Dec :5:6 - Kinemaics model of Simple Arm.sm This file is provided for educaional purposes as guidance for he use of he sofware ool. I is no guaraeed o be free from errors or ommissions. The mehods and assumpions ma no appl in all cases. Engineers and designers should see design and calculaion approval from a relevan organisaion. Covenr Universi - Nov (David Sandells) This is an eample o show ou how SMah can calculae he movemen of inemaic mechanisms. Plugins : You will need he following plugins (ools>plugins) enabled o view his file:- Mah Region, Picure Region, Plo Region, Te Region, Area Region SMah Sudio Files Plugin Special Funcions You can find plugins here:- hp://smah.info/?eensions=smahsudio_desop hp://en.smah.info/forum/af_opics_eensions.asp Ploing funcions (for use wih maing graphs) size 5 sm "+" col "Red" Seings for smbol, size and colour of he ploed joings. p vec vec vec sm size col Funcion o plo a poin based on a posiion vecor. whl vec lne vec vec, vec vec "." 5 "Blac" vec vec vec vec Funcion o plo a wheel shape (a ver big do!) based on a posiion vecor. Funcion o plo a line beween wo poins defined b vecors. / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm Definiion of Transforms Translaion from Frame o Frame is a combined ranslaion and roaion:- The following is a smbolic equaion (bold equal signs). I is for show and is no acuall calculaed T cos sin sin cos Combined Mari 6 6 Translaion Mari cos sin sin cos Roaion Mari Tr 6,, Using he funcions o creae he maricies / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm Locaion of each of he poins in he mechanism is found b moving o he righ coordinae frame and hen moving o a vecor posiion wihin ha frame of reference. If we are ineresed in he origin hen he vecor posiion is (,,,). (The era '' is needed o mae he mari calculaions wor). P Poin is origin (don' go anwhere) P Tr 6,, Poin is he end of he firs arm - ranslae o he boom of he arm and hen don' go anwhere else. In fac his is he same as a vecor (6,-,,) P Tr 6,, poin is he righ angle of he second arm Translae hen roae o ge ino he second coordinae frame (frame ) and hen go o posiion,,. This depends on he roaion angle so is no a consan and herefore is defined as a funcion of he roaion angle. P Tr 6,, poin is he end poin of he second arm Translae hen roae o ge ino he second coordinae frame (frame ) and hen go o posiion,,. As before defined as a funcion of he roaion angle. Les r using he funcion o find ou where he end poin (poin ) will be for differen roaion angles. P deg 9 P deg 9 P 8 deg 6 P 7 deg 8 7 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm Plo of he end poin (Poin ) for differen roaion angles - - - - Using graphs lie his can help visualise he movemen of he mechanism. However, ou have o add a erm for each poin ou wan o plo. -5-6 -7-8 -9-6 8 deg deg 5 deg deg 5 deg 9 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm We can use he line and poin ploing funcions above o generae daa ses ha can be ploed on he graph o show he posiion of he mechanism a a paricular poin. Arm lne P, P Arma lne P, P deg Armb lne P deg, P deg - - - - -5-6 -7-8 -9 Arm Arma Armb - 6 8 deg deg 5 deg deg 5 deg 9 5 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm As an alernaive, he graph below uses animaion o show he movemen of he mechanism. Funcions are used o allow producion of differen daa for differen angles The inpu "" is replaced b he chosen animaion lis variable (Lis). To se his up b righ clic on he plo and selecing "ani" variable under he animae sub-menu. The "Frame Rae" sub menu conrols he animaion speed ec. for,, Lis deg deg This 'for loop' ses up a lis of angles o be used in he animaed graph. The lis is shown off he page o he righ. Arm lne P, P Arma lne P, P Armb lne P, P - - - - -5-6 -7-8 9-6 8 Arm Arma Armb 6 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
More Comple mechanism Dec :5:6 - Kinemaics model of Simple Arm.sm In addiion o he roaion of he second arm we can also allow he firs arm o roae b and angle we will call α. P P α α Tr 6,, Now we need o roae he firs coordinae frame b angle α before we move down he arm. P α, α Tr 6,, P α, α Tr 6,, Wih his mechanism we can se an addiional consrain ha he end poin (poin )mus slide along he ais. Therefore as we var α we need o find he angle ha will eep he end poin on he ais (i.e. =) We can use he "Solve" funcion (Par of he special funcions plugin) o achieve his. 'Plugins' can be found hrough he SMah forum websie. The are *.dll files ha need o be copied in o he righ direcor (usuall c:\smah\smah sudio\plugins). When done, he plugin will load auomaicall he ne ime SMah is sared. We are ring o find he angle ha will produce a = coordinae. Therefore we need o define a funcion ha onl provides he value (he second elemen of he vecor resul). fn α, P α, 8.986 Eample :- P deg, 5 deg.89 fn deg, 5 deg.89 The 'solve' funcion will ell us wha values mae he funcion oupu= solve fn 5 deg,,, 8 deg, 8 deg 5.79 67.59 deg Solve finds muiple values of ha mae he fn= 7 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm This ime we need o animae boh he α and values so we will generae a separae animaion variable (ani) o conrol boh. ani.. We can hen generae a lis of α angle values for ani This means for equal o ever value in ani in urn. Lis 5 deg 5 deg α And using he solve mehod we can find a maching lis of values. Noe we have limied he angles hrough which solve can search (-,+6) which should mean ha we onl ge one value of for each value of α. You can see ha his is o in he resuls shown off he righ hand side. for ani Lis solve fn Lis,,, deg, 6 deg α Find a value of beween - and +6 degs ha maes he fn= The lines for he graph now need o funcions of boh α and. Arm α lne P, P α Arma α, lne P α, P α, Armb α, lne P α,, P α, 5 - - This graph animaes using he 'ani' variable o sep hrough each resul in he lis and displa i. - - -5-6 Lis, α Arm 6 8 Arma Lis, α Armb Lis, α Lis Lis Lis The '' is replaced b each value of 'ani' and his calls each elemen of he resul lis in urn 8 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm We can now plo he he resul daa ses agains one anoher. Arm Angle (deg) v Arm Angle (deg) - -6-96 -8 8 6 8 56 6 Lis augmen, deg deg Here calculae he posiion from he angle daa and plo ha. for ani Lis P, Lis For each pair of angles calculae he posiion (elemen '' of he vecor) End poin disance along ais v Arm angle (deg) 8 6 Plo of inpu angle v oupu posiion 8 6 8 56 6 augmen, Lis deg 9 / - David Sandells - Covenr Universi Nov. See disclaimer on Page
Dec :5:6 - Kinemaics model of Simple Arm.sm We can now ae his daa furher. Les assume ha he mechanism is driven b a moor urning arm wih Nm of orque. If he joins had zero fricion hen no energ is los in hem and herefore he oupu force mus be enough o balance he inpu energ. Energ in Torque Angle Energ ou Force disance Noe ha if he joins are no perfec hen we need o calculae he energ los along he wa and subsrac i. ani.. lengh ani If we are calculaing he difference in disance hen we will end up wih a shorer vecor. for ani Erg in N m Lis of inpu energies (Torque*Angle) for ani Frc ou Lis Erg in Lis m Lis of oupu forces (energ/disance) for ani Shorer lis of inpu angles for ploing wih which aes he average posiion beween he poins Force generaed (N) v Arm angle (deg) 8 - -8 - -6 - - -8 - A graph lie his could help wih mechanism sizing and design. If we wan a more accurae resul hen we need o add more calculaion poins. -6 8 6 8 56 6 augmen, deg Frc ou N / - David Sandells - Covenr Universi Nov. See disclaimer on Page