Finite Element Analysis of a Rotating Disk_Eliannah Hunderfund_Final Term Project

8/15/2019 Finite Element Analysis of a Rotating Disk_Eliannah Hunderfund_Final Term Project

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Finite ElementAnalysis of aRotating Disk

Eliannah HunderfundMANE 4240 – Introdution to Finite Elements!rofessor Ernesto "utierre#$Mira%eteRensselaer !olytehni Institute& Hartford& '(May )*& 20)+


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(a,le of 'ontentsA,strat-----------------------------------------------------------------------------------------------------------.

Introdution and /akground---------------------------------------------------------------------------.

ystem 1%er%ie---------------------------------------------------------------------------------------------4Figure )------------------------------------------------------------------------------------------------------4

Figure 2------------------------------------------------------------------------------------------------------4

Figure .------------------------------------------------------------------------------------------------------4

Figure 4------------------------------------------------------------------------------------------------------+

(heory 3 Methodology-------------------------------------------------------------------------------------+

Analytial olution-------------------------------------------------------------------------------------------

Figure +---------------------------------------------------------------------------------------------------------*

Figure ---------------------------------------------------------------------------------------------------------5

Figure *---------------------------------------------------------------------------------------------------------5

Figure 5---------------------------------------------------------------------------------------------------------5

'omsol Multi6hysis----------------------------------------------------------------------------------------7

Figure 7---------------------------------------------------------------------------------------------------------7

Figure )0-------------------------------------------------------------------------------------------------------7

Figure ))-------------------------------------------------------------------------------------------------------7

Figure )2-----------------------------------------------------------------------------------------------------)0Figure ).-----------------------------------------------------------------------------------------------------)0

Figure ).-----------------------------------------------------------------------------------------------------))

Figure )4-----------------------------------------------------------------------------------------------------))

Figure )+-----------------------------------------------------------------------------------------------------)2

Figure )-----------------------------------------------------------------------------------------------------)2

Figure )*-----------------------------------------------------------------------------------------------------).

Figure )5-----------------------------------------------------------------------------------------------------).

Figure )7-----------------------------------------------------------------------------------------------------)4

Results-----------------------------------------------------------------------------------------------------------)4

Figure 20-----------------------------------------------------------------------------------------------------)+

Figure 2)-----------------------------------------------------------------------------------------------------)

Figure 22-----------------------------------------------------------------------------------------------------)

)


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Figure 2.-----------------------------------------------------------------------------------------------------)*

'onlusion-----------------------------------------------------------------------------------------------------)*

A66endi8 A----------------------------------------------------------------------------------------------------)7

A66endi8 /----------------------------------------------------------------------------------------------------2)

Referenes-----------------------------------------------------------------------------------------------------22

2


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A,strat

(his 6ro9et e%aluates the :nite element analysis of stress and dis6laement

indued through entri6etal fore in a thin rotating disk of uniform thikness

and onstant angular %eloity ith a )m outer diameter and a -)m inner

diameter made of (i $4 (itanium- (he stress and dis6laement ere

determined using Ma6le softare to o,tain an e8at solution for the )D ase&

and then om6ared to the FEM solution o,tained through 'omsol

Multi6hysis softare for the )D and 2D ase-

Introdution and /akground

(his 6ro9et e%aluates a thin rotating disk of uniform thikness and onstant

angular %eloity of .000 re%;m hen designing a thin rotating disk


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r


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r

a $=

t0-)m

(he solid mehanis 6rini6le to onsider in this 6ro,lem of entri6etal

aeleration and entri6etal fore as desri,ed in Figure .& Euation ) and 2&

,elo-

Figure .

a $ r ∙ω2

E-


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Figure 4

(heory 3 Methodology

(he go%erning euation to determine the dis6laement and stress for the )D

rotating disk ill ,e the seond order dierential euation desri,ed in

Euation .-

−∂2

u

∂ r2 −

1

r

∂u

∂ r +

u

r2=

(1−v2 ) ρ ω2 r E E-


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Radial tressσ r=

E

(1−v2) G< εr+v∙ εt ¿ E-


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Figure +

(he gra6h o,tained for radial and tangential strain ith radial 6osition fromra to , on the 8$a8is and strain on the y$a8is is in Figure -

Figure

(he gra6h o,tained for radial and tangential


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this re%eals that the disk should ,e limited to less than ++*+ rot;min ith

some safety fator to 6re%ent material failure- (his relationshi6 is desri,ed

in Euation )2-

At ω ++*+ rot;min& σ vm=880 MPa σ tensile , yield E-


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Figure )2

Mesh i#e Dis6laement at r,


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An e8tremely :ne mesh as a66lied to the 2D geometry& hih is shon in

Figure )+-

Figure )+ (he 2D dis6laement as o,tained using 'omsol Multi6hysis and shos

%ery dierent results than the )D& ,eause there is no a thikness hih

must ,e onsidered- (he line gra6h shos the dis6laement of the ,ottom

surfae at y0 from Figure )+- (he ma8imum dis6laement no ours at

r0-72 and is 0-000..5025*525+.4 m- (he line gra6h for the u


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Figure )

(he on Mises tress as also o,tained from 'omsol Multi6hysis and as

gra6hed in a line gra6h and surfae gra6h in Figures )* and )5& res6eti%ely-

In the 2D surfae 6lot it is e%ident that the disk is strething to ,eome

thinner due to the entri6etal fore- (he ma8imum on Mises tress is still at

ra and is 2.*)740*-*2724 or 2.*-2 M!a-

)4


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Figure )*

Figure )5

Finally& the .D surfae 6lot of the on Mises 6lot is shon in Figure )7 andsim6ly an e86ansion of the 2D urfae !lot& a66lied around .0 degrees-

)+


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Figure )7

Results

In the results setion of this 6ro9et the %alues o,tained analytially& through

Ma6le& and using FEM& using 'omsol Multi6hysis& ill ,e om6ared- (he :rst

thing to ,e om6ared ill ,e the dis6laement u


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0 0-2 0-4 0- 0-5 ) )-0

0

0

0

0

0

0

0

0

0

Dis6laement u


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Figure 2)

(his is a 6rimary e8am6le of hen a smaller mesh si#e does not 6ro%ide

more aurate results- Although it is minisule& the 6erent error for the

e8tremely :ne mesh si#e is atually greater than the normal mesh si#e- (he

e8tremely :ne mesh as not neessary to o,tain a more aurate solution

for this sim6le ase& and sometimes inreasing mesh si#e an ause a less

fa%ora,le di%ide of the area of interest-

In Figure 22 is the om6arison ,eteen the e8at )D solution and the 2D

FEM solution for dis6laement- (his shos a .0L dierene ,eteen the )D

ma8 dis6laement and the FEM 2D ma8 dis6laement& ,ut again this is

onser%ati%e error-

MethodMa8 Dis6laementu


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tresses than e86eted- (he sha6e of the gra6h and the 6oint of ma8 stress&

at ra& is the same ,eteen the )D and 2D ases-

0-00E00

+-00E0*

)-00E05

)-+0E05

2-00E05

2-+0E05

.-00E05

Von Mises Stress

2DKonKMisesK'omsol )DKonKMisesKMa6le

r (meters)

Von Mises Stress (Pa)

Figure 2.In Figure 24 is the om6arison ,eteen the e8at )D solution and the 2D

FEM solution for on Mises tress- (his shos a 20L dierene ,eteen the

)D ma8 stress and the FEM 2D ma8 stress& ,ut this error is onser%ati%e and

ould not 6ut the alulation at risk for 6ro%iding a false 6ositi%e margin to

yield strength of safety fator-

Methodon Mises tress at r a

2DK'omsolKE8tremelyKFine 2-.*E05

)DKMa6leKE8at 2-5+E05L error of )DKMa6le om6ared to2DK'omsol )7-77

)7


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'onlusion

For the 6ro,lem of e%aluating a sim6li:ed rotating disk to :nd dis6laement and

stress& the results o,tained through this 6ro9et sho that FEM using 'omsol

Multi6hysis an o,tain an e8tremely aurate solution in om6arison- In addition&

for sim6le geometry& an e8tremely :ne mesh si#e is not neessary- It is 6ossi,le to

model the )D rotating disk in less time than it takes to o,tain the e8at solution

analytially& and hen this is e86anded to a 6ro,lem ith om6le8 geometry and

transient onditions the reuired ork in6ut to o,tain an aurate solution ill ,e

muh less in FEM softare than using analytial methods- In the ase of a rotating

disk& it as also shon that the )D ase& hih ill generally ,e uiker and sim6ler

to e%aluate& an sho onser%ati%e and somehat aurate results- For the

onnetion to an aeros6ae rotor& the sim6li:ed ase ould ,e useful as a :rst 6ass

to understand here stresses and dis6laements are in a 6reliminary design ,efore

in%esting more time into a 2D or .D analysis- In the ase of om6le8 .D geometry in

rotors& the only method for 6rediting dis6laement and stress ithout atually

testing is FEM& so its auray is im6erati%e-

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A66endi8 AMa6le 'ode In6ut

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Referenes

htt6;;solidmehanis-org;te8t;'ha6ter4K);'ha6ter4K)-htm

htt6;;-e6-r6i-edu;hartford;Psarri;M;Readings;.27K04-6df

htt6;;asm-mate,-om;searh;6ei:Material-as6Q,assnumM(!4)

htt6;;home6ages-engineering-aukland-a-n#;P6kel0)+;olidMehanis/ooks;!artKI

I;04KElastiity!olar;Elastiity!olarsK04K/odyForesRotatingDiss-6df

htt6;;-doiser,ia-n,-rs;img;doi;)4+0$++54;200;)4+0$++5400)0+A-6df

htt6;;ourses-ashington-edu;me.+4a;(hikL20>alledL20'ylinders-6df

htt6;;-h6msl-neu-edu;ontent;20)2!a6ers;Hagh6anah$AME$20)2-6df

htt6;;-solid-iei-liu-se;Eduation;(MHO);formulaKta,les;formulasKta,leKsolidKmehanis-6df

htt6s;;indigo-ui-edu;,itstream;handle;)002*;)0)5+;'hianeseKtefano-6dfQ

seuene)

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http://solidmechanics.org/text/Chapter4_1/Chapter4_1.htmhttp://www.ewp.rpi.edu/hartford/~sarric/SMS/Readings/32669_04.pdfhttp://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MTP641http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_II/04_ElasticityPolar/ElasticityPolars_04_BodyForcesRotatingDiscs.pdfhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_II/04_ElasticityPolar/ElasticityPolars_04_BodyForcesRotatingDiscs.pdfhttp://www.doiserbia.nb.rs/img/doi/1450-5584/2006/1450-55840601065A.pdfhttp://courses.washington.edu/me354a/Thick%20Walled%20Cylinders.pdfhttp://www.hpmsl.neu.edu/content/2012Papers/Haghpanah-ASME-2012.pdfhttp://www.solid.iei.liu.se/Education/TMHL61/formula_tables/formulas_table_solid_mechanics.pdfhttp://www.solid.iei.liu.se/Education/TMHL61/formula_tables/formulas_table_solid_mechanics.pdfhttps://indigo.uic.edu/bitstream/handle/10027/10185/Chianese_Stefano.pdf?sequence=1https://indigo.uic.edu/bitstream/handle/10027/10185/Chianese_Stefano.pdf?sequence=1http://solidmechanics.org/text/Chapter4_1/Chapter4_1.htmhttp://www.ewp.rpi.edu/hartford/~sarric/SMS/Readings/32669_04.pdfhttp://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MTP641http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_II/04_ElasticityPolar/ElasticityPolars_04_BodyForcesRotatingDiscs.pdfhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_II/04_ElasticityPolar/ElasticityPolars_04_BodyForcesRotatingDiscs.pdfhttp://www.doiserbia.nb.rs/img/doi/1450-5584/2006/1450-55840601065A.pdfhttp://courses.washington.edu/me354a/Thick%20Walled%20Cylinders.pdfhttp://www.hpmsl.neu.edu/content/2012Papers/Haghpanah-ASME-2012.pdfhttp://www.solid.iei.liu.se/Education/TMHL61/formula_tables/formulas_table_solid_mechanics.pdfhttp://www.solid.iei.liu.se/Education/TMHL61/formula_tables/formulas_table_solid_mechanics.pdfhttps://indigo.uic.edu/bitstream/handle/10027/10185/Chianese_Stefano.pdf?sequence=1https://indigo.uic.edu/bitstream/handle/10027/10185/Chianese_Stefano.pdf?sequence=1

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Finite Element Analysis of a Rotating Disk_Eliannah Hunderfund_Final Term Project