Two marks question answers University questions covered

1. 1
DYNAMICS OF MACHINERY – Unit 4
V.Ramakrishnan
ForcedVibration
1. Define ForcedVibration.
ForcedVibrationisthe vibrationof abodyunderthe influence of anexternal force.Whenanexternal
force is acting,the bodydoesnot vibrate withitsownnatural frequency,butvibrateswiththe
frequencyof the appliedexternal force.Ex:Aircompressors,ICengines,Machine Tool.
2. Mentionthe typesof forcedvibrations forcingfunctions.
(a) Periodicforcingfunctions;
(b) Impulsive forcingfunctions;
(c) Randomforcingfunctions.
3. Distinguishbetweenimpulsive forcingfunctionandrandom forcingfunction.
Impulsive forcingfunctionsare nonperiodicandproduce transientvibrations.Theyare quite
common,butdie out soonand hence notimportant.Ex:Rock explosion,punching.
Randomforcingfunctionsare uncommon,unpredictable,andnondeterministic.Theyproduce random
vibrations.Ex:Groundmotionduringearthquake.
4. What are the typesof forcedvibrations?
(a) Dampedvibrations(Dampingisprovided)
(b) Undamped vibrations( NodampingisprovidedC= 0)
5. Show the sketchand Free Body Diagram for forcedvibration with harmonic forcing.
x
F0 Sin ωt
k c
ForcedVibrationwithHarmonicForcing

2. 2
6. Mentionthe equationof motionof spring mass systemwith a harmonic force,F = Fo sin ωt
Equationof motionisgivenby,
𝑚 𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹0 sin( 𝜔𝑡)
Where (mx ̈) is inertia force; (c x ̇) is damping force;(k x) isspring force; and
(Fo sinωt) isthe excitationforce.
7. Explainthe terms transientvibration and steady state vibration.
The equationof displacementisgivenby,
𝑥 = 𝑋−𝜉𝜔 𝑛 𝑡
sin( 𝜔 𝑛 𝑡 + Φ) +
𝐹0 sin( 𝜔𝑡 − 𝛷)
( 𝑘 − 𝑚 𝜔2)2 + ( 𝑐 𝜔)2
The firstpart of the solutiondecayswithtime andvanishes ultimately.Soitiscalledtransientvibration.
The secondpart is a particularintegral andisa sinusoidalvibrationwithconstantamplitude.Hence itis
calledsteadystate vibration.
8. Give the relationfor Amplitude (Max.displacement) offorcedvibration(OR) Amplitude ofsteady
state response.
𝐴 =
(
𝐹0
𝑘⁄ )
√(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
2
+ +(2𝜉 𝜔
𝜔 𝑛⁄ )
2
9. Define phase lag. Give the relation for Phase Lag
Phase lagis the angle bywhichdisplacementvectorlagsforce vector.
𝛷 = tan−1
2𝜉 𝜔
𝜔 𝑛⁄
(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
10. Draw the graph displacementvstime for underdamped forcedvibration system.
Displacement vs Time Plot for under damped system
Time
Displacement
x
X 0
td
ф

3. 3
11. Define the dynamic magnifieror magnificationfactor or magnificationratio.
Dynamicmagnifieristhe ratioof amplitude of steadystate responsetozerofrequencydeflection.
Zerofrequencydeflectionisthe deflectionunderthe actionof impressedforce.
Static deflection, X0 = F0 /k, and A is also referred as X max
𝐴
𝑋0
=
1
√(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
2
+ (2𝜉 𝜔
𝜔 𝑛⁄ )
2
12. Show that amplitude offorced vibrationsdue to unbalancedrotating mass isconstant at high
speeds.
At high speeds the dimensionless amplitude tends to unity. Hence the result,
A / [mo e / m) = 1; A = mo e / m
The amplitude is constant and it is independent of frequency and damping in the system.
Note:mo isreciprocatingmassbalancedinkg (i.e.unbalancedrotatingmassinkg);e iseccentricityinm;
and m is total mass of the engine; A is amplitude in m.
13. How does unbalancedrotating mass cause forced vibration?
For example,electricmotor,turbine,otherrotatingmachinerieshave some amountof unbalance left
in themevenafter rectifyingtheirunbalance onprecisionbalancingmachines.Thisunbalancedrotating
massproduces centrifugal force whichactsasexcitingforce andcausesforcedvibrationsof the machine.
14. Whyis the radius of crank taken as eccentricityin case offorced vibrationsdue to reciprocating
unbalance?
Some amountof unbalancedreciprocatingmassinamachine causesforcedvibrationsof the machine.
Asthe stroke isproportional toradiusof crank,the radiusof crankistakenaseccentricityof reciprocating
mass. Thisunbalancedreciprocatingmassproducesinertiaforce whichacts as excitingforce and causes
forced vibration of the machine.
15. What is base excitationor support motion?
The supportitselfundergoesexcitationwithsome displacement.Forexample,considerthe movement
of automobile trailer or car on wavy road. In this case as the road is sinusoidal in shape, it is considered
that the road itself produces an excitation. This excitation of support produces a force which acts as
excitingforce onthe system.Hence the systemundergoesforcedvibration.Thisiscalledforcedvibration
due to excitation of support.
16. Forced vibrationdue to support excitation:The governingequationis
𝑚 𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝑌 √ 𝑘2 + ( 𝑐𝜔)2 sin( 𝜔𝑡 + 𝛼)
Where (m x ̈) is inertia force; (c x ̇) is damping force; (k x) is spring force; and
𝑌 √𝑘2 + ( 𝑐𝜔)2 sin( 𝜔𝑡 + 𝛼) is the support excitation force.
17. Displacementtransmissionratio (OR) Amplitude ratio
𝐴
𝑌
=
√1 + (2𝜉 𝜔
𝜔 𝑛⁄ )
2
√(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
2
+ +(2𝜉 𝜔
𝜔 𝑛⁄ )
2

4. 4
18. Define transmissibilityorisolationfactor.
It is the ratio of force transmitted to the foundation FT to the force applied Fo
ɛ =
𝐹𝑇
𝐹0
19. Mentionthe angle by which the transmitted force lags the impressedforce.
Angle of lag
ψ = (Φ– α) = tan−1
2𝜉 𝜔
𝜔 𝑛⁄
(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
− tan−1(2𝜉 𝜔
𝜔 𝑛⁄ )
20. What will happenwhen dampersare usedfor frequencyratio greater than √2?
When (ω/ωn ) > √2, transmissibility increasesas damping is increased.Hence dampers should not be
used in this range.
21. Mentionthe equationof phase angle betweenappliedforce and displacement.
𝛷 = tan−1
2𝜉 𝜔
𝜔 𝑛⁄
(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
22. Define vibrationisolation.
The processof reducingthe vibrationsof machines(andhencereducingthe transmittedforce tothe
foundation) usingvibrationisolatingmaterialssuchascork,rubberis calledvibrationisolation.
23. Mentionthe materialsusedfor vibration isolation.
Springs,Dampers, Padsof rubber,corketc.
24. What is the importance of vibrationisolation?
Vibrationsare producedinmachineshavingunbalancedmass.Forexample,inertiaforce developed
ina reciprocatingengineorcentrifugal force developedinarotatingmachinerycausesvibrations.These
vibrationswill be transmittedtothe foundationuponwhichthe machinesare mounted.Thisis
undesirableandshouldbe avoided.Hence vibrationsshouldbe eliminatedoratleastshouldbe reduced
usingvibrationisolators.

5. 5
I. GoverningEquationsfor Harmonic Motion
𝒎 𝒙̈ + 𝒄𝒙̇ + 𝒌𝒙 = 𝑭 𝟎 𝐬𝐢𝐧( 𝝎𝒕)
MagnificationFactor(MF) is
𝐴
𝑋0
=
1
√(1−( 𝜔
𝜔 𝑛⁄ )
2
)
2
++(2𝜉 𝜔
𝜔 𝑛⁄ )
2
Displacementx is
𝑥 = 𝑋−𝜉𝜔 𝑛 𝑡
sin( 𝜔 𝑛 𝑡 + Φ)+
𝐹0 sin( 𝜔𝑡−𝛷)
( 𝑘−𝑚 𝜔2 )2+( 𝑐 𝜔)2
Phase Lag 𝛷 is
𝛷 = tan−1
2𝜉 𝜔
𝜔 𝑛⁄
(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
II. GoverningEquationsfor rotating /reciprocatingunbalance
𝒎 𝒙̈ + 𝒄𝒙̇ + 𝒌𝒙 = 𝒀 √ 𝒌 𝟐 + ( 𝒄𝝎) 𝟐 𝐬𝐢𝐧( 𝝎𝒕 + 𝜶)
𝐴
(
𝑚0
𝑚
𝑒)
=
(
𝜔
𝜔 𝑛
)
2
√[1 − (
𝜔
𝜔 𝑛
)
2
]
2
+ [2𝜉 (
𝜔
𝜔 𝑛
)]
2
𝛷 = tan−1
2𝜉 𝜔
𝜔 𝑛⁄
(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
III. GoverningEquationsfor Support Vibrations
Forced vibration due to support excitation: The governing equation is
𝒎 𝒙̈ + 𝒄𝒙̇ + 𝒌𝒙 = 𝒎 𝟎 𝝎 𝟐
𝒆 𝐬𝐢𝐧( 𝝎𝒕)
Where (m x ̈) is inertia force; (c x ̇) is damping force; (k x) is spring force; and 𝑚0 𝜔2 𝑒sin( 𝜔𝑡)is the
excitation force; m0 is unbalanced reciprocating mass; m is total mass of engine including reciprocating
mass.
Displacement transmission ratio (OR) Amplitude ratio
𝐴
𝑌
=
√1 + (2𝜉 𝜔
𝜔 𝑛⁄ )
2
√(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
2
+ +(2𝜉 𝜔
𝜔 𝑛⁄ )
2
Applied Force, 𝐹0; Force transmitted, 𝐹 𝑇 = 𝐴√ 𝑘 + ( 𝑐𝜔)2
Transmissibility or isolation factor,ɛ =
𝐹𝑇
𝐹0
=
√1 + (2𝜉 𝜔
𝜔 𝑛⁄ )
2
√(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
2
+ +(2𝜉 𝜔
𝜔 𝑛⁄ )
2
Angle by which the transmitted force lags the impressed force.
Angle of lag
ψ = (Φ– α) = tan−1
2𝜉 𝜔
𝜔 𝑛⁄
(1 − ( 𝜔
𝜔 𝑛⁄ )
2
)
− tan−1(2𝜉 𝜔
𝜔 𝑛⁄ )

6. 6
DYNAMICS OF MACHINERY – Unit 4
Important Relations
1. Whatis the relationfor magnificationfactororamplituderationin the case ofa forced
vibration?
𝑴.𝑭.=
𝑨
𝑿 𝟎
=
𝟏
√[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
2. Whatis the relationfor phase lagbetween displacementandforce vector?
𝑷𝒉𝒂𝒔𝒆 𝒍𝒂𝒈, 𝝓 = 𝐭𝐚𝐧−𝟏 {
𝟐𝝃(
𝝎
𝝎 𝒏
)
[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
}
3. Whatis the non-dimensional relationforamplitudeofvibrationdueto rotating/reciprocating
unbalance?
𝑨
(
𝒎 𝟎
𝒎
𝒆)
=
(
𝝎
𝝎 𝒏
)
𝟐
√[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
4. Whatis the non-dimensional relationforamplitudeofforcedvibrationsdueto base
excitation?
𝑫𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕 𝒕𝒓𝒂𝒏𝒔𝒎𝒊𝒔𝒔𝒊𝒐𝒏,
𝑨
𝒀
=
√ 𝟏 + [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
√[ 𝟏 − (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
5. Whatis the relationfor force transmissibilityordisplacementtransmissibility?
𝑻𝒓𝒂𝒏𝒔𝒎𝒊𝒔𝒔𝒊𝒃𝒊𝒍𝒊𝒕𝒚 𝒓𝒂𝒕𝒊𝒐, 𝜺 =
𝑭 𝑻
𝑭 𝟎
=
√ 𝟏 + [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
√[ 𝟏 − (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
6. Whatis the relationfor phase lagbetween motionofmass (A) and motionofsupport (Y) (or)
FT and F0?
𝑷𝒉𝒂𝒔𝒆 𝒍𝒂𝒈, 𝝋 = 𝝓 − 𝜶 = 𝐭𝐚𝐧−𝟏 {
𝟐𝝃(
𝝎
𝝎 𝒏
)
[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
} − 𝐭𝐚𝐧−𝟏 𝟐𝝃 (
𝝎
𝝎 𝒏
)
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