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ME 440

Intermediate Vibrations  Th, January 22, 2009

© Dan Negrut, 2009ME440, UW-Madison

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Before e get started! "ast Ti#e$

ME440 "ogisti%s &y''a(us )rading s%he#e *ui%+ oerie of toi%s %oered in ME440

&tarted .hater /, 1unda#enta's of i(rations3

 Today$ W 5ssigned$ /69 7 ro('e# stated in today8s resentation

7 ro('e# stated in handout to (e e#ai'ed to you: W due in one ee+

.oering #ateria' out of /6;, /69, /6/0 E

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E>a#'e, E

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E>a#'e, E

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Non'inear &rings What if you are dea'ing ith nonlinear  srings@

)o (a%+ to the 'inear sring %ase (y %arrying out a 'o%a'

'ineariAation Wor+s on'y hen the defor#ation of the sring is s#a'' (e%ause

you 'ineariAe a(out a oint, see ne>t s'ides!:

 This idea of 'ineariAation goes ay (eyond 'ineariAing srings,a'y it to any fun%tion

Do you a'ays hae to 'ineariAe@ No, (ut then #ight not (e a('e to Cnd ana'yti%a' so'ution to your

ro('e# 1a'' (a%+ on nu#eri%a' #ethod for Cnding aro>i#ation of so'ution

n 440 though, e try to +ee things 'inear

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"inearity, We "i+e

 Fou! 5 'inear ro%ess is a ro%ess in hi%h there is a 'inear

deenden%y (eteen the inut and outut

 

Most of the ro%esses in nature are non'inear

"inear syste#s are easy to dea' ith, and oer the years theengineering %o##unity got a good understanding of this tyeof syste#s

G

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"ineariAation

o one goes a(out erfor#ing a "ineariAation3

5t the system 'ee'$ assu#tions are #ade to a''o a'i%ation of

'as that 'ead to a 'inear set of e

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1un%tion "ineariAation$ ho to

get it We 'i+e 'inear fun%tions6 o do e get one@

ntuitie'y$ aro>i#ate the deenden%y through a straight'ine

;

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Mathe#ati%a''y sea+ing! Two %on%ets are i#ortant$

/: ='a%e of 'ineariAation θr here you %arry out'ineariAation: 2: The Tay'or series e>ansion of the fun%tion f 

 The 'ineariAation of f   at θr is si#'y$

1un%tion "ineariAation$ ho to

get it

9

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E>a#'e "ineariAe sinθ: at θrIπ4

/0

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Ne Toi%$

nertia Mass: E'e#ents Dis%rete #asses$

=oint #ass

as trans'ation on'y, therefore +ineti% energy is

Kigid (ody as (oth trans'ation and rotation, therefore +ineti%

energy is

//

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E

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ELM deter#ined fo''oing four stes$

STEP 1$ dentify the dis'a%e#ent aria('e of interest

STEP 2$ Write don the deCning +ine#ati% %onstraints

STEP 3$ )et e

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E>a#'e fro# te>t(oo+:$ &tart ith a syste# ith four #asses see a::6

1ind the e

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5nother E>a#'e fro# te>t(oo+:$ 1or the /DL1 syste# (e'o, Cnd it8s e

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Ne Toi%$

Da#ing E'e#ents

n rea' 'ife, syste#s don8t i(rate foreer, or ifthey do, there shou'd (e so#ething u#ingenergy into the syste#

Energy initia''y asso%iated ith an os%i''atory#otion is gradua''y %onerted to heat andorsound  This #e%hanis# is +non as da#ing

Most %o##on da#ing #e%hanis#$ is%ous Da#ing .ou'o#( fri%tion Materia' or &o'id or ystereti% Da#ing

/G

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is%ous Da#ing

D: E>erien%ed (y syste#s i(rating in a Ouid

#ediu# su%h as air, ater, oi'

Kesistan%e oPered (y the Ouid to the #oing (ody%auses energy to (e dissiated 5#ount of energy dissiated deends on$

1'uid is%osity i(ration fre

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D Beteen =ara''e' ='ates

 The #ost %o##on da#ing for%e e>ression$

/;

"inear for#, % is a %onstant %oeQ%ient, is re'atie e'o%ity

Why this e>ression@

 Justify its use for da#ing for%e a%ting (eteen to'ates ith re'atie #otion, is%ous Ouid in (eteen

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D Beteen =ara''e' ='ates.ontd:

&y#(o's used$ µ R Ouid is%osity

τ R shear stress de6 in the Ouid 'ayer at a distan%e y of the C>ed 'ate

R 'ate re'atie horiAonta' e'o%ityS no e'o%ity in the erti%a' dire%tion u R e'o%ity of inter#ediate Ouid 'ayersS assu#ed to %hange 'inear'y

/9

Homework

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HomeworkProbem

due on Jan6 29:

Kead E>a#'e /6?9 in the te>t(oo+6 t shos that (ased on so#eassu#tions, the da#ing %oeQ%ient %an (e %o#uted as seeCgure (e'o:$

1or the assign#ent$

/: "ist a'' the assu#tions thatyou (e'iee ere #ade in order

to arrie to e>ression of %reorted a(oe

2: Whi%h one of theseassu#tions see#s to (e theea+est of the# a'' hi%h is theea+est 'in+ in the argu#ent that'eads to this a'ue of the da#6%oeP6 %:@

20

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Da#ing through 1ri%tion

More re%ise'y, through .ou'o#( fri%tion

&eera' other fri%tion #ode's are in use (eside.ou'o#( fri%tion see, for instan%e, "u)re #ode'/:

We8'' sti%+ to the .ou'o#( #ode'

Da#ing for%e is %onstant in #agnitude and oosite tore'atie e'o%ity (eteen (odies in %onta%t =roortiona' to the nor#a' %onta%t for%e (eteen (odies .aused (y ru((ing surfa%es that are dry or ithout

suQ%ient 'u(ri%ation

2//$ Ni%e aer on "u)re #ode' i#'e#entation for nu#eri%a' si#u'ation aai'a('e at$htt$s%itation6ai6orggeta(sser'et)eta(s&er'et@rogInor#a'idIJ.NDDM0000020000040002;/00000/idtyeI%isgifsIye

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yeshttp://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yes

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Da#ing through 1ri%tion

.ntd: E

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Da#ing through 1ri%tion .ntd:

1inding the so'ution is not diQ%u't, (ut tri%+y

We8'' see in .hater 2 that it shou'd assu#e an e>ression of the for#$

2?

 The so'ution 'oo+s 'i+e that for ha'f of eriod, then 5 and B %hange,sin%e the dire%tion of the for%e %hanges!

Keisited in%hater 26

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Materia' of &o'id or ystereti%

Da#ing Materia's are defor#ed, energy is a(sor(ed and dissiated (y the

#ateria'

1ri%tion (eteen interna' 'anes, hi%h s'i and s'ide as the

defor#ations ta+e 'a%e &tress-strain diagra# shos hysteresis 'oo, i6e6,

24

5rea of this 'oo denotes energy 'ost er %y%'e due to da#ing

Ku((er-'i+e #ateria's do this ithout er#anent defor#ation