NATIONAL COUNCIL OF EXAMINERS

Transcription

1 NATONAL COUNCL OF EXAMNERS FOR ENGNEERNG AND SURVEYNG FUNDAMENTALS OF ENGNEERNG SUPPLED-REFERENCE HANDBOOK Fouth Edto Ntol Coul of Emes fo Egeeg d Suveg P.O. Bo 686 Clemso, SC

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3 FUNDAMENTALS OF ENGNEERNG SUPPLED-REFERENCE HANDBOOK Fouth Edto Rev.

4 b the Ntol Coul of Emes fo Egeeg d Suveg. All ghts eseved. Fst edto 996 Fouth edto

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7 FOREWORD Dug ts August 99 Aul Busess Meetg, the Ntol Coul of Emes fo Egeeg d Suveg (NCEES) voted to mke the Fudmetls of Egeeg (FE) emto NCEES suppled-efeee emto. The dug ts August 99 Aul Busess Meetg, the NCEES voted to mke the FE emto dsple-spef emto. As esult of the 99 vote, the FE emto ws developed to test the lowe-dvso subjets of tpl bhelo egeeg degee pogm dug the mog poto of the emto, d to test the uppe-dvso subjets of tpl bhelo egeeg degee pogm dug the fteoo. The lowe-dvso subjets efe to the fst 9 semeste edt hous (fve semestes t 8 edt hous pe semeste) of egeeg ousewok. The uppe-dvso subjets efe to the emde of the egeeg ousewok. Se egees el hevl o efeee mtels, the FE Suppled-Refeee Hdbook wll be mde vlble po to the emto. The emee m use ths hdbook whle pepg fo the emto. The hdbook ots ol efeee fomuls d tbles; o emple questos e luded. M ommell vlble books ot woked emples d smple questos. A emee lso pefom self-test usg oe of the NCEES FE Smple Questos d Solutos books ( ptl emto), whh m be puhsed b llg (8) The emee s ot llowed to bg efeee mtel to the emto oom. Aothe op of the FE Suppled-Refeee Hdbook wll be mde vlble to eh emee the oom. Whe the emee depts the emto oom, the FE Suppled- Refeee Hdbook suppled the oom shll be etued to the emto potos. The FE Suppled-Refeee Hdbook hs bee peped to suppot the FE emto poess. The FE Suppled-Refeee Hdbook s ot desged to ssst ll pts of the FE emto. Fo emple, some of the bs theoes, ovesos, fomuls, d deftos tht emees e epeted to kow hve ot bee luded. The FE Suppled-Refeee Hdbook m ot lude some spel mtel equed fo the soluto of ptul questo. suh stuto, the equed spel fomto wll be luded the questo sttemet. DSCLAMER: The NCEES o evet shll be lble fo ot povdg efeee mtel to suppot ll the questos the FE emto. the teest of ostt mpovemet, the NCEES eseves the ght to evse d updte the FE Suppled-Refeee Hdbook s t deems ppopte wthout fomg teested ptes. Eh NCEES FE emto wll be dmsteed usg the ltest veso of the FE Suppled-Refeee Hdbook. So tht ths hdbook be eused, PLEASE, t the emto ste, DO NOT WRTE N THS HANDBOOK.

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9 TABLE OF CONTENTS UNTS... CONVERSON FACTORS... MATHEMATCS... STATCS... DYNAMCS... MECHANCS OF MATERALS... FLUD MECHANCS... 8 THERMODYNAMCS... 7 HEAT TRANSFER TRANSPORT PHENOMENA... 6 CHEMSTRY... 6 MATERALS SCENCE/STRUCTURE OF MATTER ELECTRC CRCUTS... 7 COMPUTERS, MEASUREMENT, AND CONTROLS ENGNEERNG ECONOMCS ETHCS CHEMCAL ENGNEERNG CVL ENGNEERNG... 9 ELECTRCAL AND COMPUTER ENGNEERNG... 8 NDUSTRAL ENGNEERNG... 5 MECHANCAL ENGNEERNG... 5 NDEX... 6 v

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11 UNTS Ths hdbook uses the met sstem of uts. Ultmtel, the FE emto wll be etel met. Howeve, uetl some of the poblems use both met d U.S. Custom Sstem (USCS). the USCS sstem of uts, both foe d mss e lled pouds. Theefoe, oe must dstgush the poud-foe (lbf) fom the poud-mss (lbm). The poud-foe s tht foe whh eletes oe poud-mss t.7 ft/s. Thus, lbf.7 lbm-ft/s. The epesso.7 lbm-ft/(lbf-s ) s desgted s g d s used to esolve epessos volvg both mss d foe epessed s pouds. Fo ste, wtg Newto's seod lw, the equto would be wtte s F m/g, whee F s lbf, m lbm, d s ft/s. Sml epessos est fo othe quttes. Ket Eeg: KE mv /g, wth KE (ft-lbf); Potetl Eeg: PE mgh/g, wth PE (ft-lbf); Flud Pessue: p ρgh/g, wth p (lbf/ft ); Spef Weght: SW ρg/g, (lbf/ft ); She Stess: τ (µ/g )(dv/d), wth she stess (lbf/ft ). ll these emples, g should be egded s ut oveso fto. t s fequetl ot wtte epltl egeeg equtos. Howeve, ts use s equed to podue osstet set of uts. Note tht the oveso fto g [lbm-ft/(lbf-s )] should ot be ofused wth the lol eleto of gvt g, whh hs dffeet uts (m/s ) d m be ethe ts stdd vlue (9.87 m/s ) o some othe lol vlue. All equtos peseted ths efeee book e met-bsed equtos. f the poblem s peseted USCS uts, t m be eess to use the ostt g the equto to hve osstet set of uts. METRC PREFXES Multple Pef Smbol 8 tto 5 femto f po p 9 o 6 mo µ mll m et de d dek d heto h klo k 6 meg M 9 gg G te T 5 pet P 8 e E COMMONLY USED EQUVALENTS gllo of wte weghs 8. lbf ub foot of wte weghs 6. lbf ub h of meu weghs.9 lbf The mss of oe ub mete of wte s, klogms TEMPERATURE CONVERSONS ºF.8 (ºC) ºC (ºF )/.8 ºR ºF K ºC 7.5 FUNDAMENTAL CONSTANTS Qutt Smbol Vlue Uts eleto hge e.6 9 C (oulombs) Fd ostt 96,85 oulombs/(mol) gs ostt met R 8, J/(kmol K) gs ostt met R 8. kp m /(kmol K) gs ostt USCS R,55 ft-lbf/(lb mole-ºr) gvtto - ewto ostt G 6.67 m /(kg s ) gvtto - ewto ostt G 6.67 N m /kg gvt eleto (stdd) met g 9.87 m/s gvt eleto (stdd) USCS g.7 ft/s mol volume (del gs), T 7.5K, p. kp V m, L/kmol speed of lght vuum 99,79, m/s

12 CONVERSON FACTORS Multpl B To Obt Multpl B To Obt e,56 sque feet (ft ) joule (J) 9.78 Btu mpee-h (A-h),6 oulomb (C) J.776 ft-lbf ågstöm (Å) mete (m) J ewto m (N m) tmosphee (tm) 76. m, meu (Hg) J/s wtt (W) tm, std 9.9, meu (Hg) tm, std.7 lbf/ bs (ps) klogm (kg).5 poud (lbm) tm, std.9 ft, wte kgf ewto (N) tm, std. 5 psl (P) klomete (km),8 feet (ft) km/h.6 mph b 5 P klopsl (kp).5 lbf/ (ps) Btu,55 joule (J) klowtt (kw). hosepowe (hp) Btu.98 klowtt-h (kwh) kw, Btu/h Btu 778 ft-lbf kw 77.6 (ft-lbf )/se Btu/h.9 hosepowe (hp) kw-hou (kwh), Btu Btu/h.9 wtt (W) kwh. hp-h Btu/h.6 ft-lbf/se kwh.6 6 joule (J) kp (K), lbf loe (g-l).968 Btu K,8 ewto (N) l.56 6 hp-h l.86 joule (J) lte (L) 6. l/se.86 wtt (W) L.6 gl (US Lq) etmete (m).8 foot (ft) L/seod (L/s).9 ft /m (fm) m.9 h () L/s 5.85 gl (US)/m (gpm) etpose (P). psl se (P s) etstokes (St) 6 m /se (m /s) mete (m).8 feet (ft) ub foot (ft ) 7.8 gllo (gl) m.9 d m/seod (m/s) 96.8 feet/m (ft/m) eletovolt (ev).6 9 joule (J) mle (sttute) 5,8 feet (ft) mle (sttute).69 klomete (km) foot (ft).8 m mle/hou (mph) 88. ft/m (fpm) ft.8 mete (m) mph.69 km/h ft-poud (ft-lbf).85 Btu mm of Hg.6 tm ft-lbf klowtt-h (kwh) mm of H O tm ft-lbf. loe (g-l) ft-lbf.56 joule (J) ewto (N).5 lbf ft-lbf/se.88 hosepowe (hp) N m.776 ft-lbf gllo (US Lq).785 lte (L) N m joule (J) gllo (US Lq). ft psl (P) tmosphee (tm) gmm (γ, Γ) 9 tesl (T) P ewto/m (N/m ) guss T P se (P s) pose (P) gm (g).5 poud (lbm) poud (lbm,vdp).5 klogm (kg) lbf.8 N hete sque metes (m ) lbf -ft.56 N m hete.7 es lbf/ (ps).68 tm hosepowe (hp). Btu/m ps.7 ft of H O hp 75.7 wtt (W) ps.6 of Hg hp, (ft-lbf)/m ps 6,895 P hp 55 (ft-lbf)/se hp-h,5 Btu d 8/π degee hp-h.98 6 ft-lbf hp-h.68 6 joule (J) stokes m /s h ().5 etmete (m) them 5 Btu of Hg. tm of Hg.6 of H O wtt (W). Btu/h of H O.76 of Hg W. hosepowe (hp) of H O.6 lbf/ (ps) W joule/se (J/s) of H O.58 tm webe/m (Wb/m ), guss

13 STRAGHT LNE The geel fom of the equto s A B C The stdd fom of the equto s m b, whh s lso kow s the slope-teept fom. The pot-slope fom s m( ) Gve two pots: slope, m ( )/( ) The gle betwee les wth slopes m d m s α t [(m m )/( m m )] Two les e pepedul f m /m The dste betwee two pots s d QUADRATC EQUATON b CONC SECTONS ( ) ( ) b ± Roots b MATHEMATCS Cse. Ellpse e < : b e ; ( h) ( k) e ( b ) ( ± e, ) ; Det: ± / e Cse. Hpebol e > : / ( h,k) ; Cete t b s the stdd fom of the equto. Whe h k, Eett: Fous: e eett os θ/(os φ) [Note: X d Y, the followg ses, e tslted es.] Cse. Pbol e : b e ( h) ( k) e ( b ) / ( ± e, ) ; Det: ± / e ( h,k) ; Cete t b s the stdd fom of the equto. Whe h k, Eett: Fous: ; Bk, R.W., A Fst Ye of College Mthemts, Copght 97 b D. Appleto-Cetu Co.,. Repted b pemsso of Pete-Hll,., Eglewood Clffs, NJ. ( k) p( h); Cete t (h, k) s the stdd fom of the equto. Whe h k, Fous: (p/,); Det: p/

14 Cse. Cle e : ( h) ( k) ; Cete t (h, k) s the geel fom of the equto wth dus ( h) ( k ) h ; k b b MATHEMATCS (otued) f b s postve, le, ete (, b). f b equls zeo, pot t (, b). f b s egtve, lous s mg. QUADRC SURFACE (SPHERE) The geel fom of the equto s ( h) ( k) (z m) wth ete t (h, k, m). thee-dmesol spe, the dste betwee two pots s Legth of the tget fom pot. Usg the geel fom of the equto of le, the legth of the tget s foud fom t ( h) ( k) b substtutg the oodtes of pot P(, ) d the oodtes of the ete of the le to the equto d omputg. Co Seto Equto The geel fom of the o seto equto s A B C D E F whee ot both A d C e zeo. f B AC <, ellpse s defed. f B AC >, hpebol s defed. f B AC, the o s pbol. f A C d B, le s defed. f A B C, stght le s defed. b s the oml fom of the o seto equto, f tht o seto hs ppl s pllel to oodte s. LOGARTHMS The logthm of to the Bse b s defed b log b (), whee b Spel deftos fo b e o b e: l, Bse e log, Bse To hge fom oe Bse to othe: log b (log )/(log b) e.g., l (log )/(log e).585 (log ) dettes d ( ) ( ) ( z ) z log b b log log ; tlog ( log ) log log log log b b ; log log / log log Bk, R.W., A Fst Ye of College Mthemts, Copght 97 b D. Appleto- Cetu Co.,. Repted b pemsso of Pete-Hll,., Eglewood Clffs, NJ.

15 TRGONOMETRY Tgoomet futos e defed usg ght tgle. s θ /, os θ / t θ /, ot θ / s θ /, se θ / dettes Lw of Ses Lw of Coses b b os A b os B b b os C s θ /s θ se θ /os θ t θ s θ/os θ ot θ /t θ s θ os θ t θ se θ ot θ s θ s (α β) s α os β os α s β os (α β) os α os β s α s β s α s α os α os α os α s α s α os α t α ( t α)/( t α) ot α (ot α )/( ot α) t (α β) (t α t β)/( t α t β) ot (α β) (ot α ot β )/(ot α ot β) s (α β) s α os β os α s β os (α β) os α os β s α s β t (α β) (t α t β)/( t α t β) ot (α β) (ot α ot β )/(ot β ot α) s (α/) os (α/) t (α/) ± ± ( os α) ( os α) ( os α) ( α) ± os s A b s B s C MATHEMATCS (otued) s α os β (/)[s (α β) s (α β)] s α s β s (/)(α β) os (/)(α β) s α s β os (/)(α β) s (/)(α β) os α os β os (/)(α β) os (/)(α β) os α os β s (/)(α β) s (/)(α β) COMPLEX NUMBERS Defto ( b) ( d) ( ) (b d) ( b) ( d) ( ) (b d) ( b)( d) ( bd) (d b) b ( b)( d ) d ( d )( d ) d ( b) ( b) ( b) ( b) b ( b)( b) b Pol Coodtes os θ; s θ; θ t (/) (os θ s θ) e θ [ (os θ s θ )][ (os θ s θ )] [os (θ θ ) s (θ θ )] ( ) [ (os θ s θ)] Eule's dett e θ os θ s θ e θ os θ s θ (os θ s θ) ( os θ s θ ) ( os θ s θ ) e osθ θ e θ, e sθ ( bd ) ( b d ) Roots f k s postve tege, omple umbe (othe th zeo) hs k dstt oots. The k oots of (os θ s θ) be foud b substtutg suessvel,,,, (k ) the fomul [ os( θ θ ) s( θ θ )] θ e θ w k θ 6 θ 6 os s k k k k ot (α/) s α s β os α os β ( os α) ( α) ± os (/)[os (α β) os (α β)] (/)[os (α β) os (α β)] 5

16 MATRCES A mt s odeed etgul of umbes wth m ows d olums. The elemet j efes to ow d olum j. Multplto f A ( k ) s m mt d B (b kj ) s s mt, the mt podut AB s m s mt C ( ) b j l l lj whee s the ommo tege epesetg the umbe of olums of A d the umbe of ows of B (l d k,,, ). Fo thd-ode detemt: b b b b b b VECTORS MATHEMATCS (otued) b j k b b Addto f A ( j ) d B (b j ) e two mtes of the sme sze m, the sum A B s the m mt C ( j ) whee j j b j. dett The mt ( j ) s sque dett mt whee fo,,, d j fo j. Tspose The mt B s the tspose of the mt A f eh et b j B s the sme s the et j A d ovesel. equto fom, the tspose s B A T. vese The vese B of sque mt A s dj( A) B A, whee A dj(a) djot of A (obted b eplg A T elemets wth the oftos, see DETERMNANTS) d A detemt of A. DETERMNANTS A detemt of ode ossts of umbes, lled the elemets of the detemt, ged ows d olums d elosed b two vetl les. detemt, the mo of gve elemet s the detemt tht ems fte ll of the elemets e stuk out tht le the sme ow d the sme olum s the gve elemet. Cosde elemet whh les the hth olum d the kth ow. The ofto of ths elemet s the vlue of the mo of the elemet (f h k s eve), d t s the egtve of the vlue of the mo of the elemet (f h k s odd). f s gete th, the vlue of detemt of ode s the sum of the poduts fomed b multplg eh elemet of some spefed ow (o olum) b ts ofto. Ths sum s lled the epso of the detemt [odg to the elemets of the spefed ow (o olum)]. Fo seodode detemt: b b b b 6 A j z k Addto d subtto: A B ( b ) ( b )j ( z b z )k A B ( b ) ( b )j ( z b z )k The dot podut s sl podut d epesets the pojeto of B oto A tmes A. t s gve b A B b b z b z A B os θ B A The oss podut s veto podut of mgtude B A s θ whh s pepedul to the ple otg A d B. The podut s j A B B A b b b The sese of A B s detemed b the ght-hd ule. A B A B s θ, whee k z z ut veto pepedul to the ple of A d B.

17 Gdet, Dvegee, d Cul φ j k z φ V j k ( V Vj Vk ) z V j k V Vj Vk z The Lpl of sl futo φ s φ φ φ φ z ( ) dettes A B B A; A (B C) A B A C A A A j j k k j j k k f A B, the ethe A, B, o A s pepedul to B. A B B A A (B C) (A B) (A C) (B C) A (B A) (C A) j j k k j k j ; j k k j k j k f A B, the ethe A, B, o A s pllel to B. φ φ ( φ) ( ) ( A) ( A) ( A) A φ PROGRESSONS AND SERES Athmet Pogesso To deteme whethe gve fte sequee of umbes s thmet pogesso, subtt eh umbe fom the followg umbe. f the dffeees e equl, the sees s thmet.. The fst tem s.. The ommo dffeee s d.. The umbe of tems s.. The lst o th tem s l. 5. The sum of tems s S. l ( )d S ( l)/ [ ( ) d]/ MATHEMATCS (otued) Geomet Pogesso To deteme whethe gve fte sequee s geomet pogesso (G.P.), dvde eh umbe fte the fst b the peedg umbe. f the quotets e equl, the sees s geomet.. The fst tem s.. The ommo to s.. The umbe of tems s.. The lst o th tem s l. 5. The sum of tems s S. l S ( )/( ); S ( l)/( ); lmt S ( ) ; < A G.P. oveges f < d t dveges f. Popetes of Sees ( z ) ; ostt ( ). A powe sees, o, whh s oveget the tevl < < (o < < ), defes futo of whh s otuous fo ll vlues of wth the tevl d s sd to epeset the futo tht tevl.. A powe sees m be dffeetted tem b tem, d the esultg sees hs the sme tevl of ovegee s the ogl sees (eept possbl t the ed pots of the tevl).. A powe sees m be tegted tem b tem povded the lmts of tegto e wth the tevl of ovegee of the sees.. Two powe sees m be dded, subtted, o multpled, d the esultg sees eh se s oveget, t lest, the tevl ommo to the two sees. 5. Usg the poess of log dvso (s fo polomls), two powe sees m be dvded oe b the othe. z 7

18 Tlo's Sees f ( ) f ( ) ( ) ( ) f ( ) ( ) f!! f! ( ) ( ) ( ) s lled Tlo's sees, d the futo f () s sd to be epded bout the pot Tlo's sees. f, the Tlo's sees equto beomes Mlu's sees. PROBABLTY AND STATSTCS Pemuttos d Combtos A pemutto s ptul sequee of gve set of objets. A ombto s the set tself wthout efeee to ode.. The umbe of dffeet pemuttos of dstt objets tke t tme s P (,)! ( )!. The umbe of dffeet ombtos of dstt objets tke t tme s w P ( ) (,)! C,! [! ( )!]. The umbe of dffeet pemuttos of objets tke t tme, gve tht e of tpe, whee,,, k d Σ, s! P( ;,, k )!!! Lws of Pobblt Popet. Geel Chte of Pobblt The pobblt P(E) of evet E s el umbe the ge of to. The pobblt of mpossble evet s d tht of evet et to ou s. Popet. Lw of Totl Pobblt P(A B) P(A) P(B) P(A, B), whee P(A B) the pobblt tht ethe A o B ou loe o tht both ou togethe, P(A) the pobblt tht A ous, P(B) the pobblt tht B ous, d P(A, B) the pobblt tht both A d B ou smulteousl. k MATHEMATCS (otued) Popet. Lw of Compoud o Jot Pobblt f ethe P(A) o P(B) s zeo, P(A, B) P(A)P(B A) P(B)P(A B), whee P(B A) the pobblt tht B ous gve the ft tht A hs oued, d P(A B) the pobblt tht A ous gve the ft tht B hs oued. f ethe P(A) o P(B) s zeo, the P(A, B). Pobblt Futos A dom vble hs pobblt ssoted wth eh of ts vlues. The pobblt s temed dsete pobblt f ssume ol the dsete vlues X, X,, X,, X N The dsete pobblt of the evet X oug s defed s P(X ). Pobblt Dest Futos f s otuous, the the pobblt dest futo f () s defed so tht ( ) f d the pobblt tht les betwee d. The pobblt s detemed b defg the equto fo f () d tegtg betwee the vlues of equed. Pobblt Dstbuto Futos The pobblt dstbuto futo F(X ) of the dsete pobblt futo P(X ) s defed b F ( X ) P( X ) P( X X ) k k Whe s otuous, the pobblt dstbuto futo F() s defed b F ( ) f ( t) dt whh mples tht F() s the pobblt tht. The epeted vlue g() of futo s defed s E { g( ) } g() t f () t dt BNOMAL DSTRBUTON P() s the pobblt tht wll ou tls. f p pobblt of suess d q pobblt of flue p, the! P( ) C(,) p q p q! ( )! whee,,,,, C(, ) the umbe of ombtos, d, p pmetes. 8

19 NORMAL DSTRBUTON (Guss Dstbuto) Ths s umodl dstbuto, the mode beg µ, wth two pots of fleto (eh loted t dste σ to ethe sde of the mode). The veges of obsevtos ted to beome omll dstbuted s eses. The vte s sd to be omll dstbuted f ts dest futo f () s gve b epesso of the fom f ( ) ( µ ) e σ π σ whee µ the populto me, σ the stdd devto of the populto, d Whe µ d σ σ, the dstbuto s lled stddzed o ut oml dstbuto. The f ( ) e, whee. π A ut oml dstbuto tble s luded t the ed of ths seto. the tble, the followg ottos e utlzed: F() the e ude the uve fom to, R() the e ude the uve fom to, d W() the e ude the uve betwee d. DSPERSON, MEAN, MEDAN, AND MODE VALUES f X, X,, X epeset the vlues of tems o obsevtos, the thmet me of these tems o obsevtos, deoted X, s defed s X ( )( X X X ) ( ) X X µ fo suffetl lge vlues of. Theefoe, fo the puposes of ths hdbook, the followg s epted: µ populto me X The weghted thmet me s w X X w, whee w X w the weghted thmet me, X the vlues of the obsevtos to be veged, d w the weght ppled to the X vlue. The ve of the obsevtos s the thmet me of the squed devtos fom the populto me. smbols, X, X,, X epeset the vlues of the smple obsevtos of populto of sze N. f µ s the thmet me of the populto, the populto ve s defed b σ ( / N )[( X µ ) ( X µ ) ( X N µ ) N ( / N ) ( X µ ), ] The stdd devto of populto s σ The smple ve s s The smple stdd devto s s The oeffet of vto CV s/ X The geomet me ( N ) ( X µ ) [ ( ) ] ( ) X X ( X X ) MATHEMATCS (otued) The oot-me-sque vlue ( ) X The med s defed s the vlue of the mddle tem whe the dt e k-odeed d the umbe of tems s odd. The med s the vege of the mddle two tems whe the kodeed dt ossts of eve umbe of tems. The mode of set of dt s the vlue tht ous wth getest feque. t-dstrbuton The vte t s defed s the quotet of two depedet vtes d whee s ut oml d s the oot me sque of othe depedet ut oml vtes; tht s, t /. The followg s the t-dstbuto wth degees of feedom: Γ () [( ) ] f t Γ π t ( ) ( ) ( ) whee t. A tble t the ed of ths seto gves the vlues of t α fo vlues of α d. Note tht vew of the smmet of the t- dstbuto, t α, t α,. The futo fo α follows: α t f α, () t dt X X A tble showg pobblt d dest futos s luded o pge the NDUSTRAL ENGNEERNG SECTON of ths hdbook. X X 9

20 MATHEMATCS (otued) GAMMA FUNCTON CONFDENCE NTERVALS Cofdee tevl fo the Me µ of Noml Dstbuto () Stdd devto σ s kow (b) Stdd devto σ s ot kow whee α t oespods to degees of feedom. Cofdee tevl fo the Dffeee Betwee Two Mes µ d µ () Stdd devtos σ d σ kow (b) Stdd devtos σ d σ e ot kow whee α t oespods to degees of feedom. Z X Z X σ µ σ α α s t X s t X α α µ Z X X Z X X σ σ µ µ σ σ α α ( ) ( ) [ ] ( ) ( ) [ ] µ µ α α S S t X X S S t X X ( ) > dt, e t o t Γ

21 MATHEMATCS (otued) UNT NORMAL DSTRBUTON TABLE f() F() R() R() W() Ftles

22 MATHEMATCS (otued) t-dstrbuton TABLE VALUES OF t α, α. α.5 α.5 α. α f f. α

23 MATHEMATCS (otued) CRTCAL VALUES OF THE F DSTRBUTON TABLE Fo ptul ombto of umeto d deomto degees of feedom, et epesets the tl vlues of F oespodg to spefed uppe tl e (α). Numeto df Deomto df

24 DFFERENTAL CALCULUS The Devtve Fo futo f (), the devtve D d/d lmt lmt the slope of the uve f(). TEST FOR A MAXMUM f () s mmum fo, f f () d f () <. TEST FOR A MNMUM f () s mmum fo, f f () d f () >. TEST FOR A PONT OF NFLECTON f () hs pot of fleto t, f f (), d f f () hges sg s eses though. The Ptl Devtve [( ) ( ) ] futo of two depedet vbles d, devtve wth espet to oe of the vbles m be foud f the othe vble s ssumed to em ostt. f s kept fed, the futo z f (, ) beomes futo of the sgle vble, d ts devtve (f t ests) be foud. Ths devtve s lled the ptl devtve of z wth espet to. The ptl devtve wth espet to s deoted s follows: The Cuvtue of A Cuve {[ f ( ) f ( ) ] ( ) } (, ) z f The uvtue K of uve t P s the lmt of ts vege uvtue fo the PQ s Q ppohes P. Ths s lso epessed s: the uvtue of uve t gve pot s the te-of-hge of ts lto wth espet to ts legth. α dα K lmt s s ds MATHEMATCS (otued) CURVATURE N RECTANGULAR COORDNATES K [ ( ) ] Whe t m be ese to dffeette the futo wth espet to the th, the otto wll be used fo the devtve. d/d K [ ( ) ] THE RADUS OF CURVATURE The dus of uvtue R t pot o uve s defed s the bsolute vlue of the epol of the uvtue K t tht pot. R ( K ) K R [ ( ) ] ( ) L'Hosptl's Rule (L'Hôptl's Rule) f the ftol futo f()/g() ssumes oe of the detemte foms / o / (whee α s fte o fte), the lmt f ( ) g( ) α s equl to the fst of the epessos f ( ) f ( ) f ( ) lmt, lmt, lmt α g α g α g ( ) ( ) whh s ot detemte, povded suh fst dted lmt ests. NTEGRAL CALCULUS The defte tegl s defed s: lmt f b ( ) f ( ) ( ) Also, fo ll. A tble of devtves d tegls s vlble o pge 5. The tegl equtos be used log wth the followg methods of tegto: A. tegto b Pts (tegl equto #6), B. tegto b Substtuto, d C. Septo of Rtol Ftos to Ptl Ftos. Wde, Thoms L., Clulus, Copght 95 b G & Comp. Dgm epted b pemsso of Smo & Shuste Publshes. d

25 5 MATHEMATCS (otued) DERVATVES AND NDEFNTE NTEGRALS these fomuls, u, v, d w epeset futos of. Also,,, d epeset ostts. All gumets of the tgoomet futos e ds. A ostt of tegto should be dded to the tegls. To vod temolog dffult, the followg deftos e followed: s u s u, (s u) /s u.. d/d. d/d. d(u)/d du/d. d(u v w)/d du/d dv/d dw/d 5. d(uv)/d u dv/d v du/d 6. d(uvw)/d uv dw/d uw dv/d vw du/d d u v v du d u dv d 7. d v 8. d(u )/d u du/d 9. d[f (u)]/d {d[f (u)]/du} du/d. du/d /(d/du) d log. u du ( log e) d u d. d( lu) du d u d u. d( ) u du ( l) d d. d(e u )/d e u du/d 5. d(u v )/d vu v du/d (l u) u v dv/d 6. d(s u)/d os u du/d 7. d(os u)/d s u du/d 8. d(t u)/d se u du/d 9. d(ot u)/d s u du/d. d(se u)/d se u t u du/d. d(s u)/d s u ot u du/d ( ) ( ) d d ( s u) du ( π s u π ) d u d ( t u) du ( π < t u < π ) d u d ( ot u) du ( < ot u < π) d d ( os u) du ( os u π) d d d d ( se u) d ( s u) d u u u u u d du d ( se u < π )( π se u < π ) u d du d ( < s u π )( π < s u π ). d f () f (). d. f() d f() d. [u() ± v()] d u() d ± v() d m m d m u() dv() u() v() v () du() 7. d l b b 8. d 9. d l. s d os. os d s s. s d. s os d. s d s os 5. os d os s 6. s os d (s )/ 7. b 8. t d l os l se 9. ot d l s l s. t d t. ot d ot. e d (/) e. e d (e / )( ). l d [l () ] ( > ) b. 7. s osb d os ( b) ( b) os ( b) ( b) ( m ) ( ) d t d t, >, > d b d b d b b t b b b l b, b ( ) ( ) ( b > ) b b ( b > ) ( b ) b

26 Nomeltue A totl sufe e P pemete V volume Pbol MENSURATON OF AREAS AND VOLUMES Cul Segmet MATHEMATCS (otued) A [ (φ s φ)]/ φ s/ {os [( d)/]} Cul Seto Ellpse P ppo P π π ( b) ( b ) ( ) λ ( ) λ ( 6 ) λ ( ) ( 5 7 ) λ A πb λ, Sphee A φ / s/ φ s/ whee λ ( b)/( b) V π / πd /6 A π πd Gek, K. & Gek R., Egeeg Fomuls, 6th Ed., Copght 967 b Gek Publshg. Dgms epted b pemsso of Kut Gek. 6

27 Pllelogm MENSURATON OF AREAS AND VOLUMES Rght Cul Coe MATHEMATCS (otued) d d d P ( b) d b b ( b ) A h b b b ( sφ) f b, the pllelogm s hombus. Regul Polgo ( equl sdes) ( osφ) ( osφ) V (π h)/ A sde e bse e π A : A b : h Rght Cul Clde h φ π/ π( ) θ π P s s [t (φ/)] A (s)/ Pbolod of Revoluto πd h V π h A sde e ed es π ( h ) Psmod πd h V 8 V (h/6)( A A A) Gek, K. & R. Gek, Egeeg Fomuls, 6th Ed., Copght b Gek Publshg. Dgms epted b pemsso of Kut Gek. 7

28 CENTRODS AND MOMENTS OF NERTA The loto of the etod of e, bouded b the es d the futo f(), be foud b tegto. da A da A A f ( ) d da f ( ) d g( )d The fst momet of e wth espet to the -s d the -s, espetvel, e: M da A M da A whe ethe o s of fte dmesos the da o da efe to the etod o of da these tegls. The momet of et (seod momet of e) wth espet to the -s d the -s, espetvel, e: da da The momet of et tke wth espet to s pssg though the e's etod s the etodl momet of et. The pllel s theoem fo the momet of et wth espet to othe s pllel wth d loted d uts fom the etodl s s epessed b pllel s Ad ple, τ da Vlues fo stdd shpes e peseted tble the DYNAMCS seto. DFFERENTAL EQUATONS A ommo lss of od le dffeetl equtos s ( ) ( ) d d b b b ( ) f ( ) d d whee b,, b,, b, b e ostts. Whe the equto s homogeeous dffeetl equto, f(), the soluto s C e C e C e C e h ( ) whee s the th dstt oot of the htest poloml P() wth P() b b b b f the oot, the Ce s epled wth C e. Hghe odes of multplt mpl hghe powes of. The omplete soluto fo the dffeetl equto s () h () p (), e whee p () s soluto wth f() peset. f f() hs tems, the esoe s mfested. Futhemoe, spef f() foms esult spef p () foms, some of whh e: 8 f() A Ae α A s ω A os ω p () MATHEMATCS (otued) B Be α, α B s ω B os ω f the depedet vble s tme t, the tset dm solutos e mpled. Fst-Ode Le Homogeeous Dffeetl Equtos Wth Costt Coeffets, whee s el ostt: Soluto, Ce t, whee C ostt tht stsfes the tl odtos. Fst-Ode Le Nohomogeeous Dffeetl Equtos d A t < τ K() t () t dt B t > ( ) KA τ s the tme ostt K s the g The soluto s t () t KA ( KB KA) ep o τ t KB KA l τ KB Seod-Ode Le Homogeeous Dffeetl Equtos wth Costt Coeffets A equto of the fom b be solved b the method of udetemed oeffets whee soluto of the fom Ce s sought. Substtuto of ths soluto gves ( b) Ce d se Ce ot be zeo, the htest equto must vsh o b The oots of the htest equto e, ± b d be el d dstt fo > b, el d equl fo b, d omple fo < b. f > b, the soluto s of the fom (ovedmped) C e Ce f b, the soluto s of the fom (tll dmped) ( C C ) e f < b, the soluto s of the fom (udedmped) e α (C os β C s β) whee α β b

29 FOURER SERES Eve futo F(t) whh hs the peod τ π/ω d stsfes et otut odtos be epeseted b sees plus ostt. F() t ( os ωt bs ωt) The bove equto holds f F(t) hs otuous devtve F (t) fo ll t. Multpl both sdes of the equto b os mωt d tegte fom to τ. τ τ F F τ () t os mωtdt ( ) τ () t os mωtdt ( ) b s ωtos mωdt] Tem-b-tem tegto of the sees be justfed f F(t) s otuous. The oeffets e τ ( τ) F( t) os ωtdt d b τ F t s ωtdt, whee τ π/ω. The ostts, b e the Foue oeffets of F(t) fo the tevl to τ, d the oespodg sees s lled the Foue sees of F(t) ove the sme tevl. The tegls hve the sme vlue ove tevl of legth τ. f Foue sees epesetg peod futo s tuted fte tem N, the me sque vlue F N of the tuted sees s gve b the Psevl elto. Ths elto ss tht the me sque vlue s the sum of the me sque vlues of the Foue ompoets, o d the RMS vlue s the defed to be the sque oot of ths qutt o F N. FOURER TRANSFORM The Foue tsfom p, oe fom of whh s F f be used to hteze bod lss of sgl models tems of the feque o spetl otet. Some useful tsfom ps e: f(t) F(ω) δ(t) u(t) τ τ ut ut e jω t o F N os mωtdt os mωtdt [ os ωtos mωtdt τ τ τ ( ) ( ) N ( ) ( ) ( b ) jωt ( ω) f ( t) e dt () t [ ( π) ] F( ω) et t τ e jωt dω (/)δ(ω) /jω s( ωτ ) τ ϖτ πδ ( ) ω ω o MATHEMATCS (otued) Some mthemtl lbetes e equed to obt the seod d fouth fom. Othe Foue tsfoms e devble fom the Lple tsfom b eplg s wth jω povded f(t), t < LAPLACE TRANSFORMS The ultel Lple tsfom p st F() s f () t e dt σ st f () t F() s e dt σ π epesets poweful tool fo the tset d feque espose of le tme vt sstems. Some useful Lple tsfom ps e [Note: The lst two tsfoms epeset the Fl Vlue Theoem (F.V.T.) d tl Vlue Theoem (.V.T.) espetvel. t s ssumed tht the lmts est.]: f(t) F(s) δ(t), mpulse t t u(t), Step t t /s t[u(t)], Rmp t t /s e αt /(s α) te αt /(s α) e αt s βt β/[(s α) β ] e αt os βt (s α)/[(s α) β ] f (t τ) f () t dt < d f () t s F () s dt ( τ ) t f dτ ( t τ ) h( t) t dτ () t (/s)f(s) H(s)X(s) e τs F(s) lmt f lmt sf () s t lmt t f () t m d f m d t DFFERENCE EQUATONS Dffeee equtos e used to model dsete sstems. Sstems whh be desbed b dffeee equtos lude ompute pogm vbles tetvel evluted loop, sequetl uts, sh flows, eusve poesses, sstems wth tme-del ompoets, et. A sstem whose put v(t) d output (t) e defed ol t the equll sped tevls t kt be desbed b dffeee equto. ( ) m s m s lmt s sf() s 9

30 MATHEMATCS (otued) Fst-Ode Le Dffeee Equto The dffeee equto P k P k ( ) A epesets the ble P of lo fte the kth pmet A. f P k s defed s (k), the model beomes (k) ( ) (k ) A Seod-Ode Le Dffeee Equto The Fbo umbe sequee be geeted b (k) (k ) (k ) whee ( ) d ( ). A ltete fom fo ths model s f (k ) f (k ) f (k) wth f () d f (). z-tsfoms The tsfom defto s The vese tsfom s gve b the otou tegl d t epesets poweful tool fo solvg le shft vt dffeee equtos. A lmted ultel lst of z- tsfom ps follows [Note: The lst two tsfom ps epeset the tl Vlue Theoem (.V.T.) d the Fl Vlue Theoem (F.V.T.) espetvel.]: f(k) F(z) δ(k), mpulse t k u(k), Step t k /( z ) β k /( βz ) (k ) z Y(z) ( ) (k ) z Y(z) ( ) ( )z (k ) zy(z) z() (k ) z Y(z) z () z() H(z)X(z) EULER'S APPROXMATON t (d /dt) NUMERCAL METHODS Newto's Method of Root Etto Gve poloml P() wth smple oots,,,, whee d P( ). A oot be omputed b the tetve lgothm wth Covegee s qudt. Newto's Method of Mmzto Gve sl vlue futo h() h(,,, ) fd veto * R suh tht h(*) h() fo ll Newto's lgothm s whee d ( ) ( ) ( ) ( ) Γ π dz z z F k f z k f z F k k k ( ) ( ) m m m h k X ( ) k f k lmt ( ) z F z lmt ( ) k f k lmt ( ) ( ) z F z z lmt ( ) ( ) m m P α α α ( ) ( ) j j j P P ( ) ( ) j j P P K K K K h h h h h h h h h h h h h h h h

31 MATHEMATCS (otued) Numel tegto Thee of the moe ommo umel tegto lgothms used to evlute the tegl b f ( ) d e: Eule's o Fowd Retgul Rule b f ( ) d f ( k ) k Tpezodl Rule fo b f ( ) ( ) f ( b) f d fo > b f ( ) d f ( ) f ( k ) f ( b) k Numel Soluto of Od Dffeetl Equtos Gve dffeetl equto d/dt f(, t) wth () o At some geel tme k t [(k ) t] (k t) t f [ (k t), k t] whh be used wth sttg odto o to solve eusvel fo ( t), ( t),, ( t). The method be eteded to th ode dffeetl equtos b estg them s fst-ode equtos. Smpso's Rule/Pbol Rule ( must be eve tege) fo fo b f b 6 b ( ) d f ( ) f f ( b) ( ) f d b f ( ) f ( k ) ( ) ( ) k,, 6, f k f b k,, 5, wth (b )/

32 STATCS FORCE A foe s veto qutt. t s defed whe ts () mgtude, () pot of pplto, d () deto e kow. RESULTANT (TWO DMENSONS) The esultt, F, of foes wth ompoets F, d F, hs the mgtude of F F, F, The esultt deto wth espet to the -s usg fou-qudt gle futos s θ t F, F The veto fom of the foe s F F F j RESOLUTON OF A FORCE F F os θ ; F F os θ ; F z F os θ z os θ F /F; os θ F /F; os θ z F z /F Septg foe to ompoets (geomet of foe s kow R z ) F (/R)F; F (/R)F; F z (z/r)f MOMENTS (COUPLES) A sstem of two foes tht e equl mgtude, opposte deto, d pllel to eh othe s lled ouple. A momet M s defed s the oss podut of the dus veto dste d the foe F fom pot to the le of to of the foe. M F; M F z zf, M zf F z, d M z F F. SYSTEMS OF FORCES F Σ F M Σ ( F ) Equlbum Requemets Σ F Σ M, CENTRODS OF MASSES, AREAS, LENGTHS, AND VOLUMES Fomuls fo etods, momets of et, d fst momet of es e peseted the MATHEMATCS seto fo otuous futos. The followg dsete fomuls e fo defed egul msses, es, legths, d volumes: Σ m /Σ m, whee m the mss of eh ptle mkg up the sstem, the dus veto to eh ptle fom seleted efeee pot, d the dus veto to the ete of the totl mss fom the seleted efeee pot. The momet of e (M ) s defed s M Σ M Σ M z Σ z The etod of e s defed s M /A wth espet to ete M /A of the oodte sstem z M z /A whee A Σ The etod of le s defed s l (Σ l )/L, whee L Σ l l (Σ l )/L z l (Σ z l )/L The etod of volume s defed s v (Σ v )/V, whee V Σ v v (Σ v )/V z v (Σ z v )/V MOMENT OF NERTA The momet of et, o the seod momet of e, s defed s da da The pol momet of et J of e bout pot s equl to the sum of the momets of et of the e bout two pepedul es the e d pssg though the sme pot. z J ( ) da p A, whee p the dus of gto (see pge ).

33 Momet of et Tsfe Theoem The momet of et of e bout s s defed s the momet of et of the e bout pllel etodl s plus tem equl to the e multpled b the sque of the pepedul dste d fom the etodl s to the s questo. d A d A, whee d, d dste betwee the two es questo,, the momet of et bout the etodl s, d, the momet of et bout the ew s. Rdus of Gto The dus of gto p,, s the dste fom efeee s t whh ll of the e be osdeed to be oetted to podue the momet of et. A; A; p Podut of et The podut of et (, et.) s defed s: da, wth espet to the -oodte sstem, z zda, wth espet to the z-oodte sstem, d z zda, wth espet to the z-oodte sstem. The tsfe theoem lso pples: fo the -oodte sstem, et., d d A whee d -s dste betwee the two es questo d d -s dste betwee the two es questo. FRCTON The lgest ftol foe s lled the lmtg fto. A futhe ese ppled foes wll use moto. F µ N, whee F fto foe, µ oeffet of stt fto, d N oml foe betwee sufes ott. SCREW THREAD Fo sew-jk, sque thed, M P t (α ± φ), whee s fo sew tghteg, s fo sew looseg, M etel momet ppled to s of sew, P lod o jk ppled log d o the le of the s, the me thed dus, α the pth gle of the thed, d µ t φ the ppopte oeffet of fto. J A STATCS (otued) BRAKE-BAND OR BELT FRCTON F F e µθ, whee F foe beg ppled the deto of mpedg moto, F foe ppled to esst mpedg moto, µ oeffet of stt fto, d θ the totl gle of ott betwee the sufes epessed ds. STATCALLY DETERMNATE TRUSS Ple Tuss A ple tuss s gd fmewok stsfg the followg odtos:. The membes of the tuss le the sme ple.. The membes e oeted t the eds b ftoless ps.. All of the etel lods le the ple of the tuss d e ppled t the jots ol.. The tuss etos d membe foes be detemed usg the equtos of equlbum. Σ F ; Σ M 5. A tuss s sttll detemte f the etos d membe foes ot be solved wth the equtos of equlbum. Ple Tuss: Method of Jots The method ossts of solvg fo the foes the membes b wtg the two equlbum equtos fo eh jot of the tuss. Σ F V d Σ F H, whee F H hozotl foes d membe ompoets d F V vetl foes d membe ompoets. Ple Tuss: Method of Setos The method ossts of dwg fee-bod dgm of poto of the tuss suh w tht the ukow tuss membe foe s eposed s etel foe. CONCURRENT FORCES A sstem of foes whee the les of to ll meet t oe pot. Two Dmesos Σ F ; Σ F Thee Dmesos Σ F ; Σ F ; Σ F z

34 KNEMATCS Veto epesetto of moto spe: Let (t) be the posto veto of ptle. The the velot s v d/dt, whee v the stteous velot of the ptle, (legth/tme) t tme The eleto s dv/dt d /dt, whee the stteous eleto of the ptle, (legth/tme/tme) Retgul Coodtes j zk v d dt j z k d dt j zk, d dt v, et. d dt, et. whee Tsvese d Rdl Compoets fo Pl Poblems DYNAMCS Tgetl d Noml Compoets Ut vetos e d e t e, espetvel, oml d tget to the pth. v v t e t (dv t /dt) e t (v t /ρ) e, whee ρ stteous dus of uvtue d v t tgetl velot Ple Cul Moto Ut vetos e d e θ e, espetvel, ole wth d oml to the posto veto. e v e θ eθ θ α e θ θ e d ( ) ( ) θ the dus, θ the gle betwee the -s d, d dt, et. d dt, et. Rotto bout the og wth ostt dus: The ut vetos e e t e θ d e e. Agul velot ω θ v t Agul eleto α ω θ s θ v ω t Tgetl eleto t α dv t /dt Noml eleto v t / ω t

35 Stght Le Moto Costt eleto equtos: s s o v o t ( o t ) / v v o o t v v o o (s s o ), whee s dste log the le tveled, s o tl dste fom og (ostt), v o tl velot (ostt), o ostt eleto, t tme, d v velot t tme t. Fo fee fllg bod, o g (dowwd) Usg vble velot, v(t) t s so v() t dt Usg vble eleto, (t) t v () t dt v o PROJECTLE MOTON ; g v v o v o os θ v v o gt v o s θ gt v o t v o t os θ v o t gt / v o t s θ gt / CONCEPT OF WEGHT W mg, whee W weght, N (lbf), m mss, kg (lbf-se /ft), d g lol eleto of gvt, m/se (ft/se ). KNETCS Newto's seod lw fo ptle ΣF d(mv)/dt, whee ΣF the sum of the ppled foes tg o the ptle, N (lbf). Fo ostt mss, ΣF mdv/dt m DYNAMCS (otued) Oe-Dmesol Moto of Ptle Whe efeg to moto the -deto, F /m, whee F the esultt of the ppled foes the -deto. F deped o t, d v geel. f F depeds ol o t, the v f the foe s ostt (depedet of tme, dsplemet, o velot), Tgetl d Noml Kets fo Pl Poblems Wokg wth the tgetl d oml detos, ΣF t m t mdv t /dt d ΣF m m (v t /ρ) mpulse d Mometum Assumg the mss s ostt, the equto of moto s mdv dt F mdv Fdt t m[ v () ( )] ( ) t v F t dt The left sde of the equto epesets the hge le mometum of bod o ptle. The ght sde s temed the mpulse of the foe F (t ) betwee t d t t. Wok d Eeg Wok W s defed s W F d (Fo ptle flow, see FLUD MECHANCS seto.) KNETC ENERGY The ket eeg of ptle s the wok doe b etel get eletg the ptle fom est to velot v. T mv / hgg the velot fom v to v, the hge ket eeg s T T mv / mv / Potetl Eeg t () t v [ F ( t ) m] o t () t v t v ( t ) o F v v /m o v v (F t t /m) t dt dt The wok doe b etel get the pesee of osevtve feld s temed the hge potetl eeg. v F t /( m) t / t 5

36 Potetl Eeg Gvt Feld U mgh, whee h the elevto bove spefed dtum. Elst Potetl Eeg Fo le elst spg wth modulus, stffess, o spg ostt k, the foe s F s k, whee the hge legth of the spg fom the udefomed legth of the spg. The potetl eeg stoed the spg whe ompessed o eteded b mout s U k / The hge of potetl eeg defomg spg fom posto to posto s U U k k Pple of Cosevto of Wok d Eeg f T d U e ket eeg d potetl eeg t stte, the fo osevtve sstems (o eeg dsspto), the lw of osevto of eeg s U T U T f fto s peset, the the wok doe b the fto foes must be outed fo. U T W U T (Ce must be eesed dug omputtos to oetl ompute the lgeb sg of the wok tem). mpt Mometum s oseved whle eeg m o m ot be oseved. Fo det etl mpt wth o etel foes m v m v m v m v, whee m, m v, v v, v the msses of the two bodes, the velotes the velotes fte mpt. Fo mpt wth dsspto of eeg, the eltve velot epesso s v v e v v e the oeffet of esttuto fo the mtels, d the subspt deotes the ompoets oml to the ple of mpt. Kowg e, the velotes fte eboud e v v mv m v befoe mpt, d ( e) ( m em ) ( e) ( em m ) m m m m v v whee e. e, pefetl elst e, pefetl plst (o eboud) DYNAMCS (otued) FRCTON The Lws of Fto e. The totl fto foe F tht be developed s depedet of the mgtude of the e of ott.. The totl fto foe F tht be developed s popotol to the oml foe N.. Fo low velotes of sldg, the totl fto foe tht be developed s ptll depedet of the velot, lthough epemets show tht the foe F eess to stt sldg s gete th tht eess to mt sldg. The fomul epessg the lws of fto s F µ N, whee µ the oeffet of fto. Stt fto wll be less th o equl to µ s N, whee µ s s the oeffet of stt fto. At the pot of mpedg moto, F µ s N Whe moto s peset F µ k N, whee µ k the oeffet of ket fto. The vlue of µ k s ofte tke to be 75% of µ s. Belt fto s dsussed the Stts seto. MASS MOMENT OF NERTA z ( ) dm A tble lstg momet of et fomuls s vlble t the ed of ths seto fo some stdd shpes. Pllel As Theoem z z md, whee z the mss momet of et bout spef s ( ths se, the z-s), z the mss momet of et bout the bod's mss ete ( ths se, pllel to the z-s), m the mss of the bod, d d the oml dste fom the mss ete to the spef s desed ( ths se, the z-s). Also, z m z, whee m the totl mss of the bod d z the dus of gto ( ths se, bout the z-s). 6

37 PLANE MOTON OF A RGD BODY Fo gd bod ple moto the - ple m F m F z α M z, whee the ete of gvt d α gul eleto of the bod. Rotto About Fed As O α ΣM O, whee O deotes the s bout whh otto ous. Fo otto bout fed s used b ostt ppled momet M α M / ω ω O (M / ) t θ θ O ω O t (M / ) t The hge ket eeg of otto s the wok doe eletg the gd bod fom ω O to ω. ω ω θ O O O Mdθ θ O Ket Eeg The ket eeg of bod ple moto s stteous Cete of Rotto The stteous ete of otto fo bod ple moto s defed s tht posto bout whh ll potos of tht bod e ottg. T ACθ ω, d v BCθ, whee b ( v v ) ω m Ψ C the stteous ete of otto, θ the ottol velot bout C, d AC, BC d detemed b the geomet of the stuto. DYNAMCS (otued) CENTRFUGAL FORCE Fo gd bod (of mss m) ottg bout fed s, the etfugl foe of the bod t the pot of otto s F mω mv /, whee the dste fom the ete of otto to the ete of the mss of the bod. BANKNG OF CURVES (WTHOUT FRCTON) t θ v /(g), whee θ the gle betwee the odw sufe d the hozotl, v the velot of the vehle, d the dus of the uve. FREE VBRATON The equto of moto s m mg k( δ st ) Fom stt equlbum mg kδ st, whee k the spg ostt, d δ st the stt defleto of the spg suppotg the weght (mg). m k, ( k m) o The soluto to ths dffeetl equto s (t) C os ( k m) t C s ( k m) t, whee (t) the dsplemet the -deto d C, C ostts of tegto whose vlues deped o the tl odtos of the poblem. The qutt k m s lled the udmped tul feque ω o ω k m kδ st k Tmosheko, S. d D.H. Youg, Egeeg Mehs, Copght 95 b MGw-Hll Comp,. Dgms epoduto pemsso pedg. 7

38 Fom the stt defleto elto ω g δ st The peod of vbto s τ π ω π m k π f the tl odtos e () d ( ) v, the (t) os ω t (v /ω ) s ω t f the tl odtos e () d ( ), the (t) os ω t, whh s the equto fo smple hmo moto whee the mpltude of vbto s. Tosol Fee Vbto θ ωθ, whee ω k t GJ L k t the tosol spg ostt GJ/L, the mss momet of et of the bod, G the she modulus, J the e pol momet of et of the oud shft oss seto, d L the legth of the oud shft. The soluto to the equto of moto s θ θ os ωt ( θ ω ) sωt, whee θ the tl gle of otto d θ the tl gul velot. δ st g DYNAMCS (otued) The peod of tosol vbto s τ π ω π L GJ The udmped ul tul feque of tosol vbto s ω GJ L 8

39 DYNAMCS (otued) 9 C b b b C b C C b C b C Fgue Ae & Cetod Ae Momet of et (Rdus of Gto) Podut of et h h h h h A bh/ b/ h/ A bh/ b/ h/ A bh/ ( b)/ h/ A bh b/ h/ A h ( b) h( b) ( b) A b s θ (b os θ)/ ( s θ)/ bh /6 b h/6 bh / b h/ bh / b h/ House, Geoge W. & Dold E. Hudso, Appled Mehs Dms, Copght 959 b D. V Nostd Comp,., Peto, NJ. Tble epted b pemsso of G.W. House & D.E. Hudso. bh /6 b h/6 bh 6 [ bh( b b )] bh 6 [ bh( b b )] bh b h bh b h J [ bh( b h )] h ( b b ) h 6( b) ( b) ( b s θ) [ b sθ( b os θ) ] ( b s θ) [ b sθ( b osθ) ] ( b sθosθ) 6 p h b h b h b h b h ( b b ) 6 8 ( b b ) 6 Abh 6 b h Abh b h Abh b h 8 7 Abh 6 b h [ ] bh ( b) [ Ah( b) ] bh ( b) h 8 Ah( b) 6 h b h b ( b h ) h h 6 ( b b ) 8( b) ( b) ( b) [ ] 7 [ ] Abh b ( sθ) ( b s θos θ) ( b os θ) ( sθ) ( b osθ) ( b osθ) 6 h 7

40 DYNAMCS (otued) Fgue Ae & Cetod Ae Momet of et (Rdus of Gto) Podut of et A π A π ( b ) A π / /(π) (θ sθ os θ)/ (θ sθ os θ)/ A b/ /5 House, Geoge W. & Dold E. Hudso, Appled Mehs Dms, Copght 959 b D. V Nostd Comp,., Peto, NJ. Tble epted b pemsso of G.W. House & D.E. Hudso. ( ) π π π π π ( ) π π s θ θ θ A os s s s θ θ θ θ θ θ A θ θ θ θ θ θ θ θ θ θ os s os s os s os s A A θ θ θ θ θ θ θ θ θ θ os s os s os s os s b b b b 5 J π π π 5 p A ( ) ( ) 5 b J b b b π π π π π ( ) ( ) ( ) 5 b b b p ( ) b A π ( ) ( ) θ θ θ θ θ θ θ θ os s os s C b C C C C b b C

41 DYNAMCS (otued) Fgue Ae & Cetod Ae Momet of et (Rdus of Gto) Podut of et A b/ /5 b/8 b /5 b /7 Ab/ b House, Geoge W. & Dold E. Hudso, Appled Mehs Dms, Copght 959 b D. V Nostd Comp,., Peto, NJ. Tble epted b pemsso of G.W. House & D.E. Hudso. 7 5 b ( ) h b bh A ( ) hb bh ( ) ( ) b h ( ) h b bh A ( ) h b bh ( ) b h b C h C b (h/b ) (h/b / ) / h b C

42 DYNAMCS (otued) Fgue Mss & Cetod Mss Momet of et (Rdus of Gto) Podut of et M ΡLA L/ z δ le dest M πρra R R z δ le dest (mss/l) M πρr A h/ z δ le dest (mss/l) M πρr / z δ mss/vol. House, Geoge W. & Dold E. Hudso, Appled Mehs Dms, Copght 959 b D. V Nostd Comp,., Peto, NJ. Tble epted b pemsso of G.W. House & D.E. Hudso. ML ML z z L L z z et. et.,, MR MR MR MR z z R R R R z et. z z z z MR, ( ) ( ) h R M MR h R M z z ( ) ( ) h R R h R z z et. et.,, ( ) vol. mss δ πρ z h R R h M ( ) ( ) ( ) h R R M R R M h R R M z z ( ) ( ) ( ) h R R R R h R R z z et. et.,, MR MR MR z z R R R z z et., C L z R C z z h R R R h C z R C z

43 UNAXAL STRESS-STRAN Stess-St Cuve fo Mld Steel MECHANCS OF MATERALS Ul Lodg d Defomto σ P/A, whee σ stess o the oss seto, P lodg, d A oss-setol e. ε δ/l, whee δ logtudl defomto d L legth of membe. P A E σ ε δ L δ PL AE The slope of the le poto of the uve equls the modulus of elstt. ENGNEERNG STRAN ε L / L, whee ε egeeg st (uts pe ut), L hge legth (uts) of membe, L ogl legth (uts) of membe, ε pl plst defomto (pemet), d ε el elst defomto (eoveble). Equlbum equemets: ΣF ; ΣM Deteme geomet omptblt wth the estts. Use le foe-defomto eltoshp; F kδ. DEFNTONS She Stess-St γ τ/g, whee γ she st, τ she stess, d G she modulus (ostt le foe-defomto eltoshp). E G, ( ν) whee E modulus of elstt Posso's to, (ltel st)/(logtudl st). THERMAL DEFORMATONS δ t αl (Τ Τ o ), whee δ t defomto used b hge tempetue, α tempetue oeffet of epso, L legth of membe, Τ fl tempetue, d Τ o tl tempetue. CYLNDRCAL PRESSURE VESSEL Cldl Pessue Vessel Fo tel pessue ol, the stesses t the sde wll e: o σt P d > σ > P o Fo etel pessue ol, the stesses t the outsde wll e: σ t tgetl (hoop) stess, σ dl stess, P tel pessue P o etel pessue sde dus o outsde dus Fo vessels wth ed ps, the l stess s: σ P o These e ppl stesses. o o σt Po d > σ whee Fl, Rhd A. & Pul K. Toj, Egeeg Mtels & The Appltos, th Ed. Copght 99 b Houghto Mffl Co. Fgue used wth pemsso. > P, o

44 Whe the thkess of the lde wll s bout oe-teth o less, of sde dus, the lde be osdeed s thwlled. whh se, the tel pessue s essted b the hoop stess P P σt d σ t t whee t wll thkess. STRESS AND STRAN Ppl Stesses Fo the spel se of two-dmesol stess stte, the equtos fo ppl stess edue to σ σ σ σ σ, σ b ± τ σ The two ozeo vlues lulted fom ths equto e tempol lbeled σ d σ b d the thd vlue σ s lws zeo ths se. Depedg o the vlues, the thee oots e the lbeled odg to the oveto: lgebll lgest σ, lgebll smllest σ, othe σ. A tpl D stess elemet s show below wth ll dted ompoets show the postve sese. Moh's Cle Stess, D To ostut Moh's le, the followg sg ovetos e used.. Tesle oml stess ompoets e plotted o the hozotl s d e osdeed postve. Compessve oml stess ompoets e egtve.. Fo ostutg Moh's le ol, sheg stesses e plotted bove the oml stess s whe the p of sheg stesses, tg o opposte d pllel fes of elemet, foms lokwse ouple. Sheg stesses e plotted below the oml s whe the she stesses fom outelokwse ouple. The le dw wth the ete o the oml stess (hozotl) s wth ete, C, d dus, R, whee σ C σ, R σ σ τ MECHANCS OF MATERALS (otued) The two ozeo ppl stesses e the: σ C R σb C R The mmum ple she stess s τ m R. Howeve, the mmum she stess osdeg thee dmesos s lws σ σ τ m. Hooke's Lw Thee-dmesol se: ε (/E)[σ v(σ σ z )] γ τ /G ε (/E)[σ v(σ z σ )] γ z τ z /G ε z (/E)[σ z v(σ σ )] γ z τ z /G Ple stess se (σ z ): ε (/E)(σ vσ ) σ ε (/E)(σ vσ ) σ ε z (/E)(vσ vσ ) τ Ul se (σ σ z ): ε, ε, ε z oml st, σ, σ, σ z oml stess, γ, γ z, γ z she st, τ, τ z, τ z she stess, E modulus of elstt, G she modulus, d v Posso's to. (σ, τ) E v v ε ε v γ σ Eε o σ Eε whee STATC LOADNG FALURE THEORES Mmum-Noml-Stess Theo The mmum-oml-stess theo sttes tht flue ous whe oe of the thee ppl stesses equls the stegth of the mtel. f σ > σ > σ, the the theo pedts tht flue ous wheeve σ S t o σ S whee S t d S e the tesle d ompessve stegths, espetvel. Mmum-She-Stess Theo The mmum-she-stess theo sttes tht eldg begs whe the mmum she stess equls the mmum she stess teso-test speme of the sme mtel whe tht speme begs to eld. f σ σ σ, the the theo pedts tht eldg wll ou wheeve τ m S / whee S s the eld stegth. v (σ, τ)

45 Dstoto-Eeg Theo The dstoto-eeg theo sttes tht eldg begs wheeve the dstoto eeg ut volume equls the dstoto eeg the sme volume whe ull stessed to the eld stegth. The theo pedts tht eldg wll ou wheeve TORSON ( σ σ ) ( σ σ ) ( σ σ ) γ φz lmt z ( φ z) ( dφ dz) The she st ves det popoto to the dus, fom zeo st t the ete to the getest st t the outsde of the shft. dφ/dz s the twst pe ut legth o the te of twst. τ φz G γ φz G (dφ/dz) T G (dφ/dz) A da GJ(dφ/dz) whee J pol momet of et (see tble t ed of DYNAMCS seto). φ L T o GJ dz TL, whee GJ φ totl gle (ds) of twst, T toque, d L legth of shft. τ φz G [T/(GJ)] T/J T GJ φ L, whee T/φ gves the twstg momet pe d of twst. Ths s lled the tosol stffess d s ofte deoted b the smbol k o. Fo Hollow, Th-Wlled Shfts T τ, whee Amt t thkess of shft wll d A m the totl me e elosed b the shft mesued to the mdpot of the wll. BEAMS Sheg Foe d Bedg Momet Sg Covetos. The bedg momet s postve f t podues bedg of the bem ove upwd (ompesso top fbes d teso bottom fbes).. The sheg foe s postve f the ght poto of the bem teds to she dowwd wth espet to the left. S MECHANCS OF MATERALS (otued) The eltoshp betwee the lod (q), she (V), d momet (M) equtos e: dv() q( ) d dm() V d V V M M [ q( ) ] V ( ) d d Stesses Bems ε /ρ, whee ρ the dus of uvtue of the defleted s of the bem d the dste fom the eutl s to the logtudl fbe questo. Usg the stess-st eltoshp σ Eε, Al Stess: σ E/ρ, whee σ the oml stess of the fbe loted -dste fom the eutl s. /ρ M/(E), whee M the momet t the seto d the momet of et of the oss-seto. σ M/, whee the dste fom the eutl s to the fbe loto bove o below the s. Let, whee dste fom the eutl s to the outemost fbe of smmetl bem seto. σ ± M/ Let S /: the, σ ± M/S, whee S the elst seto modulus of the bem membe. Tsvese she flow: q VQ/ d Tsvese she stess: τ VQ/(b), whee q she flow, τ she stess o the sufe, V she foe t the seto, b wdth o thkess of the oss-seto, d Q A' ' whee A e bove the le (o ple) upo whh the desed tsvese she stess ts d ' dste fom eutl s to e etod. Tmosheko, S. & Gleso H. MCullough, Elemets of Stegth of Mtels, 99 b K. V Nostd Co. Used wth pemsso fom Wdswoth Publshg Co. 5

46 Defleto of Bems Usg /ρ M/(E), d E d d E d M, dffeetl equto of defleto uve dm()/d V d E d dv()/d q Deteme the defleto uve equto b double tegto (ppl boud odtos pplble to the defleto d/o slope). E (d/d) M() d E [ M() d] d The ostts of tegto be detemed fom the phsl geomet of the bem. COLUMNS Fo log olums wth ped eds: Eule's Fomul π E P P tl l lodg, ubed olum legth. substtute A: P π E A ( ) whee dus of gto d / sledeess to fo the olum. Fo futhe olum desg theo, see the CVL ENG- NEERNG d MECHANCAL ENGNEERNG setos. MECHANCS OF MATERALS (otued) ELASTC STRAN ENERGY f the st ems wth the elst lmt, the wok doe dug defleto (eteso) of membe wll be tsfomed to potetl eeg d be eoveed. f the fl lod s P d the oespodg elogto of teso membe s δ, the the totl eeg U stoed s equl to the wok W doe dug lodg. U W Pδ/ The st eeg pe ut volume s u U/AL σ /E MATERAL PROPERTES Mtel Uts Steel Alumum Cst o (fo teso) Wood (F) Modulus of Mps Elstt, E GP Modulus of Mps Rgdt, G GP Posso's Rto, v.... 6

47 Bem Defleto Fomuls Spel Cses (δ s postve dowwd) L P b δ m φ m P δ 6E P δ 6E ( ), fo > ( ), fo P δ m 6E ( L ) φ m P E w δ m ( 6 L L) w δ o E δ m w o L 8E φ m w o L 6E L φ m δ m M δ o E δ m M o L E φ m M o L E 7 L R Pb/L L P M b φ m R P/L δ 6 δ 6 Pb LE Pb LE L b ( ) ( L b ) ( ), fo > Pb L b [ ( L b ) ], fo δ m 9 LE t L b φ φ ( L ) Pb 6LE Pb 6LE ( L b) w δ w o E ( L L ) δ m 5w o L 8E φ φ w o L E L R w L/ R w L/ M L R M /L R M /L M o L 6E L δ δ m M L o 9 E t L φ φ M ol 6E M ol E Cdll, S.H. & N.C. Dhl, A toduto to The Mehs of Solds, Copght 959 b the MGw-Hll Book Co.,. Tble epted wth pemsso fom MGw-Hll.

48 FLUD MECHANCS DENSTY, SPECFC VOLUME, SPECFC WEGHT, AND SPECFC GRAVTY The deftos of dest, spef volume, spef weght, d spef gvt follow: lso ρ lmt V γ lmt V γ lmt V SG γ γ w m V W V g m V ρg ρ ρ w, whee ρ dest (lso mss dest), m mss of ftesml volume, V volume of ftesml objet osdeed, γ spef weght, W weght of ftesml volume, SG spef gvt, d ρ w mss dest of wte t stdd odtos, kg/m (6. lbm/ft ). SURFACE TENSON AND CAPLLARTY Sufe teso σ s the foe pe ut ott legth σ F/L, whee σ sufe teso, foe/legth, F sufe foe t the tefe, d L legth of tefe. The pll se h s ppomted b h σ os β/(γd), whee h the heght of the lqud the vetl tube, σ the sufe teso, β the gle mde b the lqud wth the wetted tube wll, γ spef weght of the lqud, d d the dmete o the pll tube. THE PRESSURE FELD N A STATC LQUD AND MANOMETRY STRESS, PRESSURE, AND VSCOSTY Stess s defed s τ ( P) lmt F A, whee A τ (P) sufe stess veto t pot P, F foe tg o ftesml e A, A ftesml e t pot P, d τ p τ t µ (dv/d) (oe-dmesol;.e., ), whee τ d τ t the oml d tgetl stess ompoets t pot P, p the pessue t pot P, µ bsolute dm vsost of the flud N s/m [lbm/(ft-se)], dv velot t boud odto, d d oml dste, mesued fom boud. v µ/ρ, whee v kemt vsost; m /s (ft /se). Fo th Newto flud flm d le velot pofle, v() V/δ; dv/d V/δ, whee V velot of plte o flm d δ thkess of flud flm. Fo powe lw (o-newto) flud τ t K (dv/d), whee K osste de d powe lw de < pseudo plst > dltt The dffeee pessue betwee two dffeet pots s p p γ (z z ) γh Bobe, W. & R.A. Keo, Flud Mehs, Copght 98 b Joh Wle & Sos,. Dgms epted b pemsso of Wllm Bobe & Rhd A. Keo. 8

49 Fo smple momete, p o p γ h γ h Absolute pessue tmosphe pessue gge pessue edg Absolute pessue tmosphe pessue vuum gge pessue edg Aothe deve tht woks o the sme pple s the momete s the smple bomete. p tm p A p v γh p B γh FLUD MECHANCS (otued) p v vpo pessue of the bomete flud FORCES ON SUBMERGED SURFACES AND THE CENTER OF PRESSURE The pessue o pot t dste Z below the sufe s p p o γz, fo Z f the tk wee ope to the tmosphee, the effets of p o ould be goed. The oodtes of the ete of pessue CP e * ( γ z sα) ( p A) d z* ( γ sα) ( p A) whee * the -dste fom the etod (C) of e (A) to the ete of pessue, z* the z-dste fom the etod (C) of e (A) to the ete of pessue, the momet d podut of et of the d z e, p the pessue t the etod of e (A), d Z the slt dste fom the wte sufe to the etod (C) of e (A). f the fee sufe s ope to the tmosphee, the p o d p γz s α. * z ( AZ ) d z* ( AZ ) The foe o the plte be omputed s F [p A v (p p ) A v /] V f γ f j, whee F foe o the plte, p pessue t the top edge of the plte e, p pessue t the bottom edge of the plte e, A v vetl pojeto of the plte e, V f volume of olum of flud bove plte, d γ f spef weght of the flud. ARCHMEDES' PRNCPLE AND BUOYANCY. The buot foe eeted o submeged o flotg bod s equl to the weght of the flud dspled b the bod.. A flotg bod dsples weght of flud equl to ts ow weght;.e., flotg bod s equlbum. The ete of buo s loted t the etod of the submeged poto of the bod. the se of bod lg t the tefe of two mmsble fluds, the buot foe equls the sum of the weghts of the fluds dspled b the bod. ONE-DMENSONAL FLOWS The Cotut Equto So log s the flow Q s otuous, the otut equto, s ppled to oedmesol flows, sttes tht the flow pssg two pots ( d ) stem s equl t eh pot, A V A V. Q AV m ρq ρav, whee Q volumet flow te, m mss flow te, Bobe, W. & R.A. Keo, Flud Mehs, Copght 98 b Joh Wle & Sos,. Dgms epted b pemsso of Wllm Bobe & Rhd A. Keo. 9

50 A oss seto of e of flow, V vege flow velot, d ρ the flud dest. Fo sted, oe-dmesol flow, m s ostt. f, ddto, the dest s ostt, the Q s ostt. The Feld Equto s deved whe the eeg equto s ppled to oe-dmesol flows. Assumg o fto losses d tht o pump o tube ests betwee setos d the sstem, p γ V z g p, p pessue t setos d, V, V vege velot of the flud t the setos, z, z the vetl dste fom dtum to the setos (the potetl eeg), γ the spef weght of the flud, d g the eleto of gvt. FLOW OF A REAL FLUD p V p V z z h f γ g γ g The pessue dop s flud flows though ppe of ostt oss-seto d whh s held t fed elevto s h f (p p )/γ, whee p γ V z, g whee h f the hed loss, osdeed fto effet, d ll emg tems e defed bove. Flud Flow The velot dstbuto fo lm flow ul tubes o betwee ples s v v m, R whee the dste (m) fom the etele, R the dus (m) of the tube o hlf the dste betwee the pllel ples, v the lol velot (m/s) t, d v m the velot (m/s) t the etele of the dut. v m.8v, fo full tubulet flow (Re >,), v m V, fo ul tubes d v m.5v, fo pllel ples, whee V the vege velot (m/s) the dut. The she stess dstbuto s τ, τ R whee w τ d τ w e the she stesses t d d R espetvel. FLUD MECHANCS (otued) The dg foe F D o objets mmesed lge bod of flowg flud o objets movg though stgt flud s C V Dρ A FD C D the dg oeffet (see pge 6), V the velot (m/s) of the udstubed flud, d A the pojeted e (m ) of bluff objets suh s sphees, ellpsods, d dsks d pltes, ldes, ellpses, d fols wth es pepedul to the flow. Fo flt pltes pled pllel wth the flow C D./Re.5 ( < Re < 5 5 ) C D./Re /7 ( 6 < Re < 9 ) The htest legth the Reolds Numbe (Re) s the legth of the plte pllel wth the flow. Fo bluff objets, the htest legth s the lgest le dmeso (dmete of lde, sphee, dsk, et.) whh s pepedul to the flow. Reolds Numbe Re VDρ µ VD v ( ) V D ρ Re, ( ) K 8 whee ρ the mss dest, D the dmete of the ppe o dmeso of the flud stemle, µ the dm vsost, v the kemt vsost, Re the Reolds umbe (Newto flud), Re' the Reolds umbe (Powe lw flud), d K d e defed o pge 8. The tl Reolds umbe (Re) s defed to be the mmum Reolds umbe t whh flow wll tu tubulet. Hdul Gdet (Gde Le) The hdul gdet (gde le) s defed s mg le bove ppe so tht the vetl dste fom the ppe s to the le epesets the pessue hed t tht pot. f ow of pezometes wee pled t tevls log the ppe, the gde le would jo the wte levels the pezomete wte olums. Eeg Le (Beoull Equto) The Beoull equto sttes tht the sum of the pessue, velot, d elevto heds s ostt. The eeg le s ths sum o the "totl hed le" bove hozotl dtum. The dffeee betwee the hdul gde le d the eeg le s the V / g tem.

51 STEADY, NCOMPRESSBLE FLOW N CONDUTS AND PPES The eeg equto fo ompessble flow s p V p V z z h f γ g γ g f the oss-setol e d the elevto of the ppe e the sme t both setos ( d ), the z z d V V. The pessue dop p p s gve b the followg: p p γh f The D equto s L V h f f, D g whee f f(re, e/d), the fto fto, D dmete of the ppe, L legth ove whh the pessue dop ous, e oughess fto fo the ppe, d ll othe smbols e defed s befoe. A ht tht gves f vesus Re fo vous vlues of e/d, kow s Mood o Stto dgm, s vlble t the ed of ths seto o pge 5. Fto Fto fo Lm Flow The equto fo Q tems of the pessue dop p f s the Hge-Poseulle equto. Ths elto s vld ol fo flow the lm ego. πr p f πd p f Q 8µ L 8µ L Flow Noul Coduts Alss of flow oduts hvg oul oss seto uses the hdul dmete D H, o the hdul dus R H, s follows oss - setol e D R H wetted pemete Mo Losses Ppe Fttgs, Cottos, d Epsos Hed losses lso ou s the flud flows though ppe fttgs (.e., elbows, vlves, ouplgs, et.) d sudde ppe ottos d epsos. p V p V z z h f h f, fttg γ g γ g whee V h f, fttg C g Spef fttgs hve htest vlues of C, whh wll be povded the poblem sttemet. A geell epted oml vlue fo hed loss well-stemled gdul ottos s h f, fttg. V / g H FLUD MECHANCS (otued) The hed loss t ethe ete o et of ppe fom o to esevo s lso gve b the h f, fttg equto. Vlues fo C fo vous ses e show s follows. PUMP POWER EQUATON W Qγh/η, whee Q qutt of flow (m /s o fs), h hed (m o ft) the flud hs to be lfted, η effe, d W powe (wtts o ft-lbf/se). THE MPULSE-MOMENTUM PRNCPLE The esultt foe gve deto tg o the flud equls the te of hge of mometum of the flud. ΣF Q ρ V Q ρ V, whee ΣF the esultt of ll etel foes tg o the otol volume, Q ρ V the te of mometum of the flud flow eteg the otol volume the sme deto of the foe, d Q ρ V the te of mometum of the flud flow levg the otol volume the sme deto of the foe. Ppe Beds, Elgemets, d Cottos The foe eeted b flowg flud o bed, elgemet, o otto ppe le m be omputed usg the mpulse-mometum pple. p A p A os α F Qρ (V os α V ) F W p A s α Qρ (V s α ), whee F the foe eeted b the bed o the flud (the foe eeted b the flud o the bed s equl mgtude d opposte sg), F d F e the -ompoet d -ompoet of the foe, Bobe, W. & R.A. Keo, Flud Mehs, Copght 98 b Joh Wle & sos,. Dgm epted b pemsso of Wllm Bobe & Rhd A. Keo. Ved, J.K., Elemet Flud Mehs, Copght 95 b J.K. Ved. Dgms epted b pemsso of Joh Wle & Sos,.

52 p the tel pessue the ppe le, A the oss-setol e of the ppe le, W the weght of the flud, V the velot of the flud flow, α the gle the ppe bed mkes wth the hozotl, ρ the dest of the flud, d Q the qutt of flud flow. Jet Populso F Qρ(V ) F γha, whee F the populsve foe, γ the spef weght of the flud, h the heght of the flud bove the outlet, A the e of the ozzle tp, Q A gh, d V gh Defletos d Bldes FXED BLADE F Qρ(V os α V ) F Qρ(V s α ) MOVNG BLADE F Qρ(V V ) Qρ(V v)( os α) F Qρ(V V ) Qρ(V v) s α, whee v the velot of the blde. MPULSE TURBNE FLUD MECHANCS (otued) W Qρ (V v)( os α) v, whee W powe of the tube. W m Qρ (V /)( os α) Whe α 8, W m (QρV )/ (QγV )/g MULTPATH PPELNE PROBLEMS The sme hed loss ous eh bh s the ombto of the two. The followg equtos m be solved smulteousl fo V A d V B : l A VA l B VB hl f A f B D g D g ( πd ) V ( πd A ) VA ( πdb ) VB The flow Q be dvded to Q A d Q B whe the ppe htests e kow. OPEN-CHANNEL FLOW AND/OR PPE FLOW Mg's Equto V (k/)r / S /, whee k fo S uts k.86 fo USCS uts V velot (m/s, ft/se), oughess oeffet, R hdul dus (m, ft), d S slope of eeg gde le (m/m, ft/ft). Hze-Wllms Equto V k CR.6 S.5, whee C oughess oeffet k.89 fo S uts k.8 fo USCS uts Othe tems defed s bove. A Ved, J.K., Elemet Flud Mehs, Copght 95 b J.K. Ved. Dgms epted b pemsso of Joh Wle & Sos,. B

53 MACH NUMBER The speed of soud del gs s gve b krt, whee k P / v. Ths shows tht the oust velot del gs depeds ol o ts tempetue. The mh umbe M s to of the flud velot V to the speed of soud: M V/ FLUD MEASUREMENTS The Ptot Tube Fom the stgto pessue equto fo ompessble flud, ( )( ) ( ) γ V ρ po ps g po ps whee V the velot of the flud, p o the stgto pessue, d p s the stt pessue of the flud t the elevto whee the mesuemet s tke. FLUD MECHANCS (otued) Ofes The oss-setol e t the ve ott A s htezed b oeffet of otto C d gve b C A. p p Q CA g z z γ γ whee C, the oeffet of the mete, s gve b C C C v ( A ) C A V g P s V, P s P o Fo ompessble flud, use the bove ompessble flud equto f the mh umbe.. Vetu Metes Q C A p g γ z z v ( A ) γ A p Submeged Ofe opetg ude sted-flow odtos: whee, C v the oeffet of velot. The bove equto s fo ompessble fluds. Q A V CA C C g ( h h ) v A ( h h ) whh the podut of C d C v s defed s the oeffet of dshge of the ofe. Ved, J.K., Elemet Flud Mehs, Copght 95 b J.K. Ved. Dgms epted b pemsso of Joh Wle & Sos, g

54 Ofe Dshgg Feel to Atmosphee Q CA gh whh h s mesued fom the lqud sufe to the etod of the ofe opeg. DMENSONAL HOMOGENETY AND DMEN- SONAL ANALYSS Equtos tht e fom tht do ot deped o the fudmetl uts of mesuemet e lled dmesoll homogeeous equtos. A spel fom of the dmesoll homogeeous equto s oe tht volves ol dmesoless goups of tems. Bukghm's Theoem: The umbe of depedet dmesoless goups tht m be emploed to desbe pheomeo kow to volve vbles s equl to the umbe ( ), whee s the umbe of bs dmesos (.e., M, L, T) eeded to epess the vbles dmesoll. SMLTUDE ode to use model to smulte the odtos of the pototpe, the model must be geometll, kemtll, d dmll sml to the pototpe sstem. To obt dm smlt betwee two flow ptues, ll depedet foe tos tht be wtte must be the sme both the model d the pototpe. Thus, dm smlt betwee two flow ptues (whe ll possble foes e tg) s epessed the fve smulteous equtos below. F Fp F FV F FG F FE F FT p p p p p F Fp F FV F FG F FE F FT m m m m m ρv p Vlρ µ p p Vlρ µ p ρv p V V lg lg p m ρv ρv E E p m ρlv ρlv σ σ m m [ Re] [ Re] [ F] [ F] m p p [ C] [ C] p m m m [ We] p [ We] m FLUD MECHANCS (otued) whee the subspts p d m std fo pototpe d model espetvel, d F et foe, F P pessue foe, F V vsous foe, F G gvt foe, F E elst foe, F T sufe teso foe, Re Reolds umbe, We Webe umbe, C Cuh umbe, F Foude umbe, l htest legth, V velot, ρ dest, σ sufe teso, E modulus of elstt, µ dm vsost, p pessue, d g eleto of gvt. Re PROPERTES OF WATER f Tempetue C VD ρ VD µ v Spef Weght, α, kn/m Dest, ρ, kg/m Vsost, µ, P s Kemt Vsost, v 6, m /s Vpo Pessue e, pv, kp Fom "Hdul Models," A.S.C.E. Mul of Egeeg Pte, No. 5, A.S.C.E., 9. See footote. e Fom J.H. Kee d F.G. Kees, Themodm Popetes of Stem, Joh Wle & Sos, 96. f Compled fom m soues ludg those dted, Hdbook of Chemst d Phss, 5th Ed., The CRC Pess, 97, d Hdbook of Tbles fo Appled Egeeg See, The Cheml Rubbe Co., 97. Hee, f E/ 6.98 the E.98 6 kp, whle f µ.78, the µ.78 P's, d so o. Ved, J.K. d Robet L. Steet, Elemet Flud Mehs, Copght 95, Joh Wle & Sos,. Ved, J.K., Elemet Flud Mehs, Copght 95 b J.K. Ved. Dgms epted b pemsso of Joh Wle & Sos,

55 MOODY (STANTON) DAGRAM FLUD MECHANCS (otued) Repted b pemsso of ASHRAE. e, (ft) e, (mm) Rveted steel Coete Cst o.85.5 Glvzed o.5.5 Commel steel o wought o.5.6 Dw tubg f Re f 5

56 DRAG COEFFCENTS FOR SPHERES, DSKS, AND CYLNDERS FLUD MECHANCS (otued) C D, Re < Re CD FD ρv A 6 DV

57 THERMODYNAMCS PROPERTES OF SNGLE-COMPONENT SYSTEMS R R, ( mol. wt. ) Nomeltue. tesve popetes e depedet of mss.. Etesve popetes e popotol to mss.. Spef popetes e lowe se (etesve/mss). Stte Futos (popetes) Absolute Pessue, p (lbf/ o P) Absolute Tempetue, T ( R o K) Spef Volume, v (ft /lbm o m /kg) tel Eeg, u (usull Btu/lbm o kj/kg) Ethlp, h u Pv (sme uts s u) Etop, s [ Btu/(lbm- R) o kj/(kg K)] Gbbs Fee Eeg, g h Ts (sme uts s u) Helmholz Fee Eeg, u Ts (sme uts s u) Het Cpt t Costt Pessue, h p T Het Cpt t Costt Volume, Qult (pples to lqud-vpo sstems t stuto) s defed s the mss fto of the vpo phse: m g /(m g m f ), whee m g mss of vpo d m f mss of lqud. Spef volume of two-phse sstem be wtte: v v g ( )v f o v v fg v f, whee v f spef volume of stuted lqud, v g spef volume of stuted vpo, d v fg spef volume hge upo vpozto v g v f Sml epessos est fo u, h, d s: u u g ( ) u f h h g ( ) h f s s g ( ) s f Fo smple subste, spefto of two tesve, depedet popetes s suffet to f ll the est. Fo del gs, Pv RT o PV mrt, d P v /T P v /T, whee p pessue, v spef volume, m mss of gs, R gs ostt, d T tempetue. v u T P v R s spef to eh gs but be foud fom whee R the uvesl gs ostt,55 ft-lbf/(lbmol- R) 8, J/(kmol K). Fo del Gses, P v R Also, fo del Gses: h u p p T T Fo old stdd, het ptes e ssumed to be ostt t the oom tempetue vlues. tht se, the followg e tue: u v T; h P T s P l (T /T ) R l (P /P ); d s v l (T /T ) R l (v /v ). Fo het ptes tht e tempetue depedet, the vlue to be used the bove equtos fo h s kow s the me het pt ( ) d s gve b p p T T dt p T T Also, fo ostt etop poesses: P v k P v k ; T P ( k)/k ( k)/k T P T v (k ) T v (k ), whee k p / v FRST LAW OF THERMODYNAMCS The Fst Lw of Themodms s sttemet of osevto of eeg themodm sstem. The et eeg ossg the sstem boud s equl to the hge eeg sde the sstem. Het Q s eeg tsfeed due to tempetue dffeee d s osdeed postve f t s wd o dded to the sstem. Closed Themodm Sstem (o mss osses boud) Q w U KE PE whee KE hge ket eeg PE hge potetl eeg Eeg oss the boud ol the fom of het o wok. Wok be boud wok, w b, o othe wok foms (eletl wok, et.) Wok w s osdeed postve f t s outwd o wok doe b the sstem. Revesble boud wok s gve b w b P dv. 7

58 SPECAL CASES OF CLOSED SYSTEMS Costt Pessue (Chles' Lw): w b P v (del gs) T/v ostt Costt Volume: w b (del gs) T/P ostt setop (del gs), Pv k ostt: w (P v P v )/( k) R (T T )/( k) Costt Tempetue (Bole's Lw): (del gs) Pv ostt w b RTl (v /v ) RTl (P /P ) Poltop (del gs), Pv ostt: w (P v P v )/( ) Ope Themodm Sstem (llowg mss to oss the boud) Thee s flow wok (PV) doe b mss eteg the sstem. The evesble flow wok s gve b: w ev v dp KE PE Fst Lw pples whethe o ot poesses e evesble. FRST LAW (eeg ble) [ V gz ] Σm [ h V gz ] Σm h e e Q W d et ( m u ) dt s s whee W et te of et o shft wok tsfe, m s mss of flud wth the sstem, u s spef tel eeg of sstem, Q te of het tsfe (egletg ket d potetl eeg). SPECAL CASES OF OPEN SYSTEMS Costt Volume: w ev v (P P ) Costt Pessue: w ev Costt Tempetue: (del gs) Pv ostt: w ev RTl (v /v ) RTl (P /P ) setop (del gs): Pv k ostt: w ev k (P v P v )/( k) kr (T T )/( k) w ev ( ) k P RT k P e k k THERMODYNAMCS (otued) Sted-Stte Sstems The sstem does ot hge stte wth tme. Ths ssumpto s vld fo sted opeto of tubes, pumps, ompessos, thottlg vlves, ozzles, d het ehges, ludg boles d odeses. m ( h V gz ) m e ( he Ve gz e ) Q W out d m m e, whee m mss flow te (subspts d e efe to let d et sttes of sstem), g eleto of gvt, Z elevto, V velot, d w te of wok. SPECAL CASES OF STEADY-FLOW ENERGY EQUATON Nozzles, Dffuses: Velot tems e sgft. No elevto hge, o het tsfe, d o wok. Sgle mss stem. h V / h e V e / Effe (ozzle) ( h h ), whee h es ethlp t setop et stte. Tubes, Pumps, Compessos: Ofte osdeed dbt (o het tsfe). Velot tems usull be goed. Sgft wok tems. Sgle mss stem. h h e w h he Effe (tube) h h hes h Effe (ompesso, pump) he h Thottlg Vlves d Thottlg Poesses: No wok, o het tsfe, d sgle-mss stem. Velot tems ofte sgft. h h e Boles, Codeses, Evpotos, Oe Sde Het Ehge: Het tsfe tems e sgft. Fo sglemss stem, the followg pples: h q h e V Het Ehges: No het o wok. Two septe flow tes m d m : h h m h h e V es es ( ) ( ) m e e Poltop: Pv ostt w ev (P v P v )/( ) Mes, Septos, Ope o Closed Feedwte Hetes: m h m eh m m e e d 8

59 BASC CYCLES Het eges tke het Q H t hgh tempetue T H, podue et mout of wok w, d ejet het Q L t low tempetue T L. The effe η of het ege s gve b: η w/q H (Q H Q L )/Q H The most effet ege possble s the Cot Cle. ts effe s gve b: η (T H T L )/T H whee T H d T L bsolute tempetues (Kelv o Rke). The followg het-ege les e plotted o P-v d T-s dgms (see pge 5): Cot, Otto, Rke Refgeto Cles e the evese of het-ege les. Het s moved fom low to hgh tempetue equg wok W. Cles be used ethe fo efgeto o s het pumps. Coeffet of Pefome (COP) s defed s: COP Q H /W fo het pump, d s COP Q L /W fo efgetos d odtoes. Uppe lmt of COP s bsed o evesed Cot Cle: COP T H /(T H T L ) fo het pump d COP T L /(T H T L ) fo efgeto. to efgeto, Btu/h,56 W DEAL GAS MXTURES,,, osttuets. Eh osttuet s del gs. Mole Fto: N umbe of moles of ompoet. N /N; N Σ N ; Σ Mss Fto: m /m; m Σ m ; Σ Moleul Weght: M m/n Σ M Gs Costt: R R / M To ovet mole ftos to mss ftos: M ( M ) To ovet mss ftos to mole ftos: M ( M ) m RT Ptl Pessues p p ; p V Ptl Volumes m RT V Vj ;V V whee p, V, T the pessue, volume, d tempetue of the mtue. p /p V /V Othe Popetes u Σ ( u ); h Σ ( h ); s Σ ( s ) u d h e evluted t T, d s s evluted t T d p. THERMODYNAMCS (otued) PSYCHROMETRCS We del hee wth mtue of d (subspt ) d wte vpo (subspt v): p p p v Spef Humdt (bsolute humdt) ω: ω m v /m, whee m v mss of wte vpo d m mss of d. ω.6p v /p.6p v /(p p v ) Reltve Humdt φ: φ m v /m g p v /p g, whee m g mss of vpo t stuto d p g stuto pessue t T. Ethlp h: h h ωh v Dew-Pot Tempetue T dp : T dp T st t p g p v Wet-bulb tempetue T wb s the tempetue dted b themomete oveed b wk stuted wth lqud wte d ott wth movg. Humdt Volume: Volume of most /mss of d. Pshomet Cht A plot of spef humdt s futo of d-bulb tempetue plotted fo vlue of tmosphe pessue. (See ht t ed of seto.) PHASE RELATONS Clpeo Equto fo Phse Tstos: dp h fg s fg, whee dt Tv v st h fg ethlp hge fo phse tstos, v fg volume hge, s fg etop hge, T bsolute tempetue, d (dp/dt) st slope of vpo-lqud stuto le. Gbbs Phse Rule P F C, whee P umbe of phses mkg up sstem, F degees of feedom, d C umbe of ompoets sstem. fg fg 9

60 Gbbs Fee Eeg Eeg elesed o bsobed eto oug evesbl t ostt pessue d tempetue G. Helmholtz Fee Eeg Eeg elesed o bsobed eto oug evesbl t ostt volume d tempetue A. COMBUSTON PROCESSES Fst, the ombusto equto should be wtte d bled. Fo emple, fo the stohomet ombusto of methe oge: CH O CO H O Combusto A Fo eh mole of oge, thee wll be.76 moles of toge. Fo stohomet ombusto of methe : CH O (.76) N CO H O 7.5 N Combusto Eess A The eess oge ppes s oge o the ght sde of the ombusto equto. omplete Combusto Some bo s bued to ete bo moode (CO). A-Fuel Rto (A/F): A/F Stohomet (theoetl) -fuel to s the -fuel to lulted fom the stohomet ombusto equto. Peet Theoetl A Peet Eess A A F tul A F ( ) ( A F ) ( ) mss of mss of ( A F ) ( A F ) stohomet tul stohomet stohomet fuel SECOND LAW OF THERMODYNAMCS Theml Eeg Resevos S esevo Q/T esevo, whee Q s mesued wth espet to the esevo. Kelv-Plk Sttemet of Seod Lw No het ege opete le whle tsfeg het wth sgle het esevo. COROLLARY to Kelv-Plk: No het ege hve hghe effe th Cot le opetg betwee the sme esevos. THERMODYNAMCS (otued) Clusus Sttemet of Seod Lw No efgeto o het pump le opete wthout et wok put. COROLLARY: No efgeto o het pump hve hghe COP th Cot le efgeto o het pump. VAPOR-LQUD MXTURES He's Lw t Costt Tempetue At equlbum, the ptl pessue of gs s popotol to ts oetto lqud. He's Lw s vld fo low oettos;.e.,. p p h, whee h He's Lw ostt, p ptl pessue of gs ott wth lqud, mol fto of the gs the lqud, mol fto of the gs the vpo, d p totl pessue. Roult's Lw fo Vpo-Lqud Equlbum Vld fo oettos e ;.e.,. p p *, whee p ptl pessue of ompoet, mol fto of ompoet the lqud, d * p vpo pessue of pue ompoet t the tempetue of the mtue. ENTROPY ds (/T) δq ev s s (/T) δq ev eqult of Clusus φ (/T) δq (/T) δq s s sotheml, Revesble Poess s s s Q/T setop poess s ; ds A evesble dbt poess s setop. Adbt Poess δq ; s ese of Etop Pple s s s s totl totl m sstem out s out suoudgs m s ( Q T ) etel etel 5

61 THERMODYNAMCS (otued) Tempetue-Etop (T-s) Dgm Q ev T ds Etop Chge fo Solds d Lquds ds (dt/t) s s (dt/t) me l (T /T ), whee equls the het pt of the sold o lqud. evesblt w ev w tul Closed-Sstem Avlblt (o heml etos) φ (u u o ) T o (s s o ) p o (v v o ) w evesble φ φ Ope-Sstem Avlblt ψ (h h o ) T o (s s o ) V / gz w evesble ψ ψ 5

62 Cot COMMON THERMODYNAMC CYCLES THERMODYNAMCS (otued) Revesed Cot Otto (gsole ege) η k v /v Rke Refgeto (Revesed Rke Cle) p p p p η ( h h ) ( h h ) h h COP ef h h h h COP HP h h h h 5

63 THERMODYNAMCS (otued) Temp. o C T St. Pess. kp p st MP Spef Volume m /kg St. St. lqud vpo v f v g Stuted Wte - Tempetue Tble tel Eeg Ethlp kj/kg kj/kg St. St. St. lqud vpo lqud u f Evp. u fg u g h f Evp. h fg St. vpo h g St. lqud s f Etop kj/(kg K) Evp. s fg St. vpo s g

64 THERMODYNAMCS (otued) Supeheted Wte Tbles T Temp. v m /kg u kj/kg h kj/kg s kj/(kg K) v m /kg u kj/kg h kj/kg o C p. MP (5.8 o C) p.5 MP (8. o C) s kj/(kg K) St St St St p. MP (99.6 o C) p. MP (. o C) p. MP (.6 o C) p.6 MP (58.85 o C) p.8 MP (7. o C) p. MP (79.9 o C)

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