2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V
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1 ME 352 VETS 2. VETS Vecor algebra form he mahemaical foundaion for kinemaic and dnamic. Geomer of moion i a he hear of boh he kinemaic and dnamic of mechanical em. Vecor anali i he imehonored ool for decribing geomer. Vecor can be decribed eiher geomericall or algebraicall. Vecor Geomeric preenaion Vecor are denoed b bold-face characer uch a, V, ec. The magniude of a vecor, uch a, i denoed a,, V or. The angle of a vecor i denoed a which i meaured poiivel couner-clockwie (W) wih repec o a welldefined ai. I i common o conider he poiive -ai a he reference ai. In kinemaic and dnamic a vecor ma repreen poiion, veloci, acceleraion, or force/momen. F Noe: Since we canno wrie b hand in boldface, we denoe a vecor wih an over-core arrow or an under-core line, for eample or. lgebraic preenaion vecor can be projeced ono he - and -ae of a areian frame in order o form i analical repreenaion. co in co Thi repreenaion remain valid regardle of wheher he angle i in he fir, econd, hird, or fourh quadran, a long a he angle i meaured W wih repec o he poiive -ai. oaed Vecor If a vecor uch a i roaed 90 W, i will be denoed a. The roaed vecor will have he ame magniude a bu i - componen will be differen: co( + π ) 2 in( + π ) in 2 co in co Poiion vecor In general a poiion vecor decribe he poiion of one poin wih repec o anoher poin. Eiher of hee poin could be moving or aionar. When a poin move, he orienaion and/or he magniude of i poiion vecor change a well. Eample: connec wo poin ha are defined on wo eparae moving link; connec poin on he ground (aionar) o poin on a moving link. Noe: poin depiced a a mall black circle in a figure i aumed aionar (defined on he ground). P.E. Nikraveh 2-1
2 ME 352 VETS Inde of Vecor In kinemaic anali of mechanim, i i helpful o aign indice o poiion vecor. Mo commonl, an inde refer o he end poin of he vecor. The fir leer in an inde indicae he head of he vecor (he arrow) and he econd leer refer o he ail of he vecor. For eample vecor, which read poiion of relaive o, and ha read poiion of relaive o. In ome problem, for noaional implificaion, he inde ma carr a ingle number. For eample vecor 3 in he diagram i he ame a vecor. 3 Linear Veloci and cceleraion The ime derivaive of a poiion vecor repreen he veloci of he Q poin he poiion vecor repreen. For eample: d d V (veloci of relaive o ) Q d d V (veloci of relaive o ) Thi veloci i alo called linear veloci. Noe: Linear veloci i defined for a poin (no for a link or for a vecor). When he reference (he ail) poin of a poiion vecor i aionar, ha poin inde could be dropped from he inde of he veloci vecor. For eample: d d V V (a) If we conider anoher aionar poin uch a Q, we ge d d V V (b) Q Q Thi mean ha he veloci of a poin wih repec o he ground i independen of he defined reference poin on he ground. Thi vecor repreen he veloci of relaive o he ground or he abolue veloci of. The econd ime derivaive of a poiion vecor, or he ime derivaive of a veloci vecor, denoe he acceleraion of he poin he poiion vecor repreen. For eample, acceleraion of relaive o i denoed a d 2 d 2 d d V V The abolue acceleraion of (acceleraion of wih repec o he ground) i denoed a d 2 d 2 d d V V Noe: Linear acceleraion i defined for a poin (no for a link or a vecor). ngular Veloci and cceleraion The ime rae of change in he orienaion of a vecor, or a link, i defined a he angular veloci of ha vecor or link. In planar em, he angular veloci i he ime rae of roaion abou he z-ai and i i denoed a d d ω. We conider a couner-clockwie (W) roaion a poiive and a clockwie (W) roaion a negaive. Noe: ngular veloci i defined for a vecor or a link, no for a poin. P.E. Nikraveh 2-2
3 ME 352 VETS Noe: Vecor ha are defined on he ame bod eperience he ame angular veloci. For eample,,, and have he ame angular veloci ince he are defined beween he poin of he ame bod. The ime rae of change of he angular veloci of a vecor, or a link, i defined a he angular acceleraion of ha vecor or link. In planar em, he angular acceleraion i a vecor along he z-ai and i i denoed a d 2 d 2 dω d ω α. We conider a couner-clockwie (W) roaion a poiive and a clockwie (W) roaion a negaive. Poiion, Veloci, and cceleraion Vecor in Mechanim Poiion vecor mu be conruced beween well-defined poin of a mechanim. The magniude and he angle of a poiion vecor mu reveal cerain informaion abou he poiion and orienaion of a link wih repec o anoher link or wih repec o he ground. Depending on he pe of a poiion vecor, he correponding veloci and acceleraion vecor ma be decompoed differenl. The following eample how pical poiion vecor, and heir correponding veloci and acceleraion vecor ha appear in kinemaic anali of mechanim. onan Magniude, onan ngle The poiion vecor i defined beween wo non-moving poin. Q coψ Q L Q inψ, V, Q Q Q where L Q i he conan magniude and ψ i he conan angle of he ψ vecor. Zero vecor for he veloci and acceleraion hould be obviou ince boh end of he poiion vecor are fied o he ground. Variable Magniude, onan ngle Two link form a liding join. ne of he link (he rod) i fied o he ground and he oher (he block) lide on he rod. The poiion vecor i defined beween a poin on he ground and a poin on he block parallel o he liding ai. Thi vecor doe no roae ince neiher link can roae. coψ inψ ψ V coψ V inψ V V coψ, inψ (I i aumed ha he lip componen are poiive.) The veloci and acceleraion vecor are along he ai of he are called lip veloci and lip acceleraion. onan Magniude, Variable ngle The poiion vecor i defined beween wo poin on a link a co L where L i he conan lengh of he vecor. The link i free o roae. P.E. Nikraveh 2-3
4 ME 352 VETS in V ω L V co n V ω Thi vecor i perpendicular o i i called angenial veloci. V co in ω 2 n L + αl co + (I i aumed ha he angular n ω 2 α veloci and acceleraion are poiive.) The acceleraion vecor conain wo componen: one in he oppoie direcion of called normal, and one perpendicular o called angenial acceleraion. Variable Magniude, Variable ngle c block and a rod form a liding join. oh he rod and he block roae ogeher; i.e., he wo link have he ame angular veloci and he ame angular acceleraion. n co in co V V ω + co V in + V V ω V V The veloci vecor can be decompoed ino wo (I i aumed ha he lip componen, and angular veloci and acceleraion are componen: a angenial veloci perpendicular o, poiive.) and a lip veloci along he ai of. co in co in ω 2 + α + co + 2ω in n ω 2 α co c 2ωV n c The acceleraion vecor i decompoed ino four componen: a normal componen in he oppoie direcion of, a angenial componen perpendicular o, a lip componen along he ai of, and a orioli componen perpendicular o. oordinae, veloci, and acceleraion of Poin noher pe of poiion vecor decribe he coordinae of a poin wih repec o he origin of a areian reference frame. For eample, decribe he coordinae of poin wih repec o he origin. When he reference poin i he origin, for noaional implificaion, i ma be dropped from he inde; e.g., we ma a inead of. ne common ep in kinemaic anali i o deermine he veloci and acceleraion of a poin baed on ome oher known velociie and acceleraion. For hi purpoe we ar wih a poiion vecor equaion and hen ake i ime derivaive. Here we conider wo pical eample. P.E. Nikraveh 2-4
5 ME 352 VETS In he fir eample, for link, i i aumed ha he veloci and acceleraion of poin and he angular veloci and acceleraion of he link are given. We are aked o deermine he veloci and acceleraion of poin. We ar from he following poiion vecor equaion: + (a) The ime derivaive of hi equaion ield V V + V V V + V V + ω (b) uming he given angular veloci i W, he vecor ummaion in (b) can be conruced graphicall a hown. The acceleraion of i obained from he ime derivaive of (b): n ω 2 + α (c) uming he given angular veloci and acceleraion of he link are W, he acceleraion epreion in (c) can be conruced graphicall a hown. Equaion (a), (b) and (c) can alo be evaluaed algebraicall a: + L co (d) V + ω L in (e) co ω co 2 L + αl in (f) co V In he econd eample, wo link form a liding join a hown. The veloci and acceleraion of poin, he angular veloci and acceleraion of he link, and he liding veloci and acceleraion are given. We are aked o deermine he veloci and acceleraion of poin. Noe ha boh link have idenical angular veloci and alo idenical angular acceleraion. We ar wih he poiion vecor equaion, which i idenical o (a). The correponding veloci equaion i: V V + V + V V + ω + V (g) uming ha he given angular veloci and he lip veloci are boh poiive, V can be conruced graphicall a hown. The acceleraion epreion i wrien a n c ω 2 + α + + 2ωV uming ha he given angular and lip acceleraion are poiive, he acceleraion epreion in (h) can be conruced graphicall a hown. Equaion (a), (g) and (h) can alo be olved algebraicall: + co (i) (h) V V V V c V V V n V n P.E. Nikraveh 2-5
6 ME 352 VETS V + ω in co + co (j) ω co 2 + α in co + co + 2ω in co (k) Vecor Loop The poiion vecor ha are defined for kinemaic anali of a mechanim hould form one or more kinemaic loop (alo called cloed chain). an eample he vecor ha are defined for he four-bar in (a) form a loop. Thee vecor ma be direced differenl o form a loop a hown in (b) If we navigae in a loop from vecor o vecor, he vecor ha i navigaed from ail-o-head i conidered poiive and a vecor ha i navigaed from head-o-ail i conidered negaive. For eample, for he four-bar in (a), if we navigae hrough he loop in he following fahion, vecor 2 and are navigaed poiivel and 4 and 4 are navigaed negaivel (a) (b) kinemaic loop can be epreed a a vecor equaion. For eample, he vecor in he four-bar (a) form he vecor-loop equaion Similarl, he vecor in (b) form he vecor-loop equaion bvioul, here are man oher poible cenario. n of hee vecor loop equaion can be ued for kinemaic anali of he mechanim. Eample Several eample of commonl ued planar mechanim are preened here. For each mechanim a e of vecor are defined o form a vecor-loop equaion(). (a) Four-bar (b) Slider-crank (inverion 1) (c) Slider-crank (inverion 2) P.E. Nikraveh 2-6
7 ME 352 VETS (d) Slider-crank (inverion 3) (e) Slider-crank (inverion 4) (f) ffe lider-crank (inverion 1) 2 Q + Q 2 Q Q Q (g) ffe lider-crank (inverion 3) (f) ffe lider-crank (inverion 2) (h) Si-bar mechanim (a four-bar and a lider-crank) Thi mechanim conain wo independen loop: and. Therefore wo independen vecor loop equaion can be conruced: (a) (b) 4 hird loop ma be viualized a. However, hi i a redundan loop i vecor loop equaion can be obained from ubracing he econd equaion from he fir: If we combine he wo ground vecor, 2 and 4, ino one vecor we ge: (c) We onl need wo of hee hree equaion. 4 P.E. Nikraveh 2-7
8 ME 352 VETS (i) Si-bar mechanim (wo lider-crank) Thi mechanim conain wo independen loop: (a) (b) hird dependen loop can be conruced a (c) We onl need wo of hee hree equaion Noe: In eample conaining a liding join, he vecor ha connec he wo link mu be defined parallel o he ai of he liding join. Thi i a rule ha we mu follow, oherwie we can make a imple problem oo difficul o olve. For eample, for hi lider-crank, we hould no define a vecor direcl from o. Inead, we hould define wo vecor, 4 which i parallel o he liding ai and perpendicular o he liding ai. 4 Noe: If he ground vecor i no parallel o he -ai, we ma decompoe i ino wo vecor one parallel and one perpendicular o he -ai. For eample, 4 ma be replaced b Q2 + 4 Q. 4 Q2 4 Q Q P.E. Nikraveh 2-8
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