Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.


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1 Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal to and n the same drecton o the orce actng on t Thrd Law: The orces o acton and reacton between nteractng artcles are equal n magntude, collnear, and ooste n drecton Dynamcs o A Partcle A artcle (ont) Mass m ( ) Force vector Acceleraton vector r or a Equaton o moton m ( ) r = Partcle Equaton o moton m r = In exanded orm m r = m 0 0 x 0 m 0 y = 0 0 m Dynamcs o A System o Partcles A system o artcles Center o mass (centrod) s ostoned wth vector r Equaton o moton or the centrod: mr = where m m and = Follow the dervaton o the equatons o moton or the mass center o a system o artcles n Sec Observe how the nternal reacton orces between the artcles cancel each other! Poston o the center o mass 1 Resultant equaton r 1 m m r m s = 0 x x ( x) ( y) ( ) r y y m ( ) r s r a j
2 Note that r locates artcle rom the orgn o the reerence rame where s locates the artcle rom the mass center! Translatonal Equatons o Moton or A Body Translatonal equatons o moton (centrodal) or a body are m r = In exanded orm m 0 0 x (x) 0 m 0 y = 0 0 m (y) () Mass o the body: m Sum o orces actng on the body: Acceleraton o the mass center: r r O mass center C Moment o A Force Moment o a orce actng on a body at ont P s comuted as n = s P where s P locates ont P rom the mass center o the body C s P P Rotatonal Equatons o Moton or A Body The dervaton shown here s much smler that the one gven n the textbook! Assume that the body s made o nnte () number o artcles For artcle the equaton o moton s m r = +, j where, s an external orce that may exst or some o the artcles, j ; j = 1,..., are the reacton orces exerted on ths artcle by other artcles Premultly ths equaton by s : m s r = s + s, j m s (r +s ) = s + s, j x 1 r 1 y j s,j j, j We wrte ths equaton or every artcle and then sum over all the artcles:
3 m s r + m s s = s + s, j (a) (b) We examne each o the our terms searately! (a) In ths term r can be moved outsde sgma. What remans wthn sgma, accordng to m s = 0, s equal to ero! (c) (m s r) = (m s ) r = 0 (b) From Lesson 10 we have s = s, s = s + s where s the angular velocty o the body. Substtutng n (b) and reerrng to Problem 2.15 yelds ( ) m s s m s (s + s ) m s s + m s s m s s + m s s m s s + m s s We now take and out o sgma n each term m s s + m s s (c) Ths term reresents the sum o all moments actng on the body n = (d) We can show that ths term s exactly equal to ero! I we exand the terms wthn the sgmas, we can ar every two terms as shown: s, j s + s, j + s j j, ++ s, j s j, j + + (s s j ), j + Accordng to the gure the vector s, j = s s j s arallel to the reacton orce, j. Thereore, the roduct (s s j ), j s ero! Now the equaton o moton has been smled to s j j, m s s + m s s = n j We relace each sgma by ntegral (nnte number o artcles) and then ntroduce the nerta matrx as J m s s = s s dm vol. The rotatonal equaton o moton becomes J + J = n The nerta matrx J s obtaned wth resect to a reerence rame attached to the mass center o the body and remanng arallel to the nonmovng xy rame. Thereore (d ) s s, j,j
4 the comonents o J vary wth changng orentaton o the body J ' The nerta matrx J ' s dened as J ' = A T JA The comonents o ths matrx are constants they are obtaned wth resect to a rame that s attached to and rotates wth the body The rotatonal equaton o moton can also be exressed as J ' '+ ' J ' ' = n' Equatons o Moton or A Rgd Body NewtonEuler equatons Both the translatonal and rotatonal equatons are descrbed n the xy comonents m r = J + J = n mi 0 r 0 + = 0 J J n mi 0 r = (a) 0 J n J The translatonal (Newton) equatons are descrbed n the xy comonents but the rotatonal (Euler) equatons are descrbed n the comonents m r = J ' '+ ' J ' ' = n' mi 0 r 0 + = 0 J' ' ' J ' ' n' mi 0 r = (b) 0 J' ' n' ' J ' ' Reer to the textbook or urther dscusson on these equatons! When Euler equatons are descrbed n terms o ther local comonents;.e., Eq. (b), the nerta matrx J ' remans a constant. Ths could be a bg advantage over Eq. (a) where the nerta matrx J needs to be reevaluated every tme the rotatonal orentaton o the body changes! We can derve the multbody equatons o moton based on Eq. (a) or Eq. (b). I we derve the equatons n one orm, t takes a smle transormaton to obtan the equatons n the other orm! The rotatonal equatons o moton can also be derved n terms o the second tme dervatve o Euler arameters (ths s done n the textbook we wll not use ths orm n ths course!)
5 In ths course, we derve and use the equatons o moton n the orm o (a). Ths wll be consstent wth the knematc constrants rom Lesson 11. These equatons or body are exressed n comact orm as M v = g where, M = m I 0 0 J, v = r, g = n J Note that M and g n your textbook (Eqs. 8.36, 8.38, and 8.39) are dened derently! A System o Unconstraned Bodes In an unconstraned multbody system there are no knematc jonts; hence there are no knematc constrants. The srngs, damers, actuators are orce elements t s assumed that a orce element does not mose any constrants on a system. In the next lesson we wll learn how to construct the array o orces or several commonly used orce elements Equatons o moton are derved by constructng the NewtonEuler equatons or every body n the system. Assume that there are b bodes n the system: M v = g where M 1 M M = 2 M b v 1 g 1 v, v = 2 g, g = 2 v b g b A System o Constraned Bodes In a constraned multbody system one or more knematc jonts are resent; hence knematc constrants must be ncororated nto the equatons o moton. For a system o b bodes the equatons o moton are exressed as M v = g + g (c) where g (c) reresents the array o reacton orces, and g contans the srng, damer,... orces and moments. Other elements n the above equaton are the same as n the unconstraned equatons o moton How do we determne the reacton orces? Assume that the constrants and ther rst and second tme dervatves are reresented as
6 (q) = 0 Dv= 0 D v + Dv= 0 The Jacoban matrx, D, s used n determnng the array o reacton orces as g (c) = D T where contans as many coecents as the number o constrants. These coecents are called Lagrange multlers At ths ont these multlers are unknowns! In the ucomng lessons we wll learn how to determne these multlers. The equatons o moton can be wrtten as M v D T = g In Secton o the textbook you nd urther dscusson on the array o reacton orces In the ucomng lessons we wll look at the reacton orces and moments or some secc knematc jonts.
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