11-kinematics Of Rigid Bodies.pdf 1l7339

Engineering Mechanics: Dynamics Introduction • Kinematics of rigid bodies: relations between time and the positions, velocities, and accelerations of the particles forming a rigid body. • Classification of rigid body motions: - translation: • rectilinear translation • curvilinear translation - Fig (a) - rotation about a fixed axis - Fig (b) - general plane motion

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Engineering Mechanics: Dynamics Rotation About a Fixed Axis. • Consider the motion of a rigid body in a plane perpendicular to the axis of rotation. • Velocity of any point P of the slab, r r r r r v = ω × r = ωk × r v = rω • Acceleration of any point P of the slab, r r r r r r a = α × r + ω ×ω × r r r r = α k × r − ω 2r • Resolving the acceleration into tangential and normal components, r r r at = αk × r a t = rα r r an = −ω 2 r a n = rω 2 15 - 2

Engineering Mechanics: Dynamics Equations Defining the Rotation of a Rigid Body About a Fixed Axis • Motion of a rigid body rotating around a fixed axis is often specified by the type of angular acceleration. • Recall ω =

dθ dt

or

dt =

dθ

ω

dω d 2θ dω α= = 2 =ω dt dθ dt

• Uniform Rotation, α = 0:

θ = θ 0 + ωt • Uniformly Accelerated Rotation, α = constant: ω = ω0 + αt

θ = θ 0 + ω 0t + 12 α t 2 ω 2 = ω 02 + 2α (θ − θ 0 ) 15 - 3

Engineering Mechanics: Dynamics Two Rotating Bodies in

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Engineering Mechanics: Dynamics Two Rotating Bodies in -Same Velocities & Tangential Acceleration

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Engineering Mechanics: Dynamics Two Rotating Bodies in – Different Normal Accelerations

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Engineering Mechanics: Dynamics General Plane Motion

• General plane motion is neither a translation nor a rotation. • General plane motion can be considered as the sum of a translation and rotation. • Displacement of particles A and B to A2 and B2 can be divided into two parts: - translation to A2 and B1′ - rotation of B1′ about A2 to B2 15 - 7

Engineering Mechanics: Dynamics General Plane Motion

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Engineering Mechanics: Dynamics Absolute and Relative Velocity in Plane Motion

• Any plane motion can be replaced by a translation of an arbitrary reference point A and a simultaneous rotation about A. r r r vB = v A + vB A

r r r vB A = ωk × rAB

vB A = rω

r r r r vB = vA + ωk × rAB 15 - 9

Engineering Mechanics: Dynamics Absolute and Relative Velocity in Plane MotionExample

• Assuming that the velocity vA of end A is known, wish to determine the velocity vB of end B and the angular velocity ω in of vA, l, and θ. •The direction of vB and vB/A are known. Complete the velocity diagram.

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Engineering Mechanics: Dynamics 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 ω leads 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. • Angular velocity ω of the rod in its rotation about B is the same as its rotation about A. Angular velocity is not dependent on the choice of reference point. 15 - 11

Engineering Mechanics: Dynamics Instantaneous Center of Rotation in Plane Motion • Plane motion of all particles in a slab can always be replaced by the translation of an arbitrary point A and a rotation about A with an angular velocity that is independent of the choice of A. • The same translational and rotational velocities at A are obtained by allowing the slab to rotate with the same angular velocity about the point C on a perpendicular to the velocity at A. • The velocity of all other particles in the slab are the same as originally defined since the angular velocity and translational velocity at A are equivalent. • As far as the velocities are concerned, the slab seems to rotate about the instantaneous center of rotation C.

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Engineering Mechanics: Dynamics Instantaneous Center of Rotation in Plane Motion • If the velocity at two points A and B are known, the instantaneous center of rotation lies at the intersection of the perpendiculars to the velocity vectors through A and B . • If the velocity vectors are parallel, the instantaneous center of rotation is at infinity and the angular velocity is zero. • If the velocity vectors at A and B are perpendicular to the line AB, the instantaneous center of rotation lies at the intersection of the line AB with the line ing the extremities of the velocity vectors at A and B. • If the velocity magnitudes are equal, the instantaneous center of rotation is at infinity and the angular velocity is zero. 15 - 13

Engineering Mechanics: Dynamics Absolute and Relative Acceleration in Plane Motion

• Absolute acceleration of a particle of the slab, r r r aB = a A + aB A r • Relative acceleration a B A associated with rotation about A includes tangential and normal components,

(arB (arB

r r A )t = α k × rAB 2r ) = − ω rAB A n

(a B A )t = rα (a B A )n = rω 2 15 - 14