il[::"r L:: ) I R,"f" cf ehct"ffu r=,j h*ot,t U t*- - d ai onff tal D..{_ +0 4 rota;r,o_r., F-nj M qss q *rte- ang,tcl b"

Transcription

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3 futu^)r ff r'j -OX ;r ; -# Y n -- a\ JZ. : *r( rflpl = L-*l -+ tu! -t n :r^ [rr$* r ro;t t' *r*.*) =\ Nonl :' \ \ \ C --*.^)rc - =? oots + ==toj T.,l {l *- Vra \ j? te_ JLv - * r L \*&'' JL*vt-\ ( \"-f*:) )n ;r* J - r+*s5 ) rj n \ r \/ tno '. lt=rr6uj*lt- Mq a Gr,* 6* 5'*6*';*L?..'*.t / -- {Ltt'tL; ["3 + &v r'a '*'* -* r{u:''' &* + 3' *t : 4 sla,*^r)j + Et tu*-] *(*g t'o;'") -- -k JL* L*6 t''-' J''*J ro, *; t:? :::::; : tu?":'x *;: {""= **, 4 w -_ff + Lau ;:# ff

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5 ._ A:Sd.qr *t*$h oe An* V Olu" bken e,tahve +o ++!' X\4 tj8s,or tl..l Y^o/me^t * [$r) M cae'b ar W tnen-[t'q' b.l al^reo-x* '+akab obo"* l+'-" a b,* A q))7 c),ul tlorw l l4- = l'lt 7- aq9 =?uru K otqho n "$ ts Revo luhon abso t'l.,-a ' ol* ; c o L(r nery uc*h axt^ " evo\uhu'n ', a ptorre ",b sn'*& 6 = L' >o q"=: L*- t'j - o o - frn/ rl rr'erho! spaa a Srr^r-l t" u-% Su.l'- T-TJ- T= tc ) (, L0) mebal a Bodt,t o- t R!-\tf^'htm ' k^o t-rntfrfro Krg'dTb"a6 d Lr""q t ^re,[<-vv^te, O-' -Fre-el (teo),-,@* [- Lr"Est cor*hrn*rnta nu* $r* +" ll" 8^)r- = T-.., h revo\, hfln -'

6 007 The McGraw-ll Companes, nc. All rghts reserved. Eghth Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Ferdnand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Teas Tech Unverst Knematcs of Rgd Bodes n Three Dmensons

7 ghth dton Vector Mechancs for Engneers: Dnamcs Contents ntroducton Rgd Bod Angular Momentum n Three Dmensons Prncple of mpulse and Momentum Knetc Energ Sample Problem 8. Sample Problem 8. Moton of a Rgd Bod n Three Dmensons Euler s Equatons of Moton and D Alembert s Prncple Moton About a Fed Pont or a Fed As Sample Problem 8.3 Moton of a Groscope. Euleran Angles Stead Precesson of a Groscope Moton of an Asmmetrcal Bod Under No Force 007 The McGraw-ll Companes, nc. All rghts reserved. 8 -

8 ghth dton Vector Mechancs for Engneers: Dnamcs ntroducton The fundamental relatons developed for the plane moton of rgd bodes ma also be appled to the general moton of three dmensonal bodes. The relaton G whch was used to determne the angular momentum of a rgd slab s not vald for general three dmensonal bodes and moton. F ma M G G The current chapter s concerned wth evaluaton of the angular momentum and ts rate of change for three dmensonal moton and applcaton to effectve forces, the mpulse-momentum and the work-energ prncples. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-3

9 007 The McGraw-ll Companes, nc. All rghts reserved. Vector Mechancs for Engneers: Dnamcs ghth dton 8-4 Rgd Bod Angular Momentum n Three Dmensons Angular momentum of a bod about ts mass center, n n G m r r m v r Δ The component of the angular momentum, n n n n n m m m m m r r Δ Δ Δ Δ Δ dm dm dm

10 ghth dton Vector Mechancs for Engneers: Dnamcs Rgd Bod Angular Momentum n Three Dmensons Transformaton of nto G s charactered b the nerta tensor for the bod, Wth respect to the prncpal aes of nerta, The angular momentum of a rgd bod and ts angular veloct G have the same drecton f, and onl f, s drected along a prncpal as of nerta. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-5

11 ghth dton Vector Mechancs for Engneers: Dnamcs Rgd Bod Angular Momentum n Three Dmensons The momenta of the partcles of a rgd bod can be reduced to: L lnear momentum mv G angular momentum about G The angular momentum about an other gven pont O s r mv O G 007 The McGraw-ll Companes, nc. All rghts reserved. 8-6

12 ghth dton Vector Mechancs for Engneers: Dnamcs Rgd Bod Angular Momentum n Three Dmensons The angular momentum of a bod constraned to rotate about a fed pont ma be calculated from r mv O G Or, the angular momentum ma be computed drectl from the moments and products of nerta wth respect to the O frame. n r v Δm O n r r Δm 007 The McGraw-ll Companes, nc. All rghts reserved. 8-7

13 ghth dton Vector Mechancs for Engneers: Dnamcs Prncple of mpulse and Momentum The prncple of mpulse and momentum can be appled drectl to the three-dmensonal moton of a rgd bod, Sst Momenta + Sst Et mp - = Sst Momenta The free-bod dagram equaton s used to develop component and moment equatons. For bodes rotatng about a fed pont, elmnate the mpulse of the reactons at O b wrtng equaton for moments of momenta and mpulses about O. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-8

14 007 The McGraw-ll Companes, nc. All rghts reserved. Vector Mechancs for Engneers: Dnamcs ghth dton 8-9 Knetc Energ Knetc energ of partcles formng rgd bod, ) ( Δ Δ n n mv m r mv m v mv T f the aes correspond nstantaneousl wth the prncple aes, ) ( mv T Wth these results, the prncples of work and energ and conservaton of energ ma be appled to the three-dmensonal moton of a rgd bod.

15 007 The McGraw-ll Companes, nc. All rghts reserved. Vector Mechancs for Engneers: Dnamcs ghth dton 8-0 Knetc Energ Knetc energ of a rgd bod wth a fed pont, ) ( T f the aes O correspond nstantaneousl wth the prncple aes O, ) ( T

16 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. Rectangular plate of mass m that s suspended from two wres s ht at D n a drecton perpendcular to the plate. mmedatel after the mpact, determne a) the veloct of the mass center G, and b) the angular veloct of the plate. 007 The McGraw-ll Companes, nc. All rghts reserved. SOLUTON: Appl the prncple of mpulse and momentum. Snce the ntal momenta s ero, the sstem of mpulses must be equvalent to the fnal sstem of momenta. Assume that the supportng cables reman taut such that the vertcal veloct and the rotaton about an as normal to the plate s ero. Prncple of mpulse and momentum elds to two equatons for lnear momentum and two equatons for angular momentum. Solve for the two horontal components of the lnear and angular veloct vectors. 8 -

17 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. SOLUTON: Appl the prncple of mpulse and momentum. Snce the ntal momenta s ero, the sstem of mpulses must be equvalent to the fnal sstem of momenta. Assume that the supportng cables reman taut such that the vertcal veloct and the rotaton about an as normal to the plate s ero. v v v k j Snce the,, and aes are prncpal aes of nerta, G j mb ma j 007 The McGraw-ll Companes, nc. All rghts reserved. 8 -

18 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. Prncple of mpulse and momentum elds two equatons for lnear momentum and two equatons for angular momentum. Solve for the two horontal components of the lnear and angular veloct vectors. 0 mv 0 v v FΔt FΔt mv FΔt v mk m bf Δ t mb 6FΔt mb 6FΔt mab af Δ t a b j ma 6FΔt ma 007 The McGraw-ll Companes, nc. All rghts reserved. 8-3

19 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. v FΔt mk G 6FΔt a b j mab mb ma j 007 The McGraw-ll Companes, nc. All rghts reserved. 8-4

20 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. SOLUTON: The dsk rotates about the vertcal as through O as well as about OG. Combne the rotaton components for the angular veloct of the dsk. A homogeneous dsk of mass m s mounted on an ale OG of neglgble mass. The dsk rotates counterclockwse at the rate about OG. Determne: a) the angular veloct of the dsk, b) ts angular momentum about O, c) ts knetc energ, and d) the vector and couple at G equvalent to the momenta of the partcles of the dsk. Compute the angular momentum of the dsk usng prncple aes of nerta and notng that O s a fed pont. The knetc energ s computed from the angular veloct and moments of nerta. The vector and couple at G are also computed from the angular veloct and moments of nerta. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-5

21 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. SOLUTON: The dsk rotates about the vertcal as through O as well as about OG. Combne the rotaton components for the angular veloct of the dsk. j Notng that the veloct at C s ero, vc rc 0 0 j L rj L r k r L r L j 007 The McGraw-ll Companes, nc. All rghts reserved. 8-6

22 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. r L j Compute the angular momentum of the dsk usng prncple aes of nerta and notng that O s a fed pont. j k O mr O 4 ml mr r L ml mr mr m L r r L j The knetc energ s computed from the angular veloct and moments of nerta. T mr m L 4r r L r T mr 6 8 L 007 The McGraw-ll Companes, nc. All rghts reserved. 8-7

23 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8. r L j The vector and couple at G are also computed from the angular veloct and moments of nerta. mv mr k G j k mr mr r L j 4 G r mr L j 007 The McGraw-ll Companes, nc. All rghts reserved. 8-8

24 ghth dton Vector Mechancs for Engneers: Dnamcs Moton of a Rgd Bod n Three Dmensons F ma M G Angular momentum and ts rate of change are taken wth respect to centrodal aes GX Y Z of fed orentaton. Transformaton of nto G s ndependent of the sstem of coordnate aes. Convenent to use bod fed aes G where moments and products of nerta are not tme dependent. Defne rate of change of change of respect to the rotatng frame, j k G G Then, G G wth G G G 007 The McGraw-ll Companes, nc. All rghts reserved. 8-9

25 ghth dton Vector Mechancs for Engneers: Dnamcs Euler s Eqs of Moton & D Alembert s Prncple Wth and G chosen to correspond to the prncpal aes of nerta, M M M G G G G Euler s Equatons: M Sstem of eternal forces and effectve forces are equvalent for general three dmensonal moton. Sstem of eternal forces are equvalent to the vector and couple, ma and. G 007 The McGraw-ll Companes, nc. All rghts reserved. 8-0

26 007 The McGraw-ll Companes, nc. All rghts reserved. Vector Mechancs for Engneers: Dnamcs ghth dton 8 - Moton About a Fed Pont or a Fed As For a rgd bod rotaton around a fed pont, O O O O O M For a rgd bod rotaton around a fed as, j k j k j k k j M O O O O M M M

27 ghth dton Vector Mechancs for Engneers: Dnamcs Rotaton About a Fed As For a rgd bod rotaton around a fed as, M M M f smmetrcal wth respect to the plane, M 0 M 0 M f not smmetrcal, the sum of eternal moments wll not be ero, even f = 0, M M M 0 A rotatng shaft requres both statc 0 and dnamc 0 balancng to avod ecessve vbraton and bearng reactons. 007 The McGraw-ll Companes, nc. All rghts reserved. 8 -

28 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8.3 SOLUTON: Evaluate the sstem of effectve forces b reducng them to a vector ma attached at G and couple G. Rod AB wth weght W = 40 N s pnned at A to a vertcal ale whch rotates wth constant angular veloct = 5 rad/s. The rod poston s mantaned b a horontal wre BC. Determne the tenson n the wre and the reacton at A. Epressng that the sstem of eternal forces s equvalent to the sstem of effectve forces, wrte vector epressons for the sum of moments about A and the summaton of forces. Solve for the wre tenson and the reactons at A. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-3

29 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8.3 SOLUTON: 007 The McGraw-ll Companes, nc. All rghts reserved. Evaluate the sstem of effectve forces b reducng them to a vector ma attached at G and couple G. a an r Lcos 450 m/ s 40 ma N g j k G G G 0 G ml cos ml cos 0 sn 0 ml cos sn j ml cos ml G G sn cos k N mk 8-4

30 ghth dton Vector Mechancs for Engneers: Dnamcs Sample Problem 8.3 Epressng that the sstem of eternal forces s equvalent to the sstem of effectve forces, wrte vector epressons for the sum of moments about A and the summaton of forces. M 6.93J A M A eff T 40J 3.46J 800 F F eff A A J A X Y 6.93T 80K K Z 078.4K T K 0 40J 800 A 590 N 0 N 40 NJ 007 The McGraw-ll Companes, nc. All rghts reserved. 8-5

31 ghth dton Vector Mechancs for Engneers: Dnamcs Moton of a Groscope. Euleran Angles A groscope conssts of a rotor wth ts mass center fed n space but whch can spn freel about ts geometrc as and assume an orentaton. From a reference poston wth gmbals and a reference dameter of the rotor algned, the groscope ma be brought to an orentaton through a successon of three steps: a) rotaton of outer gmbal through j about AA, b) rotaton of nner gmbal through q about c) rotaton of the rotor through about CC. j, q, and are called the Euleran Angles and rate of q rate of rate of precesson nutaton spn 007 The McGraw-ll Companes, nc. All rghts reserved. 8-6

32 ghth dton Vector Mechancs for Engneers: Dnamcs Moton of a Groscope. Euleran Angles The angular veloct of the groscope, K q j k wth K snq cosq j snq q j cosq k Equaton of moton, M M M M O O O O snq q j cosq k K q j O snq q cosq q cosq q snq cosq snq cosq d cosq dt 007 The McGraw-ll Companes, nc. All rghts reserved. 8-7

33 ghth dton Vector Mechancs for Engneers: Dnamcs Stead Precesson of a Groscope Stead precesson, q,, are constant sn q k O sn q k sn q cosq k M O cosq sn q j O Couple s appled about an as perpendcular to the precesson and spn aes When the precesson and spn as are at a rght angle, q 90 j M O Groscope wll precess about an as perpendcular to both the spn as and couple as. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-8

34 ghth dton Vector Mechancs for Engneers: Dnamcs Moton of an Asmmetrcal Bod Under No Force Consder moton about ts mass center of an asmmetrcal bod under no force but ts own weght, e.g., projectles, satelltes, and space craft. G 0 constant G Defne the Z as to be algned wth G and n a rotatng aes sstem along the as of smmetr. The as s chosen to le n the Z plane. G snq G snq 0 0 G cosq G cosq q = constant and bod s n stead precesson. Note: tan tanq 007 The McGraw-ll Companes, nc. All rghts reserved. 8-9

35 ghth dton Vector Mechancs for Engneers: Dnamcs Moton of an Asmmetrcal Bod Under No Force Two cases of moton of an asmmetrcal bod whch under no force whch nvolve no precesson: Bod set to spn about ts as of smmetr, and G 0 are algned and bod keeps spnnng about ts as of smmetr. Bod s set to spn about ts transverse as, and G 0 are algned and bod keeps spnnng about the gven transverse as. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-30

36 ghth dton Vector Mechancs for Engneers: Dnamcs Moton of an Asmmetrcal Bod Under No Force The moton of a bod about a fed pont (or ts mass center) can be represented b the moton of a bod cone rollng on a space cone. n the case of stead precesson the two cones are crcular. <. Case of an elongated bod. < q and the vector les nsde the angle ZG. The space cone and bod cone are tangent eternall; the spn and precesson are both counterclockwse from the postve as. The precesson s sad to be drect. >. Case of a flattened bod. > q and the vector les outsde the angle ZG. The space cone s nsde the bod cone; the spn and precesson have opposte senses. The precesson s sad to be retrograde. 007 The McGraw-ll Companes, nc. All rghts reserved. 8-3