THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
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1 THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
2 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if the motion is one of rectiliner trnsltion or will move long congruent curves if the motion is one of curviliner trnsltion. r v A A A rb ra / B v B B r A / B remins constnt nd therefore its time derivtive is zero. Thus, ll points in the body hve the sme velocity nd the sme ccelertion.
3 2. FIXED-AXIS ROTATION Any point such s A which is not on the xis moves in circulr rc in plne norml to the xis nd hs velocity nd n ccelertion v r r r
4 3. ROTATION ABOUT A FIXED POINT When body rottes bout fixed point, the ngulr velocity vector no longer remins fixed in direction, nd this chnge clls for more generl concept of rottion. Rottion nd Proper Vectors Consider solid sphere which is cut from rigid body confined to rotte bout the fixed point O. The x-y-z xis here re ten s fixed in spce nd do not rotte with the body.
5 In prt () of the figure, two successive 90 o rottions of the sphere bout, first, the x-xis nd second, the y xis result in the motion of point which is initilly on the y-xis in position 1, to positions 2 nd 3, successively. On the other hnd, if the order of the rottions is reversed, the point undergoes no motion during the y-rottion but moves to point 3 during the 90 o rottion bout x-xis. Thus, the two cses do not produce the sme finl position, nd it is evident from this one specil exmple tht finite rottions do not generlly obey the prllelogrm lw of vector ddition nd re not commuttive. (+bb+)
6 Thus, finite rottions my not be treted s proper vectors. Infinitesiml rottions do obey the prllelogrm lw of vector ddition. Infinitesiml rottions re dq 1 nd dq 1 As result of dq 1 nd dq 2, point A hs displcement r d d r d r d q q q q Thus, two rottions re equivlent to the single rottion q 1 q 2 q d d d q q q
7 Instntneous Axis of Rottion Figure represents solid cylindricl rotor mde of cler plstic contining mny blc prticles embedded in plstics. The rotor is spinning bout its shft xis t the stedy rte 1, nd its shft is rotting bout the fixed verticl xis t the stedy rte 2. If the rotor is photogrphed t certin instnt during its motion, the picture would show one line of blc dots shrply defined, indicting tht, momentrily, their velocity ws zero. This line of points with no velocity estblishes the instntneous position of the xis of rottion O-n. Any dot on this line, such s A, would hve equl nd opposite velocity components, v 1 due to 1 nd v 2 due to 2. All other dots, such s P, would move in smll circulr rcs in plnes norml to the xis O-n. Thus, ll prticles of the body, except those on line O-n, re momentrily rotting in circulr rcs bout the instntneous xis of rottion.
8 Body nd Spce Cones The instntneous xis of rottion O-A-n genertes right-circulr cone bout the cylinder xis clled the body cone. As the two rottions continue nd the cylinder swings round the verticl xis, the instntneous xis of rottion lso genertes right-circulr cone bout the verticl xis clled the spce cone. The body cone rolls on the spce cone nd the ngulr velocity of the body is vector which lies long the common element of the two cones.
9 Angulr Accelertion The ngulr ccelertion of rigid body in three-dimensionl motion is the time derivtive of its ngulr velocity In contrst to the cse of rottion in single plne where the sclr mesures only the chnge in mgnitude of the ngulr velocity, in three-dimensionl motion the vector reflects the chnge in direction s well s its chnge in mgnitude. Thus in the figure where the tip of the ngulr velocity vector follows the spce curve p nd chnges in both mgnitude nd direction, the ngulr ccelertion becomes vector tngent to this curve in the direction of the chnge in.
10 When the mgnitude of remins constnt, the ngulr ccelertion is norml to. For this cse, if we let stnd for the ngulr velocity with the vector itself rottes (precesses) s it forms the spce cone, the ngulr ccelertion my be written The upper prt of the figure reltes the velocity of point A on rigid body to its position vector from O nd the ngulr velocity of the body. The vectors,, nd in the lower figure ber exctly the sme reltionship to ech other s do vectors v, r nd in the upper figure.
11 The one difference between the cse of rottion bout fixed xis nd rottion bout fixed point lies in the fct tht for rottion bout fixed point, the ngulr ccelertion v r r will hve component norml to due to the chnge in direction of. r Although ny point on the rottion xis n-n momentrily will hve zero velocity, it will not hve zero ccelertion s long s is chnging its direction.
12 Exmple The electric motor with n ttched dis is running t constnt low speed of 120 rev/min in the direction shown. Its housing nd mounting bse re initilly rest. The entire ssembly is next set in rottion bout the verticl Z-xis t the constnt rte N=60 rev/min with fixed ngle g of 30 o. The xes x-y-z with unit vectors K Determine () the ngulr velocity nd ngulr ccelertion of the dis, (b) the spce nd body cone, nd (c) the velocity nd ccelertion of point A t the top of the dis for the instnt shown. i, j, j cos g sing re ttched to the motor frme.
13 () the ngulr velocity nd ngulr ccelertion of the dis The rotor nd dis hve two components of ngulr velocity : o =120(2p)/60=4p rd/s bout the z-xis nd = 60(2p)/60=2p rd/s bout the Z-xis. j j K o o o 5 3 sin cos p g g The ngulr ccelertion of the dis i i j j o o 68.4 cos sin cos sin cos g g g g g
14 b) the spce nd body cone
15 (c) the velocity nd ccelertion of point A t the top of the dis r v j r O A / i v j j v p p j v r r r
16 Exmple The rotor nd shft re mounted in clevis which cn rotte bout the z-xis with n ngulr. With =0 nd q constnt, the rotor hs n ngulr velocity r 0.5i 1.2 j 1.1 o 4 j 3 rd / Find the velocity v A of point A on the rim if its position vector t the instnt is m s Wht is the rim speed v B of ny point B?
17 Exmple The pnel ssembly nd ttched x-y-z xes rotte with constnt ngulr velocity =0.6 rd/s bout the verticl z-xis. Simultneously, the pnels rotte bout the y-xis s shown with constnt o =2 rd/s. Determine the ngulr ccelertion of pnel A nd find the ccelertion of point P for the instnt when b=90 o.
18 Exmple In mnipulting the dumbbell, the jws of robotic device hve n ngulr velocity P =2 rd/s bout the xis OG with g fixed t 60 o. The entire ssembly rottes bout the verticl Z-xis t the constnt rte =0.8 rd/s. Determine the ngulr velocity nd ngulr ccelertion of the dumbbell.
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