Kinematics of rigid bodies

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1 Kinematics of igid bodies elations between time and the positions, elocities, and acceleations of the paticles foming a igid body. (1) Rectilinea tanslation paallel staight paths Cuilinea tanslation (3) (2), a 0 Rotation about a fixed axis Paallel cicles concentic cicles Cuilinea tanslation otation Plane motion

2 (4) Sliding od Rolling wheel Geneal Plane motion neithe otation no tanslation (5) Motion about a fixed point

(1) Tanslation Conside igid body in tanslation: diection of any

time, + / 0 Magnitude constant since, belong to same igid body ll

3 (1) Tanslation Conside igid body in tanslation: diection of any staight line inside the body is constant, all paticles foming the body moe in paallel lines. Fo any two paticles in the body, + Diffeentiating with espect to time, + / 0 Magnitude constant since, belong to same igid body ll paticles hae the same elocity. Diffeentiating with espect to time again, a a ll paticles hae the same acceleation.

4 a a When a igid body is in tanslation, all the points of the body hae the same elocity and same acceleation at any gien instant In cuilinea tanslation, the elocity and acceleation change in diection and in magnitude at eey instant; In ectilinea tanslation, elocity, acceleation diection ae same duing entie motion

the body otates though θ, s ( ) θ ( φ) ds dt P lim t 0 sin θ θ t ( φ) θ

5 (2) Rotation bout a Fixed xis: Velocity & cceleation Conside otation of igid body about a fixed axis The length s of the ac descibed by P when the body otates though θ, s ( ) θ ( φ) ds dt P lim t 0 sin θ θ t ( φ) θ φ sin d V dt sin Vecto diection along the otation axis, angula elocity, k θ k

6 Diffeentiating to detemine the acceleation, a d d dt dt d + dt d + dt ( ) d dt ngula acceleation, α a α x + x ( x ) Tangential acceleation component Radial acceleation component

7 Equations Defining the Rotation of a Rigid ody bout a Fixed xis α dθ dt d dt o dt 2 d θ d 2 dt dθ dθ dx/dt a d 2 x/dt 2 a (d/dx) Unifom Rotation, α 0: θ θ 0 + t Equns. Can be used only when α 0 & constant x x 0 + t Unifomly cceleated Rotation, α constant: θ 2 θ α t t α t 2 ( θ ) 2α θ at x x t + ½ at a (x-x 0 )

8 The ectangula block shown otates about the diagonal O with a constant angula elocity of 6.76 ad/s. Knowing that the otation is counteclockwise as iewed fom, detemine the elocity and acceleation of point at the instant shown. ngula elocity. Velocity of point. x R /O (i j k) x (0.127i j) -0.95i j k m/s cceleation of point. a -3.2 i 1.88j k m/s 2

The assembly shown consists of two ods and a ectangula plate CDE which ae welded togethe. The assembly otates about the axis with a constant angula elocity of 10 ad/s.

9 The assembly shown consists of two ods and a ectangula plate CDE which ae welded togethe. The assembly otates about the axis with a constant angula elocity of 10 ad/s. Knowing that the otation is counteclockwise as iewed fom, detemine the elocity and acceleation of cone E. Find /, E/ Find angula elocity ecto; ( / / l ) 1) E x E/ ; 2) a E x E

Ring has an inne adius 2 and hangs fom the hoizontal shaft as shown. Shaft otates with a constant angula elocity of 25 ad/s and no slipping occus. Knowing that 1 1.27 cm, 2 6.35 cm, and 3 8.

10 Ring has an inne adius 2 and hangs fom the hoizontal shaft as shown. Shaft otates with a constant angula elocity of 25 ad/s and no slipping occus. Knowing that cm, cm, and cm, detemine (a) the angula elocity of ing, (b) the acceleation of the points of shaft and ing which ae in contact, (c) the magnitude of the acceleation of a point on the outside suface of ing.

Geneal plane motion can be consideed as the sum of a tanslation and

12 (3) Geneal Plane Motion Geneal plane motion is neithe a tanslation no a otation. Geneal plane motion can be consideed as the sum of a tanslation and otation. Displacement of paticles 1 and 1 to 2 and 2 can be diided into two pats: - tanslation to 2 and 1 - otation of about 2 to 2 1

13 bsolute and Relatie Velocity in Plane Motion bsolute and Relatie Velocity in Plane Motion ny plane motion can be eplaced by a tanslation of an abitay efeence point and a simultaneous otation about. + +

14 Point as efeence ssuming that the elocity of end is known, wish to detemine the elocity of end and the angula elocity in tems of, l, and θ. The diection of and / ae known. Complete the elocity diagam. tanθ tanθ l l cosθ cosθ

Point as efeence Selecting point as the efeence point and soling fo the elocity of end and the angula elocity leads to an equialent elocity tiangle. / has the same magnitude but opposite sense of /.

15 Point as efeence Selecting point as the efeence point and soling fo the elocity of end and the angula elocity leads to an equialent elocity tiangle. / has the same magnitude but opposite sense of /. The sense of the elatie elocity is dependent on the choice of efeence point. ngula elocity of the od in its otation about is the same as its otation about. ngula elocity is not dependent on the choice of efeence point.

Cank The elocity is obtained fom the cank otation data. 2000 e min min 60s ( ) ( 3in. )( 209.4 ad s) The elocity diection is as shown. 2π ad 209.

16 The cank has a constant clockwise angula elocity of 2000 pm. Fo the cank position indicated, detemine (a) the angula elocity of the connecting od D, and (b) the elocity of the piston P. Cank The elocity is obtained fom the cank otation data e min min 60s ( ) ( 3in. )( ad s) The elocity diection is as shown. 2π ad ad s e in/s Connecting od D The diection of the absolute elocity D is hoizontal. The diection of the elatie elocity D is pependicula to D. Compute the angle between the hoizontal and the connecting od fom the law of sines. sin 40 8in. sinβ 3in. β 13.95

17 D D 628.3in. s sin sin50 sin76.05 D D 523.4in. s 43.6ft 495.9in. s s P D 43.6ft s D D l D D 495.9in. s l 8 in ad s D ( 62.0 ad s)k

Instantaneous Cente of Rotation in Plane Motion Plane motion of all paticles in a slab can always be eplaced by the tanslation of an abitay point and a otation

The same tanslational and otational elocities at ae obtained by allowing the slab to otate with the same angula elocity about the point C on a pependicula to the

18 Instantaneous Cente of Rotation in Plane Motion Plane motion of all paticles in a slab can always be eplaced by the tanslation of an abitay point and a otation about with an angula elocity that is independent of the choice of. The same tanslational and otational elocities at ae obtained by allowing the slab to otate with the same angula elocity about the point C on a pependicula to the elocity at. The elocity of all othe paticles in the slab ae the same as oiginally defined since the angula elocity and tanslational elocity at ae equialent. s fa as the elocities ae concened, the slab seems to otate about the instantaneous cente of otation C.

1) If the elocity ectos at and ae pependicula to the line, the instantaneous cente of otation lies at the intesection of the line with the line joining the extemities

19 How to obtain instantaneous cente of otation Fig. 1 Fig. 2 If the elocity at two points and ae known, the instantaneous cente of otation lies at the intesection of the pependiculas to the elocity ectos though and. (fig. 1) If the elocity ectos at and ae pependicula to the line, the instantaneous cente of otation lies at the intesection of the line with the line joining the extemities of the elocity ectos at and. (fig. 2) If the elocity ectos ae paallel, the instantaneous cente of otation is at infinity and the angula elocity is zeo. (fig. 1) If the elocity magnitudes ae equal, the instantaneous cente of otation is at infinity and the angula elocity is zeo. (fig. 2)

Instantaneous cente of a slab in plane motion can be located eithe on slab o on outside the slab. If on the slab, the paticle C coinciding at the cente of otation has zeo elocity at that instant.

20 Instantaneous cente of a slab in plane motion can be located eithe on slab o on outside the slab. If on the slab, the paticle C coinciding at the cente of otation has zeo elocity at that instant. This coincidence will not happen at anothe time. So elocity at time t will not be same at t+dt. The paticle coinciding with the cente of otation changes with time and the acceleation of the paticle at the instantaneous cente of otation C is not zeo. The acceleation of the paticles in the slab cannot be detemined as if the slab wee simply otating about C. The tace of the locus of the cente of otation on the body is the body centode and in space is the space centode.

21 Same poblem The cank has a constant clockwise angula elocity of 2000 pm. Fo the cank position indicated, detemine (a) the angula elocity of the connecting od D, and (b) the elocity of the piston P in/s; β The instantaneous cente of otation is at the intesection of the pependiculas to the elocities though and D. γ γ D 40 + β β C CD 8 in. sin sin sin50 C in. CD 8.44 in.

22 Detemine the angula elocity about the cente of otation based on the elocity at. D ( C) D 628.3in. s C in. D 62.0ad s Calculate the elocity at D based on its otation about the instantaneous cente of otation. D ( CD) ( 8.44 in. )( 62.0 ad s) D P D 523 in. s 43.6ft s

associated with otation about includes tangential and nomal

23 bsolute and Relatie cceleation in Plane Motion bsolute acceleation of a paticle of the slab, a a a + Relatie acceleation associated with otation about includes tangential and nomal components, a ( ) n ( ) ( ) 2 α a a n t ( ) t a a 2 α t ( ) ( ) n a a a a a a + + +

Rate of Change of ecto with espect to a Rotating Fame Fame OXYZ is fixed. Fame Oxyz otates about fixed axis O with angula elocity Ω () Vecto function Q t aies in diection and magnitude.

24 Rate of Change of ecto with espect to a Rotating Fame Fame OXYZ is fixed. Fame Oxyz otates about fixed axis O with angula elocity Ω () Vecto function Q t aies in diection and magnitude. Rate of change of Q depends on fame of efeence With espect to the otating Oxyz fame, Q Qx i + Qy j+ Qzk If Q wee fixed within Oxyz then () Q is OXYZ equialent to elocity of a point in a igid body () Q Oxyz Qxi + Qy j+ Qzk attached to Oxyz and Qx di/dt+ Qy dj/dt + Q dk/dt Q With espect to the fixed OXYZ fame, z Ω () Q OXYZ Q x i + Q y j+ Q z k + Q x di/dt + Q y dj/dt + Q z dk/dt () Q k Qxi + Qy j+ Oxyz Qz Repesent elocity of paticle, ΩxQ

25 With espect to the fixed OXYZ fame, () () Q Q +Ω Q OXYZ Oxyz This elation is useful to find ate of change of Q w..t. fixed fame of efeence OXYZ when Q is defined by its components along the otating fame Oxyz

Coiolis cceleation Fame OXY is fixed and fame Oxy otates with angula elocity Ω. Position ecto P fo the paticle P is the same in both fames but the ate of change depends on the choice of fame.

26 Coiolis cceleation Fame OXY is fixed and fame Oxy otates with angula elocity Ω. Position ecto P fo the paticle P is the same in both fames but the ate of change depends on the choice of fame. Ω + The absolute elocity of the paticle P is, P () OXY ( ) Oxy The absolute acceleation of the paticle P is, a p p Ωx + Ωx + d/dt [() Oxy ] () OXY Ω ( ) Oxy + P but, d dt ( ) OXY Ω + ( ) Oxy [( & ) Oxy ] (& ) Oxy +Ω ( & ) Oxy & & a p Ω x + Ω x (Ω x ) + 2Ω x () Oxy + () Oxy

27 a p Ω x + Ω x (Ω x ) + 2Ω x () Oxy + () Oxy Coiolis acceleation, a c a α x + x ( x ) Two ectos ae nomal to each othe, 2Ω Oxy

Fom the law of cosines, 2 2 2 R + l 2Rlcos30 0.551R 2 37.1mm Fom the law of sine, sinβ sin30 sin30 sinβ β 42. 4 R 0.

28 Disk D of the Genea mechanism otates with constant counteclockwise angula elocity D 10 ad/s. t the instant when φ 150 o, detemine (a) the angula elocity of disk S, and (b) the elocity of pin P elatie to disk S. Fom the law of cosines, R + l 2Rlcos R mm Fom the law of sine, sinβ sin30 sin30 sinβ β R Magnitude and diection of absolute elocity of pin P ae calculated fom adius and angula elocity of disk D. P R D ( 50 mm)( 10 ad s) 500mm s

29 The absolute elocity of the point P may be witten as P () OXY Ω ( ) Oxy + + P P P s The inteio angle of the ecto tiangle is γ P P sinγ s ( 500mm s) s 151.2mm s 37.1 mm sin mm s s 4.08ad s ( 500m s) cos17. P s P cosγ 6 P s ( 477m s) Coiolis acceleation, ac 2 s p/s 2 (4.08) (477) 3890 mm/s 2

r cos, and y r sin with the origin of coordinate system located at

r cos, and y r sin with the origin of coordinate system located at Lectue 3-3 Kinematics of Rotation Duing ou peious lectues we hae consideed diffeent examples of motion in one and seeal dimensions. But in each case the moing object was consideed as a paticle-like object,