREE TORSIONAL VIBRATION
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                                      ACCELERATION DUE TO GRAVITY IN N /m**2
Na=1500.##                                     SPEED OF SHAFT A IN rpm
Nb=500.##                                      SPEED OF SHAFT B IN rpm
G=Na/Nb##                                     GERA RATIO
L1=.18##                                       LENGTH OF SHAFT 1 IN m
L2=.45##                                       LENGTH OF SHAFT 2 IN m
D1=.045##                                      DIAMETER OF SHAFT 1 IN m
D2=.09##                                       DIAMETER OF SHAFT 2 IN m
C=84.*10**9##                      MODUKUS OF RIDITY OF SHAFT MATERIAL IN Pascals
Ib=1400.##                        MOMENT OF INERTIA OF PUMP IN Kg-m**2
Ia=400.##                         MOMENT OF INERTIA OF MOTOR IN Kg-m**2

##======================================================================================
J=PI*D1**4./32.##                               POLAR MOMENT OF INERTIA IN m**4
Ib1=Ib/G**2.##                     MASS MOMENT OF INERTIA OF EQUIVALENT ROTOR IN m**2
L3=G**2.*L2*(D1/D2)**4.##            ADDITIONAL LENGTH OF THE EQUIVALENT SHAFT
L=L1+L3##                         TOTAL LENGTH OF EQUIVALENT SHAFT
La=L*Ib1/(Ia+Ib1)
Fn=(C*J/(La*Ia))**.5/(2.*PI)##      FREQUENCY OF FREE TORSIONAL VIBRATION IN Hz
##===================================================================================
print'%s %.2f %s'%('FREQUENCY OF FREE TORSIONAL VIBRATION = ',Fn,' Hz')

FREQUENCY OF FREE TORSIONAL VIBRATION =  4.20  Hz


Ex13-pg300¶

In [8]:
##CHAPTER 11 ILLUSRTATION 13 PAGE NO 300
##TITLE:VIBRATIONS
##FIGURE 11.21
import math
#calculate critical speed of shaft and the range of speed
##===========================================================================================
##INPUT DATA
PI=3.147
g=9.81##                                      ACCELERATION DUE TO GRAVITY IN N /m**2
D=.015##                                        DIAMETER OF SHAFT IN m
L=1.00##                                         LENGTH OF SHAFT IN m
M=15.##                                           MASS OF SHAFT IN Kg
W=M*g
e=.0003##                                        ECCENTRICITY IN m
E=200.*10**9.##                                  YOUNGS MODUKUS OF SHAFT MATERIAL IN Pascals
f=70.*10**6.##                                        PERMISSIBLE STRESS IN N/m**2
##============================================================================================
I=PI*D**4./64.##                MOMENT OF INERTIA OF SHAFT IN m**4
DELTA=W*L**3./(192.*E*I)##      STATIC DEFLECTION IN m
Fn=.4985/(DELTA)**.5##           NATURAL FREQUENCY OF TRANSVERSE VIBRATION
Nc=Fn*60.##                   CRITICAL SPEED OF SHAFT IN rpm
M1=16.*f*I/(D*g*L)
W1=M1*g##                    ADDITIONAL LOAD ACTING
y=W1/W*DELTA##               ADDITIONAL DEFLECTION DUE TO W1
N1=Nc/(1.+e/y)**.5##              MIN SPEED IN rpm
N2=Nc/(1.-e/y)**.5##              MAX SPEED IN rpm
##===========================================================================================
print'%s %.3f %s %.3f %s %.3f %s '%('CRITICAL SPEED OF SHAFT = ',Nc,' rpm''THE RANGE OF SPEED IS FROM',N1,'rpm TO ',N2,' rpm')

CRITICAL SPEED OF SHAFT =  762.330  rpmTHE RANGE OF SPEED IS FROM 709.555 rpm TO  828.955  rpm