KINEMTICS OF RIGI OIES
Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription of the motion of rigid bodies is importnt for to resons: 1) To generte, trnsmit or control motions by using cms, gers nd linkges of vrious types nd nlyze the displcement, velocity nd ccelertion of the motion to determine the design geometry of the mechnicl prts. Furthermore, s result of the motion generted, forces my be developed hich must be ccounted for in the design of the prts. ) To determine the motion of rigid body cused by the forces pplied to it. Clcultion of the motion of rocket under the influence of its thrust nd grvittionl ttrction is n exmple of such problem.
Rigid ody ssumption rigid body is system of prticles for hich the distnces beteen the prticles nd the ngle beteen the lines remin unchnged. Thus, if ech prticle of such body is locted by position vector from reference xes ttched to nd rotting ith the body, there ill be no chnge in ny position vector s mesured from these xes. Of course this is n ideliztion since ll solid mterils chnge shpe to some extent hen forces re pplied to them.
Nevertheless, if the movements ssocited ith the chnges in shpe re very smll compred ith the movements of the body s hole, then the ssumption of rigidity is usully cceptble. For exmple, the displcements due to the flutter of n ircrft ing do not ffect the description of the ircrft s hole nd thus the rigid body ssumption is cceptble. On the other hnd, if the problem is one of describing, s function of time, the internl ing stress due to ing flutter, then the reltive motions of portions of the ing cnnot be neglected, nd the ing my not be considered s rigid body.
Plne Motion rigid body executes plne motion hen ll prts of the body move in prllel plnes. The plne of motion is considered, for convenience, to be the plne hich contins the mss center, nd e tret the body s thin slb hose motion is confined to the plne of the slb. This ideliztion dequtely describes very lrge ctegory of rigid body motions encountered in engineering. The plne motion of rigid body is divided into severl ctegories:
Trnsltion It is ny motion in hich every line in the body remins prllel to its originl position t ll times. In trnsltion, there is no rottion of ny line in the body. In rectiliner trnsltion, ll points in the body move in prllel stright lines. Rocket test sled
In curviliner trnsltion, ll points move on congruent curves. In ech of the to cses of trnsltion, the motion of the body is completely specified by the motion of ny point in the body, since ll the points hve the sme motion.
Fixed xis Rottion Rottion bout fixed xis is the ngulr motion bout the xis. ll prticles in rigid body move in circulr pths bout the xis of rottion nd ll lines in the body hich re perpendiculr to the xis of rottion rotte through the sme ngle t the sme time.
Generl Plne Motion It is the combintion of trnsltion nd rottion.
We should note tht in ech of the exmples cited, the ctul pths of ll prticles in the body re projected onto the single plne of motion. nlysis of the plne motion of rigid bodies is ccomplished either by directly clculting the bsolute displcements nd their time derivtives from the geometry involved or by utilizing the principles of reltive motion.
Rottion The rottion of rigid body is described by its ngulr motion. The figure shos rigid body hich is rotting s it undergoes plne motion in the plne of the figure. The ngulr positions of ny to lines 1 nd ttched to the body re specified by q 1 nd q mesured from ny convenient fixed reference direction. ecuse the ngle b is invrint, the reltion q = q 1 + b upon differentition ith respect to time gives q q 1 nd q q 1 or, during finite intervl, q = q 1. Thus, ll lines on rigid body in its plne of motion hve the sme ngulr displcement, the sme ngulr velocity nd the sme ngulr ccelertion.
The ngulr motion of line depends only on its ngulr position ith respect to ny rbitrry fixed reference nd on the time derivtives of the displcement. ngulr motion does not require the presence of fixed xis, norml to the plne of motion, bout hich the line nd the body rotte.
ngulr Motion Reltions The ngulr velocity nd ngulr ccelertion of rigid body in plne rottion re, respectively, the first nd second time derivtives of the ngulr position coordinte q of ny line in the plne of motion of the body. These definitions give ωdω dq q dt d dt α dq or or d q q dt q d q q dθ In ech of these reltions, the positive direction for nd, clockise or counterclockise, is the sme s tht chosen for q.
For rottion ith constnt ngulr ccelertion, the reltionships become 1 t t t q q q q Here q nd re the vlues of the ngulr position coordinte nd ngulr velocity t time t = nd t is the durtion of the motion considered. s seen, the reltionships given for the rotry motion of rigid bodies re nlogous to those derived for the prticle.
Rottion bout Fixed xis When rigid body rottes bout fixed xis, ll points other thn those on the xis move in concentric circles bout the fixed xis. Thus, for the rigid body in the figure rotting bout fixed xis norml to the plne of the figure through O, ny point such s moves in circle of rdius r. So the velocity nd the ccelertion of point cn be ritten s v r n t r r v / r v
These quntities my be expressed lterntively using the cross product reltionship of vector nottion. The vector formultion is especilly importnt in the nlysis of three dimensionl motion. The ngulr velocity of the rotting body my be expressed by the vector norml to the plne of rottion nd hving sense governed by the right hnd rule. From the definition of the vector cross product, the vector v is obtined by crossing into r. This cross product gives the correct mgnitude nd direction for. v r r The order of the vectors to be crossed must be retined. The reverse order gives r v v
The ccelertion of point is obtined by differentiting the cross product expression for, hich gives v r v r r r r v Here stnds for the ngulr ccelertion of the body. Thus, e cn rite r r r v t n
For three dimensionl motion of rigid body, the ngulr velocity vector my chnge direction s ell s mgnitude, nd in this cse, the ngulr ccelertion, hich is the time derivtive of ngulr velocity,, ill no longer be in the sme direction s.
PROLEMS 1. The ngulr velocity of ger is controlled ccording to = 1 3t, here in rd/s is positive in the clockise sense nd here t is the time in seconds. Find the net ngulr displcement q from the time t = to t = 3 s. lso find the totl number of revolutions N through hich the ger turns during the three seconds. d dq 6t dq dt dt dt q dq 3 3 3 1 3t dt, q 1t t 13 q 9 rd 3 3 3 3 9 rd
PROLEMS oes the ger stop beteen t = nd t = 3 seconds? 1 3t q 1 q 16 dq dq 7 3 1 3t dt, q 1t t 1 3 1 3t 3 rd 1revolution dt N revolutions 1 3t 3, rd t rd, 1 s ( it q 1t stops N 3 3 3 3 t 3 3 t t 7 s) rd 3 16 3.66 revolutions rd
PROLEMS. Lod is connected to double pulley by one of the to inextensible cbles shon. The motion of the pulley is controlled by cble C, hich hs constnt ccelertion of 9 cm/s nd n initil velocity of 1 cm/s, both directed to the right. etermine, ) The number of revolutions executed by the inner pulley for t = seconds. b) The velocity nd chnge in position of the lod fter seconds. c) The ccelertion of point on the rim of the inner pulley t t =. 3 cm 5 cm
) The number of revolutions executed by the inner pulley for t = seconds. v r v C O O t v 3 r r o cm o O t 1 cm / s 1 q qo ot t Number of revolutions o v r r O t t x 1 3 9 3 9 rev cm / s.4.3 1.4 rd / s rd / s.4.3() 1rd / s rd, ( c) 1 q.4().3() 1 rev rd x 1.4 rd ( c) 1.4 3 cm.3 5 cm rev
b) The velocity nd chnge in position of the lod fter seconds. v y r r 1 (5) q ( 5 cm / s )( 14. ) 7 cm ( uprds) c) The ccelertion of point on the rim of the inner pulley t t =. t t t n 9 cm / s r o.4 rd (3)(.4) 1/ 1. cm / s n o t / s 4.8 cm / s 3 cm 5 cm
PROLEMS 3. The motor shon is used to turn heel by the pulley ttched to it. If the pulley strts rotting from rest ith n ngulr ccelertion of = rd/s, determine the mgnitudes of the velocity nd ccelertion of point P on the heel, fter the heel hs turned 1 revolutions. ssume the trnsmission belt does not slip on the pulley nd the heel. q 6.83 rd s q r q r q.15 6.83(.4) q 167.6 rd O () 167.6 18.31 q 1 qo 1 rev rd rd / s C v v C P r r r 6.87(.4).75 m / s 18. 31( 15. ) (.4) 6.87 rd / s C t P n r.3.4 r.75 r rd / s.4(6.87) r 18.88 C t m / s Pt.3 m / s s q r s q r P P n Pt 18.9 m / s
bsolute Motion In the first pproch in rigid body kinemtics, the bsolute motion nlysis, e mke use of the geometric reltions hich define the configurtion of the body involved nd then proceed to tke the time derivtives of the defining geometric reltions to obtin velocities nd ccelertions. The constrined motion of connected prticles is lso n bsolute motion nlysis. For the pulley configurtions, the relevnt velocities nd ccelertions ere determined by successive differentition of the lengths of the connecting cbles. In rigid body motion, the defining geometric reltions include both liner nd ngulr vribles nd, therefore, the time derivtives of these quntities ill involve both liner nd ngulr velocities nd liner nd ngulr ccelertions.
PROLEM heel of rdius r rolls on flt surfce ithout slipping. etermine the ngulr motion of the heel in terms of the liner motion of its center O. lso determine the ccelertion of point on the rim of the heel s the point comes into contct ith the surfce on hich the heel rolls.
The heel rolls to the right from the dshed to the full position ithout slipping. The liner displcement of the center O is s, hich is lso the rc length C' long the rim on hich the heel rolls. The rdil line CO rottes to the ne position C'O' through the ngle q, here q is mesured from the verticl direction. If the heel does not slip, the rc C' must equl the distnce s, since in rigid body ll points ill hve the sme displcement. Thus, the displcement reltionship nd its to time derivtives give s s s rq vo r q r r q r o (q is in rdins)
If the heel is sloing don, ill be directed opposite to v nd nd ill hve opposite directions. When point C hs moved long its cycloidl pth to C, its ne coordintes nd their time derivtives become x s r sin q rq r sin q r x r q r q cosq r r cosq v x r q r q cosq r q sin q r r cosq r sin q 1 cosq r sin q q sin q y r r cosq r1 cosq 1 cosq v v y r q sin q v rv sin q y r q cosq r q sin q r cosq r sin q r cosq sin q
For the desired instnt of contct, q = nd x y r Thus the ccelertion of point C on the rim t the instnt of contct ith the ground depends only on r nd nd is directed tord the center of the heel. If desired, the velocity nd ccelertion of C t ny position q my be obtined by riting the expressions v xi yj nd xi yj. If the heel slips s it rolls, the foregoing reltions ill no longer be vlid.
Reltive Velocity The second pproch to rigid body kinemtics uses the principles of reltive motion. In kinemtics of prticles for motion reltive to trnslting xes, e pplied the reltive velocity eqution v v v/ to the motions of to prticles nd. We no choose to points on the sme rigid body for our to prticles. The consequence of this choice is tht the motion of one point s seen by n observer trnslting ith the other point must be circulr since the rdil distnce to the observed point from the reference point does not chnge. This observtion is the key to the successful understnding of lrge mjority of problems in the plne motion of rigid bodies.
The figure shos rigid body moving in the plne of the figure from position to '' during time t. This movement my be visulized s occurring in to prts. First, the body trnsltes to the prllel position ''' ith the displcement r. Second, the body rottes bout ' through the ngle q, from the nonrotting reference xes x'-y' ttched to the reference point ', giving rise to the displcement r / of ith respect to.
To the nonrotting observer ttched to, the body ppers to undergo fixed xis rottion bout ith executing circulr motion. Point is rbitrrily chosen s the reference point for ttchment of the nonrotting reference xes x-y. Point could hve been used just s ell, in hich cse e ould observe to hve circulr motion bout considered fixed. In this cse, the sense of the rottion, counterclockise direction, is the sme hether e choose or s the reference, nd e see tht = -. r / r /
With s the reference point, the totl displcement of is r r r / Where r / hs the mgnitude rq s q pproches zero. We note tht the reltive liner motion r / is ccompnied by the bsolute ngulr motion q, s seen from the trnslting xes x' -y'. ividing the expression for r by the corresponding time intervl t nd pssing to the limit, e obtin the reltive velocity eqution v v v/ We should note tht in this expression the distnce r beteen nd remins constnt.
The mgnitude of the reltive velocity is thus seen to be hich, ith becomes r v / Using to represent the vector, e my rite the reltive velocity s the vector t r t r v t t q / lim / lim / / q r v / r / r Therefore, the reltive velocity eqution becomes r v v
Here, is the ngulr velocity vector norml to the plne of the motion in the sense determined by the right hnd rule. It should be noted tht the direction of the reltive velocity ill lys be perpendiculr to the line joining the points nd. Interprettion of the Reltive Velocity Eqution We cn better understnd the reltive velocity eqution by visulizing the trnsltion nd rottion components seprtely. trnsltion Fixed xis rottion
trnsltion Fixed xis rottion In the figure, point is chosen s the reference point nd the velocity of is the vector sum of the trnsltionl portion v, plus the rottionl portion v r, hich hs the / mgnitude v / =r, here q, the bsolute ngulr velocity of. The fct tht the reltive liner velocity is lys perpendiculr to the line joining the to points nd is n importnt key to the solution of mny problems.
Solution of the Reltive Velocity Eqution Solution of the reltive velocity eqution my be crried out by sclr or vector lgebr or by grphic interprettion. In the sclr pproch, ech term in the reltive motion eqution my be ritten in terms i j of its nd components, from hich e ill obtin to sclr equtions.
Reltive ccelertion Consider the eqution v v v/ hich describes the reltive velocities of to points nd in plne motion in terms of nonrotting reference xes. y differentiting the eqution ith respect to time, e obtin the reltive ccelertion eqution, hich is / v v v/ or This eqution sttes tht the ccelertion of point equls the vector sum of the ccelertion of point nd the ccelertion hich ppers to hve to nonrotting observer moving ith.
Reltive ccelertion ue to Rottion If points nd re locted on the sme rigid body nd in the plne of motion, the distnce r beteen them remins constnt so tht the observer moving ith perceives to hve circulr motion bout. ecuse the reltive motion is circulr, it follos tht the reltive ccelertion term ill hve both norml component directed from tord due to the chnge of direction of v / nd tngentil component perpendiculr to due to the chnge in mgnitude of v /. Thus, e my rite, / n / t Where the mgnitudes of the reltive ccelertion components re / v/ / v r / n t / r r
In vector nottion the ccelertion components re r / r / n t In these reltionships, is the ngulr velocity nd is the ngulr ccelertion of the body. The vector locting from is r. It is importnt to observe tht the reltive ccelertion terms depend on the respective bsolute ngulr velocity nd the bsolute ngulr ccelertion. The reltive ccelertion eqution, thus, becomes r r
Interprettion of the Reltive ccelertion Eqution The mening of reltive ccelertion eqution is indicted in the figure hich shos rigid body in plne motion ith points nd moving long seprte curved pths ith bsolute ccelertions nd. Contrry to the cse ith velocities, the ccelertions nd re, in generl, not tngent to the pths described by nd hen these pths re curviliner.
The figure shos the ccelertion of to be composed of to prts: the ccelertion of nd the ccelertion of ith respect to. sketch shoing the reference point s fixed is useful in disclosing the correct sense of the to components of the reltive ccelertion term.
lterntively, e my express the ccelertion of in terms of the ccelertion of, hich puts the nonrotting reference xes on rther thn. This order gives / / Here nd its n nd t components re the negtives of nd its n nd t components / ( ). / /
Solution of the Reltive ccelertion Eqution s in the cse of the reltive velocity eqution, the reltive ccelertion eqution my be crried out by sclr or vector lgebr or by grphicl construction. ecuse the norml ccelertion components depend on velocities, it is generlly necessry to solve for the velocities before the ccelertion clcultions cn be mde.