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Kinematics of Kinematics of

Rigid BodiesRigid Bodies

Reference:Beer, Ferdinand P. et al, Vector Mechanics for Engineers : Dynamics, 8th Edition, Mc GrawHillHibbeler R.C., Engineering Mechanics: Dynamics, 11th Edition, Prentice Hall

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Introduction

• Classification of rigid body motions:

- general motion

- motion about a fixed point

- general plane motion

- rotation about a fixed axis

- curvilinear translation

- rectilinear translation

- translation:

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An example of bodies undergoing the three types of motion is shown in this mechanism.

C

E

A

D

B

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Translation

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Rotation About a Fixed Axis. Velocity

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Rotation About a Fixed Axis. Acceleration

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Rotation About a Fixed Axis.

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Equations Defining the Rotation of a Rigid Body About a Fixed Axis

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Given: Starting from rest when s = 0, pulley A (rA = 50 mm) is given a constant angular acceleration, αA = 6 rad/s2. Pulley C (rC = 150 mm) has an inner hub D (rD = 75 mm) which is fixed to C and turns with it.

Find: The speed of block B when it has risen s = 6 m.

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General Plane Motion

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Absolute and Relative Velocity in Plane Motion

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Absolute and Relative Velocity in Plane Motion

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Absolute and Relative Velocity in Plane Motion

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Sample Problem 15.2

The double gear rolls on the stationary lower rack: the velocity of its center is 1.2 m/s.

Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear.

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Sample Problem 15.3

The crank AB has a constant clockwise angular velocity of 2000 rpm.

For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P.

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Given: Block A is moving down at 2 m/s.

Find: The velocity of B at the instant θ = 45°.

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Given: Collar C is moving downward with a velocity of 2 m/s.

Find: The angular velocities of CB and AB at this instant.

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Instantaneous Center of Rotation in Plane Motion

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Instantaneous Center of Rotation in Plane Motion

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Sample Problem 15.4

The double gear rolls on the stationary lower rack: the velocity of its center is 1.2 m/s.

Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear.

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Sample Problem 15.5

The crank AB has a constant clockwise angular velocity of 2000 rpm.

For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P.

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Given: A linkage undergoing motion as shown. The velocity of the block, vD, is 3 m/s.

Find: The angular velocities of links AB and BD.

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Absolute and Relative Acceleration in Plane Motion

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Absolute and Relative Acceleration in Plane Motion

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Analysis of Plane Motion in Terms of a Parameter

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Sample Problem

The center of the double gear has a velocity and acceleration tothe right of 1.2 m/s and 3 m/s2, respectively. The lower rack is stationary.

Determine (a) the angular acceleration of the gear, and (b) the acceleration of points B, C, and D.

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Given: Point A on rod AB has an acceleration of 3 m/s2 and a velocity of 2 m/s at the instant the rod becomes horizontal.

Find: The angular acceleration of the rod at this instant.

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Given: The disk is rotating with ω = 3 rad/s, α = 8 rad/s2 at this instant.

Find: The acceleration at point B, and the angular velocity and acceleration of link AB.

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Sample Problem 15.7

Crank AG of the engine system has a constant clockwise angular velocity of 2000 rpm.

For the crank position shown, determine the angular acceleration of the connecting rod BD and the acceleration of point D.

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Sample Problem 15.7

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Sample Problem 15.8

In the position shown, crank AB has a constant angular velocity ω1 = 20 rad/s counterclockwise.

Determine the angular velocities and angular accelerations of the connecting rod BD and crank DE.

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Example Problems

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Homework

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