ME 598 – Project
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Transcript of ME 598 – Project
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ME 598 – Noise in Engineering andPhysical Systems
Project Presentation by
Bharath Raghavan
Fall !"5
Acceleration waves in
random media
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Acceleration waves
• #cceleration $aves are moving sing%lars%r&aces $ith a j%m' in 'article acceleration(
• )se&%l &or st%dying the &ormation o&shoc*$aves
• +hey are governed by the Berno%lli e,%ation
- is 'osition. / is the j%m' in 'articleacceleration. 0 and 1 re'resent the dissi'ationand elastic nonlinearity o& the material(
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Acceleration waves inhomogeneous media
• +here e-ists a 2nite distance in $hicha shoc* can be &ormed
• is the critical am'lit%de. and isthe initial am'lit%de
• 3& 4 then the $ave decayse-'onentially
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Acceleration waves in Randommedia
"( Se'aration o& Scales – +he $ave&ront thic*ness is 6
– 7haracteristic grain sie is d
( 7ase " – 6 d : ;eterministic contin%%m limit
– Fl%ct%ations are insigni2cant as the$ave&ront is oblivio%s to local materialdisorder
( 7ase – 6 is 2nite relative to grain sie
– Fl%ct%ations are signi2cant
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Acceleration waves in Randommedia
• 3nterested in Case (2).
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Fig 1. 6d a deterministic contin%%m . 6 is2nite relative to d so statistical @%ct%ations are
signi2cant(
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Stochastic BernoulliEquation
• +he case $here 0 and 1 are'ert%rbed by the same standarderoAmean $hite noise
• #dmit the &ollo$ing decom'osition
• $here S is the intensity val%e(D
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Stochastic BernoulliEquation
• S%bstit%ting the decom'osition in theBerno%lli ;E gives.
• e inter'ret the above e,%ation in theStratonovich sense to obtain the S;E
• $here is the Stratonovich ty'ediGerential o& the einer 'rocess
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Equivalent to S!E•
e $ish to %se the to "ormula tocom'%te vario%s &%nctions o& /(
• So $e need to get the e,%ivalent 3to
S;E &or the Stratonovich S;E(• e de2ne
• +o obtain
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Equivalent to S!E
• e de2ne the 3to S;E as
•
$here
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#oments equations•
e are 'rimarily interested in themoments so $e de2ne
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#nd a''lying 3toIs &orm%la
• +a*ing the e-'ectation
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$he inverse am%litude
• +he inverse am'lit%de is de2ned as
• 3t trans&orms the Jblo$A%'I o& the$ave am'lit%de / to in&nit' to oneo" erocrossing
• #''lying 3toIs &orm%la
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• e obtain
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Sim'li2ed to
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Moments e,%ation – via thes%bstit%tion
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$he inverse am%litude
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$he inverse am%litude
• Set o& e,%ations &or moments
• First moments
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#oments o" the inverseam%litude
• #nalytical e-'ressions &or &rst andsecond moment
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$here
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#ean o" the inverseam%litude
• Set
• e are interested in 2nding thecritical am'lit%de
• #lso and
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Critical Am%litude
• 7ase ":
– No shoc* &ormation as and
• 7ase :
– JBlo$ %'I as and• +he critical inverse am'lit%de is
given as
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7oncl%sions
• bservations: +he stochastic criticalam'lit%de is in general less than thedeterministic critical am'lit%de
• #dditional St%dies: – Stochastic Berno%lli e,%ation $ith t$o
inde'endent $hite noise 'ert%rbations
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Stochastic Berno%lli e,%ation $ith t$ocorrelated $hite noise 'ert%rbations
– Fo**erAPlanc* e,%ation and 'robability o&shoc* &ormation
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