ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms ENGI9496 Lecture Notes Multport Moels n Mechancs (New text Secton 4..3; Secton 9.1 generalzes to 3D moton) Defntons Generalze coornates any set of coornates that can be use to completely escrbe the confguraton of a system. set of generalze coornates s typcally not unque. Generalze coornates can nclue postons, angles, veloctes, electrcal fluxes, pressures, etc. Degrees of freeom (DOF) mnmum number of generalze coornates neee to completely escrbe the confguraton of a system. For example f you know the crank angle of a sler crank, you can compute all other lnk postons n terms of that angle. Inertal (prmtve) coornate a coornate escrbng the absolute poston of a boy s centre of mass, or ts absolute poston. q k : vector of the mnmal (smallest possble) set of generalze coornates m( q k ) = q, where q = number of DOF q I : vector of the nertal coornates m( q I ) = p = 3n for D systems, or 6n for 3D systems, where n = number of boes typcally a maxmal (largest possble) set of generalze coornates v : vector of nertal (prmtve) veloctes I q I Constrant an algebrac equaton relatng generalze coornates. For D systems escrbe by nertal coornates, there shoul be 3n-q constrants, an 6n-q constrants for 3D systems. pproach to Mechancal System Moelng One approach to moelng multboy mechancal systems, whch s easy to unerstan an mplement n bon graphs, s to efne a set of maxmal nertal coornates (sometmes calle prmtve coornates ), an constran them usng velocty constrant equatons. s we ve seen prevously, f you constran the velocty noes n a bon graph, then the force or torque equatons (Newton s Laws) wll be automatcally satsfe. We wll restrct ourselves to planar moton (D), an conser absolute coornates as well as boy-fxe coornates, whch requre gyrator elements to capture nner prouct terms of Euler s Equatons as escrbe n Secton 9.1) 1
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Summary of Newton s Laws for Invual Lnks bsolute Coornates Conser a boy wth mass an rotatonal nerta, subect to external forces at ponts an B. Centre of gravty s G. For multboy systems, the easest formulaton s one where all boes contrbute three nertal coornates (x,y,θ). To convert ths to a penulum, make velocty of pont equal to zero (through flow sources, or approxmately zero usng parastc sprngs). Ths wll create pn forces at. For such a penulum: 1 egree of freeom (DOF); knowng θ, you can compute x an y; shoul be 3 1 = poston constrant equatons there wll be much ervatve causalty, whch can be remove usng parastc elements fferentate poston constrants to get velocty constrants (mplement n bon graphs usng TF or MTF elements) reversng causalty of MTF elements may create the rsk of sngulartes (vson by zero)
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Generc Rg Boy Bon Graph Usng two of the above boes, you can easly create a contnuous ouble penulum two boes gve x 3 = 6 nertal coornates (all measure wth respect to a sngle orgn) two egrees of freeom (system confguraton can be escrbe by the angles θ an θ 3. therefore there must be 6 = 4 poston constrant equatons Develop a bon graph of the ouble-penulum. Then, turn t nto a sler-crank by affxng D to a mass that can only sle. 3
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Double-Penulum Bon Graph 4
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Extenson to Sler-Crank 5
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Boy-Fxe Coornates We now conser an x-y axs system that s fxe to, an rotates wth, the boy. We must now stngush between absolute, or nertal coornates (resolvng vectors n fxe x-y rectons); an boy-fxe coornates (resolvng vectors n rectons relatve to the boy). When we raw bon graphs, gven that bon graphs are a velocty-base formulaton (as oppose to poston-base), we wll fn ourselves expressng absolute velocty vectors n boy-fxe coornates. Boy-fxe coornates ntrouce a complcaton, as wll be escrbe below. Generc Rg Boy In the ppenx to ths ocument, some revew of coornate transformatons, an ervatve of a vector resolve nto a rotatng coornate frame, s gven. Development of Bon Graph 6
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Development of Rg Boy Bon Graph, Boy-Fxe Coornates (cont ) 7
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Smulaton Exercse: Shop Crane Conser a shop crane such as those sol at Prncess uto: To evelop a ynamc, mult-boy smulaton of ths, we wll take avantage of both methos of rg boy bon graphs absolute an boy-fxe coornates. 8
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms Shop Crane Bon Graph, Boy-Fxe Coornates 9
ENGI 9496 Moelng an Smulaton Boy Fxe Vectors Date: ppenx - Boy Fxe (Rotatng) Coornate Frames To ths pont, we have expresse vectors n x an y components - an arbtrary vector s the sum of an x component multple by a fxe unt vector an a y component tmes a fxe unt vector. xî yĵ To fferentate the vector, we techncally nee to use the prouct rule of calculus. However, snce an are fxe, they have no ervatve an the ervatve of s smply: xî yĵ In robotcs an avance ynamcs, especally 3D knematcs, t s customary to efne vector components along reference frames that are affxe to a boy an rotate wth that boy, nstea of efnng components n fxe an rectons. In Fg. 1 below, the relatve poston of B wth respect to, can be expresse n ether absolute or boy-fxe coornates: Fg. 1 î, î, ĵ ĵ unt vectors fxe to lnk, fxe unt vectors n x, y rotatng wth angular velocty ω rectons sn B r / B (1) B We can fferentate r B/ to get tangental velocty easly by fferentatng - components. We can also, by nspecton, wrte v B/ n boy-fxe components (tangental velocty has magntue B, recton perpencular to lne B). From Eq'ns (1), (), we can see that sn sn (3) v B / B r B / B sn B sn B () In matrx form: sn sn sn sn î î R 1 ĵ ĵ î î R1 ĵ ĵ 1
ENGI 9496 Moelng an Smulaton Boy Fxe Vectors Date: The matrces R 1 an R 1 are calle "rotaton matrces", because they allow us to convert a vector expresse n coornates of one frame, nto a vector expresse n coornates of a secon frame, where the frst an secon frames are rotate by an angle. Gven a vector expresse n frame 1 (fxe -) or frame components, as ncate by left superscrpt: R R 1 1 1 1 It s apparent from the structure of the matrx equatons on page 1 that T R R R 1 1 1 1 Ths s a property of rotaton matrces, that they are orthogonal - ther nverse s equal to ther transpose. s an example of rotaton matrces n acton, transform the tangental velocty v B/ from frame to frame 1 coornates: B B B B v B B v B B sn sn 0 sn sn 0 / 1 / Boy fxe frames can make t easer to express poston vectors. In Fg. 1, the poston of B wth respect to s smply B n the recton, regarless of the orentaton of the boy. When we fferentate vectors expresse n boy-fxe frames, however, we have to take nto account the fact that the unt vectors, because they change recton, now contrbute a ervatve. If we fferentate Eq'ns (3), we see that k k sn sn sn sn You shoul be able to vsualze the cross proucts usng the rght-han rule. Thus, the ervatve of a rotatng unt vector s equal to the cross prouct of the frame's angular velocty an the unt vector tself. Ths means that the ervatve of a vector expresse wth respect to a rotatng frame has a toal ervatve as gven below. These results exten to the three mensonal case. Gven ĵ b î a b a b a b a b a
ENGI 9496 Moelng an Smulaton Boy Fxe Vectors Date: a b a b rel where the frst term s the ervatve as seen by an observer movng an rotatng wth the coornate frame. The frst term s the rate of change of the components. The secon term, whch has the orgnal vector crosse wth the angular velocty of the reference frame, s an aonal rate of change arsng from the rotaton of the unt vectors. Ths term woul be seen only by an external non-rotatng observer. Ths has mportant mplcatons n 3D knetcs an the ervaton of gyroscopc torques. s wll be seen, there s conserable avantage n usng reference frames wth angular velocty equal to (or nearly equal to) the boy to whch they are attache. ω (4) rel 3