OINTCOORDINATE FORMULATION Two or more poits ca be used to describe a rigid body. This will elimiate the eed to defie rotatioal coordiates for the body i z r i i, j r j j rimary oits: The coordiates of primary poits appear explicitly i the formulatio of the costraits ad equatios of otio; e.g., poit ad j. rimary Coordiates, Velocities, Acceleratios: x i % x i % x i % r i = " y i &, r i = y i " &, r i = y i " & z i ' z i ' z i ' rimary Costrait: (r i r j T (r i r j i, j ( = 0 Secodary (Norimary oits: These coordiates are defied as a fuctio of the primary coordiates; e.g., poit. Describig A Body by its rimary oits Two rimary oits: z r r, Three rimary oits: z r r r,,, 6 coordiates; costrait 9 coordiates; costraits age OINTCOORDINATE FORMULATION. E. Nikravesh
Four rimary oits: z r r r r coordiates; 6 costraits The four poits are ot i the same plae. Locatig A Secodary oit Two rimary oits: z r, r a r A =, (a r a r a ad a are directioal costats. + a A Three rimary oits: s s A s, s, A oit A does ot have to be o the plae of the three primary poits. r A = r + a (r r + a (r r + a (r r (r r How do we determie a, a ad a? Assume r, r, r, ad r A are kow iitially; Compute: s, = r r, s, = r r, s A = r A r s = s, s, Describe: s A = a s, + a s, + a s Defie: S [ s, s, s], a { a a a } T s A = Sa Solve: a = S s A Note: This process performed oly oce age OINTCOORDINATE FORMULATION. E. Nikravesh
Four rimary oits: s, s, s A s, A r A = r + a (r r + a (r r + a (r r How do we determie a, a ad a? Use s, to replace vector the threepoit case. Assume r, r, r, r, ad r A are kow iitially; Compute:, = r i r ; i =,,, s A = r A r Describe: s A = a s, + a s, + a s, Defie: S [ s, s, s], a { a a a } T, s A = Sa Solve: a = S s A Kiematic Joits Shared rimary oits Spherical Joit: Revolute Joit: (i (j Uiversal Joit: (i (j T = 0 Cylidrical Joit: (i d s i = 0, d = 0 (j rismatic Joit: (i a i d b j (j = 0, d = 0, a i T b j cost. = 0 age OINTCOORDINATE FORMULATION. E. Nikravesh
NoShared rimary oits Spherical Joit: Revolute Joit: l k (j (i r l i r k j = 0 Other joits follow a similar procedure. (i m (j Assume that poit m is shared = 0 Kiematic Costraits Two Coicidig oits: r i l r j k = 0 Two erpedicular Vectors: T = 0 Two arallel Vectors: s i = 0, s i d = 0 Note: Defie additioal vectors, the use scalar product twice istead of vector product. Kiematics of A Multibody System (i (j s m (m d (k s ( (l p : Number of primary poits v : Number of coordiates; velocities; acceleratios; v = p c : Number of primary ad kiematic costraits Array of Coordiates: r ( v Array of Velocities: r ( v Array of Acceleratios: r ( v age OINTCOORDINATE FORMULATION. E. Nikravesh
ositio Costraits: (r = 0 ( c Velocity Costraits: " Dr = 0 ( c Acceleratio Costraits: " D r + D r = 0 ( c Force Distributio Geeral: (i A force acts at poit o body i; Fid a equivalet set of forces actig o the primary poits that defie body i f j j = = f j = s i j = f f s s s s.... Assume that all forces are parallel to the origial force: f j = j f i ; j =,..., The, j " = j = " j s j f = s i i j = Sice these equatios must be valid for ay applied force o the body, we get: " j = ad j j= j= " = These equatios are solved for j ; j =,..., (costats. The ukow coefficiets are a fuctio of the positio of poit, ad ot a fuctio of the magitude or the directio of the applied force. age 5 f f OINTCOORDINATE FORMULATION. E. Nikravesh
Two rimary oits: " % '* & + ( (,. = * + Mass Distributio,. Three rimary oits: (i &* % " " " (, % ( + % (, '. *.,, / = ", + /,, 0, 0 Four rimary oits: (i ' + & &" " " " &, & %& (. * * * * / + / " 0 =, 0. Geeral: s s s s.... A body is defied by poits. We distribute the mass to the poits such that: m j j = = m i ( equatio m j = 0 ( equatios j = T m j = J i (6 equatios j = These are 0 equatios, therefore we ca have up to 0 ukows. Four rimary oits: Defie six secodary poits betwee each two primary poits: 6 9 7 0 8 5 0 m j j = = m i ( equatio s 5 = (s + s age 6 OINTCOORDINATE FORMULATION. E. Nikravesh
0 j = 0 m j = 0 ( equatios " m j = J i (6 equatios j = 0 equatio 0 ukow masses Three rimary oits: Defie three secodary poits. All 6 poits ad the mass ceter are i the same plae; j = 0; j =,, 6 6 equatio 6 ukow masses " Two rimary oits: Defie oe secodary poit. All poits ad the mass ceter are alog the same lie; j = " j = 0; j =,, 6 5 " equatio ukow masses Special Case I: Two primary poits have equal distaces from the mass ceter, therefore the secodary poit is positioed at the mass ceter. The legth betwee the two primary poits deoted as, the m = m = j i /, m = m i j i / Special Case II: For j i = m i / we get m = m = m i / 6, m = m i / age 7 OINTCOORDINATE FORMULATION. E. Nikravesh
Equatios of Motio ad Iertia Matrix Two rimary oit: ositio Costraits: Velocity Costraits: (r r T (r r (, = 0 (r r T ( r r = 0 r + r r = 0 r + r r = 0 Acceleratio Costraits: (r r T ( r r = ( r r T ( r r r + r r = 0 Equatios of Motio: m I 0 0 ' r + r. r I ' f + 0 m & I 0 &( r,. r. r &'/ + I &(, = f (, " 0 0 m I& %* r " 0.I&* / % * 0 We elimiate the secodary poit from the equatios by applyig the coordiate trasformatio process to obtai: ositio Costraits: Velocity Costraits: (r r T (r r (, = 0 (r r T ( r r = 0 Acceleratio Costraits: (r r T ( r r = ( r r T ( r r Equatios of Motio: (m + m I m I &' r * & m ( + I (m + m " I & r, r r ' &. = f * ( + " r r % f, %& Special Case II: m = m = m / 6 ad m = m / Mass matrix becomes: m I m 6 I & m 6 I m & " I & % Three rimary oits: 5 6 age 8 OINTCOORDINATE FORMULATION. E. Nikravesh
After elimiatig the secodary poits from the equatios of motio, we will have: ositio Costraits: Velocity Costraits: (r r T (r r (, (r r T ( r r = 0 ( = 0 ( = 0 ( = 0 (r r T (r r (, (r r T (r r (, Acceleratio Costraits: (r r T ( r r = ( r r T ( r r (r r T ( r r = ( r r T ( r r (r r T ( r r = 0 (r r T ( r r = 0 (r r T ( r r = ( r r T ( r r Differetial Equatios of Motio: m, I m, I m, I ' r + " r r 0 r r %( f m, I m, I m, & I& ( r,. r r r r ' + 0 ' ( + + *. = f + *. " m, I m, I m, I& %* r 0 r r r r ' + &,( + + /, f + / where, m, = m + m + m 6, m, = m + m + m 5, m, = m + m5 + m 6 m, = m, = m, m, = m, = m 6, m, = m, = m5 Four rimary oits: 6 5 There are six primary costraits. The differetial equatios of motio are: ( m, I m, I m, I m, I ' r + s, s, s, + " 0 0 0 % ( + + + f m, I m, I m, I m, & I& r s, 0 0 s, s, ' 0 ' + ( + f + + m, I m, I m, I m, I& (,. r & 0 s, 0 s, 0 s, '* (. = * ' + + f. + + " m, I m, I m, I m, I% & * r 0 0 s, 0 s, s, &' + ( 5 + + f + + +, /, + ( 6 / + where,, j = r i r j, m, = m + m5 + m 8 + m 0, m, = m + m 5 + m 6 + m 9, m, = m + m 6 + m 7 + m 0, m, = m + m7 + m 8 + m 9, m, = m, = m 5, m, = m, = m0 9 0 7 8 age 9 OINTCOORDINATE FORMULATION. E. Nikravesh
m, = m, = m 6 A Multibody System, m, = m, = m 8, m, = m, = m 9, m, = m, = m 7 Costructio of the iertia matrix ad the force vector is described through a example. Example: Two bodies are coected by a spherical joit; there are seve primary poits. The mass matrix ad the array of forces are: " m, I m, I m, I m, I % " ' m, I m, I m, I m, I ' m, I m, I m, I m, I ' ' M 7 = m, I m, I m, I m, I m,5 I m,6 I m,7 I', f 7 = m 5, I m 5,5 I m 5,6 I m 5,7 ' I ' m 6, I m 6,5 I m 6,6 I m 6,6 I' ' m 7, I m 7,5 I m 7,6 I m 7,7 I& ' % where: m, = m,, i + m j, f = f i + f j (i (j 5 7 6 f f f f f 5 f 6 f 7 & ' ( Geeral Form ositio Costraits: (r = 0 ( c Velocity Costraits: " Dr = 0 ( c Acceleratio Costraits: " D r + D r = 0 ( c Equatios of Motio: M r D T " = f ( v = p age 0 OINTCOORDINATE FORMULATION. E. Nikravesh