Locion is relive Coordine Sysems The posiion o hing is sed relive o noher hing (rel or virul) review o he physicl sis h governs mhemicl represenions Reerence oec mus e deined Disnce mus e nown Direcion mus e nown Vecor Vecors Physicl - mgniude nd direcion - mgniude my e velociy, orce, ec. Spil - mgniude is disnce Displcemen - spil rnslion o single oec. The displcemen o n oec cn e given wihou respec o noher oec. Posiion - spil relion (locion) o n oec wih respec o noher (reerence oec + spil vecor) Trnslion - Reles he chnging posiion o n oec. y e displcemen o he oec or is reerence. hemicl Represenion- Vried uses Which o he ollowing cn e descried wih vecors?? Temperure Elecric Field Liquid Flow Re Volge ss Posiion Spil Vecor ddiion Spil Reerence Frme (Bsis) represenion o lengh nd direcion D C B Reerence Direcions Deined uni lengh (disnce) Origin (reerence poin ) xes hrough origin in reerence direcions posiive disinguished rom negive Spil (Rel Physicl) Bsis Vecors Uni displcemen long xes Liner sum o sis vecors descries posiion/direcion 1
cs P i is xis i is sis vecor x is sclr muliple Posiion Reerence Relion Beween Coordine Frmes o x i OPP x i + y + x i y [ ] i Coordine represenion o Posiion P [P] POP (x,y, ) (x, y, ) lerne Reerence Frmes Vecor Descripion Relion Beween Coordine Frmes cs o x y P i Posiion Coordines P (x,y, ) (x, y, ) P (x,y, ) (x, y, ) So Posiion coordines re dependen on he reerence rme chosen When ming mesuremens, he spil reerence rme mus e undersood or speciied 1
Swichingvecordescripionseweennonligned coordinermes cs o,o P cs (x,y, ) (x,y, ) i i ( OP ) [ i ][ P ] [ i ][ P] [ i ] x y i 1i + 2 +3 ndsince 1i + 2 +3 c 1i +c 2 +c3 iollowsh 1 1 c1 x 2 2 c2 y 3 3 c3 [ i ][ P] so P i P ( OP ) [ i ] [ ] [ ] Deiloprevious rnsormioneween roed coordinermes wihcommon origin ( OP ) [ i ][ P ] [ i ][ P] x i 1i + 2 +3 i y, u i + + [ ] 1 2 3 c i +c +c 1 2 3 ( 1 i 2 3 ) ( 1 i 2 3 ) ( 1 i 2 3) ( x+y +xc ) + ( x +y +c ) + ( x +y +c) x + + +y + + + c +c +c i 1 1 1 2 2 2 3 3 3 [ i ] [ i ] so c x+y+xc x +y +c 1 1 1 2 2 2 x +y +c 3 3 3 c c 1 1 1 2 2 2 c 3 3 3 x x y [ i ][ P] [ P] 2 2 c2 y i [ P] R [ P] 1 1 1 c 3 3 3 2
lernive represenionoprevious rnsormioneween roedcoordine rmes wihcommonorigin ( OP ) [ P] [ P] i [ P] [ ] i i i i 1i+ 2 +3 x y x i+y +, u 1i + 2 +3 c i +c +c x + + +y + + + c +c +c 1 2 3 ( i ) ( i ) ( i ) 1 2 3 1 2 3 1 2 3 ( x+y+xc ) + ( x +y +c ) + ( x +y +c) i 1 1 1 2 2 2 3 3 3 [ x+y+xc 1 1 1, x 2 +y 2 +c 2, x 3 +y 3 +c 3] 1 2 3 [ x y ] 1 2 3 ( OP ) [ P] [ P] [ x y ] 1 2 3 [ P] so c c c 1 2 3 1 2 3 c c c 1 2 3 i ( i ) ( ) ( ) i [ ] P R T i 3
chines nd erology Bsics o odel Building chine n ssemlge o componens designed o ssis in desired cion eing perormed on n oec. Sysem which uses energy in ny orm o move or ler n oec. From he Gree 'Μεχοσ - mechos mening expedien chine Tool chine which uses ny o vriey o ools o chnge he physicl shpe or oher inrinsic se prmeer o worpiece. Common operions include cuing, urning, oring, drilling ec. Oen incorpores proes, cuors, rnsducers, nd sges o cilie is operion. chine Design Frme - Bsic srucure o mchine which provides suppor o ll elemens. Dynmic Elemens- moors, llscrews, crriges Provide cuion Forces Consrining Elemens - guidewys, llscrews Limis Degrees o Freedom erology Elemens encoders, scles, redheds Provide Posiioning Reerence Srucurl Loop The ssemly chin rom one mchine elemen o noher which provides he physicl suppor nd consrin or ech elemen. Crriges, guidewys (wys), clmping ixures, oolholder, spindle, erings From worpiece o ool erology Loop/Lins The ph h weves hrough he mchine elemens h provides/ecs mchine's posiioning reerence. y inerlce hrough he spindle, quill, le, crriges, scles, llscrews, rme ec. Cn e sepred o some exen rom he srucurl nd dynmic loops nd hus no e suec o disored rme cused y cuion orces. Ph my inerlce hrough some supporing componens nd e eced y heir herml expnsion, orce disorions, nd mislignmens. 1
Dynmic Loops/Lins The ssemly chin rom one mchine elemen o noher which direcs he orces producing moion hroughou he mchine. For mchine ool, he dynmic loops my involve such hings s he moors, cuors, spindle, crriges, rme, erings, llscrews, worpiece, ool, nd clmping ixure. Frme Tool nd Holder Worpiece Frme Tool nd Holder Worpiece Frme Tool nd Holder Worpiece 2
Trnsormion Developmen on Two Crrige chine + + + + F F F F F + R + R F F F F BF B soh + R R F F F BF B 1 R + R BF 1 R + R BF F F F B F F (physicl model) F (coord represenion) B 1 R + BF TF RF TF B esuremens wr rme Trnsormion Developmen on Two Crrige chine + + + + B B B + R B B FB F + R + B B FB F F + R + R B B FB F F + R + R B B FB F F T + R T + R B B FB F F esuremen wr B(le) 3