Failure Analysis in a Cracked Cantilever Beam Under

8/10/2019 Failure Analysis in a Cracked Cantilever Beam Under

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Failure analysis in a crackedFailure analysis in a crackedcantilever beam undercantilever beam under

dynamic conditionsdynamic conditions

Batch number : 16Batch number : 16


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Current research going in this area :Current research going in this area :


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How we are going toHow we are going to

approach ?approach ?

We analyze this project by including the concepts of vibrationsWe analyze this project by including the concepts of vibrationsand finite element methodsand finite element methods

Purpose of using vibrationsPur

pose of using vibrations

y using vibrations we will understand dynamic behavior ofy using vibrations we will understand dynamic behavior of

given systemgiven system y plotting graph between amplitude vs! time we cany plotting graph between amplitude vs! time we can

observe where the critical failure occursobserve where the critical failure occurs

we will plot this graph by using mat lab!we will plot this graph by using mat lab!

purpose of using finite element methods:

purpose of using finite element methods:

y using fem the given system is divided into numbers ofy using fem the given system is divided into numbers of

elements and then we will calculate mass and stiffness matriceselements and then we will calculate mass and stiffness matrices

which are then idealized into a e"uivalent mass and springwhich are then idealized into a e"uivalent mass and spring

systemsystem


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#teps involved to achieve the given project#teps involved to achieve the given project

#tep $: observing the free vibrations of given#tep $: observing the free vibrations of givencantilever beam under the presence of damping withcantilever beam under the presence of damping with

out any crac%out any crac%

#tep &: observing the forced vibrations of given#tep &: observing the forced vibrations of givensystem under damping with out any crac%system under damping with out any crac%

#tep ': observing the forced vibrations of given#tep ': observing the forced vibrations of givensystem under damping by locating a crac% at somesystem under damping by locating a crac% at some

point in the beampoint in the beam


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#o far we covered in this#o far we covered in thisproject:project:

(irstly we assumed a cantilever beam )freely(irstly we assumed a cantilever beam )freelyvibrating* with out any crac% and then we calculatedvibrating* with out any crac% and then we calculated

e"uivalent mass and stiffness matrices which aree"uivalent mass and stiffness matrices which are

idealized into mass and spring systemidealized into mass and spring system

With help of e"uivalent mass and spring system weWith help of e"uivalent mass and spring system we

calculated different natural fre"uencies and thencalculated different natural fre"uencies and then

plotted a graph between amplitude vs! time by usingplotted a graph between amplitude vs! time by usingmat lab program code!mat lab program code!


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#teps involved to plot the graph between#teps involved to plot the graph between

amplitude vs! timeamplitude vs! time

+"uivalent mass and spring system!+"uivalent mass and spring system!


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Cantilever beam with square cross sectionCantilever beam with square cross section

Dimensions assumed:Dimensions assumed: Length o the beam ! 1"""mmLength o the beam ! 1"""mm

#oment o inertia ! $e6mm%$#oment o inertia ! $e6mm%$

&rea o cross section! 6'()mm%(&rea o cross section! 6'()mm%(

#aterial assumed#aterial assumed #ild steel "*(+ carbon#ild steel "*(+ carbon

,!("" -.a,!("" -.a

Density ! /)0" kgm%2Density ! /)0" kgm%2


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,pplying elimination approach u$-u&-. we get a &/&,pplying elimination approach u$-u&-. we get a &/&matri0 in reduced formmatri0 in reduced form

$!.e1.!..2/ 34!5$'4 6$!&'&'7 6$!&'&' .!&&89/3u'$!.e1.!..2/ 34!5$'4 6$!&'&'7 6$!&'&' .!&&89/3u'u9 - .u9 - .

From u=mu!1 we get u2 and u$From u=mu!1 we get u2 and u$

Finally we get dis.lacement matri uFinally we get dis.lacement matri u

!trans.ose o 5" " "*1$' 1*"0228!trans.ose o 5" " "*1$' 1*"0228

>ote: the above matri is or only one>ote: the above matri is or only one

value o lambda*value o lambda* ?ince there are our ty.es o degrees o?ince there are our ty.es o degrees o

reedom@ our ty.es o res.onses arereedom@ our ty.es o res.onses are

.ossible@ here we calculated only or one.ossible@ here we calculated only or one

value o res.onse*value o res.onse*


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;raph between amplitude vs! time;raph between amplitude vs! time

(inally we re"uired to draw response graph(inally we re"uired to draw response graphbetween amplitude and time for freebetween amplitude and time for free

vibrating systemvibrating system Please note that an assumption of zero damping is typically not accurate!Please note that an assumption of zero damping is typically not accurate!

his resistance will damp the vibration and dissipate energy7 the oscillatorymotion caused by the initial disturbance will eventually be reduced to zero!motion caused by the initial disturbance will eventually be reduced to zero!

therefore we consider damping to certain e0tent by introducingtherefore we consider damping to certain e0tent by introducingdamping factor as zi-.!!in this way the resultant motion isdamping factor as zi-.!!in this way the resultant motion is

oscillatory with decreasing amplitudeoscillatory with decreasing amplitude

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Failure Analysis in a Cracked Cantilever Beam Under