OPTIMIZATION OF TOW-PLACED, TAILORED COMPOSITE LAMINATES
|
|
- Myron Morris
- 5 years ago
- Views:
Transcription
1 6 H INERNAIONAL CONFERENCE ON COMPOSIE MAERIALS OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES Adiana W. Blom* Mosafa M. Abdalla* Zafe Güdal* *Delf Univesi of echnolog he Nehelands Kewods: vaiable siffness ow placemen cuved fibes seamline hickness ovelap Absac Fibe-einfoced composies ae usuall designed ug consan fibe oienaion in each pl. In ceain cases howeve a vaing fibe angle migh be favoable fo sucual pefomance. his possibili can be full uilized ug ow placemen echnolog. Because of he fibe angle vaiaion ow-placed couses ma ovelap and pl hickness will build up on he suface. his hickness build up affecs manufacuing ime sucual esponse and suface quali of he finished poduc. his pape will pesen a mehod fo designing composie plaes and shells ug vaing fibe angles. he hickness build-up is pediced as funcion of pl angle vaiaion ug a smeaed appoach. I is found ha he hickness build-up is no unique and depends on he chosen sa locaions of fibe couses. Opimal fibe couses ae fomulaed in ems of minimizing he maimum pl hickness maimizing suface smoohness o combining hese objecives wih and wihou peiodic bounda condiions. Inoducion In indus fibe-einfoced composies ae usuall designed ug a consan fibe oienaion in each pl. he fibe angles in hese laminaes ae picall 0 90 and ±45 degees. adiionall he choice of hese la-ups was moivaed b manufacuabili while nowadas la-ups wih changing o even nonconvenional fibe angles ae avoided because of he lack of allowables. Howeve eseach on composies wih a vaing in-plane fibe oienaion has shown ha vaiable siffness can be beneficial fo sucual pefomance [-] because vaiable-siffness laminaes ae able o edisibue he loading as opposed o consansiffness laminaes. In mos cases cuvilinea fibe pahs manufacued b ow placemen ae used o consuc he vaiable-siffness laminaes [458-02]. Jegle aing and Güdal [8-0] designed vaiable-siffness fla plaes wih holes and demonsaed hei effeciveness b building and eg seveal specimens. Due o fibe angle vaiaion a ow-placed shell picall ehibis gaps and/o ovelaps beween adjacen couses and pl hickness will change along he suface [8-02]. he amoun of gap/ovelap affecs sucual esponse manufacuing ime and suface quali of he finished poduc. his pape pesens a mehod fo designing composie plaes and shells ug vaing fibe angles. he hickness build-up is pediced as funcion of pl angle vaiaion ug a seamline analog. I is found ha he hickness build-up is no unique and depends on he chosen sa locaions of fibe couses. Opimal disibuions of fibe couses ae fomulaed in ems of minimizing he maimum pl hickness o maimizing suface smoohness eihe wih o wihou peiodic bounda condiions. Subsequenl he discee hickness build-up esuling fom he ow placemen pocess will be shown as compaison o he smeaed hickness appoimaion. Finall a numbe of applicaions fo he developed mehods and suggesions fo fuue eseach will be given. 2 Seamline Analog Fo he consucion of discee fibe pahs a seamline analog is being used. In his case
2 ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal each seamline epesens he ceneline of a couse o if he couse widh is made infiniel small each seamline will epesen a gle fibe. Mahemaicall a seamline is epesened b a seam funcion ( ) C () which connecs all he poins wih a consan value C. Fo a given fibe angle vaiaion () he seamlines can be found b solving he following paial diffeenial equaion: d ds d d + ds ds + 0 (2) A unique soluion fo he seam funcion (and hus he locaion of he seam lines) depends on he bounda condiions. Befoe seeking a soluion o he seam funcion addiional consideaions elevan o he phsical epesenaion of he fibe pahs ae in ode. ae discee a fis appoimaion o he amoun of ovelap could be made b smeaing ou his discee ovelap o fom a coninuous hickness disibuion. In his case he smeaed hickness will be invesel popoional o he disance beween adjacen couses which can be eplained as follows. If a numbe of N couses wih a given widh w c and hickness has a fied volume V and if hese successive couses ae placed close han he widh of he couses hen he oal widh coveed is less hen N w c and he hickness has o be inceased in ode o mainain he same maeial volume V. When he disance beween wo seamlines is dn hen / dn (as eplained above). Since d/dn and d beween wo seamlines is consan accoding o Eq.. he hickness will be popoional o n as follows: (3) dn n ( d ) d If d is assumed o be hen. which can be used o deive a diec coelaion beween he hickness disibuion and he fibe angle vaiaion: ( ) n s ln (4) Fig.. Definiions As saed ealie he seamlines epesen he cenal pah of a finie widh couse. Unless he seamlines ae paallel he successive couses will alwas ovelap each ohe when no gaps ae allowed beween hem (o alenaivel he gaps will fom beween he passes if wo successive finie widh passes ae no allowed o ovelap). he amoun of ovelap depends on he disance beween he couse cenelines: when he disance is deceased he ovelap aea is inceased. Alhough in eali hese ovelaps in which s and n epesen he angen and nomal ve o a seamline especivel as shown in Fig.. he phsical eplanaion of Eq. 4 is ha he change in hickness along a seamline depends on he change of he fibe oienaion pependicula o ha seamline. Since boh ve s and n depend on he given fibe angle disibuion () he onl unknown in Eq. 4 is he hickness. he hickness can now be deemined b solving his equaion bu ce i is a diffeenial equaion bounda condiions ae needed in ode o obain a unique soluion. In accodance wih seamline heo bounda condiions ae onl needed a he inflow bounda whee he inflow bounda is defined b: 2
3 OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES s N 0 (5) whee s is he veco angen o he seamline and N is he ouwad nomal veco o he bounda as shown in Fig.. B changing he hickness a he inflow bounda he hickness disibuion inside he domain and a he ouflow boundaies will change. 3. Deemining Bounda Condiions hee eis an infinie numbe of possible bounda condiions fo which he hickness disibuion associaed wih he seamlines can be found bu he mos difficul pa is o find he ones ha ae phsicall sensible fo he poblem in hand. In his pape he bounda condiions ae esablished such ha he fulfill a ceain opimali cieion. he opimali cieia demonsaed in his pape ae minimum maimum hickness maimum smoohness and combinaions of hese wo. In addiion o he opimali cieion consains such as a minimum of one o peiodic bounda condiions can be enfoced as well. 3. Geneal Soluion B ug he following change of vaiables: τ ln Eq. 4 becomes: s τ n (6) Above equaion is solved numeicall b disceizing he deivaives so ha i is wien as: [ M ] τ B (7) whee [M] is he mai ha epesens he lef hand side of Eq. 6 τ is he veco ha epesens τ a eve gid poin and B is he veco ha epesens he igh hand side of Eq. 6 as well as he bounda condiions. If he hickness a he inflow boundaies is assumed o be one evewhee (τ 0) a nominal soluion can be found fo τ which will be efeed o as τ 0. A geneal soluion of Eq. 7 can be epessed as: [ ] τ in τ τ 0 + (8) whee each column j in mai [] epesens he influence of bounda gid poin j on he hickness disibuion in he complee domain while saisfing Eq. 7. Since hese columns ae independen of each ohe and ce Eq. 7 is a linea equaion an linea combinaion of hese columns also epesens a soluion as given b Eq. 8. he enies in τ in all ende he hickness a a gle poin on he inflow bounda. B subsiuing Eq. 8 in Eq. 7 he hickness can be opimized fo one of he cieia menioned ealie b ug τ in as design vaiables. Ofen i is desied o have a leas one lae of maeial evewhee so ha no gaps eis. heefoe i is equied ha he hickness ove he enie domain is a leas one ( τ 0) in all opimizaions descibed below. 3.2 Minimized Maimum hickness Minimizing he maimum hickness of he plae is he fis opimali cieion ha will be elaboaed on in his pape. his cieion is elevan fo judging how pacical he esuling hickness disibuion would be in a eal life sucue as well as fo deemining if i is possible o manufacue a plae wih consan hickness fo he given fibe angle vaiaion. If he hickness build-up is oo sevee (i.e. if one poin is 00 o 000 imes hicke han anohe poin) i will no be applicable o ealisic sucues. In ode o solve he min-ma poblem he bound fomulaion as inoduced b Olhoff [3] is used. his fomulaion inoduces a new vaiable α which epesens he maimum hickness and which also seves as he new objecive funcion fo he minimizaion. Addiionall a consain on he hickness a each gid poin is being inoduced so ha he hickness neve eceeds he minimized maimum hickness α: 3
4 ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal s.. F min τ α α i 2... i N g (9) whee N g is he numbe of gid poins. he design vaiables ha esul fom he opimizaion ae subsiued in Eq. 8 and hen he hickness disibuion is found b changing τ i vaiables again: e. i 3.3 Maimized Smoohness Anohe possible opimizaion objecive is o maimize he smoohness of he hickness disibuion of he composie panel. Alhough in eali he change in hickness will alwas be discee due o he discee naue of ow couses i would sill be desiable fo pl dops/ovelaps o be disibued houghou he panel ahe han o be concenaed a paicula egions. In ode o achieve his smoohness is defined as he nom of he ae of change of hickness. Smoohness is maimized b minimizing he H -nom of he hickness: min τ [ K]τ (0) 2 whee [K] is he mai ha disceizes he Laplacian. Subsiuing he epession fo τ (Eq. 8) in Eq. 0 gives: 2 τ [ K] τ 2τ 0 [ K ] τ 0 + τ 0 [ K ] + τ in [ ] [ K ][ ] τ in 2 τ in () he fis em in his equaion is consan so ha he objecive funcion o be minimized is: wih F [ K ] τ in f τ in min 2 τ in (2) [ K ] [ ] [ K ][ ] f τ 0 [ K ][ ] (3) he minimum of Eq. 2 can be found b diffeeniaing i and equaing i o zeo so ha: [ K ] in f τ (4) his is a linea ssem ha can be solved fo τ in. Howeve he [K ]-mai is one ime gula and heefoe one en of τ in is given an assumed value so ha he ssem can be solved. Afe he soluion is subsiued in Eq. 8 a consan can be added o τ such ha he condiion of τ 0 is being me (his will no change he H -nom bu will change he absolue value of hickness). 3.4 Combined Objecive Funcion Since boh minimizing he maimum hickness and maimizing he smoohness ae valid opimizaion cieia designes migh conside combining he wo in ode o obain a bee design. Depending on he designe diffeen weighs can be assigned o he individual cieia. he objecive funcions of Eq. 9 and Eq. 2 can hen be combined o fom a new objecive funcion: F min ( w) α + w α 2τ in 2τ in [ K ] τ in f τ in [ K ] τ in f τ in (5) In his equaion w is he weighing funcion ha indicaes he impoance of he smoohness in he opimizaion. Fuhemoe he wo objecive funcions ae nomalized b α * and τ 2 in [ K ] τ in f τ in especivel whee α * is he minimum maimum hickness obained fom Eq. 9 and τ in ae he design vaiables fo maimum smoohness as obained b Eq Peiodic Bounda condiions he pesen fomulaion would also be valid fo clindical shells ce poins on a clindical suface ae in one o one coespondence o poins on a ecangula panel. Neveheless an impoan diffeence eiss; in 4
5 OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES he case of a clindical shell he soluion mus be peiodic. When he pl angle vaiaion is peiodic coninui in hickness is obained b including he hickness peiodici consains in he ealie descibed opimizaion ouines. Fo peiodici in -diecion his akes he fom: ( 0) τ ( b) 0 l τ (6) i i whee l is he lengh and b is he widh of he panel. 4. Discee Fibe Couses Once he smeaed hickness disibuion is obained hough one of he opimizaions descibed above he coesponding seam funcion can be obained b inegaing n ove dn: ( ) ndn d d dn + d dn d + i d d d d dn dn (7) he deivaives of wih espec o and can be epessed as funcions of s and n as follows: s s + (8) Since s 0 and n he combinaion of Eqs. 8 and 9 will give: ( ) ( ) ( ) d ( ) ( ) d (9) Boh () and () ae known funcions so ha () can be solved. B ploing he conou lines of his funcion a fied values d fom each ohe he seamlines ae found ha could epesen he cenelines of he acual fibe couses. he consan of inegaion will deemine he eac locaion of he fibe couses which can be used fo saggeing in case of muliple plies wih he same fibe angle disibuion. Once he couse cenelines ae known discee couses can be placed on op of hem and he discee hickness disibuions can be found. 5. Resuls o illusae he diffeences beween he vaious opimali cieia descibed in secion 3 an eample panel is analzed which has he following linea angle vaiaion in -diecion: ( ) l (20) such ha he fibe oienaion is a -30 degees a he lef side of he panel ( 0) and -60 degees a he igh side of he panel ( l ). he lengh o widh aio of he panels is 3. Fig. 2a. shows he hickness disibuion of a panel fo which he maimum hickness is minimized. he hickness along he lef and op edge is one which means ha hee ae no ovelaps on hese sides. he maimum hickness occus in he lowe igh cone of he panel. he hickness disibuion of a panel wih maimized smoohness is pesened in Fig. 2b. Compaed o he fis panel he maimum hickness is inceased b appoimael 20 pecen while smoohness is impoved b 40 pecen. In Fig. 2c. he smeaed hickness fo he combined objecive wih w 0.5 is ploed. Since boh maimum hickness and smoohness ae included in he objecive funcion he incease in maimum hickness is onl 7 pecen while he impovemen in smoohness is 30 pecen when compaed o he fis panel. Finall a panel wih peiodic bounda condiions is shown in Fig. 2d. he maimum hickness of his panel is moe han 40 pecen lage han he minimum maimum hickness and also smoohness is deceased. he discee hickness disibuions coesponding o he fou smeaed hickness disibuions of Fig. 2. ae shown in Fig. 3. he widh of hese couses was assumed o be /6 of 5
6 ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal Fig. 2. hickness disibuion fo vaious opimizaion cieia Fig. 3. Discee hickness build-up (black lae whie 2 laes) he panel widh. Fig. 3a. cleal shows he leas amoun of ovelap. If a laminae wih consan hickness was desied he fibe pahs obained b his opimizaion can be used as basic pahs and he ovelaps could be eliminaed b cuing individual ows on he sides of he couses. he smoohness of he laminae in Fig. 3b. is no ve appaen unil muliple plies ae sacked on op of each ohe and saggeed wih espec o each ohe. he combined objecive laminae of Fig. 3c. is indeed in beween he laminaes of Fig. 3a. and Fig. 3b. Finall he elaivel lage hickness build-up of he laminae wih peiodic bounda condiions is anslaed in lage ovelap aeas as shown in Fig. 3d. 6. Fuue Reseach In his pape i was successfull demonsaed ha a seam line analog can be used o pedic and influence he hickness disibuion in a laminae wih vaiable fibe angles. Diffeen objecives fo opimizaion 6
7 OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES wee consideed and in he fuue ohes such as minimum volume migh be eploed as well. In addiion o one-pl designs he developed mehods could be used o design complee laminaes wih boh vaing fibe angles and vaing hickness. One appoach would be o design he laminae ug laminaion paamees and hickness [674] as spaiall vaing design vaiables and hen as a pos-pocesg sep muliple plies wih vaing fibe angles and hei coesponding hickness disibuions could be fi in ode o mach boh he desied laminaion paamees and hickness disibuion as close as possible. Finall a simila mehod will be developed fo cuved sufaces in ode o epand he applicabili of he mehod. Refeences [] He M.W. and Chaee R.F. he use of cuvilinea fibe foma in composie sucue design. Poceedings of he 30 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence New Yok NY pape no [2] He M.W. and Lee H.H. he use of cuvilinea fibe foma o impove buckling esisance of composie plaes wih cenal holes. Composie Sucues Vol. 8 pp [3] Nagenda S. Kodialam A. Davis J.E. and Pahasaah V.N. Opimizaion of ow fibe pahs fo composie design. Poceedings of he 36 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence New Oleans LA pape no [4] Waldha C.J. and Güdal Z. Analsis of ow placed paallel fibe vaiable siffness laminaes. Poceedings of he 37 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Sal Lake Ci U pape no [5] Panas L. Oal S. and Cehan U. Opimum design of composie sucues wih cuved fibe couses. Composie Science and echnolog Vol. 63 pp [6] Seoodeh S. Abdalla M.M. and Güdal Z. Design of vaiable siffness laminaes ug laminaion paamees. Composies Pa B: Engineeing Vol. 37 pp [7] Abdalla M.M. Seoodeh S. and Güdal Z. Design of vaiable siffness composie panels fo maimum fundamenal fequenc ug laminaion paamees. Poceedings of he Euopean Confeence on Spacecaf Sucues Maeials and Mechanical eg Noodwijk he Nehelands [8] aing B.F. and Güdal Z. Design and manufacuing of elasicall ailoed ow-placed plaes. NASA/CR [9] aing B.F. and Güdal Z. Auomaed finie elemen analsis of elasicall-ailoed plaes. NASA/CR [0] Jegle D.C. aing B.F. and Güdal Z. ow-seeed panels wih holes subjeced o compession o shea loading. Poceedings of he 36 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Au X pape no [] Blom A.W. Seoodeh S. Hol J.M.A.M. and Güdal Z. Design of vaiable-siffness conical shells fo maimum fundamenal fequenc. Poceedings of he 3 d Euopean Confeence on Compuaional Mechanics: Solids Sucues and Coupled Poblems in Engineeing Lisbon Pougal June [2] Blom A.W. aing B.F. Hol J.M.A.M. and Güdal Z. Pah definiions fo elasicall ailoed conical shells. Poceedings of he 47 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Newpo RI pape no [3] Olhoff N. Mulicieion sucual opimizaion via bound fomulaion and mahemaical pogamming. Sucual Opimizaion Vol pp [4] Seoodeh S. Blom A.W. Abdalla M.M. and Güdal Z. Geneaing cuvilinea fibe pahs fom laminaion paamees disibuion. Poceedings of he 47 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Newpo RI pape no Appendi A he deivaives of he seam funcion ae: he second deivaives of he seam funcion ae: + Coninui of he second deivaives of he seam funcion implies ha so ha: 7
8 ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal 8 ( ) ( ) 0 + Reaangemen gives: + + (A) Ug he following definiions: ( ) s n ln Eq. A can be wien as: ( ) n s ln
MECHANICS OF MATERIALS Poisson s Ratio
Fouh diion MCHANICS OF MATRIALS Poisson s Raio Bee Johnson DeWolf Fo a slende ba subjeced o aial loading: 0 The elongaion in he -diecion is accompanied b a conacion in he ohe diecions. Assuming ha he maeial
More informationOn The Estimation of Two Missing Values in Randomized Complete Block Designs
Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.
More informationOn Control Problem Described by Infinite System of First-Order Differential Equations
Ausalian Jounal of Basic and Applied Sciences 5(): 736-74 ISS 99-878 On Conol Poblem Descibed by Infinie Sysem of Fis-Ode Diffeenial Equaions Gafujan Ibagimov and Abbas Badaaya J'afau Insiue fo Mahemaical
More informationThe sudden release of a large amount of energy E into a background fluid of density
10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid
More informationLecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain
Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as
More information, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t
Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission
More informationGeneral Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security
1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,
More informationLecture 22 Electromagnetic Waves
Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should
More informationSTUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE
More informationMonochromatic Wave over One and Two Bars
Applied Mahemaical Sciences, Vol. 8, 204, no. 6, 307-3025 HIKARI Ld, www.m-hikai.com hp://dx.doi.og/0.2988/ams.204.44245 Monochomaic Wave ove One and Two Bas L.H. Wiyano Faculy of Mahemaics and Naual Sciences,
More informationMATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH
Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias
More informationComputer Propagation Analysis Tools
Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion
More informationChapter 7. Interference
Chape 7 Inefeence Pa I Geneal Consideaions Pinciple of Supeposiion Pinciple of Supeposiion When wo o moe opical waves mee in he same locaion, hey follow supeposiion pinciple Mos opical sensos deec opical
More informationMEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING
MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens
More informationOrthotropic Materials
Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε
More informationSections 3.1 and 3.4 Exponential Functions (Growth and Decay)
Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens
More informationAN EVOLUTIONARY APPROACH FOR SOLVING DIFFERENTIAL EQUATIONS
AN EVOLUTIONARY APPROACH FOR SOLVING DIFFERENTIAL EQUATIONS M. KAMESWAR RAO AND K.P. RAVINDRAN Depamen of Mechanical Engineeing, Calicu Regional Engineeing College, Keala-67 6, INDIA. Absac:- We eploe
More informationAn Automatic Door Sensor Using Image Processing
An Auomaic Doo Senso Using Image Pocessing Depamen o Elecical and Eleconic Engineeing Faculy o Engineeing Tooi Univesiy MENDEL 2004 -Insiue o Auomaion and Compue Science- in BRNO CZECH REPUBLIC 1. Inoducion
More informationCombinatorial Approach to M/M/1 Queues. Using Hypergeometric Functions
Inenaional Mahemaical Foum, Vol 8, 03, no 0, 463-47 HIKARI Ld, wwwm-hikaicom Combinaoial Appoach o M/M/ Queues Using Hypegeomeic Funcions Jagdish Saan and Kamal Nain Depamen of Saisics, Univesiy of Delhi,
More informationInternational Journal of Pure and Applied Sciences and Technology
In. J. Pue Appl. Sci. Technol., 4 (211, pp. 23-29 Inenaional Jounal of Pue and Applied Sciences and Technology ISS 2229-617 Available online a www.ijopaasa.in eseach Pape Opizaion of he Uiliy of a Sucual
More informationLow-complexity Algorithms for MIMO Multiplexing Systems
Low-complexiy Algoihms fo MIMO Muliplexing Sysems Ouline Inoducion QRD-M M algoihm Algoihm I: : o educe he numbe of suviving pahs. Algoihm II: : o educe he numbe of candidaes fo each ansmied signal. :
More information( ) exp i ω b ( ) [ III-1 ] exp( i ω ab. exp( i ω ba
THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 26 III. EVIEW OF BASIC QUANTUM MECHANICS : TWO -LEVEL QUANTUM SYSTEMS : The lieaue of quanum opics and lase specoscop abounds wih discussions
More informationLecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation
Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion
More informationToday - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations
Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy
More information7 Wave Equation in Higher Dimensions
7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,
More informationExponential and Logarithmic Equations and Properties of Logarithms. Properties. Properties. log. Exponential. Logarithmic.
Eponenial and Logaihmic Equaions and Popeies of Logaihms Popeies Eponenial a a s = a +s a /a s = a -s (a ) s = a s a b = (ab) Logaihmic log s = log + logs log/s = log - logs log s = s log log a b = loga
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]
ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,
More informationPressure Vessels Thin and Thick-Walled Stress Analysis
Pessue Vessels Thin and Thick-Walled Sess Analysis y James Doane, PhD, PE Conens 1.0 Couse Oveview... 3.0 Thin-Walled Pessue Vessels... 3.1 Inoducion... 3. Sesses in Cylindical Conaines... 4..1 Hoop Sess...
More informationFig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial
a b c Fig. S. The anenna consucion: (a) ain geoeical paaees, (b) he wie suppo pilla and (c) he console link beween wie and coaial pobe. Fig. S. The anenna coss-secion in he y-z plane. Accoding o [], he
More informationChapter Finite Difference Method for Ordinary Differential Equations
Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence
More informationFINITE DIFFERENCE APPROACH TO WAVE GUIDE MODES COMPUTATION
FINITE DIFFERENCE ROCH TO WVE GUIDE MODES COMUTTION Ing.lessando Fani Elecomagneic Gou Deamen of Elecical and Eleconic Engineeing Univesiy of Cagliai iazza d mi, 93 Cagliai, Ialy SUMMRY Inoducion Finie
More informationApplication of Bernoulli wavelet method for numerical solution of fuzzy linear Volterra-Fredholm integral equations Abstract Keywords
Applicaion o enoulli wavele mehod o numeical soluion o uzz linea Volea-edholm inegal equaions Mohamed A. Ramadan a and Mohamed R. Ali b a Depamen o Mahemaics acul o Science Menouia Univesi Egp mamadan@eun.eg;
More informationThe Production of Polarization
Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview
More informationNUMERICAL SIMULATION FOR NONLINEAR STATIC & DYNAMIC STRUCTURAL ANALYSIS
Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada NUMERICAL SIMULATION FOR NONLINEAR STATIC & DYNAMIC STRUCTURAL ANALYSIS ABSTRACT
More information[ ] 0. = (2) = a q dimensional vector of observable instrumental variables that are in the information set m constituents of u
Genealized Mehods of Momens he genealized mehod momens (GMM) appoach of Hansen (98) can be hough of a geneal pocedue fo esing economics and financial models. he GMM is especially appopiae fo models ha
More informationRisk tolerance and optimal portfolio choice
Risk oleance and opimal pofolio choice Maek Musiela BNP Paibas London Copoae and Invesmen Join wok wih T. Zaiphopoulou (UT usin) Invesmens and fowad uiliies Pepin 6 Backwad and fowad dynamic uiliies and
More informationTwo-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch
Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion
More informationProjection of geometric models
ojecion of geomeic moels Copigh@, YZU Opimal Design Laboao. All ighs eseve. Las upae: Yeh-Liang Hsu (-9-). Noe: his is he couse maeial fo ME55 Geomeic moeling an compue gaphics, Yuan Ze Univesi. a of his
More informationCS 188: Artificial Intelligence Fall Probabilistic Models
CS 188: Aificial Inelligence Fall 2007 Lecue 15: Bayes Nes 10/18/2007 Dan Klein UC Bekeley Pobabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Given a join disibuion, we can
More informationThe shortest path between two truths in the real domain passes through the complex domain. J. Hadamard
Complex Analysis R.G. Halbud R.Halbud@ucl.ac.uk Depamen of Mahemaics Univesiy College London 202 The shoes pah beween wo uhs in he eal domain passes hough he complex domain. J. Hadamad Chape The fis fundamenal
More informationLecture 17: Kinetics of Phase Growth in a Two-component System:
Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien
More informationAn Open cycle and Closed cycle Gas Turbine Engines. Methods to improve the performance of simple gas turbine plants
An Open cycle and losed cycle Gas ubine Engines Mehods o impove he pefomance of simple gas ubine plans I egeneaive Gas ubine ycle: he empeaue of he exhaus gases in a simple gas ubine is highe han he empeaue
More informationWORK POWER AND ENERGY Consevaive foce a) A foce is said o be consevaive if he wok done by i is independen of pah followed by he body b) Wok done by a consevaive foce fo a closed pah is zeo c) Wok done
More information156 There are 9 books stacked on a shelf. The thickness of each book is either 1 inch or 2
156 Thee ae 9 books sacked on a shelf. The hickness of each book is eihe 1 inch o 2 F inches. The heigh of he sack of 9 books is 14 inches. Which sysem of equaions can be used o deemine x, he numbe of
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationReinforcement learning
Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback
More informationRelative and Circular Motion
Relaie and Cicula Moion a) Relaie moion b) Cenipeal acceleaion Mechanics Lecue 3 Slide 1 Mechanics Lecue 3 Slide 2 Time on Video Pelecue Looks like mosly eeyone hee has iewed enie pelecue GOOD! Thank you
More informationVariance and Covariance Processes
Vaiance and Covaiance Pocesses Pakash Balachandan Depamen of Mahemaics Duke Univesiy May 26, 2008 These noes ae based on Due s Sochasic Calculus, Revuz and Yo s Coninuous Maingales and Bownian Moion, Kaazas
More informationA Weighted Moving Average Process for Forecasting. Shou Hsing Shih Chris P. Tsokos
A Weighed Moving Aveage Pocess fo Foecasing Shou Hsing Shih Chis P. Tsokos Depamen of Mahemaics and Saisics Univesiy of Souh Floida, USA Absac The objec of he pesen sudy is o popose a foecasing model fo
More informationDiscretization of Fractional Order Differentiator and Integrator with Different Fractional Orders
Inelligen Conol and Auomaion, 207, 8, 75-85 hp://www.scip.og/jounal/ica ISSN Online: 253-066 ISSN Pin: 253-0653 Disceizaion of Facional Ode Diffeeniao and Inegao wih Diffeen Facional Odes Qi Zhang, Baoye
More informationOverview. Overview Page 1 of 8
COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIORNIA DECEMBER 2001 COMPOSITE BEAM DESIGN AISC-LRD93 Tecnical Noe Compac and Noncompac Requiemens Tis Tecnical Noe descibes o e pogam cecks e AISC-LRD93 specificaion
More informationRepresenting Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example
C 188: Aificial Inelligence Fall 2007 epesening Knowledge ecue 17: ayes Nes III 10/25/2007 an Klein UC ekeley Popeies of Ns Independence? ayes nes: pecify complex join disibuions using simple local condiional
More informationME 304 FLUID MECHANICS II
ME 304 LUID MECHNICS II Pof. D. Haşme Tükoğlu Çankaya Uniesiy aculy of Engineeing Mechanical Engineeing Depamen Sping, 07 y du dy y n du k dy y du k dy n du du dy dy ME304 The undamenal Laws Epeience hae
More informationProjection of geometric models
ojecion of geomeic moels Eie: Yeh-Liang Hsu (998-9-2); ecommene: Yeh-Liang Hsu (2-9-26); las upae: Yeh-Liang Hsu (29--3). Noe: This is he couse maeial fo ME55 Geomeic moeling an compue gaphics, Yuan Ze
More informationNumerical solution of fuzzy differential equations by Milne s predictor-corrector method and the dependency problem
Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Numeical soluion o uzz dieenial equaions b Milne s pedico-coeco meod and e dependenc poblem Kanagaajan K Indakuma S Muukuma
More informationLecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light
Lecue 5 Chape 3 lecomagneic Theo, Phoons, and Ligh Gauss s Gauss s Faada s Ampèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d qv A q A J ds ds In mae fields ae defined hough ineacion
More informationElastic-Plastic Deformation of a Rotating Solid Disk of Exponentially Varying Thickness and Exponentially Varying Density
Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong Elasic-Plasic Defomaion of a Roaing Solid Dis of Exponenially Vaying hicness Exponenially
More informationUnsupervised Segmentation of Moving MPEG Blocks Based on Classification of Temporal Information
Unsupevised Segmenaion of Moving MPEG Blocs Based on Classificaion of Tempoal Infomaion Ofe Mille 1, Ami Avebuch 1, and Yosi Kelle 2 1 School of Compue Science,Tel-Aviv Univesiy, Tel-Aviv 69978, Isael
More informationProbabilistic Models. CS 188: Artificial Intelligence Fall Independence. Example: Independence. Example: Independence? Conditional Independence
C 188: Aificial Inelligence Fall 2007 obabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Lecue 15: Bayes Nes 10/18/2007 Given a join disibuion, we can eason abou unobseved vaiables
More informationME 3560 Fluid Mechanics
ME3560 Flid Mechanics Fall 08 ME 3560 Flid Mechanics Analsis of Flid Flo Analsis of Flid Flo ME3560 Flid Mechanics Fall 08 6. Flid Elemen Kinemaics In geneal a flid paicle can ndego anslaion, linea defomaion
More informationStress Analysis of Infinite Plate with Elliptical Hole
Sess Analysis of Infinie Plae ih Ellipical Hole Mohansing R Padeshi*, D. P. K. Shaa* * ( P.G.Suden, Depaen of Mechanical Engg, NRI s Insiue of Infoaion Science & Technology, Bhopal, India) * ( Head of,
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationr r r r r EE334 Electromagnetic Theory I Todd Kaiser
334 lecoagneic Theoy I Todd Kaise Maxwell s quaions: Maxwell s equaions wee developed on expeienal evidence and have been found o goven all classical elecoagneic phenoena. They can be wien in diffeenial
More informationControl Volume Derivation
School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass
More informationEFFECT OF PERMISSIBLE DELAY ON TWO-WAREHOUSE INVENTORY MODEL FOR DETERIORATING ITEMS WITH SHORTAGES
Volume, ssue 3, Mach 03 SSN 39-4847 EFFEC OF PERMSSBLE DELAY ON WO-WAREHOUSE NVENORY MODEL FOR DEERORANG EMS WH SHORAGES D. Ajay Singh Yadav, Ms. Anupam Swami Assisan Pofesso, Depamen of Mahemaics, SRM
More informationA Numerical Hydration Model of Portland Cement
A Numeical Hydaion Model of Poland Cemen Ippei Mauyama, Tesuo Masushia and Takafumi Noguchi ABSTRACT : A compue-based numeical model is pesened, wih which hydaion and micosucual developmen in Poland cemen-based
More informationOnline Completion of Ill-conditioned Low-Rank Matrices
Online Compleion of Ill-condiioned Low-Rank Maices Ryan Kennedy and Camillo J. Taylo Compue and Infomaion Science Univesiy of Pennsylvania Philadelphia, PA, USA keny, cjaylo}@cis.upenn.edu Laua Balzano
More informationOn the Semi-Discrete Davey-Stewartson System with Self-Consistent Sources
Jounal of Applied Mahemaics and Physics 25 3 478-487 Published Online May 25 in SciRes. hp://www.scip.og/jounal/jamp hp://dx.doi.og/.4236/jamp.25.356 On he Semi-Discee Davey-Sewason Sysem wih Self-Consisen
More informationP h y s i c s F a c t s h e e t
P h y s i c s F a c s h e e Sepembe 2001 Numbe 20 Simple Hamonic Moion Basic Conceps This Facshee will:! eplain wha is mean by simple hamonic moion! eplain how o use he equaions fo simple hamonic moion!
More informationEnergy dispersion relation for negative refraction (NR) materials
Enegy dispesion elaion fo negaive efacion (NR) maeials Y.Ben-Ayeh Physics Depamen, Technion Isael of Technology, Haifa 3, Isael E-mail addess: ph65yb@physics.echnion,ac.il; Fax:97 4 895755 Keywods: Negaive-efacion,
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationOrdinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationThe k-filtering Applied to Wave Electric and Magnetic Field Measurements from Cluster
The -fileing pplied o Wave lecic and Magneic Field Measuemens fom Cluse Jean-Louis PINÇON and ndes TJULIN LPC-CNRS 3 av. de la Recheche Scienifique 4507 Oléans Fance jlpincon@cns-oleans.f OUTLINS The -fileing
More informationCircular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can.
1 Cicula Moion Radians One evoluion is equivalen o 360 0 which is also equivalen o 2π adians. Theefoe we can say ha 360 = 2π adians, 180 = π adians, 90 = π 2 adians. Hence 1 adian = 360 2π Convesions Rule
More informationQuantum Algorithms for Matrix Products over Semirings
CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2017, Aicle 1, pages 1 25 hp://cjcscsuchicagoedu/ Quanum Algoihms fo Maix Poducs ove Semiings Fançois Le Gall Haumichi Nishimua Received July 24, 2015; Revised
More informationEngineering Accreditation. Heat Transfer Basics. Assessment Results II. Assessment Results. Review Definitions. Outline
Hea ansfe asis Febua 7, 7 Hea ansfe asis a Caeo Mehanial Engineeing 375 Hea ansfe Febua 7, 7 Engineeing ediaion CSUN has aedied pogams in Civil, Eleial, Manufauing and Mehanial Engineeing Naional aediing
More informationAuthors name Giuliano Bettini* Alberto Bicci** Title Equivalent waveguide representation for Dirac plane waves
Auhos name Giuliano Beini* Albeo Bicci** Tile Equivalen waveguide epesenaion fo Diac plane waves Absac Ideas abou he elecon as a so of a bound elecomagneic wave and/o he elecon as elecomagneic field apped
More informationResearch on the Algorithm of Evaluating and Analyzing Stationary Operational Availability Based on Mission Requirement
Reseach on he Algoihm of Evaluaing and Analyzing Saionay Opeaional Availabiliy Based on ission Requiemen Wang Naichao, Jia Zhiyu, Wang Yan, ao Yilan, Depamen of Sysem Engineeing of Engineeing Technology,
More informationChapter 2: The Derivation of Maxwell Equations and the form of the boundary value problem
Chape : The eiaion of Mawell quaions and he fom of he bounda alue poblem In moden ime, phsics, including geophsics, soles eal-wold poblems b appling fis pinciples of phsics wih a much highe capabili han
More informationDepartment of Chemical Engineering University of Tennessee Prof. David Keffer. Course Lecture Notes SIXTEEN
D. Keffe - ChE 40: Hea Tansfe and Fluid Flow Deamen of Chemical Enee Uniesi of Tennessee Pof. Daid Keffe Couse Lecue Noes SIXTEEN SECTION.6 DIFFERENTIL EQUTIONS OF CONTINUITY SECTION.7 DIFFERENTIL EQUTIONS
More informationr P + '% 2 r v(r) End pressures P 1 (high) and P 2 (low) P 1 , which must be independent of z, so # dz dz = P 2 " P 1 = " #P L L,
Lecue 36 Pipe Flow and Low-eynolds numbe hydodynamics 36.1 eading fo Lecues 34-35: PKT Chape 12. Will y fo Monday?: new daa shee and daf fomula shee fo final exam. Ou saing poin fo hydodynamics ae wo equaions:
More informationWavefront healing operators for improving reflection coherence
Wavefon healing opeaos fo impoving eflecion coheence David C. Henley Wavefon healing ABSTRACT Seismic eflecion image coninuiy is ofen advesely affeced by inadequae acquisiion o pocessing pocedues by he
More informationMeasures the linear dependence or the correlation between r t and r t-p. (summarizes serial dependence)
. Definiions Saionay Time Seies- A ime seies is saionay if he popeies of he pocess such as he mean and vaiance ae consan houghou ime. i. If he auocoelaion dies ou quickly he seies should be consideed saionay
More informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationTechnical Report Doc ID: TR March-2013 (Last revision: 23-February-2016) On formulating quadratic functions in optimization models.
Technical Repor Doc ID: TR--203 06-March-203 (Las revision: 23-Februar-206) On formulaing quadraic funcions in opimizaion models. Auhor: Erling D. Andersen Convex quadraic consrains quie frequenl appear
More informationMolecular Evolution and Phylogeny. Based on: Durbin et al Chapter 8
Molecula Evoluion and hylogeny Baed on: Dubin e al Chape 8. hylogeneic Tee umpion banch inenal node leaf Topology T : bifucaing Leave - N Inenal node N+ N- Lengh { i } fo each banch hylogeneic ee Topology
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationIMPROVED DESIGN EQUATIONS FOR ASYMMETRIC COPLANAR STRIP FOLDED DIPOLES ON A DIELECTRIC SLAB
IMPROVED DESIGN EQUATIONS FOR ASYMMETRIC COPLANAR STRIP FOLDED DIPOLES ON A DIELECTRIC SLAB H.J. Visse* * Hols Cene TNO P.O. Bo 855 565 N Eindhoven, The Nehelands E-mail: hui.j.visse@no.nl eywods: Coupled
More informationApplication of De-Laval Nozzle Transonic Flow Field Computation Approaches
Wold Academ of Science, Engineeing and Technolog Vol:7, No:, 0 Applicaion of De-Laval Nozzle Tansonic Flow Field Compuaion Appoaches A. Haddad, H. Kbab Inenaional Science Index, Mechanical and Mechaonics
More informationProblem Set 7-7. dv V ln V = kt + C. 20. Assume that df/dt still equals = F RF. df dr = =
20. Assume ha df/d sill equals = F + 0.02RF. df dr df/ d F+ 0. 02RF = = 2 dr/ d R 0. 04RF 0. 01R 10 df 11. 2 R= 70 and F = 1 = = 0. 362K dr 31 21. 0 F (70, 30) (70, 1) R 100 Noe ha he slope a (70, 1) is
More information( ) c(d p ) = 0 c(d p ) < c(d p ) 0. H y(d p )
8.7 Gavimeic Seling in a Room Conside a oom of volume V, heigh, and hoizonal coss-secional aea A as shown in Figue 8.18, which illusaes boh models. c(d ) = 0 c(d ) < c(d ) 0 y(d ) (a) c(d ) = c(d ) 0 (b)
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More information2-d Motion: Constant Acceleration
-d Moion: Consan Acceleaion Kinemaic Equaions o Moion (eco Fom Acceleaion eco (consan eloci eco (uncion o Posiion eco (uncion o The eloci eco and posiion eco ae a uncion o he ime. eloci eco a ime. Posiion
More informationExtremal problems for t-partite and t-colorable hypergraphs
Exemal poblems fo -paie and -coloable hypegaphs Dhuv Mubayi John Talbo June, 007 Absac Fix ineges and an -unifom hypegaph F. We pove ha he maximum numbe of edges in a -paie -unifom hypegaph on n veices
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationConvective Heat Transfer (6) Forced Convection (8) Martin Andersson
Convecive Hea Tansfe (6) Foced Convecion (8) Main Andesson Agenda Convecive hea ansfe Conini eq. Convecive dc flow (inodcion o ch. 8) Convecive hea ansfe Convecive hea ansfe Convecive hea ansfe f flid
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More information