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Vehicle Dynamics – Lecture-2

A course on Vehicle Dynamics

By

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Prof. Sarvesh Mahajan BITS, PILANI

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Learning !jec"ives #

Brief History of Vibration

Importance of Study of Vibration

Basic Concept of Vibration

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Classification of Vibration

Vibration #nalysis $rocedure

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Brief $is"ory of Vi!ra"ion #

$eople became interested in %ibration &hen they created the first musical

instruments' probably' &histles or drums(

)he *ree+ philosopher and mathematician $ytha!oras ,.2 /0 B(C( is considered

to be the first person to in%esti!ate musical sounds on a scientific basis(


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#lthou!h the concept of pitch &as de%eloped by the time of $ytha!oras' the

relation bet&een the pitch and the fre3uency &as not understood until the time of

*alileo in the si4teenth century(

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Brief $is"ory of Vi!ra"ion # In about // B(C(' in a treatise called Introduction to Harmonics, Euclid, wrote

briefly about music &ithout any reference to the physical nature of sound(

China e4perienced many earth3ua+es in ancient times'5han! Hen!' &ho ser%ed as a historian and astronomer

in the second century' percei%ed a need to de%elop

an instrument to measure earth3ua+es precisely(


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( (

*alileo &as inspired to study the beha%iour of a

simple pendulum by obser%in! the pendulum

7o%ements of a lamp in a church in $isa

He described the dependence of the fre3uency

of %ibration on the len!th of a simple pendulum'

alon! &ith the phenomenon of sympathetic %ibrations ,resonance(

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Brief $is"ory of Vi!ra"ion #

otable amon! the many contributions of 8aylei!h is the method of findin! the

fundamental fre3uency of %ibration of a conser%ati%e system by ma+in! use of the

principle of conser%ation of ener!y no& +no&n as 8aylei!h s method(

)his method pro%ed to be a helpful techni3ue for the solution of difficult %ibration

problems( #n e4tension of the method' &hich can be used to find multiple natural

-


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' (

In 1:/2 "rahm in%esti!ated the importance of torsional %ibration study in the

desi!n of the propeller shafts of steamships(

#mon! the modern contributers to the theory of %ibrations' the names of Stodola'De La%al' )imoshen+o' and 7indlin are notable(

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Im%or"ance of S"u&y of Vi!ra"ion #

7ost human acti%ities in%ol%e %ibration in one form or other( "or e4ample' &e hear

because our eardrums %ibrate and see because li!ht &a%es under!o %ibration(

Breathin! is associated &ith the %ibration of lun!s and &al+in! in%ol%es ,periodicoscillatory motion of le!s and hands( Human speech re3uires the oscillatory

motion of laryn!es ,and ton!ues(


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applications of %ibration' such as the desi!n of machines' foundations' structures'en!ines' turbines' and control systems(

7ost prime mo%ers ha%e %ibrational problems due to the inherent unbalance in

the en!ines(

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In turbines' %ibrations cause spectacular mechanical failures( <n!ineers ha%e not

yet been able to pre%ent the failures that result from blade and dis+ %ibrations in

turbines(

the structures desi!ned to support hea%y centrifu!al machines' li+e motors and

turbines' or reciprocatin! machines' li+e steam and !as en!ines and reciprocatin!


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' ( '

machine component sub=ected to %ibration can fail because of material fati!ueresultin! from the cyclic %ariation of the induced stress(

"urthermore' the %ibration causes more rapid &ear of machine parts such as

bearin!s and !ears and also creates e4cessi%e noise( In machines' %ibration canloosen fasteners such as nuts( In metal cuttin! processes' %ibration can cause

chatter' &hich leads to a poor surface finish(

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Basic 'once%" of Vi!ra"ion #

#ny motion that repeats itself after an

inter%al of time is called vibration or oscillation.

)he s&in!in! of a pendulum and the motion of apluc+ed strin! are typical e4amples of %ibration(

# %ibratory system' in !eneral' includes -


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a means or stor n! potent a ener!y spr n! '

a means for storin! +inetic ener!y ,mass'

and a means by &hich ener!y

is !radually lost ,damper(

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Num!er of Degree of (ree&om #

)he minimum number of independent coordinates re3uired to determine

completely the positions of all parts of a system at any instant of time defines the

number of de!rees of freedom of the system(

#ll systems sho&n belo& and $endulum sho&n in abo%e slide are ha%in! sin!le

de!ree of freedom


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#ll systems sho&n belo& and $endulum sho&n in abo%e slide are ha%in! t&o

de!ree of freedom


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#ll systems sho&n belo& and $endulum sho&n in abo%e slide are ha%in! three

de!ree of freedom


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Some systems' especially those in%ol%in! continuous elastic members' ha%e an

infinite number of de!rees of freedom( # Cantile%er beam is ha%in! infinite no( of

De!ree of "reedom' Since the beam has an infinite number of mass points' &e

need an infinite number of coordinates to specify its deflected confi!uration(


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Systems &ith a finite number of de!rees of freedom are called &iscre"e or lum%e&

%arame"er sys"ems' and those &ith an infinite number of de!rees of freedom are

called con"inuous or &is"ri!u"e& sys"ems.

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'lassifica"ion of Vi!ra"ion #

Vibration can be classified in many &ays but here &e &ill discuss some important

classifications –

(ree an& (orce& vi!ra"ions #

(ree Vi!ra"ion. If a system' after an initial disturbance' is left to %ibrate on its o&n'


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(

system( )he oscillation of a simple pendulum is an e4ample of free %ibration(

(orce& Vi!ra"ion. If a system is sub=ected to an e4ternal force ,often' a repeatin!

type of force' the resultin! %ibration is +no&n as forced %ibration( )he oscillation

that arises in machines such as diesel en!ines is an e4ample of forced %ibration(

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Dam%e& an& )n&am%e& vi!ra"ions #

If no ener!y is lost or dissipated in friction or other resistance durin! oscillation'

the %ibration is +no&n as un&am%e& vi!ra"ion(

If any ener!y is lost in this &ay' ho&e%er' it is called &am%e& vi!ra"ion(

#


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If all the basic components of a %ibratory system the sprin!' the mass' and the

damper beha%e linearly' the resultin! %ibration is +no&n as linear vi!ra"ion( If'

ho&e%er' any of the basic components beha%e nonlinearly' the %ibration is called

nonlinear vi!ra"ion(

If the %ibration is linear' the principle of superposition holds' and the mathematical

techni3ues of analysis are &ell de%eloped( "or nonlinear %ibration' the

superposition principle is not %alid' and techni3ues of analysis are less &ell +no&n(

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De"erminis"ic an& non*&e"erminis"ic vi!ra"ions #

If the %alue or ma!nitude of the e4citation ,force or motion actin! on a %ibratorysystem is +no&n at any !i%en time' the e4citation is called deterministic( )he

resultin! %ibration is +no&n as &e"erminis"ic vi!ra"ion.


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n some cases' t e e4c tat on s non e"erm n s" c or ran om> t e %a ue o t e

e4citation at a !i%en time cannot be predicted( In these cases' a lar!e collection of

records of the e4citation may e4hibit some statistical re!ularity( It is possible to

estimate a%era!es such as the mean and mean s3uare %alues of the e4citation(

<4amples of random e4citations are &ind %elocity' road rou!hness' and !round

motion durin! earth3ua+es(

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De"erminis"ic an& non*&e"erminis"ic vi!ra"ions #


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aett zg 524-l2

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