Ch01_ Introduction 2015

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Mechanical

VibrationsSingiresu S.

Rao

SI Edition

Chapter 1

Fundamentals of

Vibration


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© 2005 Pearson Education South Asia Pte Ltd. 2

1. Fundamentals of Vibration2. Free Vibration of Single !F S"stems

#. $armonicall" E%cited Vibration

&. Vibration under 'eneral Forcing(onditions

5. )*o !F S"stems

+. ,ultidegree of Freedom S"stems-. etermination of atural Fre/uencies

and ,ode Shaes

Course Outline


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. (ontinuous S"stems

. Vibration (ontrol

10. Vibration ,easurement and Alications

Course Outline


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1.1 Preliminar" 3emar4s

1.2 rief $istor" of Vibration

1.# 6mortance of the Stud" of Vibration

1.& asic (oncets of Vibration

1.5 (lassification of Vibration

1.+ Vibration Anal"sis Procedure1.- Sring Elements

(hater !utline


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1. ,ass or 6nertia Elements

1. aming Elements

1.10 $armonic ,otion

1.11 $armonic Anal"sis

(hater !utline


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1.1 Preliminar" 3emar4s

7E%amination of 8ibration9s imortant role7 Vibration anal"sis of an engineering s"stem

7 efinitions and concets of 8ibration7 (oncet of harmonic anal"sis for general

eriodic motions


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1.# 6mortance of the Stud" of Vibration

7:h" stud" 8ibration;Vibrations can lead to e%cessi8e deflections

and failure on the machines and structures)o reduce 8ibration through roer design of

machines and their mountings)o utili *elding rocesses)o stimulate earth/ua4es for geological

research and conduct studies in design ofnuclear reactors


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Vibration ? an" motion that reeats itself afteran inter8al of time

Vibrator" S"stem consists of@1 sring or elasticit"

2 mass or inertia# damer

6n8ol8es transfer of otential energ" to 4inetic

energ" and 8ice 8ersa


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egree of Freedom (d.o.f.) ?min. no. of indeendent coordinates re/uiredto determine comletel" the ositions of allarts of a s"stem at an" instant of time

E%amles of single degreeBofBfreedoms"stems@


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E%amles of single degreeBofBfreedoms"stems@


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E%amles of )*o degreeBofBfreedom s"stems@


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E%amles of )hree degree of freedom s"stems@


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Free Vibration@ A s"stem is left to 8ibrate on its o*n after aninitial disturbance and no e%ternal force acts onthe s"stem. E.g. simle endulum

Forced Vibration@ A s"stem that is subCected to a reeatinge%ternal force. E.g. oscillation arises from dieselengines

Resonance occurs *hen the fre/uenc" of thee%ternal force coincides *ith one of thenatural fre/uencies of the s"stem


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Dndamed Vibration@:hen no energ" is lost or dissiated in frictionor other resistance during oscillations

amed Vibration@

:hen any energ" is lost or dissiated infriction or other resistance during oscillations

Linear Vibration@

:hen all basic comonents of a 8ibrator"s"stem= i.e. the sring= the mass and thedamer beha8e linearl"


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1.+ Vibration Anal"sis Procedure

Ste 1@ ,athematical ,odeling

Ste 2@ eri8ation of 'o8erning E/uations

Ste #@ Solution of the 'o8erning E/uations

Ste &@ 6nterretation of the 3esults


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1.+ Vibration Anal"sis Procedure

E%amle of the modeling of a forging hammer@

E l 1 1


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E%amle 1.1,athematical ,odel of a ,otorc"cle


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Dsing mathematical model to reresent theactual 8ibrating s"stemE.g. 6n figure belo*= the mass and daming

of the beam can be disregarded the s"stem

can thus be modeled as a sringBmasss"stem as sho*n.

1. ,ass or 6nertia Elements


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1. aming Elements

Viscous aming@

aming force is roortional to the 8elocit" ofthe 8ibrating bod" in a fluid medium such as air=*ater= gas= and oil.

(oulomb or r" Friction aming@aming force is constant in magnitude butoosite in direction to that of the motion of the8ibrating bod" bet*een dr" surfaces

,aterial or Solid or $"steretic aming@Energ" is absorbed or dissiated b" materialduring deformation due to friction bet*eeninternal lanes


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E%amle 1.10 E/ui8alent Sring and aming(onstants of a ,achine )ool Suort


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E%amle 1.10 Solution


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Let the total forces acting on all the srings and allthe damers be F s and F d = resecti8el" see Fig.1.#-d. )he force e/uilibrium e/uations can thusbe e%ressed as


E.1)(4,3,2,1;

4,3,2,1;

==

==

i xc F

i xk F

idi

i si

E.2)(4321

4321

d d d d d

s s s s s

F F F F F

F F F F F

+++=

+++=


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E.3)( xc F

xk F

eqd

eq s

=

=

*here F s

+ F d

= W = *ith W denoting the total8ertical force including the inertia force acting onthe milling machine. From Fig. 1.#-d= *e ha8e

E/uation E.2 along *ith E/s. E.1 and E.#=

"ield

E.4)(4

4

4321

4321

cccccc

k k k k k k

eq

eq

=+++=

=+++=


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( )1.1kx F =

F ? sring force=k ? sring stiffness or sring constant= and x ? deformation dislacement of one end

*ith resect to the other

1.- Sring Elements

Linear sring is a t"e of mechanical lin4 that isgenerall" assumed to ha8e negligible mass anddaming

Spring force is gi8en b"@


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Work done (U) in deforming a sring or thestrain otential energ" is gi8en b"@

( )2.12

1 2kxU =

1.- Sring Elements


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1.- Sring Elements

Static deflection of a beam at the free end is

gi8en b"@

Spring onstant is gi8en b"@

( )6.13

3

EI

Wl st =δ

W ? mg is the *eight of the mass m= E ? Goung9s ,odulus= and I ? moment of inertia of crossBsection of beam

( )7.13

3

l

EI W k

st

==

δ


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1.- Sring Elements

(ombination of Srings@

!) Springs in parallel H if *e ha8e n sringconstants k 1= k 2= I= k n in parallel = then thee/ui8alent sring constant k e/ is@

( )11.121 ... neq k k k k +++=


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1.- Sring Elements

(ombination of Srings@

") Springs in series H if *eha8e n sring constants k 1=k 2= I= k n in series= then the

e/ui8alent sring constant4e/ is@

( )17.11

...

111

21 neqk k k k +++=


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( )30.1sinsin t A A x ω θ ==

( )31.1cos t Adt

dxω ω =

( )32.1sin 222

2

xt Adt

xd ω ω ω −=−=


Periodic ,otion@ motion reeated after e/ual

inter8als of time$armonic ,otion@ simlest t"e of eriodic

motion

islacement x@ (on #ori$ontal a%is)

Velocit"@

Acceleration@


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(omle% number reresentation of harmonic

motion@

*here i ? J –1) and a and b denote the real and

imaginar" x and y comonents of & =resecti8el".

( )35.1iba X +=



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efinitions of )erminolog"@

Amlitude A is the ma%imum dislacementof a 8ibrating bod" from its e/uilibriumosition

Period of oscillation T is time ta4en tocomlete one c"cle of motion

Fre/uenc" of oscillation f is the no. ofc"cles er unit time

( )59.12

ω

∏

=T

( )60.12

1

π

ω ==

T f



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efinitions of )erminolog"@

atural fre/uenc" is the fre/uenc" *hich as"stem oscillates *ithout e%ternal forces

Phase angle φ is the angular differencebet*een t*o s"nchronous harmonic motions

( )

( ) ( )62.1sin

61.1sin

22

11

φ ω

ω

+=

=

t A x

t A x

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Ch01_ Introduction 2015