Mathematical Modeling

Transcription

1 ME pplie Engineering nlsis Chper Mhemicl Moeling Professor Ti-Rn Hsu, Ph.D. Deprmen of Mechnicl n erospce Engineering Sn Jose Se Universi Sn Jose, Cliforni, US Jnur

2 Chper Lerning Ojecives Mhemicl moeling in engineering nlsis Elemens of mhemicl moeling Funcions n vriles Differeniion n erivives Inegrion n pplicion of inegrion in engineering nlsis Differenil equions in mhemicl moeling

3 Mhemicl moeling involves: Trnsling phsicl siuion ino mhemicl epressions I is similr cion of wriing MUSIC from he meloies in he mins of gre composers, e.g., Beehoven, Mozr, ec.

4 Engineers uies inclue: CRETION, DECISION MKING, n PROBLEM SOLVING Performing ech of hese uies involves process in reching soluions The sujecs in hese processes re rnsle ino FUNCTIONS, n he fcors h ffecing he vlues of hese sujecs re VRIBLES. VRIBLES inclue: Spil vriles: represene coorine ssems wih se reference poins. Commonl use coorine ssems re: (,, z) in recngulr coorines, or (r, θ, z) in clinricl polr coorines Temporl vrile: ime (),,z n re INDEPENDENT vriles The process for soluions is o inclue he FUNCTIONS wih VRIBLE in PPROPRITE MTH MODELS, n rech mh soluions

5 Emple for FUNCTIONS n Vriles: he funcion P The posiion of cruising vehicle P (he funcion) chnges wih ime (he vrile): Siuion : Cruising consn spee long fl srigh ro: Posiion of he cr ime X() X Siuion : Cruising uphill given ime : X(,) Siuion : Cruising long wining rugge ro: S(,,z,) z

6 Frequen Funcions in ME Engineering nlses Mss (m), weigh (W), Lengh (L), re (), Volume (V) of solis Forces (F) Sress (σ), Srin (ε) in eforme solis Disnce rvele moving rigi o (S) Temperure in solis n fluis (T) Veloci of rigi o or flui (V) Properies of Funcions Funcions m chnge heir vlues wih he chnge of inepenen vriles (spil n emporl) - So, funcions re epenen vriles The vlue of funcion is CONSTNT-epening on he vlues of he ssocie inepenen vriles.

7 The Derivives Funcions represen phsicl quniies in engineering nlsis These phsicl quniies chnge heir vlues wih he chnge of ssocie inepenen vriles, e.g., (,,z,) Chnge of phsicl quniies (i.e., he funcions) cn e CONTINUOUS, or INCREMENTL : Posiion of he cr ime X() X X() Coninuous vriion: Rel X() Incremenl vriion: Unrel

8 Definiion of Derivives Mhemicl epression represening he RTE of CONTINUOUS VRITION of funcions Re of coninuous vriion cn e viewe s vriion of funcion wih INFINITESIMLLY SMLL incremens of he ssocie inepenen vriles: n/or n/or z n/or P() Coninuous vriion: Rel P () Incremenl vriion: Unrel I II III IV >> lim The re of chnge of P(): P lim P ( + ) P( ) is he DERIVTIVE of funcion P() (.) P The res of chnge of P(): I P II P III P No single erivive for ll possile! IV

9 Orers of Derivives ) ( he firs ( s ) orer erivive ) ( ) ( he secon ( n ) orer erivive ) ( ) ( he hir ( r ) orer erivive 4 4 ) ( ) ( he fourh (4 h ) orer erivive ME nlses lmos never involve erivives wih orers higher hn 4

10 Phsicl Represenions of Higher Orer Derivives Deflecion Curve of Ben Bem: Deflecion () ( ) C ( ) The slope of he eflecion curve of he en em evlue locion. Bening momen wih C eing consn. α ( ) Sher force, wih α eing consn.

11 The Inegrls Inegrion is reverse process of iffereniion P() Differeniions evlue he re of chnge of funcion vlues wih infiniesiml incremens of he ssocie vriles, e.g., he re of chnge funcion P() eween n + is: P() P() Inegrion SUMS UP he funcion vlues oine in ll infiniesiml incremens of vriles ssocie wih he funcion, e.g., P() ( ) + P Inegrl, I Sumof P() over rnge of rnge rngep( ) Elemen res: Funcion vlue Funcion Rnge P( )

12 Inegrion of Funcion P() eween n P() Mhemicl Formulion for re Uner he Curve Represene Funcion (): () k- Elemen k k P() Funcion vlue P( ) Elemen re, k P k k Funcion vlue P( ) Becuse he incremen is so smll, he re of Elemen k uner he curve cn e me o equl: k k k Consequenl he ol re uner he curve eween X n is: n k n k l im k k ( ) (.) n k ε k- ε k NOTE: You nee o formule he funcion () o oin he re Eq. (.)

13 Emple.: Deermine he re of ringle: C 4 unis unis B Sep : To se he plne in coorine ssem wih reference O: C B (,4) Sep : Deermine he funcion (): () () () (,) OR (,) (4,) Sep : Use Equion (.) o eermine he re uner funcion (): ( + 4) ( + 4 ) 4 ( ) uni squre Re: Emples.4 (p. ) n.5 (p. )

14 verge of CONTINUOUS vring phsicl quni represene funcion) Funcion () represens he vriion of phsicl phenomenon: () () () re v - The verge vlue of funcion () eween n is oine : v ( ) v re Emple. Deermine he verge emperure of fricion process (p. )

15 Plne re Bone Two Curves () Funcion () Funcion () The re efine he wo funcions eween limis n is: ( ) ( ) [ ( ) ( ) ] (.)

16 Emple (No in he prine lecure noes) Deermine he re of ple wih geomer efine wo curves of ellipse n rc wih imensions shown elow: Hlf-ellipse Hlf-circle 4m m We m eermine he re one hese wo curves in n (,) coorine ssem s shown (using he geomeric smmer ou he -coorine) () for ellipse () for circle

17 Fining he funcions for he ellipicl pr: () for ellipse () for circle The equion for circle is: Thus, he funcion ( ) The equion for n ellipse shown in he figure lef is: + from which, we ge: ( ) Thus, we hve: ( ) + r wih r rius m The re is using Equion (.): 4 4 ( ) ( ) 4 The ove inegrl cn e evlue eiher using he CRC Mh Tles, or clculor o e: 4.7 m. This les o he ol re efine he hlf-ellipse n circle To e: 9.4 m

18 Volume of Solis of Revoluion Solis of revoluion: Solis wih heir geomer smmericl o n is of revoluion. The re commonl use in mchine componen esign. Emples; Cliners; cones, wine n coke oles Mhemicll, he re efine s: f() (is of revoluion) Soli Volume Lengh: The volume of revoluion ou he -is cn e oine : V v π [ f ( ) ] π (.4)

19 The volume of revoluion ou he -is cn lso e oine : g() V π c c [ g( ) ] π (.5) c Emple.6: Deermine he volume of righ-cone using he inegrion meho. () r 4 () Funcion ().5 h 8 unis 8 is of revoluion Use Equion (.4) o eermine he volume of revoluion: V 8 8 [ ( ) ] π (.5) 4. π π 67

20 Emple.8 (p. 7) Deermine he volume of wine h cn e conine in ole shown elow Secion Geomer: cm z cm i 5 z 4r + 74 from curve fiing Secion 4 cm Secion 4 cm 8 cm i r () Inerior profile of wine ole Wine ole is soli of revoluion, wih he coorine z eing he is of revoluion Becuse he is of revoluion coincies wih vericl coorine, we will use Eq. (.5) o ge he volume of he esigne secions The volume of Secion n re righ cliners. There is no nee o use inegrion meho π V 4 l Volume of Secion : ( ) Volume of Secion : π V l.785 () cm 4 cm

21 Volume of Secion he curve secion: z z (r,z8) 5 z -4r + 74 z r(z) (r4,z4) r B using Equion (.5): z V π 76 4 π [ r( z) ] z π r z z 6. cm The ol volume insie he wine ole is: V V + V + V cm

22 Cenroi of Plne res Cenroi is he locion in plne soli, e.g., ples, which siues he cener of grvi I is n imporn prmeer in rigi-o nmic nlsis n compuer-ie-esign Cenroi locion: (, ) elemen re (, ) re, Define: re momen ou he -is: re momen ou he -is: M M Then, he cenroi locion is: M n M Epressions for re momens n cenroi locion: M M [ ( ) ] ( ) (.7) M M [ ( ) ] ( ) (.7)

23 Emple.9 Deermine he locion of he cenroi in ple of semi-circulr geomer - Funcion () The funcion () cn e erive from he equion of circle: + in he forms: ( ) or ( ) B using Equion (.7), we hve: leing o: ( ) π π + 4 Sin The locion is ecuse of he smmer of geomer ou -is

24 Cenroi of Plne res Enclose Muliple Funcions No ville in he prine noes () () () () B c res of iniviul elemens: c c ( ) ( ) ( ) Clcule cenrois of iniviul elemens using Equions (.7) n (.7): (, ) (, ) (, ) for Elemen for Elemen for Elemen (, ) The cenroi for he plne efine cn e oine he following epressions: n

25 Cenroi of Plne res Enclose Muliple Funcions No ville in he prine noes Emples: Deermine he locions of he cenroi in he following ples: ringulr ple join in lrge mechnism Hlf-ellipse Hlf-circle rooic rm specil ple

26 Differenil Equions in Mhemicl Moeling Wh re ifferenil equions? Equions involving erivives (of ifferen orers) How ifferenil equions re erive? The re erive from he lws of phsics Wh re he lws of phsics relevn o engineering pplicions? Funmenl lws of Phsics: Conservion of energ The firs lw of hermonmics Conservion of momenum Conservion of mss pplicion lws of phsics in ME: Newon s lws for soli mechnics (sic n nmic) Fourier lw for he conucion in solis Newon s cooling lw for convecive he rnsfer in fluis Bernoullis lw for flui nmics