DYNAMICS. Kinematics of Rigid Bodies VECTOR MECHANICS FOR ENGINEERS: Tenth Edition CHAPTER

Tenth E CHTER 15 VECTOR MECHNICS FOR ENGINEERS: YNMICS Ferdinnd. eer E. Russell Johnston, Jr. hillip J. Cornwell Lecture Notes: rin. Self Cliforni olytechnic Stte Uniersity Kinemtics of Rigid odies 013 The McGrw-Hill Compnies, Inc. ll rights resered.

Contents Introduction Trnsltion Rottion bout Fixed xis: Velocity Rottion bout Fixed xis: ccelertion Rottion bout Fixed xis: Representtie Slb Equtions efining the Rottion of Rigid ody bout Fixed xis Smple roblem 5.1 Generl lne Motion bsolute nd Reltie Velocity in lne Motion Smple roblem 15. Smple roblem 15.3 Instntneous Center of Rottion in lne Motion Smple roblem 15.4 Smple roblem 15.5 013 The McGrw-Hill Compnies, Inc. ll rights resered. bsolute nd Reltie ccelertion in lne Motion nlysis of lne Motion in Terms of rmeter Smple roblem 15.6 Smple roblem 15.7 Smple roblem 15.8 Rte of Chnge With Respect to Rotting Frme Coriolis ccelertion Smple roblem 15.9 Smple roblem 15.10 Motion bout Fixed oint Generl Motion Smple roblem 15.11 Three imensionl Motion. Coriolis ccelertion Frme of Reference in Generl Motion Smple roblem 15.15 15 -

pplictions bttering rm is n exmple of curiliner trnsltion the rm stys horizontl s it swings through its motion. 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 3

pplictions How cn we determine the elocity of the tip of turbine blde? 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-4

pplictions lnetry ger systems re used to get high reduction rtios with minimum weight nd spce. How cn we design the correct ger rtios? 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 5

pplictions iomedicl engineers must determine the elocities nd ccelertions of the leg in order to design prostheses. 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 6

Introduction Kinemtics of rigid bodies: reltions between time nd the positions, elocities, nd ccelertions of the prticles forming rigid body. Clssifiction of rigid body motions: - trnsltion: rectiliner trnsltion curiliner trnsltion - rottion bout fixed xis - generl plne motion - motion bout fixed point - generl motion 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-7

Trnsltion 013 The McGrw-Hill Compnies, Inc. ll rights resered. Consider rigid body in trnsltion: - direction of ny stright line inside the body is constnt, - ll prticles forming the body moe in prllel lines. For ny two prticles in the body, r r r ifferentiting with respect to time, r r r r ll prticles he the sme elocity. ifferentiting r r with r respect to time gin, r ll prticles he the sme ccelertion. 15-8

Rottion bout Fixed xis. Velocity Consider rottion of rigid body bout fixed xis Velocity ector dr dt of the prticle is tngent to the pth with mgnitude ds dt s r sin ds dt lim t0 t r sin r sin The sme result is obtined from dr r dt k k ngulr elocity 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-9

Concept Quiz Wht is the direction of the elocity of point on the turbine blde? ) b) c) d) y x L r kˆ Li 013 The McGrw-Hill Compnies, Inc. ll rights resered. L ˆj ˆ 15-10

013 The McGrw-Hill Compnies, Inc. ll rights resered. enth Rottion bout Fixed xis. ccelertion 15-11 ifferentiting to determine the ccelertion, r dt d dt dr r dt d r dt d dt d k k k ngulr ccelertion dt d rdil ccelertion component ccelertion component tngenti l r r r r ccelertion of is combintion of two ectors,

013 The McGrw-Hill Compnies, Inc. ll rights resered. enth Rottion bout Fixed xis. Representtie Slb 15-1 Consider the motion of representtie slb in plne perpendiculr to the xis of rottion. Velocity of ny point of the slb, r r k r ccelertion of ny point of the slb, r r k r r Resoling the ccelertion into tngentil nd norml components, r r r r k n n t t

Concept Quiz Wht is the direction of the norml ccelertion of point on the turbine blde? ) b) c) d) n n 013 The McGrw-Hill Compnies, Inc. ll rights resered. r ˆ ( Li ) ˆ L i n y x L 15-13

Equtions efining the Rottion of Rigid ody bout Fixed xis Motion of rigid body rotting round fixed xis is often specified by the type of ngulr ccelertion. Recll d dt d dt or dt d d dt d d Uniform Rottion, = 0: 0 t Uniformly ccelerted Rottion, = constnt: t 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-14 0 0 t 0 0 1 t 0

Smple roblem 5.1 SOLUTION: ue to the ction of the cble, the tngentil elocity nd ccelertion of re equl to the elocity nd ccelertion of C. Clculte the initil ngulr elocity nd ccelertion. Cble C hs constnt ccelertion of 9 in/s nd n initil elocity of 1 in/s, both directed to the right. etermine () the number of reolutions of the pulley in s, (b) the elocity nd chnge in position of the lod fter s, nd (c) the ccelertion of the point on the rim of the inner pulley t t = 0. 013 The McGrw-Hill Compnies, Inc. ll rights resered. pply the reltions for uniformly ccelerted rottion to determine the elocity nd ngulr position of the pulley fter s. Elute the initil tngentil nd norml ccelertion components of. 15-15

Smple roblem 5.1 013 The McGrw-Hill Compnies, Inc. ll rights resered. SOLUTION: The tngentil elocity nd ccelertion of re equl to the elocity nd ccelertion of C. 0 0 1in. s C t C 9in. s r t r 0 0 0 r 0 1 3 4rd s 9 r t 3rd s 3 pply the reltions for uniformly ccelerted rottion to determine elocity nd ngulr position of pulley fter s. t 4rd s t N y 3rd s s 10rd s 0 0 14 rd 1 rd t 4rd s s 1 3rd s s 1re 14 number of rd r r 5 in. 10rd s 5 in. 14 rd res N.3re y 50in. s 70 in. 15-16

Smple roblem 5.1 Elute the initil tngentil nd norml ccelertion components of. 9in. s t C r 3 in. 4rd s 48in n 0 s in. s 48in. s t 9 n Mgnitude nd direction of the totl ccelertion, t 9 48 n 48.8in. s tn 48 9 n t 79. 4 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-17

Group roblem Soling Elute the initil tngentil nd norml ccelertion components of. 9in. s t C r 3 in. 4rd s 48in n 0 s in. s 48in. s t 9 n Mgnitude nd direction of the totl ccelertion, t 9 48 n 48.8in. s tn 48 9 n t 79. 4 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-18

Group roblem Soling SOLUTION: Using the liner elocity nd ccelertions, clculte the ngulr elocity nd ccelertion. series of smll mchine components being moed by coneyor belt pss oer 6-in.-rdius idler pulley. t the instnt shown, the elocity of point is 15 in./s to the left nd its ccelertion is 9 in./s to the right. etermine () the ngulr elocity nd ngulr ccelertion of the idler pulley, (b) the totl ccelertion of the mchine component t. Using the ngulr elocity, determine the norml ccelertion. etermine the totl ccelertion using the tngentil nd norml ccelertion components of. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-19

Group roblem Soling Find the ngulr elocity of the idler pulley using the liner elocity t. r 15 in./s (6 in.) r 9 in./s (6 in.).50 rd/s Find the ngulr elocity of the idler pulley using the liner elocity t. 1.500 rd/s Find the norml ccelertion of point. n r (6 in.)(.5 rd/s) n 37.5 in./s = 15 in/s t = 9 in/s Wht is the direction of the norml ccelertion of point? ownwrds, towrds the center 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-0

Group roblem Soling t = 9 in/s Find the totl ccelertion of the mchine component t point. n = 37.5 in./s t 9.0 in./s n 37.5 in./s Clculte the mgnitude 9.0 37.5 38.6 in./s t = 9 in/s Clculte the ngle from the horizontl 37.5 rctn 76.5 9.0 Combine for finl nswer o n = 37.5 in/s 38.6 in./s 76.5 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-1

Golf Robot Not ours mybe Tom Mse hs pic? golf robot is used to test new equipment. If the ngulr elocity of the rm is doubled, wht hppens to the norml ccelertion of the club hed? If the rm is shortened to ¾ of its originl length, wht hppens to the tngentil ccelertion of the club hed?, If the speed of the club hed is constnt, does the club hed he ny liner ccelertion? 013 The McGrw-Hill Compnies, Inc. ll rights resered. -

Exmple Generl lne Motion The knee hs liner elocity nd ccelertion from both trnsltion (the runner moing forwrd) s well s rottion (the leg rotting bout the hip). 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 3

Generl lne Motion Generl plne motion is neither trnsltion nor rottion. Generl plne motion cn be considered s the sum of trnsltion nd rottion. isplcement of prticles nd to nd cn be diided into two prts: - trnsltion to nd 1 - rottion of bout to 1 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-4

013 The McGrw-Hill Compnies, Inc. ll rights resered. enth bsolute nd Reltie Velocity in lne Motion 15-5 ny plne motion cn be replced by trnsltion of n rbitrry reference point nd simultneous rottion bout. r r k r k

bsolute nd Reltie Velocity in lne Motion ssuming tht the elocity of end is known, wish to determine the elocity of end nd the ngulr elocity in terms of, l, nd. The direction of nd / re known. Complete the elocity digrm. tn tn l l cos cos 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-6

bsolute nd Reltie Velocity in lne Motion Selecting point s the reference point nd soling for the elocity of end nd the ngulr elocity leds to n equilent elocity tringle. / hs the sme mgnitude but opposite sense of /. The sense of the reltie elocity is dependent on the choice of reference point. ngulr elocity of the rod in its rottion bout is the sme s its rottion bout. ngulr elocity is not dependent on the choice of reference point. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-7

Smple roblem 15. SOLUTION: The displcement of the ger center in one reolution is equl to the outer circumference. Relte the trnsltionl nd ngulr displcements. ifferentite to relte the trnsltionl nd ngulr elocities. The double ger rolls on the sttionry lower rck: the elocity of its center is 1. m/s. etermine () the ngulr elocity of the ger, nd (b) the elocities of the upper rck R nd point of the ger. The elocity for ny point on the ger my be written s Elute the elocities of points nd. k r 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-30

Smple roblem 15. SOLUTION: The displcement of the ger center in one reolution is equl to the outer circumference. For x > 0 (moes to right), < 0 (rottes clockwise). x r x r 1 y x ifferentite to relte the trnsltionl nd ngulr elocities. r 1 r 1 1.m s 0.150m k 8rd sk 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-31

Smple roblem 15. For ny point on the ger, k r Velocity of the upper rck is equl to elocity of point : R k r 1.m s i 8rd s k 0.10 m 1.m si 0.8m si m si R 013 The McGrw-Hill Compnies, Inc. ll rights resered. j Velocity of the point : k r 1.m si 8rd sk 0.150 mi 1.m si 1.m s 1.697m s j 15-3

Smple roblem 15.3 SOLUTION: Will determine the bsolute elocity of point with The elocity is obtined from the gien crnk rottion dt. The crnk hs constnt clockwise ngulr elocity of 000 rpm. For the crnk position indicted, determine () the ngulr elocity of the connecting rod, nd (b) the elocity of the piston. The directions of the bsolute elocity nd the reltie elocity re determined from the problem geometry. The unknowns in the ector expression re the elocity mgnitudes nd which my be determined from the corresponding ector tringle. 013 The McGrw-Hill Compnies, Inc. ll rights resered. The ngulr elocity of the connecting rod is clculted from. 15-33

Smple roblem 15.3 SOLUTION: Will determine the bsolute elocity of point with The elocity 000 is obtined from the crnk rottion dt. re min min 60s rd 09.4 rd re 3in. 09.4 rd s The elocity direction is s shown. The direction of the bsolute elocity is horizontl. The direction of the reltie elocity is perpendiculr to. Compute the ngle between the horizontl nd the connecting rod from the lw of sines. sin 40 8in. sin 3in. 13.95 s 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-34

Smple roblem 15.3 etermine the elocity mgnitudes from the ector tringle. 68.3in. s sin 53.95 sin 50 sin76.05 nd 53.4in. s 43.6ft 495.9in. s s 43.6ft s l 495.9in. l 8 in. 6.0 rd s s 6.0 rd sk 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-35

Group roblem Soling In the position shown, br hs n ngulr elocity of 4 rd/s clockwise. etermine the ngulr elocity of brs nd E. Which of the following is true? ) The direction of is b) The direction of is c) oth ) nd b) re correct 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 36

Group roblem Soling In the position shown, br hs n ngulr elocity of 4 rd/s clockwise. etermine the ngulr elocity of brs nd E. SOLUTION: The displcement of the ger center in one reolution is equl to the outer circumference. Relte the trnsltionl nd ngulr displcements. ifferentite to relte the trnsltionl nd ngulr elocities. The elocity for ny point on the ger my be written s Elute the elocities of points nd. k r 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 37

Group roblem Soling y x etermine the ngulr elocity of brs nd E. How should you proceed? = 4 rd/s etermine with respect to, then work your wy long the linkge to point E. Write in terms of point, clculte. r / (4 rd/s) k r (7 in.) i r ( 4 k) ( 7 i) / / (8 in./s) j oes it mke sense tht is in the +j direction? 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 38

, Group roblem Soling y x = 4 rd/s etermine with respect to. / k r (8 in.) j r 8 j ( k) ( 8 j) / 8j 8 i etermine with respect to E, then equte it to eqution boe. k r (11 in.) i (3 in.) j E E / E r ( k ) ( 11i 3 j) E / E E 11 j 3 i E E Equting components of the two expressions for j: 8 11.5455 rd/s E E 3 i: 8 3 8 E 013 The McGrw-Hill Compnies, Inc. ll rights resered. E.55 rd/s 0.955 rd/s - 39

Instntneous Center of Rottion in lne Motion lne motion of ll prticles in slb cn lwys be replced by the trnsltion of n rbitrry point nd rottion bout with n ngulr elocity tht is independent of the choice of. The sme trnsltionl nd rottionl elocities t re obtined by llowing the slb to rotte with the sme ngulr elocity bout the point C on perpendiculr to the elocity t. The elocity of ll other prticles in the slb re the sme s originlly defined since the ngulr elocity nd trnsltionl elocity t re equilent. s fr s the elocities re concerned, the slb seems to rotte bout the instntneous center of rottion C. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-40

Instntneous Center of Rottion in lne Motion If the elocity t two points nd re known, the instntneous center of rottion lies t the intersection of the perpendiculrs to the elocity ectors through nd. If the elocity ectors re prllel, the instntneous center of rottion is t infinity nd the ngulr elocity is zero. If the elocity ectors t nd re perpendiculr to the line, the instntneous center of rottion lies t the intersection of the line with the line joining the extremities of the elocity ectors t nd. If the elocity mgnitudes re equl, the instntneous center of rottion is t infinity nd the ngulr elocity is zero. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-41

Instntneous Center of Rottion in lne Motion The instntneous center of rottion lies t the intersection of the perpendiculrs to the elocity ectors through nd. C l sin C l cos l cos tn The elocities of ll prticles on the rod re s if they were rotted bout C. The prticle t the center of rottion hs zero elocity. The prticle coinciding with the center of rottion chnges with time nd the ccelertion of the prticle t the instntneous center of rottion is not zero. The ccelertion of the prticles in the slb cnnot be determined s if the slb were simply rotting bout C. The trce of the locus of the center of rottion on the body is the body centrode nd in spce is the spce centrode. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-4

Instntneous Center of Rottion in lne Motion t the instnt shown, wht is the pproximte direction of the elocity of point G, the center of br? G ) b) c) d) 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-43

Smple roblem 15.4 SOLUTION: The point C is in contct with the sttionry lower rck nd, instntneously, hs zero elocity. It must be the loction of the instntneous center of rottion. etermine the ngulr elocity bout C bsed on the gien elocity t. The double ger rolls on the sttionry lower rck: the elocity of its center is 1. m/s. Elute the elocities t nd bsed on their rottion bout C. etermine () the ngulr elocity of the ger, nd (b) the elocities of the upper rck R nd point of the ger. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-44

Smple roblem 15.4 SOLUTION: The point C is in contct with the sttionry lower rck nd, instntneously, hs zero elocity. It must be the loction of the instntneous center of rottion. etermine the ngulr elocity bout C bsed on the gien elocity t. 1.m s r 8rd s r 0.15 m Elute the elocities t nd bsed on their rottion bout C. R r 0.5 m8rd s R m si 013 The McGrw-Hill Compnies, Inc. ll rights resered. r 0.15 m r 0.11 m 0.11 m8rd s 1.697m s 1. i 1. j m s 15-45

Smple roblem 15.5 SOLUTION: The crnk hs constnt clockwise ngulr elocity of 000 rpm. For the crnk position indicted, determine () the ngulr elocity of the connecting rod, nd (b) the elocity of the piston. etermine the elocity t from the gien crnk rottion dt. The direction of the elocity ectors t nd re known. The instntneous center of rottion is t the intersection of the perpendiculrs to the elocities through nd. etermine the ngulr elocity bout the center of rottion bsed on the elocity t. Clculte the elocity t bsed on its rottion bout the instntneous center of rottion. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-46

Smple roblem 15.5 SOLUTION: From Smple roblem 15.3, 403.9i 481.3 j in. s 13.95 68.3in. s The instntneous center of rottion is t the intersection of the perpendiculrs to the elocities through nd. 40 53.95 90 76.05 C C 8 in. sin 76.05 sin 53.95 sin50 C 10.14 in. C 8.44 in. etermine the ngulr elocity bout the center of rottion bsed on the elocity t. C 68.3in. s C 10.14 in. Clculte the elocity t bsed on its rottion bout the instntneous center of rottion. C 8.44 in. 6.0rd s 6.0rd s 53in. s 43.6ft s 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-47

Instntneous Center of Zero Velocity Wht hppens to the loction of the instntneous center of elocity if the crnkshft ngulr elocity increses from 000 rpm in the preious problem to 3000 rpm? Wht hppens to the loction of the instntneous center of elocity if the ngle is 0? 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 48

Group roblem Soling In the position shown, br hs n ngulr elocity of 4 rd/s clockwise. etermine the ngulr elocity of brs nd E. 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 49

Group roblem Soling Wht is the elocity of? Wht direction is the elocity of? Wht direction is the elocity of? ( ) (0.5 m)(4 rd/s) 1m/s = 4 rd/s Find 1 0.06 m tn 1.8 0.15 m 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 50

Group roblem Soling Locte instntneous center C t intersection of lines drwn perpendiculr to nd. Find distnces C nd C C 100 mm 0.1 m 0.1 m C 0.5 m tn tn 1.8 0.5 m 0.5 m C 0.693 m cos cos1.8 Clculte ( C) 1m/s (0.5 m) 4 rd/s Find E 0.5 m ( C) (4 rd/s) cos 1 m/s 0.15 m ( E) E ; ; cos cos E E 6.67 rd/s 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 51

bsolute nd Reltie ccelertion in lne Motion s the bicycle ccelertes, point on the top of the wheel will he ccelertion due to the ccelertion from the xle (the oerll liner ccelertion of the bike), the tngentil ccelertion of the wheel from the ngulr ccelertion, nd the norml ccelertion due to the ngulr elocity. 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 5

bsolute nd Reltie ccelertion in lne Motion bsolute ccelertion of prticle of the slb, Reltie ccelertion ssocited with rottion bout includes tngentil nd norml components, k r r t r n r t n 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-53

bsolute nd Reltie ccelertion in lne Motion Gien nd, determine nd. n t Vector result depends on sense of nd reltie mgnitudes of n nd the Must lso know ngulr elocity. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-54

bsolute nd Reltie ccelertion in lne Motion Write x components: 0 l sin l cos y components: l cos l sin Sole for nd. in terms of the two component equtions, 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-55

nlysis of lne Motion in Terms of rmeter In some cses, it is dntgeous to determine the bsolute elocity nd ccelertion of mechnism directly. x l sin y l cos x l cos y l sin l cos l sin x l sin l cos y l cos l sin l sin l cos l cos l sin 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-56

Concept Question You he mde it to the kickbll chmpionship gme. s you try to kick home the winning run, your mind nturlly drifts towrds dynmics. Which of your following thoughts is TRUE, nd cuses you to shnk the bll horribly stright to the pitcher? ) Energy will not be consered when I kick this bll ) In generl, the liner ccelertion of my knee is equl to the liner ccelertion of my foot C) Throughout the kick, my foot will only he tngentil ccelertion. ) In generl, the ngulr elocity of the upper leg (thigh) will be the sme s the ngulr elocity of the lower leg 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 57

Smple roblem 15.6 SOLUTION: The expression of the ger position s function of is differentited twice to define the reltionship between the trnsltionl nd ngulr ccelertions. The center of the double ger hs elocity nd ccelertion to the right of 1. m/s nd 3 m/s, respectiely. The lower rck is sttionry. The ccelertion of ech point on the ger is obtined by dding the ccelertion of the ger center nd the reltie ccelertions with respect to the center. The ltter includes norml nd tngentil ccelertion components. etermine () the ngulr ccelertion of the ger, nd (b) the ccelertion of points, C, nd. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-58

Smple roblem 15.6 SOLUTION: The expression of the ger position s function of is differentited twice to define the reltionship between the trnsltionl nd ngulr ccelertions. x r1 r r 1 1 r 1 1.m s 0.150 m 8 rd s r 1 r1 r 1 3m s 0.150 m k 0rd s k 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-59

Smple roblem 15.6 k r r t The ccelertion of ech point is obtined by dding the ccelertion of the ger center nd the reltie ccelertions with respect to the center. 3m s i 0rd s k 0.100 m j 8rd s 0.100 m 3m s i m s i 6.40m s j The ltter includes norml nd tngentil ccelertion components. n j 5 m s i 6.40m s j 8.1m s 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-60

013 The McGrw-Hill Compnies, Inc. ll rights resered. enth Smple roblem 15.6 15-61 j i i j j k i r r k C C C C s 9.60m s 3m s 3m 0.150 m s 8rd 0.150 m s 0rd s 3m j c s 9.60m i j i i i k i r r k s 9.60m s 3m s 3m 0.150m s 8rd 0.150m s 0rd s 3m s 1.95m s 3m.6m 1 j i s

Smple roblem 15.7 SOLUTION: The ngulr ccelertion of the connecting rod nd the ccelertion of point will be determined from t n The ccelertion of is determined from the gien rottion speed of. Crnk G of the engine system hs constnt clockwise ngulr elocity of 000 rpm. For the crnk position shown, determine the ngulr ccelertion of the connecting rod nd the ccelertion of point. The directions of the ccelertions,, nd re t n determined from the geometry. Component equtions for ccelertion of point re soled simultneously for ccelertion of nd ngulr ccelertion of the connecting rod. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-6

Smple roblem 15.7 The directions of the ccelertions determined from the geometry. i, nd t n From Smple roblem 15.3, = 6.0 rd/s, = 13.95 o. 8 ft 6.0rd s 563ft s n 1 563ft s cos13.95i sin13. 95 j n 8 ft 0. 667 t 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-64 1 The direction of ( / ) t is known but the sense is not known, 0.667 sin 76.05i cos76. 05 j t, re

Smple roblem 15.7 Component equtions for ccelertion of point re soled simultneously. x components: t n,96cos40 563cos13.95 0.667 sin13. 95 y components: 10 0 10,96sin 40 563sin13.95 0.667 cos13. 95 9940 rd s k 990ft s i 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-65

Smple roblem 15.8 SOLUTION: The ngulr elocities re determined by simultneously soling the component equtions for In the position shown, crnk hs constnt ngulr elocity 1 = 0 rd/s counterclockwise. etermine the ngulr elocities nd ngulr ccelertions of the connecting rod nd crnk E. The ngulr ccelertions re determined by simultneously soling the component equtions for 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-66

Smple roblem 15.8 SOLUTION: The ngulr elocities re determined by simultneously soling the component equtions for E r Ek 17i 17 j 17 Ei 17 E j r 0k 8i 14 j 80i 160 j r k 1i 3 j 3 i 1 j x components: y components: 17 80 3 E 17 160 1 E 9.33rd sk 11.9rd sk E 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-67

Smple roblem 15.8 013 The McGrw-Hill Compnies, Inc. ll rights resered. The ngulr ccelertions re determined by simultneously soling the component equtions for E r Er Ek 17 Ei 17 E j 170i 170 j r r 0 0 8i 14 j 300i 5600 j r r k 1i 3 j 9.33 1i 3 j 3 i 1 j 10,30i 580 j 17i 17 j 11.9 17i 17 j x components: 17 E 3 15, 690 y components: 17 E 1 6010 645rd s k 809rd s k E 15-68

Group roblem Soling Knowing tht t the instnt shown br hs constnt ngulr elocity of 4 rd/s clockwise, determine the ngulr ccelertion of brs nd E. Which of the following is true? ) The direction of is b) The ngulr ccelertion of must lso be constnt c) The direction of the liner ccelertion of is 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 69

Group roblem Soling SOLUTION: The ngulr elocities were determined in preious problem by simultneously soling the component equtions for Knowing tht t the instnt shown br hs constnt ngulr elocity of 4 rd/s clockwise, determine the ngulr ccelertion of brs nd E. The ngulr ccelertions re now determined by simultneously soling the component equtions for the reltie ccelertion eqution. 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 70

Group roblem Soling From our preious problem, we used the reltie elocity equtions to find tht: = 4 rd/s E.55 rd/s 0.955 rd/s We cn now pply the reltie ccelertion eqution with 0 nlyze r / / r r / (4) ( 7 ) 11 in./s r i i nlyze r r / r / 11 i k ( 8 j) (0.95455) ( 8 j) (11 8 ) i 7.89j 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 71

Group roblem Soling nlyze r E = 4 rd/s E r / E Er / E k ( 11i 3 j) (.5455) ( 11i 3 j) E 11 j 3 i 71.75i 19.439j E E ( 3 71.75) i (11 19.439) j E E From preious pge, we hd: (11 8 ) i 7.89j Equte like components of j: 7.89 (11 19.439) E i: 11 8 [ (3)(.498) 71.75] E.498 rd/s 4.1795 rd/s 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 7

Concept Question If the clockwise ngulr elocity of crnkshft is constnt, which of the following sttement is true? ) The ngulr elocity of is constnt b) The liner ccelertion of point is zero c) The ngulr elocity of is counterclockwise d) The liner ccelertion of point is tngent to the pth 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 73

pplictions Rotting coordinte systems re often used to nlyze mechnisms (such s musement prk rides) s well s wether ptterns. 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 74

Rte of Chnge With Respect to Rotting Frme With respect to the rotting Oxyz frme, Q Qxi Qy j Qzk Q i Q j Q k Q Oxyz x y z Frme OXYZ is fixed. Frme Oxyz rottes bout fixed xis O with ngulr elocity Vector function Q t ries in direction nd mgnitude. With respect to the fixed OXYZ frme, Q Q i Q j Q k Q i Q OXYZ x y z x y j Q Q xi Q y j Q zk Q Oxyz rte of chnge with respect to rotting frme. If Q were fixed within Oxyz then Q OXYZ is equilent to elocity of point in rigid body ttched to Oxyz nd Q i Q j Q k Q With respect to the fixed OXYZ frme, Q Q Q OXYZ Oxyz x y z z k 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-75

Coriolis ccelertion Frme OXY is fixed nd frme Oxy rottes with ngulr elocity. osition ector r for the prticle is the sme in both frmes but the rte of chnge depends on the choice of frme. The bsolute elocity of the prticle is r r r OXY Oxy Imgine rigid slb ttched to the rotting frme Oxy or F for short. Let be point on the slb which corresponds instntneously to position of prticle. r F Oxy elocity of long its pth on the slb ' bsolute elocity of point on the slb bsolute elocity for the prticle my be written s F 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-76

Coriolis ccelertion r r F Oxy bsolute ccelertion for the prticle is d r r OXY r Oxy dt but, r r r OXY Oxy r Oxy r Oxy r Oxy d dt r r Oxy r r Oxy r Oxy Utilizing the conceptul point on the slb, r r F bsolute ccelertion for the prticle becomes F r Oxy F c r Coriolis ccelertion c Oxy F 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-77

Coriolis ccelertion Consider collr which is mde to slide t constnt reltie elocity u long rod O. The rod is rotting t constnt ngulr elocity. The point on the rod corresponds to the instntneous position of. bsolute ccelertion of the collr is F where r r r F Oxy 0 u c F c r c The bsolute ccelertion consists of the rdil nd tngentil ectors shown 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-78

Coriolis ccelertion t t, t t t, u u 013 The McGrw-Hill Compnies, Inc. ll rights resered. Chnge in elocity oer t is represented by the sum of three ectors RR TT T T TT is due to chnge in direction of the elocity of point on the rod, TT lim t0 t recll, lim r t0 t r r r r RR nd T T result from combined effects of reltie motion of nd rottion of the rod RR T T r lim u t t t lim 0 t0 t t u u u recll, u c F c 15-79

Concept Question You re wlking with constnt elocity with respect to the pltform, which rottes with constnt ngulr elocity w. t the instnt shown, in which direction(s) will you experience n ccelertion (choose ll tht pply)? ) +x b) -x c) +y d) -y e) ccelertion = 0 y x r r r Oxy r Oxy 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 80

Smple roblem 15.9 isk of the Gene mechnism rottes with constnt counterclockwise ngulr elocity = 10 rd/s. t the instnt when = 150 o, determine () the ngulr elocity of disk S, nd (b) the elocity of pin reltie to disk S. SOLUTION: The bsolute elocity of the point my be written s s Mgnitude nd direction of elocity of pin re clculted from the rdius nd ngulr elocity of disk. irection of elocity of point on S coinciding with is perpendiculr to rdius O. irection of elocity s of with respect to S is prllel to the slot. Sole the ector tringle for the ngulr elocity of S nd reltie elocity of. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-81

Smple roblem 15.9 SOLUTION: The bsolute elocity of the point my be written s s Mgnitude nd direction of bsolute elocity of pin re clculted from rdius nd ngulr elocity of disk. R 50 mm10 rd s 500mm s irection of elocity of with respect to S is prllel to slot. From the lw of cosines, r R l Rl cos30 0.551R r 37.1mm From the lw of cosines, sin sin 30 sin 30 sin 4. 4 R r 0.74 The interior ngle of the ector tringle is 90 4.4 30 17. 6 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-8

Smple roblem 15.9 irection of elocity of point on S coinciding with is perpendiculr to rdius O. From the elocity tringle, r sin s 500mm s s 151.mm s 37.1 mm sin17.6 151.mm s s 4.08rd sk s cos 500m scos17. 6 477m s cos 4.4i sin 4. 4 j s 500 mm s 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-83

Smple roblem 15.10 SOLUTION: The bsolute ccelertion of the pin my be expressed s s The instntneous ngulr elocity of isk S is determined s in Smple roblem 15.9. c In the Gene mechnism, disk rottes with constnt counterclockwise ngulr elocity of 10 rd/s. t the instnt when j = 150 o, determine ngulr ccelertion of disk S. The only unknown inoled in the ccelertion eqution is the instntneous ngulr ccelertion of isk S. Resole ech ccelertion term into the component prllel to the slot. Sole for the ngulr ccelertion of isk S. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-84

Smple roblem 15.10 SOLUTION: bsolute ccelertion of the pin my be expressed s s c From Smple roblem 15.9. 4.4 S 4.08rd sk 477mm s cos 4.4i sin 4.4 j s Considering ech term in the ccelertion eqution, R 500mm10rd s 5000mm 5000mm s cos30i sin 30 j n r S cos 4.4i sin 4.4 j t rs sin 4.4i cos 4.4 j 37.1mm sin 4.4i cos 4.4 j t n S note: S my be positie or negtie t s 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-85

Smple roblem 15.10 The direction of the Coriolis ccelertion is obtined by rotting the direction of the reltie elocity s by 90 o in the sense of S. c S s sin 4.4i cos 4.4 j 4.08rd s 477mm s sin 4.4i cos 4.4 j 3890mm s sin 4.4i cos 4.4 j The reltie ccelertion the slot. s must be prllel to Equting components of the ccelertion terms perpendiculr to the slot, 37.1 S S 3890 5000cos17.7 0 33rd s S 33rd sk 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-86

Group roblem Soling The sleee C is welded to n rm tht rottes bout sttionry point with constnt ngulr elocity = (3 rd/s) j. In the position shown rod F is being moed to the left t constnt speed u=16 in./s reltie to the sleee. etermine the ccelertion of oint. 013 The McGrw-Hill Compnies, Inc. ll rights resered. SOLUTION: The bsolute ccelertion of point my be expressed s ' C c etermine the ccelertion of the irtul point. Clculte the Coriolis ccelertion. dd the different components to get the oerll ccelertion of point. 15-87

Group roblem Soling Gien: u= 16 in./s, = (3 rd/s) j. Find: Write oerll expression for r r r r o ny of the terms go to zero? Oxy r r r r Oxy etermine the norml ccelertion term of the irtul point r (3 rd/s) j (3 rd/s) j[ (5 in.) j (1 in.) k] (108 in./s ) k Oxy Oxy where r is from to 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 88

Group roblem Soling r r r r etermine the Coriolis ccelertion of point Oxy Oxy C / F (3 rd/s) j(16 in./s) k (96 in./s ) i dd the different components to obtin the totl ccelertion of point / F C (108 in./s ) k 0 (96 in./s ) i (96 in./s ) i (108 in./s ) k 013 The McGrw-Hill Compnies, Inc. ll rights resered. - 89

Group roblem Soling In the preious problem, u nd were both constnt. Wht would hppen if u ws incresing? ) The x-component of would increse b) The y-component of would increse c) The z-component of would increse d) The ccelertion of would sty the sme Wht would hppen if ws incresing? ) The x-component of would increse b) The y-component of would increse c) The z-component of would increse d) The ccelertion of would sty the sme 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-90

Motion bout Fixed oint The most generl displcement of rigid body with fixed point O is equilent to rottion of the body bout n xis through O. With the instntneous xis of rottion nd ngulr elocity the elocity of prticle of the body is dr r dt nd the ccelertion of the prticle is d r r. dt The ngulr ccelertion represents the elocity of the tip of. s the ector moes within the body nd in spce, it genertes body cone nd spce cone which re tngent long the instntneous xis of rottion., ngulr elocities he mgnitude nd direction nd obey prllelogrm lw of d. They re ectors. 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-91

013 The McGrw-Hill Compnies, Inc. ll rights resered. enth Generl Motion 15-9 For prticles nd of rigid body, rticle is fixed within the body nd motion of the body reltie to X Y Z is the motion of body with fixed point r Similrly, the ccelertion of the prticle is r r Most generl motion of rigid body is equilent to: - trnsltion in which ll prticles he the sme elocity nd ccelertion of reference prticle, nd - of motion in which prticle is ssumed fixed.

Concept Question The figure depicts model of coster wheel. If both 1 nd re constnt, wht is true bout the ngulr ccelertion of the wheel? ) It is zero. b) It is in the +x direction c) It is in the +z direction d) It is in the -x direction e) It is in the -z direction 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-93

Smple roblem 15.11 SOLUTION: With 1 0.30 j 0.50k r 1cos30i sin30 j 10.39i 6 j The crne rottes with constnt ngulr elocity 1 = 0.30 rd/s nd the boom is being rised with constnt ngulr elocity = 0.50 rd/s. The length of the boom is l = 1 m. etermine: ngulr elocity of the boom, ngulr ccelertion of the boom, elocity of the boom tip, nd ccelertion of the boom tip. ngulr elocity of the boom, 1 ngulr ccelertion of the boom, 1 Oxyz 1 Velocity of boom tip, r ccelertion of boom tip, r r r 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-94

Smple roblem 15.11 1 0.30 j r 10.39i 6 j 0.50k SOLUTION: ngulr elocity of the boom, 1 0.30rd s j 0.50rd sk ngulr ccelertion of the boom, 1 Oxyz 0.30rd s j 0.50rd 1 Velocity of boom tip, i r 0 sk 10.39 j 0.3 6 k 0.5 0 0.15rd 3.54m si 5.0m s j 3.1m sk s i 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-95

Smple roblem 15.11 ccelertion of boom tip, r r r i j k i j 0.15 0 0 0 0.30 10.39 6 0 3 5.0 3.1 0.90k 0.94i.60i 1.50 j 0.90k k 0.50 3.54m s i 1.50m s j 1.80m s k 1 0.30 j r 10.39i 6 j 0.50k 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-96

Three-imensionl Motion. Coriolis ccelertion With respect to the fixed frme OXYZ nd rotting frme Oxyz, Q Q Q OXYZ Oxyz Consider motion of prticle reltie to rotting frme Oxyz or F for short. The bsolute elocity cn be expressed s r r Oxyz F The bsolute ccelertion cn be expressed s r r r Oxyz r Oxyz p F c r Coriolis ccelertion c Oxyz F 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-97

Frme of Reference in Generl Motion Consider: - fixed frme OXYZ, - trnslting frme X Y Z, nd - trnslting nd rotting frme xyz or F. With respect to OXYZ nd X Y Z, r r r The elocity nd ccelertion of reltie to X Y Z cn be found in terms of the elocity nd ccelertion of reltie to xyz. r r F r r F xyz r r c xyz xyz 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-98

Smple roblem 15.15 SOLUTION: efine fixed reference frme OXYZ t O nd moing reference frme xyz or F ttched to the rm t. For the disk mounted on the rm, the indicted ngulr rottion rtes re constnt. etermine: the elocity of the point, the ccelertion of, nd ngulr elocity nd ngulr ccelertion of the disk. With of the moing reference frme coinciding with, the elocity of the point is found from F The ccelertion of is found from F The ngulr elocity nd ngulr ccelertion of the disk re F F c 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-99

Smple roblem 15.15 SOLUTION: efine fixed reference frme OXYZ t O nd moing reference frme xyz or F ttched to the rm t. r Li Rj r Rj 1 j k F With of the moing reference frme coinciding with, the elocity of the point is found from F F r 1 j Li Rj 1L k r k Rj R i F Ri 1 L k 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-100

013 The McGrw-Hill Compnies, Inc. ll rights resered. enth Smple roblem 15.15 15-101 The ccelertion of is found from c F i L Lk j r 1 1 1 j R R i k r F F F k R R i j c 1 1 F Rk Rj Li 1 1 ngulr elocity nd ccelertion of the disk, F k j 1 k j j 1 1 F i 1

Group roblem Soling The crne shown rottes t the constnt rte 1 = 0.5 rd/s; simultneously, the telescoping boom is being lowered t the constnt rte = 0.40 rd/s. Knowing tht t the instnt shown the length of the boom is 0 ft nd is incresing t the constnt rte u= 1.5 ft/s determine the ccelertion of oint. SOLUTION: efine moing reference frme xyz or F ttched to the rm t. The ccelertion of is found from ' F c The ngulr elocity nd ngulr ccelertion of the disk re F F 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-10

Group roblem Soling Gien: 1 = 0.5 rd/s, = -0.40 rd/s. L= 0 ft, u= 1.5 ft/s Find:. Eqution of oerll ccelertion of r r r r o ny of the terms go to zero? Oxy r r r r Let the unextending portion of the boom be rotting frme of reference. Wht re nd? i1j 1j Oxy (0.40 rd/s) i(0.5 rd/s) j. Oxy Oxy k 1 (0.10 rd/s ) k. i 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-103

Group roblem Soling r r r r Oxy etermine the position ector r / r r / (0 ft)(sin 30j cos30 k) (10 ft) j(10 3 ft) k Oxy Find r r 0 0 0.10 (1 ft/s ) i i j k Find r 0 10 10 3 r (0.40i 0.5 j) (0.40i 0.5 j) (10 j 10 3 k) 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-104

Group roblem Soling r r r r Oxy etermine the position ector r / r r / (0 ft)(sin 30j cos30 k) (10 ft) j(10 3 ft) k Oxy Find r r 0 0 0.10 (1 ft/s ) i Find r i j k 0 10 10 3 r (0.40i 0.5 j) (0.40i 0.5 j) (10 j 10 3 k) (1 ft/s ) i (1.6 ft/s ) j (3.8538 ft/s ) k 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-105

Group roblem Soling r r r r Oxy Oxy etermine the Coriolis ccelertion first define the reltie elocity term / u(sin 30j cos30 k) F (1.5 ft/s)sin30 j (1.5 ft/s)cos30k Clculte the Coriolis ccelertion Ω ()(0.40i 0.5 j) (1.5sin 30j 1.5cos30 k) F / (0.6459 ft/s ) i (1.0393 ft/s ) j (0.6 ft/s ) k dd the terms together (.65 ft/s ) i (.64 ft/s ) j (3.5 ft/s ) k 013 The McGrw-Hill Compnies, Inc. ll rights resered. 15-106