Chapter 6 Free Vibration

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FREE VIBRATION

Presented by

Tuan Mohd Hafeez binTuan Ibrahim


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Introduction Vibration refers to mechanical oscillations about an equilibrium

point. The oscillations may be periodic such as the motion of a

pendulum or random such as the movement of a tire on a gravelroad.

Vibration is occasionally "desirable". For example the motion of a

tuning fork, the reed in a woodwind instrument or harmonica, or the

cone of a loudspeaker is desirable vibration, necessary for the

correct functioning of the various devices. ore often, vibration is undesirable, wasting energy and creating

unwanted sound ! noise. For example, the vibrational motions of

engines, electric motors, or any mechanical device in operation are

typically unwanted. uch vibrations can be caused by imbalances in

the rotating parts, uneven friction, the meshing of gear teeth, etc.#areful designs usually minimi$e unwanted vibrations.


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Why Do We Care About Vibration?

Vibration is%

&asted energy

' ma(or cause of component failure #ause of aircraft noise which contributes to crew and

passenger discomfort

For some cases, vibration are desirable


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Desirable Vibration

MRI UltrasoundAtomic Force Microscopy

Time keeping


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Undesirable Vibrations

Tacoma Narrows Bridge (19!"


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#art$%uake&induced 'irational Failure)

*aiti +!1!

,ic$uan +!!-

.$ili +!1!

http://www.cbsnews.com/2300-202_162-10002626-12.htmlhttp://www.time.com/time/photogallery/0,29307,1953257_2024509,00.html


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'iration .ontrol and Isolation


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Terinolo!y )eriod *T + ! time for harmonic motion to complete cycle. -nit is second

*s+

T /π 0ω s #ycle ! #omplete movement in period of time. )eak to )eak 1 The distance from the top of the positive peak to bottom of

the negative peak. )eak 1 The measurement from the $ero line to the top of the positive peak. 'verage *'V2+ 1 .345 of peak. 6oot ean quare *6+ 1 .575 of peak.


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Frequency

The rate of mechanical oscillation in a period of time.

Frequency can be expressed in one of the followingunits%

6) 1 6evolutions per inute

#) 1 #ycles per inute

#) 1 #ycles per econd 8$ 1 8ert$, 8$ 1

#ycle per econd *to convert from 8$ to 6) or#), apply the following formula% 8$ 9 37 6).


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Ty"es o# $ibration %&co"es o# 'ecture(

Vibration

-ndamped

Force VibrationFree Vibration

:amped


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F6;; Vibration ! Free vibration occurs when a mechanical system is set

off with an initial input and then allowed to vibrate freely.


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Damped

Undamped

,implest system) one /degree o0 0reedom

Free Vibration


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&i"le )aronic *otion %&)*(

=scillation motion may repeat itself periodically.


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#rank of radius A rotates about the point O and at the

end of crankP

&hen crank rotates with angular velocity w , the endpoint S of the slotted link and hence the mass m of thespring1mass system are displaced from their middleposition by an amount x *in time t+ given by%

> ' sin ? ' sin @t

&here ? A7B, /57B, x 7C ? D7B, x 1'

*E.+


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The velocity of the mass m at time t is given by%

The acceleration is

x A sin θ A sin ωt


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8armonic motion can be represented conveniently


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From description above, oscillation of mass are inharmonic motion and can be expressed as%

z A cos ωt B sin ωt*E.3+

&here% A and B are constant number found from the initialcondition of the system and the ω is the frequency ofoscillation in rad0s.


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Unda"ed Free Vibration + ,endulu&yste

The natural frequency of the system can be determined in two ways%

i. GewtonHs Iaw of otion

ii. )rinciple of #onservation of ;nergy


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,endulu &yste - Ne.ton/s 'a. o#*otion


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,endulu &yste - ,rinci"le o# Con$ersiono# Ener!y


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Unda"ed Free Vibration - *ass &"rin!&yste


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*ass &"rin! &yste - Ne.ton/s 'a. o#*otion


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*ass &"rin! &yste - ,rinci"le o#Conser$ation o# Ener!y


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Unda"ed Free Vibration - Inertia &ha#t0Disc &yste


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Inertia &ha#t0Disc &yste - Ne.ton/s 'a.o# *otion


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Inertia &ha#t0Disc &yste - ,otential Ener!y*ethod


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E1ui$alence *ethod


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E1ui$alence 'en!th &yste


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&"rin! In &eries


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&"rin! In ,arallel


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E1ui$alence Distributed *ass


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E1ui$alence *ass ,oint


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E1ui$alence ,endulu &yste


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l


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E2a"le

2iven one pendulum system with the rope length is half

meter. Find the natural frequency in rad0s and periodictime in second.

l


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E2a"le


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E l


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E2a"le

DA*,ED FREE VIBRATION


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Introduction


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'iscous 2amping


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.oulom or dry 0riction damping

8ere the damping is constant in magnitude but opposite in direction

to that of the vibration body.

Chapter 6 Free Vibration

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