PLANE KINETICS OF RIGID BODIES

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Bruce Harris
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1 PLANE KNETCS OF RD BODES

2 The knetcs of rgd bodes treats the relatonshps between the external forces actng on a body and the correspondng translatonal and rotatonal motons of the body. n the knetcs of the partcle, we found that two force equatons of moton were requred to defne the plane moton of a partcle whose moton has two lnear components.

3 For the plane moton of a rgd body, an addtonal equaton s needed to specfy the state of rotaton of the body. Thus, two force and one moment equatons or ther equvalent are requred to determne the state of rgd-body plane moton.

ENERAL EQUATONS OF MOTON n our study of Statcs, a general system of forces actng on a rgd body may be replaced by a resultant force appled at a chosen pont and a correspondng couple.

4 ENERAL EQUATONS OF MOTON n our study of Statcs, a general system of forces actng on a rgd body may be replaced by a resultant force appled at a chosen pont and a correspondng couple. By replacng the external forces by ther equvalent force-couple system n whch the resultant force acts through the mass center, we may vsualze the acton of the forces and the correspondng dynamc response. Dynamc response

5 F ma M H a) Relevant free-body dagram (FBD) b) Equvalent force-couple system wth resultant force appled through c) Knetc dagram whch represents the resultng dynamc effects

6 PLANE MOTON EQUATONS Fgure shows a rgd body movng wth plane moton n the x-y plane. The mass center has an acceleraton and the body has an angular velocty k and an angular acceleraton k. a The angular momentum about the mass center for the representatve partcle m : H m v : poston vector relatve to of partcle m v cos snj k H m Velocty of partcle m

7 The angular momentum about the mass center for the rgd body: s a constant property of the body and s a measure of the rotatonal nerta or resstance to change n rotatonal velocty due to the radal dstrbuton of mass around the z-axs through. (MASS MOMENT OF NERTA of the body the about z-axs through ) H k H k m k m H m j m j k m j m H H 1 sn cos sn cos sn cos sn cos 1 kgm dm m n m n dt d dt d dt H M H M M

8 Analyss Procedure n the soluton of force-mass-acceleraton problems for the plane moton of rgd bodes, the followng steps should be taken after the condtons and requrements of the problem are clearly n mnd. 1) Knematcs : Frst, dentfy the class of moton and then solve any needed lnear or angular acceleratons whch can be determned from gven knematc nformaton. ) Dagrams: Always draw the complete free-body dagram and knetc dagram. M 3) Apply the three equatons of moton. ( F ma, )

9 Mass Moments of nerta Mass moment of nerta of dm about the axs OO, d: O r dm d r dm O Total mass moment of nerta of mass m : d r dm s always postve and ts unts s kg. m.

10 Transfer of axes for mass moment of nerta: f the moment of nerta of a body s known about an axs passng through the mass center, t may be determned easly about any parallel axs. O md d O

11 Mass Moments of nerta for Some Common eometrc Shapes Thn bar Thn crcular plate Thn rectangular plate

12 Radus of yraton, k: The radus of gyraton k of a mass m about an axs for whch the moment of nerta s s defned as k m k m Thus k s a measure of the dstrbuton of mass of a gven body about the axs n queston, and ts defnton s analogous to the defnton of the radus of gyraton for area moments of nerta. The moment of nerta of a body about a partcular axs s frequently ndcated by specfyng the mass of the body and the radus of gyraton of the body about the axs. When the expressons for the rad of gyraton are used, the equaton becomes k k d

13 1) TRANSLATON a) Rectlnear Translaton: FBD 0 0 Knetc Dagram F F 1 A m d P F n x A m d P ma x x F 3 F external force Fx max e. f. ma M 0 e. f. M A 0 e. f. M P maxd e. f.

14 b) Curvlnear Translaton: FBD 0 0 Knetc Dagram F F 3 F 1 m n F n t A d A B d B m ma t ma n m r n t mr Fn ma n M e. f. e. f. F ma M A ma nd t t e. f. e. f. M B matd B e. f. 0 ( A 0) + +

15 ) FXED-AXS ROTATON For ths moton, all ponts n the body descrbe crcles about the rotaton axs, and all lnes of the body have the same angular velocty and angular acceleraton. The acceleraton components of the mass center n n-t coordnates: a t r a n r Equatons of Moton FBD Knetc Dagram F F t ma mr F n m r M * **

16 For fxed-axs rotaton, t s generally useful to apply a moment equaton drectly about the rotaton axs O. M mr O a t Usng transfer-of-axs relaton for mass moments of nerta; O mr O mr M O mr mrr O O M O o For the case of rotaton axs through ts mass center : a 0 and F 0 FBD Knetc Dagram M

17 3) ENERAL PLANE MOTON The dynamcs of general plane moton of a rgd body combnes translaton and rotaton. FBD Knetc Dagram Equatons of moton: F ma M n some cases, t may be more convenent to use the alternatve moment relaton about any pont P. M P mad

18 PROBLEMS 1. The unform 30-kg bar OB s secured to the acceleratng frame n the 30 o poston from the horzontal by the hnge at O and roller at A. f the horzontal acceleraton of the frame s a=0 m/s, compute the force F A on the roller and the x- and y-components of the force supported by the pn at O.

19 PROBLEMS. The block A and attached rod have a combned mass of 60 kg and are confned to move along the 60 o gude under the acton of the 800 N appled force. The unform horzontal rod has a mass of 0 kg and s welded to the block at B. Frcton n the gude s neglgble. Compute the bendng moment M exerted by the weld on the rod at B.

20 SOLUTON FBD x Knetc Dagram m T a x =60a x x N 60 o W=60(9.81) N a x F x ma x 4.84 m / s (9.81)sn 60 60a x B y FBD of rod KD of rod m 1 a x =0a x B x M W 1 =0(9.81) N M B ma x M 196 m / s d M 0(9.81)0.7 (0)(4.94)(0.7sn 60)

PROBLEMS 3. The parallelogram lnkage shown moves n the vertcal plane wth the unform 8 kg bar EF attached to the plate at E by a pn whch s welded both to the plate and to the bar.

21 PROBLEMS 3. The parallelogram lnkage shown moves n the vertcal plane wth the unform 8 kg bar EF attached to the plate at E by a pn whch s welded both to the plate and to the bar. A torque (not shown) s appled to lnk AB through ts lower pn to drve the lnks n a clockwse drecton. When q reaches 60 o, the lnks have an angular acceleraton an angular velocty of 6 rad/s and 3 rad/s, respectvely. For ths nstant calculate the magntudes of the force F and torque M supported by the pn at E.

22 PROBLEMS 4. The unform 100 kg log s supported by the two cables and used as a batterng ram. f the log s released from rest n the poston shown, calculate the ntal tenson nduced n each cable mmedately after release and the correspondng angular acceleraton of the cables.

SOLUTON FBD T A T B man +n KD +n +t W=100(9.81) N ma t When t starts to move, v=0, =0 but 0 r 0 Fn 0 T cos 30 0.. A TB mg TA TB d k Ft mat mg sn30 mat at 4.905 m / s a t d. k. r 4.

23 SOLUTON FBD T A T B man +n KD +n +t W=100(9.81) N ma t When t starts to move, v=0, =0 but 0 r 0 Fn 0 T cos A TB mg TA TB d k Ft mat mg sn30 mat at m / s a t d. k. r rad / s Length of the cables The moton of the log s curvlnear translaton. T A M d. k N 0 T B T A N sn 60(1.5) T B a n sn 60(0.5) 0 3T A +t T B * *

24 PROBLEMS 5. An 18 kg trangular plate s supported by cables AB and CD. When the plate s n the poston shown, the angular velocty of the cables s 4 rad/s ccw. At ths nstant, calculate the acceleraton of the mass center of the plate and the tenson n each of the cables. A C 4 cm 60 B 60 D 10 cm 0 cm 0 cm Answer: a 6.3 m / s TAB N TCD N

25 PROBLEMS 6. The unform 8 kg slender bar s hnged about a horzontal axs through O and released from rest n the horzontal poston. Determne the dstance b from the mass center to O whch wll result n an ntal angular acceleraton of 16 rad/s, and fnd the force R on the bar at O just after release.

26 PROBLEMS 7. The sprng s uncompressed when the unform slender bar s n the vertcal poston shown. Determne the ntal angular acceleraton a of the bar when t s released from rest n a poston where the bar has been rotated 30 o clockwse from the poston shown. Neglect any sag of the sprng, whose mass s neglgble.

27 SOLUTON Unstrecthed length of the sprng: 5 l o (l / 4) l l When q=30 o, length of the sprng: When q=30 o, sprng force: F sprng 3 l sprng 5 k l l 3 l kl 5 3 (n compresson) O t W +t 60 o O +n 30 o 60 o. l F sprng l sprng 30 o +t ma t +n ma n m M O l mg cos 60 F l 0 4 ma k m t sprng g l l 1 1 ml ma t l 4 l 4 O n

28 PROBLEMS 8. n the mechansm shown, the flywheel has a mass of 50 kg and radus of gyraton about ts center of 160 mm. Unform connectng rod AB has a mass of 10 kg. Mass of the pston B s 15 kg. Flywheel s rotatng by the couple T ccw at a constant rate 50 rad/s. When q=53 o determne the angular velocty and angular acceleraton of the connectng rod AB ( AB ve AB ). What are the forces transmtted by the pns at A and B? Neglect the frcton. Take sn 53=0.8, cos 53=0.6.

29 PROBLEMS 9. Member AB s beng rotated at a constant angular velocty of = 10 rad/s n ccw drecton by a torque (not seen n the fgure). Rotaton of AB actvates the 6 kg rod BC, whch causes the 3 kg gear D to move. The radus of gyraton of the gear about C s 00 mm. The radus of gear D s gven as r = 50 mm. For the nstant represented determne the forces actng at pns B and C.

30 PROBLEMS 10. The unbalanced 0 kg wheel wth the mass center at has a radus of gyraton about of 0 mm. The wheel rolls down the 0 o nclne wthout slppng. n the poston shown. The wheel has an angular velocty of 3 rad/s. Calculate the frcton force F actng on the wheel at ths poston.

31 SOLUTON eneral Moton FBD mg KD y mk 0(0.0) kgm a o r 0. 5 x = x F N a a ma ao a / O 0. 5 k j Fx max mg sn 0 F ef F Fy ma y N mg cos ef N M N( 0.075) F(0.5) ef a x a y k 3k F N rad / s N N

32 PROBLEMS 11. The unform 15 kg bar s supported on the horzontal surface at A by a small roller of neglgble mass. f the coeffcent of knetc frcton between end B and the vertcal surface s 0.30, calculate the ntal acceleraton of end A as the bar s released from rest n the poston shown.

Rotation Kinematics, Moment of Inertia, and Torque

Rotation Kinematics, Moment of Inertia, and Torque Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute