E g n i g n i e n e e r e in i g n g M e M c e h c a h n a i n c i s c : s D yn y a n m a i m c i s 15-1

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1 Engineeing Mechanics: Dynamics Intoduction Kinematics of igid bodies: elations between time and the positions, velocities, and acceleations of the paticles foming a igid body. Classification of igid body motions: - tanslation: ectilinea tanslation cuvilinea tanslation - Fig (a) - otation about a fixed axis - Fig (b) - geneal plane motion 15-1

2 Engineeing Mechanics: Dynamics Engineeing Mechanics: Dynamics Rotation About a Fixed Axis. Conside the motion of a igid body in a plane pependicula to the axis of otation. Velocity of any point P of the slab, v k v 15 - Acceleation of any point P of the slab, k a α α + Resolving the acceleation into tangential and nomal components, α α a a a k a n n t t

3 Engineeing Mechanics: Dynamics Equations Defining the Rotation of a Rigid Body About a Fixed Axis Motion of a igid body otating aound a fixed axis is often specified by the type of angula acceleation. Recall α dθ o dt dt d d θ d dt dt dθθ dθ Unifom Rotation, α 0: θ θ + t 0 Unifomly Acceleated Rotation, α constant: + αt 0 0 θ θ + t αt ( θ ) + α θ

4 Engineeing Mechanics: Dynamics Two Rotating Bodies in Contact 15-4

5 Engineeing Mechanics: Dynamics Two Rotating Bodies in Contact-Same Velocities & Tangential Acceleation 15-5

6 Engineeing Mechanics: Dynamics Two Rotating Bodies in Contact Diffeent Nomal Acceleations 15-6

7 Engineeing Mechanics: Dynamics 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 A and B to A and B can be divided into two pats: - tanslation to A and B 1 - otation of B about A to B

8 Engineeing Mechanics: Dynamics Geneal Plane Motion 15-8

9 Engineeing Mechanics: Dynamics Absolute and Relative Velocity in Plane Motion Any plane motion can be eplaced by a tanslation of an abitay efeence point A and a simultaneous otation about A. vb va + vb A v k v B A AB B A v B v + k A AB 15-9

10 Engineeing Mechanics: Dynamics Absolute and Relative Velocity in Plane Motion- Example Assuming that the velocity v A of end A is known, wish to detemine the velocity v B of end B and the angula velocity in tems of v A, l, and θ. The diection of v B and v B/A ae known. Complete the velocity diagam

11 Engineeing Mechanics: Dynamics Absolute and Relative Velocity in Plane Motion Selecting point B as the efeence point and solving fo the velocity v A of end A and the angula velocity leads to an equivalent velocity tiangle. v A/B has the same magnitude but opposite sense of v B/A. The sense of the elative velocity is dependent on the choice of efeence point. Angula velocity of the od in its otation about B is the same as its otation about A. Angula velocity is not dependent on the choice of efeence point

12 Engineeing Mechanics: Dynamics 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 A and a otation about A with an angula velocity that is independent of the choice of A. The same tanslational and otational velocities at A ae obtained by allowing the slab to otate with the same angula velocity about the point C on a pependicula to the velocity at A. The velocity of all othe paticles in the slab ae the same as oiginally defined since the angula velocity and tanslational velocity at A ae equivalent. As fa as the velocities ae concened, the slab seems to otate about the instantaneous cente of otation C. 15-1

13 Engineeing Mechanics: Dynamics Instantaneous Cente of Rotation in Plane Motion If the velocity at two points A and B ae known, the instantaneous cente of otation lies at the intesection of the pependiculas to the velocity vectos though A and B. If the velocity vectos ae paallel, the instantaneous cente of otation is at infinity and the angula velocity is zeo. If the velocity vectos at A and B ae pependicula to the line AB, the instantaneous cente of otation lies at the intesection of the line AB with the line joining the extemities of the velocity vectos at A and B. If the velocity magnitudes ae equal, the instantaneous cente of otation is at infinity and the angula velocity is zeo

14 Engineeing Mechanics: Dynamics Absolute and Relative Acceleation in Plane Motion Absolute acceleation of a paticle of the slab, a a + a B A B A Relative acceleation a B A associated with otation about A includes tangential and nomal components, ( ab A ) α k t AB ( ab A ) α t ( ab A ) n AB a ( ) B A n 15-14

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Uniform Rectilinear Motion Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 11-1 Engineeing Mechanics : Dynamics

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