Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Fednand P. Bee E. Russell Johnston, J. Lectue Notes: J. Walt Ole Teas Tech Unvest Knematcs of Rgd Bodes n Thee Dmensons 003 The McGaw-ll Companes, nc. All ghts eseved.
Edton Vecto Mechancs fo Engnees: Dnamcs Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd Bod n Thee Dmensons Eule s Equatons of Moton and D Alembet s Pncple Moton About a Fed Pont o a Fed As Sample Poblem 8.3 Moton of a Goscope. Eulean Angles Stead Pecesson of a Goscope Moton of an Asmmetcal Bod Unde No Foce 003 The McGaw-ll Companes, nc. All ghts eseved. 8 -
Edton Vecto Mechancs fo Engnees: Dnamcs ntoducton F ma M G G The fundamental elatons developed fo the plane moton of gd bodes ma also be appled to the geneal moton of thee dmensonal bodes. The elaton G whch was used to detemne the angula momentum of a gd slab s not vald fo geneal thee dmensonal bodes and moton. The cuent chapte s concened wth evaluaton of the angula momentum and ts ate of change fo thee dmensonal moton and applcaton to effectve foces, the mpulse-momentum and the wok-eneg pncples. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-3
003 The McGaw-ll Companes, nc. All ghts eseved. Vecto Mechancs fo Engnees: Dnamcs Seventh Edton 8-4 Rgd Bod Angula Momentum n Thee Dmensons Angula momentum of a bod about ts mass cente, ( ) ( ) [ ] n n G m m v The component of the angula momentum, ( ) ( ) [ ] ( ) ( ) [ ] ( ) n n n n n m m m m m ( ) dm dm dm
Edton Vecto Mechancs fo Engnees: Dnamcs Rgd Bod Angula Momentum n Thee Dmensons Tansfomaton of nto Gs chaacteed b the neta tenso fo the bod, Wth espect to the pncpal aes of neta, 0 0 0 0 0 0 The angula momentum of a gd bod and ts angula veloct have G the same decton f, and onl f, s dected along a pncpal as of neta. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-5
Edton Vecto Mechancs fo Engnees: Dnamcs Rgd Bod Angula Momentum n Thee Dmensons The momenta of the patcles of a gd bod can be educed to: L lnea momentum mv G angula momentum about G The angula momentum about an othe gven pont O s mv O G 003 The McGaw-ll Companes, nc. All ghts eseved. 8-6
003 The McGaw-ll Companes, nc. All ghts eseved. Vecto Mechancs fo Engnees: Dnamcs Seventh Edton 8-7 Rgd Bod Angula Momentum n Thee Dmensons The angula momentum of a bod constaned to otate about a fed pont ma be calculated fom O G mv O, the angula momentum ma be computed dectl fom the moments and poducts of neta wth espect to the O fame. ( ) ( ) [ ] n n O m m v
Edton Vecto Mechancs fo Engnees: Dnamcs Pncple of mpulse and Momentum The pncple of mpulse and momentum can be appled dectl to the theedmensonal moton of a gd bod, Sst Momenta Sst Et mp - Sst Momenta The fee-bod dagam equaton s used to develop component and moment equatons. Fo bodes otatng about a fed pont, elmnate the mpulse of the eactons at O b wtng equaton fo moments of momenta and mpulses about O. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-8
003 The McGaw-ll Companes, nc. All ghts eseved. Vecto Mechancs fo Engnees: Dnamcs Seventh Edton 8-9 Knetc Eneg Knetc eneg of patcles fomng gd bod, ) ( n n mv m mv m v mv T f the aes coespond nstantaneousl wth the pncple aes, ) ( mv T Wth these esults, the pncples of wok and eneg and consevaton of eneg ma be appled to the theedmensonal moton of a gd bod.
003 The McGaw-ll Companes, nc. All ghts eseved. Vecto Mechancs fo Engnees: Dnamcs Seventh Edton 8-0 Knetc Eneg Knetc eneg of a gd bod wth a fed pont, ) ( T f the aes O coespond nstantaneousl wth the pncple aes O, ) ( T
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. SOLUTON: Appl the pncple of mpulse and momentum. Snce the ntal momenta s eo, the sstem of mpulses must be equvalent to the fnal sstem of momenta. Rectangula plate of mass m that s suspended fom two wes s ht at D n a decton pependcula to the plate. mmedatel afte the mpact, detemne a) the veloct of the mass cente G, and b) the angula veloct of the plate. Assume that the suppotng cables eman taut such that the vetcal veloct and the otaton about an as nomal to the plate s eo. Pncple of mpulse and momentum elds to two equatons fo lnea momentum and two equatons fo angula momentum. Solve fo the two hoontal components of the lnea and angula veloct vectos. 003 The McGaw-ll Companes, nc. All ghts eseved. 8 -
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. SOLUTON: Appl the pncple of mpulse and momentum. Snce the ntal momenta s eo, the sstem of mpulses must be equvalent to the fnal sstem of momenta. Assume that the suppotng cables eman taut such that the vetcal veloct and the otaton about an as nomal to the plate s eo. v v v k j Snce the,, and aes ae pncpal aes of neta, G j mb ma j 003 The McGaw-ll Companes, nc. All ghts eseved. 8 -
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. Pncple of mpulse and momentum elds two equatons fo lnea momentum and two equatons fo angula momentum. Solve fo the two hoontal components of the lnea and angula veloct vectos. 0 v mv 0 v F t v ( F t m)k mv F t m bf t mb 6F t mb af t ma ( 6F t ma) 6F t mab ( a b j ) 003 The McGaw-ll Companes, nc. All ghts eseved. 8-3
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. v ( F t m)k G 6F t mab mb ( a b j ) ma j 003 The McGaw-ll Companes, nc. All ghts eseved. 8-4
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. SOLUTON: The dsk otates about the vetcal as though O as well as about OG. Combne the otaton components fo the angula veloct of the dsk. A homogeneous dsk of mass m s mounted on an ale OG of neglgble mass. The dsk otates counte-clockwse at the ate about OG. Detemne: a) the angula veloct of the dsk, b) ts angula momentum about O, c) ts knetc eneg, and d) the vecto and couple at G equvalent to the momenta of the patcles of the dsk. Compute the angula momentum of the dsk usng pncple aes of neta and notng that O s a fed pont. The knetc eneg s computed fom the angula veloct and moments of neta. The vecto and couple at G ae also computed fom the angula veloct and moments of neta. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-5
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. SOLUTON: The dsk otates about the vetcal as though O as well as about OG. Combne the otaton components fo the angula veloct of the dsk. j Notng that the veloct at C s eo, vc C 0 0 ( j ) ( L j ) L k ( ) L ( L) j 003 The McGaw-ll Companes, nc. All ghts eseved. 8-6
003 The McGaw-ll Companes, nc. All ghts eseved. Vecto Mechancs fo Engnees: Dnamcs Seventh Edton 8-7 Sample Poblem 8. Compute the angula momentum of the dsk usng pncple aes of neta and notng that O s a fed pont. ( ) j L k j O ( ) ( )( ) ( ) 0 0 4 4 m ml L m ml m ( )( ) j L m L m O 4 The knetc eneg s computed fom the angula veloct and moments of neta. ( ) ( )( ) [ ] 4 L m L m T 8 6 L m T
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8. ( L) j The vecto and couple at G ae also computed fom the angula veloct and moments of neta. mv m G j k m m L 4 ( ) j G k m j L 003 The McGaw-ll Companes, nc. All ghts eseved. 8-8
Edton Vecto Mechancs fo Engnees: Dnamcs Moton of a Rgd Bod n Thee Dmensons F ma M G Angula momentum and ts ate of change ae taken wth espect to centodal aes GX Y Z of fed oentaton. Tansfomaton of nto Gs ndependent of the sstem of coodnate aes. Convenent to use bod fed aes G whee moments and poducts of neta ae not tme dependent. Defne ate of change of change of wth G espect to the otatng fame, j ( ) k G G Then, ( ) Ω Ω G G G G 003 The McGaw-ll Companes, nc. All ghts eseved. 8-9
Edton Vecto Mechancs fo Engnees: Dnamcs Eule s Eqs of Moton & D Alembet s Pncple Wth Ω and G chosen to coespond to the pncpal aes of neta, M G ( ) G Ω G G Eule s Equatons: M M M ( ) ( ) ( ) Sstem of etenal foces and effectve foces ae equvalent fo geneal thee dmensonal moton. Sstem of etenal foces ae equvalent to the vecto and couple, ma and. G 003 The McGaw-ll Companes, nc. All ghts eseved. 8-0
003 The McGaw-ll Companes, nc. All ghts eseved. Vecto Mechancs fo Engnees: Dnamcs Seventh Edton 8 - Moton About a Fed Pont o a Fed As Fo a gd bod otaton aound a fed pont, ( ) O O O O O M Ω Fo a gd bod otaton aound a fed as, ( ) ( ) ( ) ( ) ( ) α j k j k j k k j M O O O O α α α M M M
Edton Vecto Mechancs fo Engnees: Dnamcs Rotaton About a Fed As Fo a gd bod otaton aound a fed as, M α M M α α f smmetcal wth espect to the plane, M 0 M 0 M α f not smmetcal, the sum of etenal moments wll not be eo, even f α 0, M M M A otatng shaft eques both statc 0and dnamc ( 0) balancng to avod ecessve vbaton and beang eactons. ( ) 0 003 The McGaw-ll Companes, nc. All ghts eseved. 8 -
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8.3 SOLUTON: Evaluate the sstem of effectve foces b educng them to a vecto attached ma at G and couple G. Rod AB wth weght W 0 kg s pnned at A to a vetcal ale whch otates wth constant angula veloct 5 ad/s. The od poston s mantaned b a hoontal we BC. Detemne the tenson n the we and the eacton at A. 003 The McGaw-ll Companes, nc. All ghts eseved. Epessng that the sstem of etenal foces s equvalent to the sstem of effectve foces, wte vecto epessons fo the sum of moments about A and the summaton of foces. Solve fo the we tenson and the eactons at A. 8-3
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8.3 SOLUTON: Evaluate the sstem of effectve foces b educng them to a vecto attached ma at G and couple G. a an ( Lcosβ) ( 35m s ) ma G G (0 kg)( 35m/s j ml cos β ml cos β ( ) G G G ) ( 700 N) k G 0 sn β 0 ml ( cosβ sn β j ) ( ml cosβ ) 0 ml sn β cosβ k ( 935N m)k 003 The McGaw-ll Companes, nc. All ghts eseved. 8-4
Edton Vecto Mechancs fo Engnees: Dnamcs Sample Poblem 8.3 Epessng that the sstem of etenal foces s equvalent to the sstem of effectve foces, wte vecto epessons fo the sum of moments about A and the summaton of foces. M.08J ( ) eff ( T ) 0.6 ( 96J ).04J ( 700 ) A M A (.08T 8) K ( 808 935)K 935K F ( F ) eff A A J X Y A Z K T 856N 856 96J 700 A 8 ( 44 N) ( 96 N)J 003 The McGaw-ll Companes, nc. All ghts eseved. 8-5
Edton Vecto Mechancs fo Engnees: Dnamcs Moton of a Goscope. Eulean Angles A goscope conssts of a oto wth ts mass cente fed n space but whch can spn feel about ts geometc as and assume an oentaton. Fom a efeence poston wth gmbals and a efeence damete of the oto algned, the goscope ma be bought to an oentaton though a successon of thee steps: ) otaton of oute gmbal though ϕ about AA, ) otaton of nne gmbal though θ about BB, 3) otaton of the oto though ψ about CC. ϕ, θ, and ψ ae called the Eulean Angles and φ ate of pecesson θ ate of nutaton Ψ ate of spn 003 The McGaw-ll Companes, nc. All ghts eseved. 8-6
Edton Vecto Mechancs fo Engnees: Dnamcs Moton of a Goscope. Eulean Angles The angula veloct of the goscope, φ K θ j Ψ k wth K snθ cosθ j φ snθ θ j ( Ψ φ cosθ )k Equaton of moton, M O M M M ( ) O Ω O O φ snθ θ j cos Ω φk θ j O ( Ψ φ θ) ( φ snθ θφ cosθ ) θ ( Ψ φ cosθ ) ( ) θ φ snθ cosθ φ snθ ( Ψ φ cosθ ) d dt ( Ψ φ cosθ ) k 003 The McGaw-ll Companes, nc. All ghts eseved. 8-7
Edton Vecto Mechancs fo Engnees: Dnamcs Stead Pecesson of a Goscope Stead pecesson, θ, φ, ψ ae constant φ sn θ k O φ sn θ k Ω φ sn θ φ cosθ k M O Ω ( φ cosθ) φ sn θ j Couple s appled about an as pependcula to the pecesson and spn aes O When the pecesson and spn as ae at a ght angle, θ 90 Ψφ j M O Goscope wll pecess about an as pependcula to both the spn as and couple as. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-8
Edton Vecto Mechancs fo Engnees: Dnamcs Moton of an Asmmetcal Bod Unde No Foce Consde moton about ts mass cente of an asmmetcal bod unde no foce but ts own weght, e.g., pojectles, satelltes, and space caft. G 0 constant G Defne the Z as to be algned wth and G n a otatng aes sstem along the as of smmet. The as s chosen to le n the Z plane. G sn θ G sn θ 0 0 G cosθ G cosθ θ constant and bod s n stead pecesson. Note: tan γ tan θ 003 The McGaw-ll Companes, nc. All ghts eseved. 8-9
Edton Vecto Mechancs fo Engnees: Dnamcs Moton of an Asmmetcal Bod Unde No Foce Two cases of moton of an asmmetcal bod whch unde no foce whch nvolve no pecesson: Bod set to spn about ts as of smmet, 0 and ae algned G and bod keeps spnnng about ts as of smmet. Bod s set to spn about ts tansvese as, 0 and ae G algned and bod keeps spnnng about the gven tansvese as. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-30
Edton Vecto Mechancs fo Engnees: Dnamcs Moton of an Asmmetcal Bod Unde No Foce The moton of a bod about a fed pont (o ts mass cente) can be epesented b the moton of a bod cone ollng on a space cone. n the case of stead pecesson the two cones ae ccula. <. Case of an elongated bod. γ < θ and the vecto les nsde the angle ZG. The space cone and bod cone ae tangent etenall; the spn and pecesson ae both counteclockwse fom the postve as. The pecesson s sad to be dect. >. Case of a flattened bod. γ > θ and the vecto les outsde the angle ZG. The space cone s nsde the bod cone; the spn and pecesson have opposte senses. The pecesson s sad to be etogade. 003 The McGaw-ll Companes, nc. All ghts eseved. 8-3