( ) Two-Dimensional Experimental Kinematics. Notes_05_02 1 of 9. Digitize locations of landmarks { r } Pk

Transcription

1 Notes_05_0 o 9 Two-Dmesoal Expermetal Kematcs Dgtze locatos o ladmarks { r } o body or pots to at gve tme t ll pots must be attached to body Use ladmark weghtg actor = pot k s avalable at tme t. Use = 0 pot k ot avalable at gve tme t. r r r r at gve tme t. Determe { } { } { } { } Mea values mea { r } { r } / / / / / k mea } } mea { r } { r } mea { r } { r } mea { r } { r } S = = mea T mea ( ({ r } { r } ) ({ r } { r } ) Velocty ω T ({ r } ) [ R]{ r } { r } mea 0 ( ) / S or [ R] = 0 [ ω ] = [ R] ω ot s the stataeous ceter o rotato or body wth respect to the groud. Note that the s ot attached to the body. ot o the body cocdet wth has zero velocty. ( ) or ay pot attached to body { r } = [ ω ]{ r } { r} mea mea _ { r} = { r } [ ω ] } or } _ at = 0

2 Notes_05_0 o 9 ccelerato ω mea ( ) / S T ({ r } ) [ R]{ r } { r } ω ω [ ] = ω [ ] [ β] = [ ω ] + [ ω R ] = ω ω ω ot I s the stataeous accelerato pole or body. Note that the I s ot attached to the body. ot o the body cocdet wth I has zero accelerato. I ( ) } = [ β]{ r } { r} or ay pot attached to body I mea mea _ I { r} = { r } [ β] } or { r } _ at = 0 OLD Jerk ω = ω + T ({ r } ) [ R]{ r } { r } mea ( ) / S ω ω ω ω [ ω ] = ω [ ] [ η] = [ ω ] + [ ω ] + [ ω R ] = ω ω ω ω ot IJ s the stataeous jerk pole or the body. Note that the IJ s ot attached to the body. ot o the body cocdet wth IJ has zero jerk. IJ ( ) } = [ η]{ r } { r} or ay pot attached to the body IJ mea mea _ IJ { r} = { r } [ η] { r } or { r} _ at = 0 Sap ω = 6ω ω + T mea ({ r } ) [ R]{ ( r } { r } ) / S [ ω ] = ω [ R] [ σ] = [ ω ] + 6[ ω ] + 4[ ω ] + [ ω ] + [ ω ] 4 4ω ω ω + ω ω ω ω [ ] = σ ω ω ω ω ω ω + ω 4 4

3 Notes_05_0 o 9 ot IS s the stataeous sap pole or the body. Note that the IS s ot attached to the body. ot o the body cocdet wth IS has zero sap. IS } = [ σ]{ ( r } { r} ) or ay pot attached to the body IS mea mea _ IS { r} = { r } [ σ] { r } or { r } _ at = 0 etrode Locato o the measured relatve to groud chages as body moves ad sweeps a locus called the xed cetrode or body. The tme dervatve o the locus descrbes how the moves. Trackg the locato o the relatve to a coordate rame xed to body provdes a locus called the movg cetrode. Moto o body may be characterzed as pure rollg o the movg cetrode o the xed cetrode because body stataeously has zero velocty at each locato o the. ( mea )/ ω mea } = [ ω]{ r } [ β] } The secod tme dervatve o the locato o the also chages as body moves. Frst ad secod tme dervatves o posto alog a locus may be combed to determe curvature κ o the xed cetrode. I body s part o a mechasm wth moblty o oe, curvature o the cetrode at each locato wll be varat to speed o the mechasm. ω mea mea mea { r} } + { r } + { r} / ω 0 ω ω ω ω ω 0 ω ωω ω ω ω 0 ω ω 0 κ = T ( { r} ) [ R] } )/ } T ( ) } ) / Relatve velocty For plaar moto, the relatve agular velocty o body j wth respect to body s the derece betwee the two agular veloctes. The relatve stataeous ceter o rotato (R) or body j about body descrbes a uque pot that has zero relatve velocty betwee the two bodes. Note that locato o the R s measured wth respect to the groud. ω j _ wrt _ = ω j ω ( ) ωj _ wrt _ { r } = [ ω ]{ r } [ ω ]{ r } j _ wrt _ j j /

4 Notes_05_0 4 o 9 Rgd Body Determe the velocty o pot o rgd body lk. The rgd body ad the velocty vectors are draw to scale. Lk s NOT ped to the groud. Show your work. Xc = 7 mm Yc = mm VB = 5.7 cm/sec X = 4 mm Y = 56 mm B XB = 90 mm YB = 80 mm 0º 5º V = 9. cm/sec 5.56 mmps 9.50 mmps B } = { r } 8.8 mmps = 8.88 mmps ω V B_wrt_ 9.8 mmps 58.8 mmps _ B } B _ wrt = } { r } = = mmps 7. B B _ wrt _ } = ω ( B) B = 5.77 mm ω.4 rad / sec W orm = 6 mm 65 mm { r } _ wrt _ = { r } _ wrt _ = ω [ R]{ r } _ wrt _ 80.7 mmps = 9.87 mmps V _wrt_ ω.65 mmps mmps _ wrt _ } = } + { r } = = 58.8 mmps 87. 4

5 Notes_05_0 5 o 9 Rgd Body Determe the velocty o pot o rgd body lk. The rgd body ad the velocty vectors are draw to scale. Lk s NOT ped to the groud. Show your work. Xc = 7 mm Yc = mm VB = 5.7 cm/sec X = 4 mm Y = 56 mm B XB = 90 mm YB = 80 mm 0º 5º V = 9. cm/sec pot = = { r } = { r } pot B = = { } { } use lmkd per attached code 4 mm 8.8 mmps = rom above 56 mm 8.88 mmps 90 mm 5.56 mmps r = r = 80 mm 9.50 mmps ω = +.4 rad/sec { } r = mm.97 { r } { r } = { r } { r}.6 mmps ( ) = = 58.8 mmps mm 0 ω = 4 mm ω mmps

6 Notes_05_0 6 o 9 % rbk.m - rgd body kematcs % HJSIII, 4.0. clear % costats Rmat = [ 0 - ; 0 ]; % puts = [ ]; r = [ 4 90 ; ]; rd = [ ; ]; rdd = zeros(,); rddd = zeros(,); % call ucto [ w, r, wd, ri, wdd, rij, rd, kappa ] = lmkd(, r, rd, rdd, rddd ); w r % d velocty o r = [ 7 ]'; rd = w * Rmat * ( r - r ) % bottom o rbk

7 Notes_05_0 7 o 9 % t_lmkd.m - test D kematcs rom ladmark moto % HJSIII, 4.0. clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % example puts - web cutter our bar % Haug page 97 - ot ME 58 web cutter % B = [ ]; r = [ ; ]; rd = [ ; ]; rdd = [ ; ]; rddd = [ ; ]; % expected outputs w_test = ; r_test = [ ; ]; wd_test = ; ri_test = [ ;.0846 ]; wdd_test = 6.8; rij_test = [.8647 ;.9747 ]; rd_test = [ ; ]; kappa_test = ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % test ucto [ w, r, wd, ri, wdd, rij, rd, kappa ] = lmkd(, r, rd, rdd, rddd ); w r wd ri wdd rij rd kappa % bottom o t_lmkd

8 Notes_05_0 8 o 9 ucto [ w, r, wd, ri, wdd, rij, rd, kappa ] = lmkd(, r, rd, rdd, rddd ) % D stataeous kematcs o a rgd body rom ladmark moto % HJSIII, 4.0. % % USGE % ucto [ w, r, wd, ri, wdd, rij, rd, kappa ] = lmkd(, r, rd, rdd, rddd ) % INUTS % - x vector o weghts - (j)= meas data vald, (j)=0 meas data ot avalable % r - x matrx o x,y ladmark locato % rd - x matrx o x,y ladmark velocty % rdd - x matrx o x,y ladmark accelerato % rddd - x matrx o x,y ladmark jerk % % OUTUTS % r - x locato o stataeous ceter o rotato % w - agular velocty % ri - x locato o stataeous accelerato pole % wd - agular accelerato % rij - x locato o stataeous jerk pole % wdd - agular jerk % rd - x tme dervatve o locato o stataeous ceter % kappa - curvature o cetrode % costats Rmat = [ 0 - ; 0 ]; % mea values [ r, ] = sze( r ); mat = dag( ); s = sum( ' ); rm = sum( mat*r' )' /s; rdm = sum( mat*rd' )' /s; rddm = sum( mat*rdd' )' /s; rdddm = sum( mat*rddd' )' /s; % cetered locato rc = r - rm*oes(,); S = trace( rc * mat * rc' ); % velocty vmat = rd * mat * rc'; w = ( vmat(,) - vmat(,) ) /S; wsk = w * Rmat; r = rm - v(wsk) * rdm; % accelerato amat = rdd * mat * rc'; wd = ( amat(,) - amat(,) ) /S; wdsk = wd * Rmat; beta = wdsk + wsk*wsk; ri = rm - v(beta) * rddm; % jerk jmat = rddd * mat * rc'; wdd = w*w*w + ( jmat(,) - jmat(,) ) /S; wddsk = wdd * Rmat; eta = wddsk + *wsk*wdsk + wsk*wsk*wsk; rij = rm - v(eta) * rdddm; % sap %rddddm = sum( mat*rdddd' )' /s; %smat = rdddd * mat * rc'; %wddd = 6*w*w*wd + ( smat(,) - smat(,) ) /S; %wdddsk = wddd * Rmat; %sgma = wdddsk + 6*wsk*wsk*wdsk + 4*wsk*wddsk + *wdsk*wdsk + wsk*wsk*wsk*wsk; %ris = rm - v(sgma) * rddddm; % cetrode rd = ( wsk*rddm - beta*rdm ) /w/w; rd = orm( rd ); sk = [ 0 (w*wdd-*wd*wd) ; -(w*wdd-*wd*wd) 0 ];

9 Notes_05_0 9 o 9 sk = [ w*w*w *w*wd ; -*w*wd w*w*w ]; sk = [ 0 -w*w ; w*w 0 ]; rdd = ( sk*rdm + sk*rddm + sk*rdddm ) /w/w/w; kappa = ( rd()*rdd() - rd()*rdd() ) /rd/rd/rd; % bottom o lmkd