Seventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University

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1 Seenth 15Fedinand P. ee E. Russell Johnston, J. Kinematics of Lectue Notes: J. Walt Ole Texas Tech Uniesity Rigid odies CHPTER VECTOR MECHNICS FOR ENGINEERS: YNMICS 003 The McGaw-Hill Companies, Inc. ll ights eseed. Seenth Vecto Mechanics fo Enginees: ynamics Contents Intoduction Tanslation Rotation bout a Fixed xis: Velocity Rotation bout a Fixed xis: cceleation Rotation bout a Fixed xis: Repesentatie Slab Equations efining the Rotation of a Rigid ody bout a Fixed xis Sample Poblem 5.1 Geneal Plane Motion bsolute and Relatie Velocity in Plane Motion Sample Poblem 15. Sample Poblem 15.3 Instantaneous Cente of Rotation in Plane Motion Sample Poblem 15.4 Sample Poblem The McGaw-Hill Companies, Inc. ll ights eseed. bsolute and Relatie cceleation in Plane Motion nalysis of Plane Motion in Tems of a Paamete Sample Poblem 15.6 Sample Poblem 15.7 Sample Poblem 15.8 Rate of Change With Respect to a Rotating Fame Coiolis cceleation Sample Poblem 15.9 Sample Poblem Motion bout a Fixed Point Geneal Motion Sample Poblem Thee imensional Motion. Coiolis cceleation Fame of Refeence in Geneal Motion Sample Poblem

2 Seenth Vecto Mechanics fo Enginees: ynamics Intoduction Kinematics of igid bodies: elations between time and the positions, elocities, and acceleations of the paticles foming a igid body. Classification of igid body motions: - tanslation: ectilinea tanslation cuilinea tanslation - otation about a fixed axis - geneal plane motion - motion about a fixed point - geneal motion 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Tanslation 003 The McGaw-Hill Companies, Inc. ll ights eseed. 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, + iffeentiating with espect to time, & & + & & ll paticles hae the same elocity. iffeentiating with espect to time again, & & + & & a a ll paticles hae the same acceleation. 15-4

3 Seenth Vecto Mechanics fo Enginees: ynamics Rotation bout a Fixed xis. Velocity Conside otation of igid body about a fixed axis Velocity ecto d dt of the paticle P is tangent to the path with magnitude ds dt s ( P) θ ( sinφ ) θ ds dt lim t 0 θ ( sinφ ) & θ sinφ t The same esult is obtained fom d ω dt ω ωk & θ k angula elocity 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Rotation bout a Fixed xis. cceleation iffeentiating to detemine the acceleation, d d a ( ω ) dt dt dω d + ω dt dt dω + ω dt dω α angula acceleation dt αk & ωk && θ k cceleation of P is combination of two ectos, a α + ω ( ω ) α tangential acceleation component ω ( ω ) adial acceleation component 003 The McGaw-Hill Companies, Inc. ll ights eseed

4 Seenth Vecto Mechanics fo Enginees: ynamics Rotation bout a Fixed xis. Repesentatie Slab Conside the motion of a epesentatie slab in a plane pependicula to the axis of otation. Velocity of any point P of the slab, ω ωk ω cceleation of any point P of the slab, a α + ω ( ω ) αk ω Resoling the acceleation into tangential and nomal components, at αk at α a ω a ω n n 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Equations efining the Rotation of a Rigid ody bout a Fixed xis Motion of a igid body otating aound a fixed axis is often specified by the type of angula acceleation. Recall dθ dθ ω o dt dt ω dω d θ dω α ω dt dt dθ Unifom Rotation, α 0: θ θ + ωt 0 Unifomly cceleated Rotation, α constant: ω ω + αt α( θ ) θ0 003 The McGaw-Hill Companies, Inc. ll ights eseed θ θ + ω t ω ω + 1 αt 4

5 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 5.1 Cable C has a constant acceleation of 9 in/s and an initial elocity of 1 in/s, both diected to the ight. etemine (a) the numbe of eolutions of the pulley in s, (b) the elocity and change in position of the load afte s, and (c) the acceleation of the point on the im of the inne pulley at t 0. ue to the action of the cable, the tangential elocity and acceleation of ae equal to the elocity and acceleation of C. Calculate the initial angula elocity and acceleation. pply the elations fo unifomly acceleated otation to detemine the elocity and angula position of the pulley afte s. Ealuate the initial tangential and nomal acceleation components of. 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem The McGaw-Hill Companies, Inc. ll ights eseed. The tangential elocity and acceleation of ae equal to the elocity and acceleation of C. ( ) 0 ( ) 0 1in. s ( ) a C t ac 9in. s ( ) 0 ω ( a 0 ) t α ( ) 0 1 ( a ) 9 ω0 4ad s α t 3ad s 3 3 pply the elations fo unifomly acceleated otation to detemine elocity and angula position of pulley afte s. s + ( 3ad s )( s) 10ad s ( 4ad s)( s) + 1 ( 3ad s )( s) ω ω0 + αt 4ad θ ω 1 0t + αt 14 ad 1e N ( 14 ad) numbe of es π ad ω ( 5 in. )( 10ad s) y θ ( 5 in. )( 14 ad) N.3e 50in. s y 70 in

6 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 5.1 Ealuate the initial tangential and nomal acceleation components of. a a 9in. s ( ) t C ( a ) ω ( 3 in. )( 4ad s) 48in n 0 s ( a ) in. s ( a ) 48in. s t 9 n Magnitude and diection of the total acceleation, 003 The McGaw-Hill Companies, Inc. ll ights eseed. a ( a ) + ( a ) t n tanφ ( a ) ( a ) 48 9 n t a 48.8in. s φ Seenth Vecto Mechanics fo Enginees: ynamics 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. isplacement of paticles and to and can be diided into two pats: - tanslation to and 1 - otation of about to The McGaw-Hill Companies, Inc. ll ights eseed

7 Seenth Vecto Mechanics fo Enginees: ynamics 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. + ωk ω + ωk 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics bsolute and Relatie Velocity in Plane Motion ssuming that the elocity of end is known, wish to detemine the elocity of end and the angula elocity ω in tems of, l, and θ. The diection of and / ae known. Complete the elocity diagam. tanθ tanθ cosθ lω ω l cosθ 003 The McGaw-Hill Companies, Inc. ll ights eseed

8 Seenth Vecto Mechanics fo Enginees: ynamics bsolute and Relatie Velocity in Plane Motion Selecting point as the efeence point and soling fo the elocity of end and the angula elocity ω 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 ω 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. 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15. The double gea olls on the stationay lowe ack: the elocity of its cente is 1. m/s. etemine (a) the angula elocity of the gea, and (b) the elocities of the uppe ack R and point of the gea. The displacement of the gea cente in one eolution is equal to the oute cicumfeence. Relate the tanslational and angula displacements. iffeentiate to elate the tanslational and angula elocities. The elocity fo any point P on the gea may be witten as + + ωk P P P Ealuate the elocities of points and. 003 The McGaw-Hill Companies, Inc. ll ights eseed

9 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15. y x The displacement of the gea cente in one eolution is equal to the oute cicumfeence. Fo x > 0 (moes to ight), ω < 0 (otates clockwise). x θ π π x θ iffeentiate to elate the tanslational and angula elocities. 1 ω 1.m s ω 0.150m 1 1 ω ωk ( 8ad s)k 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15. Fo any point P on the gea, P + + ωk P P Velocity of the uppe ack is equal to elocity of point : R + ωk ( 1.m s) i + ( 8ad s) k ( 0.10 m) j ( 1.m s) i + ( 0.8m s)i ( m s)i R 003 The McGaw-Hill Companies, Inc. ll ights eseed. Velocity of the point : + ωk ( 1.m s) i + ( 8ad s) k ( m)i ( 1.m s) i + ( 1.m s) 1.697m s j

10 Seenth Vecto Mechanics fo Enginees: ynamics bsolute and Relatie cceleation in Plane Motion bsolute acceleation of a paticle of the slab, a a + a Relatie acceleation a associated with otation about includes tangential and nomal components, a αk ( a ) α ( ) t ( a ) ω n t ( a ) ω n 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics bsolute and Relatie cceleation in Plane Motion Gien a and, detemine a andα. a a + a a + a + a ( ) ( ) n t Vecto esult depends on sense of a and a elatie magnitudes of ( a ) n and the Must also know angula elocity ω. 003 The McGaw-Hill Companies, Inc. ll ights eseed

11 Seenth Vecto Mechanics fo Enginees: ynamics bsolute and Relatie cceleation in Plane Motion Wite a a + a + x components: 0 ω a + l sinθ l α cosθ + y components: a lω cosθ lα sinθ Sole fo a and α. in tems of the two component equations, 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics nalysis of Plane Motion in Tems of a Paamete In some cases, it is adantageous to detemine the absolute elocity and acceleation of a mechanism diectly. x l sinθ a x& l & θ cosθ lω cosθ && x l & θ sinθ + l && θ cosθ lω sinθ + lα cosθ y l cosθ a y& l & θ sinθ lω sinθ && y l & θ cosθ l && θ sinθ lω cosθ lα sinθ 003 The McGaw-Hill Companies, Inc. ll ights eseed

12 Seenth Vecto Mechanics fo Enginees: ynamics 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. 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Instantaneous Cente of Rotation in Plane Motion 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. If the elocity ectos ae paallel, the instantaneous cente of otation is at infinity and the angula elocity is zeo. 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. If the elocity magnitudes ae equal, the instantaneous cente of otation is at infinity and the angula elocity is zeo. 003 The McGaw-Hill Companies, Inc. ll ights eseed

13 Seenth Vecto Mechanics fo Enginees: ynamics Instantaneous Cente of Rotation in Plane Motion The instantaneous cente of otation lies at the intesection of the pependiculas to the elocity ectos though and. ω ( ) ( ) C ω l sinθ C l cosθ l cosθ tanθ The elocities of all paticles on the od ae as if they wee otated about C. The paticle at the cente of otation has zeo elocity. The paticle coinciding with the cente of otation changes with time and the acceleation of the paticle at the instantaneous cente of otation 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. 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.4 The double gea olls on the stationay lowe ack: the elocity of its cente is 1. m/s. etemine (a) the angula elocity of the gea, and (b) the elocities of the uppe ack R and point of the gea. The point C is in contact with the stationay lowe ack and, instantaneously, has zeo elocity. It must be the location of the instantaneous cente of otation. etemine the angula elocity about C based on the gien elocity at. Ealuate the elocities at and based on thei otation about C. 003 The McGaw-Hill Companies, Inc. ll ights eseed

14 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.4 The point C is in contact with the stationay lowe ack and, instantaneously, has zeo elocity. It must be the location of the instantaneous cente of otation. etemine the angula elocity about C based on the gien elocity at. 1.m s ω ω 8ad s 0.15 m Ealuate the elocities at and based on thei otation about C. R ω ( 0.5 m)( 8ad s) ( 0.15 m) 0.11m ω ( 0.11m)( 8ad s) R ( m s)i 1.697m s ( 1.i + 1. j )( m s) 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.5 The cank has a constant clockwise angula elocity of 000 pm. Fo the cank position indicated, detemine (a) the angula elocity of the connecting od, and (b) the elocity of the piston P. etemine the elocity at fom the gien cank otation data. The diection of the elocity ectos at and ae known. The instantaneous cente of otation is at the intesection of the pependiculas to the elocities though and. etemine the angula elocity about the cente of otation based on the elocity at. Calculate the elocity at based on its otation about the instantaneous cente of otation. 003 The McGaw-Hill Companies, Inc. ll ights eseed

15 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.5 Fom Sample Poblem 15.3, 403.9i j in. s ( )( ) β in. s The instantaneous cente of otation is at the intesection of the pependiculas to the elocities though and. γ 40 + β γ 90 β C C 8 in. sin sin sin50 C in. C 8.44 in. etemine the angula elocity about the cente of otation based on the elocity at. ω ( C) ω 68.3in. s C in. Calculate the elocity at based on its otation about the instantaneous cente of otation. C ω 8.44 in. 6.0ad s ( ) ( )( ) P ω 6.0ad s 53 in. s 43.6ft s 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.6 The expession of the gea position as a function of θ is diffeentiated twice to define the elationship between the tanslational and angula acceleations. The cente of the double gea has a elocity and acceleation to the ight of 1. m/s and 3 m/s, espectiely. The lowe ack is stationay. etemine (a) the angula acceleation of the gea, and (b) the acceleation of points, C, and. The acceleation of each point on the gea is obtained by adding the acceleation of the gea cente and the elatie acceleations with espect to the cente. The latte includes nomal and tangential acceleation components. 003 The McGaw-Hill Companies, Inc. ll ights eseed

16 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.6 The expession of the gea position as a function of θ is diffeentiated twice to define the elationship between the tanslational and angula acceleations. x a 1 θ & θ ω 1 ω 1 1 & θ 1α 1 a 1.m s m 8 ad s 3m s α m k ( )k α α 0ad s 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.6 a a a + a a + αk + ( a ) + ( a ) ω t The acceleation of each point is obtained by adding the acceleation of the gea cente and the elatie acceleations with espect to the cente. n The latte includes nomal and tangential acceleation components. ( 3m s ) i ( 0ad s ) k ( 0.100m) j ( 8ad s) ( m) ( 3m s ) i + ( m s ) i ( 6.40m s )j j ( 5 m s ) i ( 6.40m s ) j a 8.1m s a 003 The McGaw-Hill Companies, Inc. ll ights eseed

17 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.6 a C a + a a + αk ω C C C ( 3m s ) i ( 0ad s ) k ( 0.150m) j ( 8ad s) ( m) ( 3m s ) i ( 3m s ) i + ( 9.60m s )j a c ( 9.60m s )j a a + a a + αk ω ( 3m s ) i ( 0ad s ) k ( 0.150m) i ( 8ad s) ( 0.150m) i ( 3m s ) i + ( 3m s ) j + ( 9.60m s )i a ( 1.6m s ) i + ( 3m s ) j a 1.95m s 003 The McGaw-Hill Companies, Inc. ll ights eseed j Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.3 The cank has a constant clockwise angula elocity of 000 pm. Fo the cank position indicated, detemine (a) the angula elocity of the connecting od, and (b) the elocity of the piston P. 003 The McGaw-Hill Companies, Inc. ll ights eseed. Will detemine the absolute elocity of point with + The elocity is obtained fom the gien cank otation data. The diections of the absolute elocity and the elatie elocity ae detemined fom the poblem geomety. The unknowns in the ecto expession ae the elocity magnitudes and which may be detemined fom the coesponding ecto tiangle. The angula elocity of the connecting od is calculated fom

18 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.3 Will detemine the absolute elocity of point with + The elocity is obtained fom the cank otation data. e min π ad ω ad s min 60s e ω 3in ad s ( ) ( )( ) The elocity diection is as shown. The diection of the absolute elocity is hoizontal. The diection of the elatie elocity is pependicula to. Compute the angle between the hoizontal and the connecting od fom the law of sines. sin 40 sin β 8in. 3in. β The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.3 etemine the elocity magnitudes fom the ecto tiangle. 68.3in. s sin sin50 sin76.05 and + ω 53.4in. s 43.6ft s 495.9in. s lω 495.9in. s l 8 in. 6.0 ad s P ω 43.6ft s ( 6.0 ad s)k 003 The McGaw-Hill Companies, Inc. ll ights eseed

19 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.7 The angula acceleation of the connecting od and the acceleation of point will be detemined fom a a + a a + a + a ( ) ( ) t n The acceleation of is detemined fom the gien otation speed of. Cank G of the engine system has a constant clockwise angula elocity of 000 pm. Fo the cank position shown, detemine the angula acceleation of the connecting od and the acceleation of point. The diections of the acceleations a, ( a ), and ( a ) ae t n detemined fom the geomety. Component equations fo acceleation of point ae soled simultaneously fo acceleation of and angula acceleation of the connecting od. 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.7 The angula acceleation of the connecting od and the acceleation of point will be detemined fom a a + a a + a + a ( ) ( ) t n The acceleation of is detemined fom the gien otation speed of. ω α a 000pm 09.4ad s constant 0 ω ( 3 ft)( 09.4ad s) 10,96ft s 1 ( 10,96ft s )( cos40 i sin j ) a The McGaw-Hill Companies, Inc. ll ights eseed

20 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.7 The diections of the acceleations a detemined fom the geomety. a ma i ( a ), and ( a ) t n 003 The McGaw-Hill Companies, Inc. ll ights eseed , Fom Sample Poblem 15.3, ω 6.0 ad/s, β o. ( a ) ( ) ( 8 ω ft )( 6.0ad s) 563ft s n 1 ( a ) ( 563ft s )( cos13.95 i + sin j ) n ( a ) ( ) α ( 8 ft) α α t 1 The diection of (a / ) t is known but the sense is not known, a 0.667α ± sin i ± cos j ( ) ( )( ) t ae Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.7 Component equations fo acceleation of point ae soled simultaneously. a a + a a + x components: ( a ) + ( a ) t n a,96cos cos α sin y components: ,96sin sin α cos ( 9940ad s ) α k a ( 990ft s )i 003 The McGaw-Hill Companies, Inc. ll ights eseed

21 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.8 The angula elocities ae detemined by simultaneously soling the component equations fo + In the position shown, cank has a constant angula elocity ω 1 0 ad/s counteclockwise. etemine the angula elocities and angula acceleations of the connecting od and cank E. The angula acceleations ae detemined by simultaneously soling the component equations fo a a + a 003 The McGaw-Hill Companies, Inc. ll ights eseed Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.8 The angula elocities ae detemined by simultaneously soling the component equations fo + ωe ωek 17ω Ei 17ω E j ω 0k ( 8i + 14 j ) 80i j ω ωk ( 1i + 3 j ) 3ω i + 1ω j ( 17i + 17 j ) x components: y components: 17ω 80 3 E ω 17 ω E ω ωe ω ( 9.33ad s) k ( 11.9ad s)k 003 The McGaw-Hill Companies, Inc. ll ights eseed

22 Seenth Vecto Mechanics fo Enginees: ynamics Sample Poblem 15.8 The angula acceleations ae detemined by simultaneously soling the component equations fo a a + a 003 The McGaw-Hill Companies, Inc. ll ights eseed a αe ωe αek 17α Ei 17α E j + 170i 170 j a α ω 0 ( 0) ( 8i + 14 j ) 300i j a α ω α k ( 1i + 3 j ) ( 9.33) ( 1i + 3 j ) 3α i + 1α j 10,30i 580 j ( 17i + 17 j ) ( 11.9) ( 17i + 17 j ) x components: 17α E + 3α 15, 690 y components: 17α E 1α 6010 α 645 ad s k α 809ad s ( ) ( )k E Seenth Vecto Mechanics fo Enginees: ynamics Motion bout a Fixed Point 003 The McGaw-Hill Companies, Inc. ll ights eseed. The most geneal displacement of a igid body with a fixed point O is equialent to a otation of the body about an axis though O. With the instantaneous axis of otation and angula elocity ω, the elocity of a paticle P of the body is d ω dt and the acceleation of the paticle P is dω a α + ω ( ω ) α. dt The angula acceleation epesents the elocity of the tip of α ω. s the ecto ω moes within the body and in space, it geneates a body cone and space cone which ae tangent along the instantaneous axis of otation. ngula elocities hae magnitude and diection and obey paallelogam law of addition. They ae ectos

23 Seenth Vecto Mechanics fo Enginees: ynamics Geneal Motion Fo paticles and of a igid body, + Paticle is fixed within the body and motion of the body elatie to X Y Z is the motion of a body with a fixed point + ω Similaly, the acceleation of the paticle P is a a + a a + α + ω ω ( ) Most geneal motion of a igid body is equialent to: - a tanslation in which all paticles hae the same elocity and acceleation of a efeence paticle, and - of a motion in which paticle is assumed fixed. 003 The McGaw-Hill Companies, Inc. ll ights eseed