KINEMTICS OF RIGID ODIES
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.
Description 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.
Plne Motion ll prts of the body move in prllel plnes. The plne motion of rigid body is divided into severl ctegories: 1. Trnsltion. Rottion 3. Generl Motion
1. 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. 1. Rectiliner Trnsltion: ll points in the body move in prllel stright lines. Rocket test sled
. 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. C C
3. Generl Plne Motion It is the combintion of trnsltion nd rottion.
Crnk (Krnk) (Rottion) Piston (Trnsltion) O Connecting rod (Generl Motion) hinge
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 during finite intervl, D q 1 = D q 1. ll lines on rigid body in its plne of motion hve the sme ngulr displcement, the sme ngulr velocity nd the sme ngulr ccelertion.
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 dq q dt d dt ωdω αdθ or or d q q dt qd q qdθ
For rottion ith constnt ngulr ccelertion, the reltionships become q q 0 t 0 0 t 0 q q 1 t 0
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 using the cross product reltionship of vector nottion, r r v k k, n t r v r r dt dr r dt d r dt d v dt d
PROLEMS 1. The ngulr velocity of ger is controlled ccording to = 1 3t, here in rds nd t is the time in seconds. Find the net ngulr displcement Dq from the time t = 0 to t = 3 s. lso find the totl number of revolutions N through hich the ger turns during the three seconds. SOLUTION dq dq dt dt q 0 dq 3 0 3 1 3t dt, q 1t t 13 Dq 9 rd 3 3 3 0 3 3 9 rd
SOLUTION Does the ger stop beteen t = 0 nd t = 3 seconds? 1 3t 0 1 3t t s ( it stops t t s) q 0 0 1 q dq dq 16 7 3 rd 0 3 1 3t dt q 1t t 1 3 1 3t dt q 1 1t 3 3 3 t 3 0 3 3 7 rd 3 16 rd 1revolution rd N revolutions 3 rd N 3.66 revolutions
PROLEMS. The belt-driven pulley nd ttched disk re rotting ith incresing ngulr velocity. t certin instnt the speed v of the belt is 1.5 ms, nd the totl ccelertion of point is 75 ms. For this instnt determine () the ngulr ccelertion of the pulley nd disk, (b) the totl ccelertion of point, nd (c) the ccelertion of point C on the belt.
C.5 0.075 300 37.5 30.5 30 0.075 0.5 0.075 300 300 0.15 45 45 60 75 60 0.15 0 0 0.075 1.5? c)? b)? ) 75 1.5 s m r s m s m r s m r s rd R s m s m R s rd r v s m s m v C C C n t t t n SOLUTION
PROLEMS 3. The design chrcteristics of ger-reduction unit re under revie. Ger is rotting clockise (c) ith speed of 300 revmin hen torque is pplied to ger t time t= s to give ger counterclockise (cc) ccelertion hich vries ith time for durtion of 4 seconds s shon. Determine the speed N of ger hen t=6 s.
SOLUTION s rd rev N s t 10 60 300 min 300 The velocities of gers nd re sme t the contct point. min 414.59 43.415 6 ) 6 ( 86.83 0 0 6 6 0 rev N s rd b b s t s t t s rd t t dt t d dt d t s rd b b v v t
bsolute Motion In this pproch, 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.
PROLEM 1) heel of rdius r rolls on flt surfce ithout slipping. Determine 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.
PROLEM ) Motion of the equilterl tringulr plte C in its plne is controlled by the hydrulic cylinder D. If the piston rod in the cylinder is moving uprd t the constnt rte of 0.3 ms during n intervl of its motion, clculte for the instnt hen q=30 o the velocity nd ccelertion of the center of the roller in the horizontl guide nd the ngulr velocity nd ngulr ccelertion of edge C.
PROLEM 3) Derive n expression for the uprd velocity v of the cr hoist in terms of q. The piston rod of the hydrulic cylinder is extending t the rte. s
PROLEM 4) Clculte the ngulr velocity of the slender br s fuction of the distnce x nd the constnt ngulr velocity o of the drum.
Reltive Motion chnge. 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
The figure shos rigid body moving in the plne of the figure from position to during time Dt. This movement my be visulized s occurring in to prts. First, the body trnsltes to the prllel position ith the displcement. Second, the body rottes bout through the ngle Dq, from the nonrotting reference xes x -y ttched to the reference point, giving rise to the displcement D of ith respect to. r Dr D r
With s the reference point, the totl displcement of is Where D r D r Dr Dr hs the mgnitude rdq s Dq pproches zero. Dividing the time intervl Dt nd pssing to the limit, e obtin the reltive velocity eqution v v v 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 dt d r t r t r v t t q q D D D D D D 0 0 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 xid rottion
In the figure, point is chosen s the reference point nd the velocity of is the vector sum of the trnsltionl portion the rottionl portion q v =r, here v r, plus, hich hs the mgnitude, the bsolute ngulr velocity of. The reltive liner velocity is lys perpendiculr to the line joining the to points nd. v
Reltive ccelertion Eqution of reltive velocity is v v v y differentiting the eqution ith respect to time, e obtin the reltive ccelertion eqution, hich is or v v v 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.
If points nd re locted on the sme rigid body, the distnce r beteen them remins constnt. ecuse the reltive motion is circulr, 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. Thus, e my rite, v n t Where the mgnitudes of the reltive ccelertion components re v r v r n t r
In vector nottion the ccelertion components re r r t n The reltive ccelertion eqution, thus, becomes r r
The figure shos the ccelertion of to be composed of to prts: the ccelertion of nd the ccelertion of ith respect to.
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.
PROLEMS 1. The center O of the disk hs the velocity nd ccelertion shon. If the disk rolls ithout slipping on the horizontl surfce, determine the velocity of nd the ccelertion of for the instnt represented.
PROLEMS. If the velocity of point is 3 ms to the right nd is constnt for n intervl including the position shon, determine the tngentil ccelertion of point long its pth nd the ngulr ccelertion of the br.
PROLEMS 3. The flexible bnd F ttched to the sector t E is given constnt velocity of 4 ms s shon. For the instnt hen D is perpendiculr to O, determine the ngulr ccelertion of D.
PROLEMS 4. mechnism for pushing smll boxes from n ssembly line onto conveyor belt is shon ith rm OD nd crnk C in their verticl positions. For the configurtion shon, crnk C hs constnt clockise ngulr velocity of rds. Determine the ccelertion of E.
PROLEMS 5. t given instnt, the ger hs the ngulr motion shon. Determine the ccelertion of points nd on the link nd the link s ngulr ccelrtion t this instnt.
PROLEMS 6. The center O of the disk rolling ithout slipping on the horizontl surfce hs the velocity nd ccelertion shon. Rdius of the disk is 4.5 cm. Clculte the velocity nd ccelertion of point. v o =45 cms o =90 cms 37 o O 4 cm 4.5 cm y 6 cm 1 y x 4 x x= cm 10 cm