Course Outline. 1. MATLAB tutorial 2. Motion of systems that can be idealized as particles

Transcription

1 Couse Outlne. MATLAB tutoal. Moton of systems that can be dealzed as patcles Descpton of moton, coodnate systems; Newton s laws; Calculatng foces equed to nduce pescbed moton; Devng and solvng equatons of moton 3. Consevaton laws fo systems of patcles Wo, powe and enegy; Lnea mpulse and momentum Angula momentum 4. Vbatons Exam topcs Chaactestcs of vbatons; vbaton of fee DOF systems Vbaton of damped DOF systems Foced Vbatons 5. Moton of systems that can be dealzed as gd bodes Descpton of otatonal moton nematcs; geas, pulleys and the ollng wheel Inetal popetes of gd bodes; momentum and enegy Dynamcs of gd bodes

2 Couse Outlne. MATLAB tutoal. Moton of systems that can be dealzed as patcles Descpton of moton, coodnate systems; Newton s laws; Calculatng foces equed to nduce pescbed moton; Devng and solvng equatons of moton 3. Consevaton laws fo systems of patcles Wo, powe and enegy; Lnea mpulse and momentum Angula momentum 4. Vbatons Chaactestcs of vbatons; vbaton of fee DOF systems Vbaton of damped DOF systems Foced Vbatons 5. Moton of systems that can be dealzed as gd bodes Descpton of otatonal moton nematcs; geas, pulleys and the ollng wheel Inetal popetes of gd bodes; momentum and enegy Dynamcs of gd bodes

3 Patcle Dynamcs concept checlst Undestand the concept of an netal fame Be able to dealze an engneeng desgn as a set of patcles, and now when ths dealzaton wll gve accuate esults Descbe the moton of a system of patcles (eg components n a fxed coodnate system; components n a pola coodnate system, etc) Be able to dffeentate poston vectos (wth pope use of the chan ule!) to detemne velocty and acceleaton; and be able to ntegate acceleaton o velocty to detemne poston vecto. Be able to descbe moton n nomal-tangental and pola coodnates (eg be able to wte down vecto components of velocty and acceleaton n tems of speed, adus of cuvatue of path, o coodnates n the cylndcal-pola system). Be able to convet between Catesan to nomal-tangental o pola coodnate descptons of moton Be able to daw a coect fee body dagam showng foces actng on system dealzed as patcles Be able to wte down Newton s laws of moton n ectangula, nomal-tangental, and pola coodnate systems Be able to obtan an adonal moment balance equaton fo a gd body movng wthout otaton o otatng about a fxed axs at constant ate. Be able to use Newton s laws of moton to solve fo unnown acceleatons o foces n a system of patcles Use Newton s laws of moton to deve dffeental equatons govenng the moton of a system of patcles Be able to e-wte second ode dffeental equatons as a pa of fst-ode dffeental equatons n a fom that MATLAB can solve

4 Inetal fame non acceleatng, non otatng efeence fame Patcle pont mass at some poston n space Poston Vecto Velocty Vecto ( t) = x( t) + y( t) + z( t) v() t = v () t + v () t + v () t x y z d dx dy dz = ( x+ y+ z) = + + dx dy dz vx() t = vy() t = vz() t = Decton of velocty vecto s paallel to path Magntude of velocty vecto s dstance taveled / tme Acceleaton Vecto O (t) (t+) d dv dvy () () () () x dv a t = a z x t + ay t + az t = ( vx+ vy+ vz) = + + Also Patcle Knematcs dv dv () x y dv () () z x y z d x d y d z a t = = a t = = a t = = dv dvy () x dv a () () z x t = vx ay t = vy az t = vz dx dy dz v(t) path of patcle

5 Patcle Knematcs Staght lne moton wth constant acceleaton = X + Vt + at = ( V + at) = a v a Tme/velocty/poston dependent acceleaton use calculus t t = X + v() t v = V + a() t v dv g() t a = = f ( v) dv = g() t f () v V xt () dx g() t v = = f ( x) dx = g() t f ( x) X t t dv = ax ( ) dv dx dv = ax ( ) v = ax ( ) dx dx vt () xt () V vdv = a( x) dx

6 Gaphcal x-v-a elatons Math Revew: Gaphcal poston-velocty-acceleaton elatons Speed s the slope of the dstance-v-tme cuve Dstance s the aea unde the speed-v-tme cuve dx () = vt v dx = t v t v() t Acceleaton s the slope of the speed-v-tme cuve Speed s the aea unde the acceleaton-v-tme cuve dv () = at v dv = t v t a() t

7 Patcle Knematcs Ccula Moton at const speed θ = ωt s= Rθ V = ωr ( cosθ snθ ) R( sn cos ) = R + v= ω θ+ θ = Vt V a= ω R(cosθ+ sn θ) = ω Rn= n R Geneal ccula moton ( cosθ snθ ) R( sn cos ) = R + v= ω θ+ θ = Vt a= Rα( snθ+ cos θ) Rω (cosθ+ sn θ) dv V = αrt + ω Rn= t + n R ω = dθ / α = dω / = d θ / s = Rθ V = ds / = Rω snθ t cosθ R n θ=ωt Rsn θ Rcosθ snθ t cosθ R n θ Rsn θ Rcosθ

8 Patcle Knematcs Moton along an abtay path v= Vt dv V a= t + n R t R n s Radus of cuvatue R = d x d y + ds ds = xs ( ) + y(s) Pola Coodnates e θ e d dθ v= e + e θ d dθ d θ d dθ a= e + + e θ θ

9 Usng Newton s laws Calculatng foces equed to cause pescbed moton Idealze system Fee body dagam Knematcs (descbe moton usually goal s to fnd fomula fo acceleaton) F=ma fo each patcle. M G = (fo steadly o non-otatng gd bodes o fames only ths s a specal case of the momentangula momentum fomula fo gd bodes) Solve fo unnown foces o acceleatons (ust le statcs)

10 Usng Newton s laws to deve equatons of moton. Idealze system. Intoduce vaables to descbe moton (often x,y coods, but we wll see othe examples) 3. Wte down, dffeentate to get a 4. Daw FBD 5. F = ma 6. If necessay, elmnate eacton foces 7. Result wll be dffeental equatons fo coods defned n (), e.g. d x dx m + λ + x = Y snωt 8. Identfy ntal conons, and solve ODE

11 Moton of a poectle n eaths gavty d = X + Y + Z t = Vx Vy Vz = + + X V ( ) ( ) = X + Vxt + Y + Vyt + Z + Vzt gt ( V ) ( V ) ( V gt) v= + + a= g x y z

12 Reaangng dffeental equatons fo MATLAB Example d y dy + ζωn + ωny = Intoduce v = dy / Then d y v v = ζωnv ωny Ths has fom dw y = f(, t w) w = v

13 Couse Outlne. MATLAB tutoal. Moton of systems that can be dealzed as patcles Descpton of moton, coodnate systems; Newton s laws; Calculatng foces equed to nduce pescbed moton; Devng and solvng equatons of moton 3. Consevaton laws fo systems of patcles Wo, powe and enegy; Lnea mpulse and momentum Angula momentum 4. Vbatons Chaactestcs of vbatons; vbaton of fee DOF systems Vbaton of damped DOF systems Foced Vbatons 5. Moton of systems that can be dealzed as gd bodes Descpton of otatonal moton nematcs; geas, pulleys and the ollng wheel Inetal popetes of gd bodes; momentum and enegy Dynamcs of gd bodes

14 Consevaton Consevaton laws fo Laws patcles: concept Concept checlst Checlst Know the defntons of powe (o ate of wo) of a foce, and wo done by a foce Know the defnton of netc enegy of a patcle Undestand powe-wo-netc enegy elatons fo a patcle Be able to use wo/powe/netc enegy to solve poblems nvolvng patcle moton Be able to dstngush between consevatve and non-consevatve foces Be able to calculate the potental enegy of a consevatve foce Be able to calculate the foce assocated wth a potental enegy functon Know the wo-enegy elaton fo a system of patcles; (enegy consevaton fo a closed system) Use enegy consevaton to analyze moton of consevatve systems of patcles Know the defnton of the lnea mpulse of a foce Know the defnton of lnea momentum of a patcle Undestand the mpulse-momentum (and foce-momentum) elatons fo a patcle Undestand mpulse-momentum elatons fo a system of patcles (momentum consevaton fo a closed system) Be able to use mpulse-momentum to analyze moton of patcles and systems of patcles Know the defnton of esttuton coeffcent fo a collson Pedct changes n velocty of two colldng patcles n D and 3D usng momentum and the esttuton fomula Know the defnton of angula mpulse of a foce Know the defnton of angula momentum of a patcle Undestand the angula mpulse-momentum elaton Be able to use angula momentum to solve cental foce poblems/mpact poblems

15 Wo-Enegy elatons fo a sngle patcle Rate of wo done by a foce (powe developed by foce) P = F v O F P v Total wo done by a foce W t F v W = = F d O F(t) P Knetc enegy T= mv = mv + v + v ( ) x y z v Powe-netc enegy elaton Wo-netc enegy elaton dt P = W = F d = T T O P

16 Potental Enegy Potental enegy of a consevatve foce (pa) U( ) = F d+ constant F = gad( U ) Type of foce Gavty actng on a patcle nea eaths suface Gavtatonal foce exeted on mass m by mass M at the ogn Potental enegy U = mgy GMm U = m F y F m O F P Foce exeted by a spng wth stffness and unstetched length L Foce actng between two chaged patcles U = ( L ) F F QQ +Q +Q U = 4πε Foce exeted by one molecule of a noble gas (e.g. He, A, etc) on anothe (Lennad Jones potental). a s the equlbum spacng between molecules, and E s the enegy of the bond. 6 a a U = E F

17 Enegy Relaton fo a Consevatve System Intenal Foces: (foces exeted by one pat of the system on anothe) Extenal Foces: (any othe foces) R F System s consevatve f all ntenal foces ae consevatve foces (o constant foces) m F m 4 R R 3 R 3 R m F R 3 R 3 F 3 m 3 t Enegy elaton fo a consevatve system m 4 F F m 3 m m 3 F = t t = t Total KE T Total PE U TOT TOT Extenal Powe P () t t Extenal wo W P() t = t F m F Total KE T Total PE U m 4 F 3 m m 3 TOT TOT ( ) TOT TOT TOT TOT W = T + U T + U Specal case zeo enal wo: TOT + TOT = TOT + TOT T U T U KE+PE = constant

18 Impulse-Momentum fo a sngle patcle Defntons Lnea Impulse of a foce Lnea momentum of a patcle t I= F() t t p= mv O F(t) m v Impulse-Momentum elatons d F = p I= p p t=t p=p O F(t) m t=t v p=p

19 Impulse-Momentum Impulse-momentum fo fo a system a system of patcles of patcles R F v Foce exeted on patcle by patcle Extenal foce on patcle Velocty of patcle m F m 4 R R 3 R 3 R m F R 3 R 3 F 3 m 3 Total Extenal Foce Total Extenal Impulse m 4 F F m 3 F I TOT TOT () t t = F t TOT () t F m 4 m F 3 Impulse-momentum fo the system: F TOT d = p TOT I = p p TOT TOT TOT t = t m m 3 F Total momentum p TOT F t = t m m 3 Total momentum p TOT Specal case zeo enal mpulse: p = p TOT TOT (Lnea momentum conseved)

20 Collsons v x A A v x A A * v x B v x B B B Momentum Resttuton fomula mv + mv = mv + mv A B A B A x B x A x B x ( ) v v = ev v B A B A m v = v ( + e) v v m + m ( ) B B A B A A m v = v + ( + e) v v m + m B ( ) A A B B A A B A v A v A A B n v B B B v Momentum Resttuton fomula m v + m v = m v + m v B A B A B A B A B A B A B A ( ) ( ) ( e) ( ) v v = v v + v v n n m m + m m m + m ( ) v B B A B A = v ( + e) v v n n B A ( ) v A A B B A = v + ( + e) v v n n B A

21 Angula Impulse-Momentum Equatons fo a Patcle patcle Angula Impulse Angula Momentum Impulse-Momentum elatons Specal Case t A= () t F() t t h= p= mv d F= h x O y (t) F(t) z A= h h A= h = h Angula momentum conseved Useful fo cental foce poblems (when foces on a patcle always act though a sngle pont, eg planetay gavty)

22 Couse Outlne. MATLAB tutoal. Moton of systems that can be dealzed as patcles Descpton of moton, coodnate systems; Newton s laws; Calculatng foces equed to nduce pescbed moton; Devng and solvng equatons of moton 3. Consevaton laws fo systems of patcles Wo, powe and enegy; Lnea mpulse and momentum Angula momentum 4. Vbatons Chaactestcs of vbatons; vbaton of fee DOF systems Vbaton of damped DOF systems Foced Vbatons 5. Moton of systems that can be dealzed as gd bodes Descpton of otatonal moton nematcs; geas, pulleys and the ollng wheel Inetal popetes of gd bodes; momentum and enegy Dynamcs of gd bodes

23 Fee Vbatons vbatons concept checlst You should be able to:. Undestand smple hamonc moton (ampltude, peod, fequency, phase). Undestand the moton of a vbatng spng-mass system (and how the moton s pedcted) 3. Calculate natual fequency of a degee of feedom lnea system (Deve EOM and use the solutons gven on the handout) 4. Calculate the ampltude and phase of an undamped DOF lnea system fom the ntal conons 5. Undestand the concept of natual fequences and mode shapes fo vbaton of a geneal undamped lnea system 6. Combne sees and paallel spngs to smplfy a system 7. Use enegy to deve an equaton of moton fo a DOF consevatve system 8. Analyze small ampltude vbaton of a nonlnea system (eg pendulum) by lneazng EOM wth Taylo sees 9. Undestand natual fequency, damped natual fequency, and Dampng facto fo a dsspatve DOF vbatng system. Know fomulas fo nat feq, damped nat feq and dampng facto fo spng-mass system n tems of,m,c. Undestand undedamped, ctcally damped, and ovedamped moton of a dsspatve DOF vbatng system. Be able to detemne dampng facto fom a measued fee vbaton esponse (wll be coveed n lectue) 3. Be able to pedct moton of a feely vbatng DOF system gven ts ntal velocty and poston, and apply ths to desgn-type poblems

24 Typcal vbaton esponse Revew Fee vbatons Peod, fequency, angula fequency ampltude Dsplacement o Acceleaton y(t) Peod, T Pea to Pea Ampltude A tme Smple Hamonc Moton ( ω φ) xt ( ) = X + Xsn t+ vt ( ) = Vcos( ωt+ φ) at ( ) = Asn ( ωt+ φ) V = ω X A= ω V Fee Vbaton of Undamped DOF systems Fee -> No tme dependent enal foces Undamped -> No enegy loss DOF -> one vaable descbes system s, d m

25 Revew Fee vbatons Hamonc Oscllato Deve EOM (F=ma) md s + s= L Compae wth standad dffeental equaton x= s C = L x = s n ω = m Soluton n st ( ) = L + ( s L) + v / ω sn( ω t+ φ) n Natual Fequency ω = n m

26 Calculatng natual fequences fo DOF systems Use F=ma (o enegy) to fnd the equaton of moton Fo an undamped system the equaton wll loo le d y A + By = D Handout onlne gves soluton to ω d x n + x= C Reaange you equaton to loo le ths y m,l Ad y B ω d x n D + y = B + x= C A ωn ω = B = n C = D/ B B A

27 Revew Natual Fequences and Mode Shapes Geneal system does not always vbate hamoncally All unfoced undamped systems vbate hamoncally at specal fequences, called Natual Fequences of the system The system wll vbate hamoncally f t s eleased fom est wth a specal set of ntal dsplacements, called Mode Shapes o Vbaton Modes. x x m m

28 Countng degees of feedom and vbaton modes # DOF = no. coodnates equed to descbe moton D system # DOF = *p + 3*-c 3D system # DOF = 3*p+6*-c # Vbaton modes = # DOF - # tanslaton/otaton gd body modes Examples of D constants

29 Tcs fo calculatng Revew nat feqs of undamped systems Usng enegy consevaton to fnd EOM ds ( KE + PE = m + s L ) = const d ds d s ds KE PE m s L ds ( + ) = + ( ) = m + s = L Combnng spngs s, d m Paallel (foces add) = + eff Sees (lengths add) = + eff These ae all n paallel eff = +

30 Calculatng the natual fequency of a nonlnea system Nonlnea systems Sometmes EOM has fom m d y + f( y) = We cant solve ths n geneal Instead, assume y s small, and note f () = (because acceleaton must be zeo fo y= fo vbatons to be possble Smplfy usng Taylo expanson of f: d y df m + f() + y+... = dy m d y df + y = dy y= y= Thee ae shot-cuts to dong the Taylo expanson

31 Canoncal damped vbaton poblem EOM d s ds m + c + s = L Standad Fom n d x ς dx + + x= C ω ω c ωn = ς = C = L m m n Damped vbatons wth s= s = v t = x s ds dx x= x = v t = s=l +x, L m c ω = ω ζ d n Ovedamped ς > Ctcally Damped ς = Undedamped ς < Ovedamped ς > Ctcally Damped Undedamped ς < v + ( ςω )( ) ( )( ) ( ) exp( ) n + ωd x C v + ςω exp( ) n ωd x C xt = C+ ςωnt ω exp( ω) ωd ωd { } ς = [ ω ] x( t) = C+ ( x C) + v + ( x C) t exp( ω t) n v + ςω ( ) ( ) exp( ) ( )cos n x C xt = C+ ςωnt x C ω+ snω ω d n

32 Applcaton of damped vbatons Calculatng natual fequency and dampng facto fom a vbaton measuement Dsplacement x(t ) x(t ) x(t ) x(t 3 ) tme t t t t 3 t 4 Measue log decement: T xt ( ) δ = log n xt ( n) Measue peod: T Then δ 4 π + ς = ω δ n = 4π + δ T

33 Foced Vbatons concept checlst You should be able to:. Be able to deve equatons of moton fo spng-mass systems subected to enal focng (seveal types) and solve EOM by compang to soluton tables. Undestand (qualtatvely) meanng of tansent and steady-state esponse of a foced vbaton system 3. Undestand the meanng of Ampltude and phase of steady-state esponse of a foced vbaton system 4. Undestand ampltude-v-fequency fomulas (o gaphs), esonance, hgh and low fequency esponse fo 3 systems 5. Detemne the ampltude of steady-state vbaton of foced spng-mass systems. 6. Use foced vbaton concepts to desgn engneeng systems

34 EOM EOM fo foced foced spng-mass vbatng systems L x(t), L m λ c F(t)=F sn ωt Extenal focng d x ς dx + + x = KF snω t ω ω n n c ωn =, ς =, K = m m L, L c λ y(t)=y snωt L, L c λ x(t) x(t) m m ω m Base Exctaton Roto Exctaton d x ς dx ς dy + + x= K y+ ω ωn ωn n c ωn =, ς =, K = m m y(t)=y snωt d x dx K d y Y n + ς ω ωn + = n = n x K snωt ω ω ω c m ω n = ς = K = m+ m m m+ m ( + m )

35 Steady-state soluton fo enal focng foce L x(t), L m c λ F(t)=F sn ωt d x ς dx + + x = KF() t c ω ωn ωn =, ς =, K = m m n ( ω φ) xt ( ) = X sn t+ = ( ωω,, ) tan n n ζ = φ= / ω / ωn X KM F M M max ζ ( ω / ωn) + ( ςω / ωn) ςω / ω System vbates at same fequency as foce Ampltude depends on focng fequency, nat fequency, and dampng coeft.

36 L Steady-state soluton fo Base base exctaton x(t) y(t)=y snωt, L m c λ d x ς dx ς dy + + x= K y+ ω ωn ωn n ( ) = sn ( ω + φ) / { + ( ςω / ωn ) } xt X t ( ω / ωn) + ( ςω / ωn) c ωn =, ς =, K = m m 3 3 ςω / ω = ( ωω,, ) tan n n ζ = φ= / ( 4 ς ) ω / ωn X KM Y M M max ζ

37 Steady state soluton to Steady-state soluton fo oto exctaton Canoncal oto excted system (steady state soluton) d x ς dx K d y + + x= C ω n ωn ωn c y = Y snωt ω n c m = ζ = K = m+ m m ( + m) m+ m p ( ) sn ( ω φ) x t = X t+ X = KY M ( ωω, n, ζ) M M max n ω / ω ςω / ω = = ( ω / ωn) + ( ςω / ωn) ζ φ tan n / ω / ωn

38 Couse Outlne. MATLAB tutoal. Moton of systems that can be dealzed as patcles Descpton of moton, coodnate systems; Newton s laws; Calculatng foces equed to nduce pescbed moton; Devng and solvng equatons of moton 3. Consevaton laws fo systems of patcles Wo, powe and enegy; Lnea mpulse and momentum Angula momentum 4. Vbatons Chaactestcs of vbatons; vbaton of fee DOF systems Vbaton of damped DOF systems Foced Vbatons 5. Moton of systems that can be dealzed as gd bodes Descpton of otatonal moton nematcs; geas, pulleys and the ollng wheel Inetal popetes of gd bodes; momentum and enegy Dynamcs of gd bodes

39 Rgd Body Dynamcs - Roadmap. Descbng moton of a gd body Rotaton tenso (matx) Angula Velocty Vecto Spn tenso (matx). Analyzng moton n systems of gd bodes Relatng velocty/acceleaton of two ponts on a gd body Mechansms Geas, pulleys and ollng wheels 3. Lnea/Angula Momentum and Knetc Enegy of a gd body Rgd body as an nfnte numbe of patcles Calculatng neta tensos Momentum and enegy of a otatng body 4. Dynamcs of gd bodes Toques Foce lnea momentum and moment angula momentum elatons Examples Usng enegy and momentum fo gd bodes

40 Dynamcs of Rgd Bodes concept checlst Rgd Body Dynamcs Concept checlst. Undestand and manpulate otaton tensos n D and 3D. Undestand angula velocty and acceleaton vectos; be able to ntegate / dffeentate angula veloctes / acceleatons fo plana moton. 3. Undestand fomulas elatng velocty/acceleaton of two ponts on a gd body 4. Undestand constants at onts and contacts between gd bodes 5. Be able to elate veloctes, acceleatons, o angula veloctes/acceleatons of two membes n a system of lns o gd bodes 6. Be able to analyze moton n systems of geas 7. Undestand fomulas elatng velocty/angula velocty and acceleaton/angula acceleaton of a ollng wheel 8. Be able to calculate the cente of mass and mass moments of neta of smple shapes; use paallel axs theoem to shft axs of neta o calculate mass moments of neta fo a set of gd bodes connected togethe 9. Undestand how to calculate the angula momentum and netc enegy of a gd body. Undestand the meanng of a foce couple o pue moment/toque. Undestand the foce-lnea momentum and moment-angula momentum fomulas F= Ma G F+ Q = Ma + I α z G G Gzz z. Undestand the specal case of these equatons fo fxed axs otaton 3. Be able to use dynamcs equatons and nematcs equatons to calculate acceleatons / foces n a system of plana gd bodes subected to foces 4. Undestand powe/wo/potental enegy of a gd body; use enegy methods to analyze moton n a system of gd bodes 5. Use angula momentum to analyze moton of gd bodes

41 Rotatons Rotaton tenso (matx) D otatons cosθ snθ R = snθ cosθ = Rp ( p ) B A B A A p B -p A B B - A A B θ 3D otaton though θ about axs paallel to unt vecto n= nx+ ny+ nz R cos ( cos ) ( cos ) sn ( cos ) sn xx Rxy Rxz θ + θ nx θ nn x y θnz θ nn x z + θn y R = Ryx Ryy Ryz = ( cos θ) nn x y + snθnz cos θ + ( cos θ) ny ( cos θ) nn y z snθn x Rzx Rzy Rzz ( cos θ) nn x z sn θny ( cos θ) nn y z + snθnx cos θ + ( cosθ) n z + cosθ = R + R + R xx yy zz ( Rzy Ryz ) ( Rxz Rzx ) ( Ryx Rxy ) n = + + snθ n Sequence of otatons () () R = R R Othogonalty T T RR = R R = I T R and R epesent opposte otatons

42 Rotatonal Moton Angula velocty vecto:. Decton paallel to otaton axs (RH scew ule). Magntude angle (adans) tuned pe sec dθ ω= n= ωn= ωx+ ωy+ ωz dω Angula acceleaton vecto: α = n Spn Tenso d W= R R T d ωz ωy W = ωz ωx ωy ωx R = WR = n Axs of otaton Wu ω u fo all vectos u ω Fo plana moton: dθ dω d θ z z = αz = = dθ d θ ω = α = W dθ / = dθ /

43 Rgd Body Dynamcs - Roadmap. Descbng moton of a gd body Rotaton tenso (matx) Angula Velocty Vecto Spn tenso (matx). Analyzng moton n systems of gd bodes Relatng velocty/acceleaton of two ponts on a gd body Mechansms Geas, pulleys and ollng wheels 3. Lnea/Angula Momentum and Knetc Enegy of a gd body Rgd body as an nfnte numbe of patcles Calculatng neta tensos Momentum and enegy of a otatng body 4. Dynamcs of gd bodes Toques Foce lnea momentum and moment angula momentum elatons Examples Usng enegy and momentum fo gd bodes

44 Rgd body Rgd nematcs Body Knematcs man concepts Rgd body nematcs fomulas Veloctes of two ponts on a gd body elated by v v = ω ( ) B A B A Acceleatons of two ponts on a gd body elated by { } a a = α ( ) + ω ω ( ) Fo D poblems B A B A B A v v = ω ( ) a a B A z B A B A = αz ( B A) ωz( B A) Constants at connectons A B va = vb A B a A = a B A B n v n = v n aa n = ab n A B No slp v A n = v Slp va n = vb n B Tangental accels equal Accels abtay

45 Veloctes at C ae equal Belt speed s constant Planetay geas (solve wth otatng fame) Geas, Belts and the ollng wheel ω ω B = A ω ω B = ω ω A R R R R ω ω A B A B R = R zr zpc S zs zpc R ωa A ωa A R A R A ω zs C Sun gea R S ωb R B ωb Planet Cae ω zpc B ω zp R B B Planet gea R R ω zr Wheel ollng wthout slp Rng gea C s statonay so v v = ω ( ) v = ω R a = α R O C z O C xo z xo z ωz R O C v xo

46 Wheels ollng and sldng on a statonay suface ωz R O C v xo v v = ω ( ) v = v + ω R C O z C O xc xo z Wheel ollng wthout slp v vxo + ωzr= C = Both FBDs coect O O T < µ N Bacspn v xc > v xo + ω R> z T N N T O N T T = µ N Topspn v xc < v xo + ω R< z O T T = µ N N

47 Rgd Body Dynamcs - Roadmap. Descbng moton of a gd body Rotaton tenso (matx) Angula Velocty Vecto Spn tenso (matx). Analyzng moton n systems of gd bodes Relatng velocty/acceleaton of two ponts on a gd body Mechansms Geas, pulleys and ollng wheels 3. Lnea/Angula Momentum and Knetc Enegy of a gd body Rgd body as an nfnte numbe of patcles Calculatng neta tensos Momentum and enegy of a otatng body 4. Dynamcs of gd bodes Toques Foce lnea momentum and moment angula momentum elatons Examples Usng enegy and momentum fo gd bodes

48 Calculatng the momentum and enegy of a gd body Pelmnay: Momentum and Enegy fo a System of Patcles Total mass M Cente of mass = m Mass moment of neta matx G = m M I v G d = d d d d d d y + z x y x z G = m dxdy dx + dz dyd z dxdz dydz dx + dy G m 3 m d 3 d 4 d G m 4 d m Lnea Momentum p= mv = MvG Angula Momentum patcles h= m v = Mv + I ω patcles G G G v vg ω IGω patcles Knetc Enegy T = m = M + ( ) We use the same dea to calculate the momentum and enegy of a gd body. The sums become ntegals ove an nfnte numbe of nfntesmal patcles

49 Inetal Popetes Inetal Popetes of Rgd Bodes Total mass M Cente of mass = ρ dv Mass moment of neta V G = M ρdv V v G d = G G d dv dy + dz dxdy dxd z d d d d d d ρdv 3D: I D: G = x y x + z y z V dxdz dydz dx + dy Paallel Axs Theoem dy + dz dxdy dxdz 3D: IO = IG + M dxdy dx + dz dyd z D: d d d d d + d x z y z x y Rotaton fomula fo neta matx I G T G = RI R di G d= = d + d + d G G x y z d= = d + d + d O G x y z = WI I W G I d d µ da Gzz = ( x + y ) A I I M d d Ozz = Gzz + ( x + y ) a I G b G d O - G O O - G O c da I G

50 Mass Moments of Ineta Psm M = ρabc c a b b + c M a + c a + b Sold Sphee 4 3 M = πρa 3 a Ma 5 Sold Cylnde M = πρa L a L/ L/ + 3 a / L ML + 3 a / L 6 a / L Sold Ellpsod 4 M = πρabc 3 a b c b + c M a + c 5 a + b Sold Cone π M = ρ a h 3 a h/4 h + h / (4 a ) 3Ma + h / (4 a ) Hollow Cylnde M = πρ( b a ) L a b L/ L/ L + 3( a + b ) M L + 3( a + b ) 6( a + b )

51 Calculatng mass moments of neta by summaton (Illustated wth D example, same dea wos n 3D) () G G G d = + G () G (3) G3 d 3 d To fnd poston of COM and neta of a complex shape, use: Total mass M = m+ m + m3 Cente of mass ( ) G = mg+ mg + m3g3 M Mass moment of neta (use paallel axs theoem and add all sectons) I = I + md + I + m d + I + m d Gzz Gzz Gzz G3zz 3 3 d s the dstance of the COM of the th secton fom the combned COM at G

52 Momentum and Enegy Equatons Momentum and Enegy of a gd body 3D: Lnea Momentum Angula Momentum p= Mv G h= G MvG + IGω T = M vg + ω IGω Knetc Enegy ( ) G d dv Lnea Momentum p= Mv G D: Angula Momentum Knetc Enegy h= Mv + I ω G G Gzz z T = M v + I ω G Gzz z G d da Specal Case: Rotaton about a fxed pont O - G O O - G O Angula Momentum h= I h= IOzzωz Oω Angula Momentum Knetc Enegy T = ω ( I ) Knetc Enegy Gω T = I ω Ozz z

53 Rgd Body Dynamcs - Roadmap. Descbng moton of a gd body Rotaton tenso (matx) Angula Velocty Vecto Spn tenso (matx). Analyzng moton n systems of gd bodes Relatng velocty/acceleaton of two ponts on a gd body Mechansms Geas, pulleys and ollng wheels 3. Lnea/Angula Momentum and Knetc Enegy of a gd body Rgd body as an nfnte numbe of patcles Calculatng neta tensos Momentum and enegy of a otatng body 4. Dynamcs of gd bodes Toques Foce lnea momentum and moment angula momentum elatons Examples of solutons to D poblems Usng enegy and momentum fo gd bodes

54 Toques (Couples, o pue moments ) Toque A toque s a otatonal foce: Causes otaton wthout tanslaton Toque s a vecto: Q= Qx+ Qy+ Qz Toque has unts of Nm 3D Toque D Toque Two non-collnea equal and opposte foces exet a toque F d F Q = Fd Powe of a toque P = Q ω Wo done by a toque Fo D: W θ t = Q ω W= Qd z θ Q = Q z θ

55 D neta, paallel axs theoem Inetal Popetes Total mass M Cente of mass = µ da Mass moment of neta A G = M µda A v G d = G Gzz = ( x + y ) µ = G = dx + dy A I d d da d G d da µ : Mass/unt aea Paallel Axs Theoem I I M d d Ozz = Gzz + ( x + y ) O - G O

56 D Momentum and enegy Momentum and Enegy of a gd body Lnea Momentum Angula Momentum Knetc Enegy p= Mv G h= Mv + I ω G G Gzz z T = M v + I ω G Gzz z G d da Specal Case: Rotaton about a fxed pont (to use these O must be statonay on the gd body) Angula Momentum h= I ω Ozz Knetc Enegy T = I ω z Ozz z O - G O

57 D equatons of moton fo gd bodes Analyzng D moton of a gd body F () ω = ω z, α = αz Lnea Momentum F = dp F= Ma G G M,I Gzz F () Q = Q z Angula Momentum F+ Q = z dh (about ogn) F+ Q = Ma + I α z G G Gzz z Specal Case: Rotaton about a fxed pont (O must be statonay) F+ Q = I α z Ozz z O - G O

58 D Knematcs fomulas Knematcs Fomulas ωz Wheel ollng wthout slp on statonay suface v v = ω ( ) O C z O C v = ω R xo xo z a = α R z R O C v xo Geneal v v = ω ( ) B A z B A a a = B A α ( B A) ω ( B A)

59 Analyzng moton of gd bodes Calculatng foces o acceleatons Idealze system Fee body dagam fo each gd body F= MaG fo each gd body. F+ Q z = G M ag + I Gzz α z fo each gd body Use nematcs equatons to elate ag, αz fo each gd body R C v v = ω ( ) B A z B A B A = α ( B A) ω ( B A) a a v v = ω ( ) O C z O C v = ω R xo xo z a = α R z ωz R O C v xo Solve fo unnown foces o acceleatons

60 Enegy equaton fo systems of gd bodes M Extenal Powe P () t t Extenal wo W P() t = t M M 4 4 F F F 3 M F 3 t t = t = t Total KE T Total PE U M M 3 TOT TOT F F M 3 M Total KE T Total PE U TOT TOT W = ( T + U ) ( T + U ) TOT TOT TOT TOT W = ( T + U ) = ( T + U ) TOT TOT TOT TOT Same as systems of patcles, but we now use the gd body fomula fo KE T = M vg + ω IGω T = M v + I ω G Gzz z

61 Angula Momentum equaton fo systems of gd bodes Extenal Moment Extenal Angula Impulse F F + Q t A = F + Q t F F3 F3 G 3 G G3 F F t = t Total Angula Momentum h TOT t = t Total Angula Momentum dh F + Q= A = h h TOT TOT TOT Same as systems of patcles, but we now use the gd body fomula fo AM h= mvg + IGω = m G + IGzzωz h v h TOT