CHAPTER VECTOR MECHANICS FOR ENGINEERS: 15DYNAMICS
Seventh EditionCHAPTER15VECTOR MECHANICS FOR ENGINEERS:DYNAMICSFerdinand P. BeerE. Russell Johnston, Jr.Lecture Notes:J. Walt OlerTexas Tech UniversityKinematics ofRigid Bodies 2003 The McGraw-Hill Companies, Inc. All rights reserved.SeventhEditionVector Mechanics for Engineers: DynamicsContentsIntroductionTranslationRotation About a Fixed Axis: VelocityRotation About a Fixed Axis: AccelerationRotation About a Fixed Axis:Representative SlabEquations Defining the Rotation of a RigidBody About a Fixed AxisSample Problem 5.1General Plane MotionAbsolute and Relative Velocity in PlaneMotionSample Problem 15.2Sample Problem 15.3Instantaneous Center of Rotation in PlaneMotionSample Problem 15.4Sample Problem 15.5Absolute and Relative Acceleration in PlaneMotionAnalysis of Plane Motion in Terms of aParameterSample Problem 15.6Sample Problem 15.7Sample Problem 15.8Rate of Change With Respect to a RotatingFrameCoriolis AccelerationSample Problem 15.9Sample Problem 15.10Motion About a Fixed PointGeneral MotionSample Problem 15.11Three Dimensional Motion. CoriolisAccelerationFrame of Reference in General MotionSample Problem 15.15 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 21
SeventhEditionVector Mechanics for Engineers: DynamicsIntroduction Kinematics of rigid bodies: relations betweentime and the positions, velocities, andaccelerations of the particles forming a rigidbody. Classification of rigid body motions:- translation: rectilinear translation curvilinear translation- rotation about a fixed axis- general plane motion- motion about a fixed point- general motion 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 3SeventhEditionVector Mechanics for Engineers: DynamicsTranslation Consider rigid body in translation:- direction of any straight line inside thebody is constant,- all particles forming the body move inparallel lines. For any two particles in the body,rr rrB rA rB A Differentiating with respect to time,rr rrr&B r&A r&B A r&ArrvB v AAll particles have the same velocity. Differentiating with respect to time again,&rr&B &rr&A &rr&B A r&r&ArraB a AAll particles have the same acceleration. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 42
SeventhEditionVector Mechanics for Engineers: DynamicsRotation About a Fixed Axis. Velocity Consider rotation of rigid body about afixed axis AA’rr Velocity vector v dr dt of the particle P istangent to the path with magnitude v ds dt s ( BP ) θ (r sin φ ) θv ds θ rθ& sin φ lim (r sin φ )dt t 0 t The same result is obtained fromrr dr r rv ω rdtrrrω ω k θ&k angular velocity 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 5SeventhEditionVector Mechanics for Engineers: DynamicsRotation About a Fixed Axis. Acceleration Differentiating to determine the acceleration,rr dv d v r (ω r )a dt dtrrdω r r dr r ω dtdtrdω r r r r ω vdtrdω r α angular acceleration dtrrr α k ω& k θ&&k Acceleration of P is combination of twovectors,r r r r r ra α r ω (ω r )r rα r tangential acceleration componentr r rω (ω r ) radial acceleration component 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 63
SeventhEditionVector Mechanics for Engineers: DynamicsRotation About a Fixed Axis. Representative Slab Consider the motion of a representative slab ina plane perpendicular to the axis of rotation. Velocity of any point P of the slab,r rr r rv ω r ωk rv rω Acceleration of any point P of the slab,r r r r r ra α r ω (ω r )r rr α k r ω 2r Resolving the acceleration into tangential andnormal components,r rrat αk ra t rαr2ran ω ra n rω 2 2003 The McGraw-Hill Companies, Inc Companies, Inc. All rights reserved.vAv A cosθv B A lωω vAl cosθ15 - 147
SeventhEditionVector Mechanics for Engineers: DynamicsAbsolute and Relative Velocity in Plane Motion Selecting point B as the reference point and solving for the velocity vA of end Aand the angular velocity ω leads to an equivalent velocity triangle. vA/B has the same magnitude but opposite sense of vB/A. The sense of therelative velocity is dependent on the choice of reference point. Angular velocity ω of the rod in its rotation about B is the same as its rotationabout A. Angular velocity is not dependent on the choice of reference point. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 15SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.2SOLUTION: The displacement of the gear center inone revolution is equal to the outercircumference. Relate the translationaland angular displacements. Differentiateto relate the translational and angularvelocities.The double gear rolls on thestationary lower rack: the velocity ofits center is 1.2 m/s. The velocity for any point P on the gearmay be written asDetermine (a) the angular velocity ofthe gear, and (b) the velocities of theupper rack R and point D of the gear. 2003 The McGraw-Hill Companies, Inc. All rights reserved.rrrvP v A vPAr rr v A ωk rPAEvaluate the velocities of points B and D.15 - 168
SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.2SOLUTION: The displacement of the gear center in one revolution isequal to the outer circumference.For xA 0 (moves to right), ω 0 (rotates clockwise).xAθ 2π r2πyx A r1θDifferentiate to relate the translational and angularvelocities.xv A r1ωω vA1.2 m s r10.150 mrrrω ωk (8 rad s )k 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 17SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.2rrr For any point P on the gear, vP v A vPAVelocity of the upper rack is equal tovelocity of point B:r rrrrvR vB v A ωk rB Arrr (1.2 m s )i (8 rad s )k (0.10 m ) jrr (1.2 m s )i (0.8 m s )irrvR (2 m s )ir rr v A ωk rPAVelocity of the point D:r rrrvD v A ωk rD Arrr (1.2 m s )i (8 rad s )k ( 0.150 m )irrrvD (1.2 m s )i (1.2 m s ) jvD 1.697 m s 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 189
SeventhEditionVector Mechanics for Engineers: DynamicsAbsolute and Relative Acceleration in Plane Motion Absolute acceleration of a particle of the slab,rrraB a A aB Ar Relative acceleration a B A associated with rotation about A includestangential and normal components,r(arB A ) α k rrB A (a B A ) rα(raBtt)A n r ω 2 rB A(a B A )n rω 2 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 19SeventhEditionVector Mechanics for Engineers: DynamicsAbsolute and Relative Acceleration in Plane Motionrr Given a A and v A ,rrdetermine a B and α .rrraB a A aB Arrr a A (a B A ) (a Bn)A tr Vector result depends on sense of a A and therelative magnitudes of a A and (a B A )n Must also know angular velocity ω. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 2010
SeventhEditionVector Mechanics for Engineers: DynamicsAbsolute and Relative Acceleration in Plane Motionrrr Write a B a A a B x components: Ain terms of the two component equations,0 a A lω 2 sin θ lα cos θy components: a B lω 2 cos θ lα sin θ Solve for aB and α. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 21SeventhEditionVector Mechanics for Engineers: DynamicsAnalysis of Plane Motion in Terms of a Parameter In some cases, it is advantageous to determine theabsolute velocity and accelera at B and D based on their rotationabout C.vR vB rBω (0.25 m )(8 rad s )rrvR (2 m s )irD (0.15 m ) 2 0.2121 mvD rDω (0.2121 m )(8 rad s )vD 1.697 m srrrvD (1.2i 1.2 j )(m s ) 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 27SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.5SOLUTION: Determine the velocity at B from thegiven crank rotation data.The crank AB has a constant clockwiseangular velocity of 2000 rpm.For the crank position indicated,determine (a) the angular velocity ofthe connecting rod BD, and (b) thevelocity of the piston P. The direction of the velocity vectors at Band D are known. The instantaneouscenter of rotation is at the intersection ofthe perpendiculars to the velocitiesthrough B and D. Determine the angular velocity about thecenter of rotation based on the velocityat B. Calculate the velocity at D based on itsrotation about the instantaneous centerof rotation. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 2814
SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.5SOLUTION: From Sample Problem 15.3,rrrvB (403.9i 481.3 j )(in. s )vB 628.3 in. sβ 13.95 The instantaneous center of rotation is at the intersectionof the perpendiculars to the velocities through B and D.γ B 40 β 53.95 γ D 90 β 76.05 CD8 in.BC sin 76.05 sin 53.95 sin50 BC 10.14 in. CD 8.44 in. Determine the angular velocity about the center ofrotation based on the velocity at B.vB (BC )ω BDω BD vB 628.3 in. s BC 10.14 in.ω BD 62.0 rad s Calculate the velocity at D based on its rotation aboutthe instantaneous center of rotation.vD (CD )ω BD (8.44 in.)(62.0 rad s )vP vD 523 in. s 43.6 ft s 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 29SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.6SOLUTION: The expression of the gear position as afunction of θ is differentiated twice todefine the relationship between thetranslational and angular accelerations.The center of the double gear has avelocity and acceleration to the right of1.2 m/s and 3 m/s2, respectively. Thelower rack is stationary. The acceleration of each point on thegear is obtained by adding theacceleration of the gear center and therelative accelerations with respect to thecenter. The latter includes normal andtangential acceleration components.Determine (a) the angular accelerationof the gear, and (b) the acceleration ofpoints B, C, and D. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 3015
SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.6SOLUTION: The expression of the gear position as a function of θis differentiated twice to define the relationshipbetween the translational and angular accelerations.x A r1θv A r1θ& r1ωω vA1.2 m s 8 rad sr10.150 ma A r1θ&& r1αα aA3 m s2 r10.150 mrr()rα α k 20 rad s 2 k 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 31SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.6 The acceleration of each pointis obtained by adding theacceleration of the gear centerand the relative accelerationswith respect to the center.The latter includes normal andtangential accelerationcomponents.() ()rrrrrra B a A aB A a A a B A aB Atnr rrr a A α k rB A ω 2 rB Arrrr 3 m s 2 i 20 rad s 2 k (0.100 m ) j (8 rad s )2 ( 0.100 m ) jrrr 3 m s 2 i 2 m s 2 i 6.40 m s 2 j(() () ()) (()) ()rrraB 5 m s 2 i 6.40 m s 2 j 2003 The McGraw-Hill Companies, Inc. All rights reserved.aB 8.12 m s 215 - 3216
SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.6r rrr a A α k rC A ω 2 rC Arrrr 3 m s 2 i 20 rad s 2 k ( 0.150 m ) j (8 rad s )2 ( 0.150 m ) jrrr 3 m s 2 i 3 m s 2 i 9.60 m s 2 jrrac 9.60 m s 2 jr rrrrrraD a A aD A a A α k rD A ω 2 rD Arrrr 3 m s 2 i 20 rad s 2 k ( 0.150 m ) i (8 rad s )2 ( 0.150m ) irrr 3 m s 2 i 3 m s 2 j 9.60 m s 2 irrraD 12.6 m s 2 i 3 m s 2 jaD 12.95 m s 2rrraC a A aCA(() () ((() () ()() ())() () ( 2003 The McGraw-Hill Companies, Inc. All rights reserved.)))15 - 33SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.3SOLUTION: Will determine the absolute velocity ofpoint D withrrrvD vB vD Br The velocity v B is obtained from thegiven crank rotation data.velocity vrDThe crank AB has a constant clockwise The directions of the absoluterand the relative velocity v D B areangular velocity of 2000 rpm.determined from the problem geometry.For the crank position indicated, The unknowns in the vector expressiondetermine (a) the angular velocity ofare the velocity magnitudes v D and v D Bthe connecting rod BD, and (b) thewhich may be determined from thevelocity of the piston P.corresponding vector triangle. The angular velocity of the connectingrod is calculated from v D B . 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 3417
SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.3SOLUTION: Will determine the absolute velocity of point D withrrrvD vB vD Br The velocity vB is obtained from the crank rotation data.ω AB 2000 rev min 2π rad 209.4 rad smin 60 s rev vB ( AB )ω AB (3 in.)(209.4 rad s )The velocity direction is as shown.r The direction of the absolute velocity vD is horizontal.rThe direction of the relative velocity vD B isperpendicular to BD. Compute the angle between thehorizontal and the connecting rod from the law of sines.sin 40 sin β 8 in.3 in.β 13.95 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 35SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.3 Determine the velocity magnitudes vD and vDfrom the vector triangle.BvD B628.3 in. svD sin 53.95 sin 50 sin76.05 vD 523.4 in. s 43.6 ft srrrvD vB vDBvDB 495.9 in. svDB lω BDvD495.9 in. sB 8 in.l 62.0 rad sω BD vP vD 43.6 ft s 2003 The McGraw-Hill Companies, Inc. All rights reserved.rrω BD (62.0 rad s )k15 - 3618
SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.7SOLUTION: The angular acceleration of theconnecting rod BD and the accelerationof point D will be determined fromrrrrrraD aB aD B aB aD B aD B()t ()n The acceleration of B is determined fromthe given rotation speed of AB. The directions of the accelerationsrrra D , a D B , and a D B aretndetermined from the geometry.Crank AG of the engine system has aconstant clockwise angular velocity of2000 rpm.(For the crank position shown,determine the angular acceleration ofthe connecting rod BD and theacceleration of point D.)() Component equations for accelerationof point D are solved simultaneously foracceleration of D and angularacceleration of the connecting rod. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 37SeventhEditionVector Mechanics for Engineers: DynamicsSample Problem 15.7SOLUTION: The angular acceleration of the connecting rod BD andthe acceleration of point D will be determined fromrrrrrra D a B aD B aB aD B aD B()t ()n The acceleration of B is determined from the given rotationspeed of AB.ω AB 2000 rpm 209.4 rad s constantα AB 02aB rω AB ((123 ft )(209.4 rad s)2 10,962 ft s2)rrraB 10,962 ft s 2 ( cos 40 i sin 40 of rotation. Angular velocities have magnitude and direction andobey parallelogram law of addition. They are vectors. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 4422
SeventhEditionVector Mechanics for Engineers: DynamicsGeneral Motion For particles A and B of a rigid body,rrrvB v A vB A Particle A is fixed within the body and motion ofthe body relative to AX’Y’Z’ is the motion of abody with a fixed pointrrr rv B v A ω rB A Similarly, the acceleration of the particle P isrrraB a A aB Arr rr r r a A α rB A ω (ω rB A ) Most general motion of a rigid body is equivalent to:- a translation in which all particles have the samevelocity and acceleration of a reference particle A, and- of a motion in which particle A is assumed fixed. 2003 The McGraw-Hill Companies, Inc. All rights reserved.15 - 4523
Absolute and Relative Velocity in Plane Motion Selecting point B as the reference point and solving for the velocity vA of end A and the angular velocity ωleads to an equivalent velocity triangle. vA/B has the same magnitude but opposite sense of vB/A. The sense of the relative velocity is dependent on the choice of reference point.
Why Vector processors Basic Vector Architecture Vector Execution time Vector load - store units and Vector memory systems Vector length - VLR Vector stride Enhancing Vector performance Measuring Vector performance SSE Instruction set and Applications A case study - Intel Larrabee vector processor
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
PEAK PCAN-USB, PEAK PCAN-USB Pro, PEAK PCAN-PCI, PEAK PCAN-PCI Express, Vector CANboard XL, Vector CANcase XL, Vector CANcard X, Vector CANcard XL, Vector CANcard XLe, Vector VN1610, Vector VN1611, Vector VN1630, Vector VN1640, Vector VN89xx, Son-theim CANUSBlight, Sontheim CANUSB, S
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
10 tips och tricks för att lyckas med ert sap-projekt 20 SAPSANYTT 2/2015 De flesta projektledare känner säkert till Cobb’s paradox. Martin Cobb verkade som CIO för sekretariatet för Treasury Board of Canada 1995 då han ställde frågan
service i Norge och Finland drivs inom ramen för ett enskilt företag (NRK. 1 och Yleisradio), fin ns det i Sverige tre: Ett för tv (Sveriges Television , SVT ), ett för radio (Sveriges Radio , SR ) och ett för utbildnings program (Sveriges Utbildningsradio, UR, vilket till följd av sin begränsade storlek inte återfinns bland de 25 största
Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
