ME 6590 Multbody Dynamcs Connectn Jonts Part I o Connectn jonts constran te relatve moton between adjonn bodes n a multbody system. Jonts rane rom allown no relatve moton (a rd jont) to allown all motons (a ree jont). o Wen derees o reedom are removed, constrant orces or torques result. Wen derees o reedom are not removed, orces, or torques at a jont may or may not be zero. For example, bodes may ave nterconnectve sprns dampers tat do not remove derees o reedom but do restrct te moton by applyn loads assocated wt te relatve motons. o A lstn o common connectn jonts s sown n te ollown table. Jont Derees o Freedom d 0 Hne (revolute) ( rotaton) Spercal (ball & socket) 3 (3 rotaton) wo-anle (unversal) ( rotaton) Prsmatc (slder) ( translaton) Cylndrcal ( translaton, rotaton) Free 6 (3 translaton, 3 rotaton) o For every deree o reedom tat a jont elmnates, a constrant equaton must be wrtten. e ollown pararaps outlne te orm o te constrant equatons or some o te common jonts. Spercal Jont: Absolute Coordnates o A spercal (or ball--socket) jont allows two bodes to sare a common pont, but stll rotate reely relatve to eac oter. Usn absolute coordnates, te constrant equaton can be wrtten as pg p G q r Kamman ME 6590 Multbody Dynamcs pae: /6
or pg p G q r 0 () o Eq. () represents a set o tree scalar constrant equatons tat elmnate te tree translatonal derees o reedom between te bodes. o For ncorporaton nto a set o equatons o moton, te constrant equatons may be derentated twce so tey are n te orm o second order derental equatons. Usn xed-rame anular velocty components ves 0 vg v G q r vg v G q r vg v G q r vg v G q r o Derentatn aan ves 0 ag ag q q r r () Calculaton o te elements o q r were dscussed n earler notes. o Usn body-rame anular velocty components ves 0 vg v G q r vg v G q r vg v G q r Derentatn aan ves or 0 ag a G q q r r Kamman ME 6590 Multbody Dynamcs pae: /6
0 ag a G q q r r (3) Spercal Jont: elatve Coordnates o Consder now te use o relatve coordnates to descrbe te postons o ponts wtn te multbody system as sown n te ure. o o dene a spercal jont tat attaces te ponts Q O, te constrant equatons are smply sss (4) 0 Hne (evolute) Jont: Absolute Coordnates o Lke te spercal jont, te ne jont connects two bodes at a snle pont, so te translatonal constrants are as ven n Eq. (). o In addton, te ne jont also restrcts te relatve rotatonal moton o te bodes by elmnatn two o te tree rotatonal derees o reedom. o Consder te two bodes sown n te daram. Let be a vector xed n parallel to te ne jont, let be vectors xed n tat are perpendcular to te ne axs (, ence, ). en, te rotatonal constrant can be expressed drectly n terms o te anular veloctes as ollows 0 0 0 (5) 0 (6) Kamman ME 6590 Multbody Dynamcs pae: 3/6
o Intal condtons are used to ensure te alnment o wt. o Usn xed-rame anular velocty components, te rst o Eqs. (5) (6) may be wrtten 0 0 o ese equatons can be derentated to ve 0 0 So, nally, te two derentated constrant equatons are 0 0 o Usn body-rame anular velocty components ves 0 0 o ese equatons can be derentated to ve (7) (8) Kamman ME 6590 Multbody Dynamcs pae: 4/6
0 0 (9) smlarly, 0 (0) Hne (evolute) Jont: elatve Coordnates o e translaton constrants are te same as te spercal jont as ven n Eq. (4). o o derve te addtonal rotatonal constrant equatons, we start wt te same set-up used or absolute coordnates. at s, let be a vector xed n parallel to te ne jont, let be vectors xed n tat are perpendcular to te ne jont (, ence, ). en, te rotatonal constrants can be expressed drectly n terms o te anular veloctes as ollows ˆ 0 or ˆ 0 () Kamman ME 6590 Multbody Dynamcs pae: 5/6
ˆ 0 or o As beore, ntal condtons are used to ensure te alnment o ˆ 0 () o Derentatn te above equatons usn components o wt ˆ ves. ˆ 0 (3) ˆ 0 (4) o Derentatn te above equatons usn components o ves ˆ 0 (5) ˆ 0 (6) ˆ Kamman ME 6590 Multbody Dynamcs pae: 6/6