CHAPTER VECTOR MECHANICS FOR ENGINEERS: 15DYNAMICS

[Pages:23]Seventh Edition

CHAPTER VECTOR MECHANICS FOR ENGINEERS:

15 DYNAMICS

Ferdinand P. Beer E. Russell Johnston, Jr.

Lecture Notes: J. Walt Oler Texas Tech University

Kinematics of Rigid Bodies

Seventh Edition

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Vector Mechanics for Engineers: Dynamics

Contents

Introduction Translation Rotation About a Fixed Axis: Velocity Rotation About a Fixed Axis: Acceleration Rotation About a Fixed Axis:

Representative Slab Equations Defining the Rotation of a Rigid

Body About a Fixed Axis Sample Problem 5.1 General Plane Motion Absolute and Relative Velocity in Plane

Motion Sample Problem 15.2 Sample Problem 15.3 Instantaneous Center of Rotation in Plane

Motion Sample Problem 15.4 Sample Problem 15.5

Absolute and Relative Acceleration in Plane Motion

Analysis of Plane Motion in Terms of a Parameter

Sample Problem 15.6 Sample Problem 15.7 Sample Problem 15.8 Rate of Change With Respect to a Rotating

Frame Coriolis Acceleration Sample Problem 15.9 Sample Problem 15.10 Motion About a Fixed Point General Motion Sample Problem 15.11 Three Dimensional Motion. Coriolis

Acceleration Frame of Reference in General Motion Sample Problem 15.15

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Vector Mechanics for Engineers: 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

- rotation about a fixed axis - general plane motion - motion about a fixed point - general motion

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Vector Mechanics for Engineers: Dynamics

Translation

? Consider rigid body in translation: - direction of any straight line inside the body is constant, - all particles forming the body move in parallel lines.

? FrorBr a=nryrAtw+ orrBpaArticles in the body,

? Differentiating with respect to time, rr&B = rr&A + rr&B A = rr&A vrB = vrA

All particles have the same velocity.

? Differentiating with respect to time again, &rr&B = &rr&A + &rr&B A = r&r&A arB = arA

All particles have the same acceleration.

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Vector Mechanics for Engineers: Dynamics

Rotation About a Fixed Axis. Velocity

? Consider rotation of rigid body about a fixed axis AA'

? Velocity vector vr = drr dt of the particle P is tangent to the path with magnitude v = ds dt

s = (BP) = (r sin )

v = ds = lim (r sin ) = r& sin

dt t0

t

? The same result is obtained from vr = drr = r ? rr r = dtkr = &kr = angular velocity

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Vector Mechanics for Engineers: Dynamics

Rotation About a Fixed Axis. Acceleration

? Difarfe=rednvrtia=tindg(tvo?drert)ermine the acceleration,

=

ddtr

?

dt rr +

r

?

drr

=

ddtr

?

rr

+

r

?

dt vr

dt

?

dr = r = angular acceleration

dt

=

r k

=

&

r k

=

&&kr

? Acceleration of P is combination of two vectors, ar = r ? rr + r ? (r ? rr) r ? rr = tangential acceleration component r ? (r ? rr) = radial acceleration component

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Vector Mechanics for Engineers: Dynamics

Rotation About a Fixed Axis. Representative Slab

? Consider the motion of a representative slab in a plane perpendicular to the axis of rotation.

? Velocity of any point P of the slab,

vr

=

r

?

rr

=

r k

?

rr

v = r

? Acceleration of any point P of the slab,

ar = r ? rr + r ? (r ? rr)

=

r k

?

rr

-

2

rr

? Resolving the acceleration into tangential and

normal components,

aarrtn

= kr ? rr = - 2rr

at = r an = r 2

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Vector Mechanics for Engineers: 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 = d dt

or

dt

=

d

=

d dt

=

d 2 dt 2

=

d d

? Uniform Rotation, = 0: =0 +t

? Uniformly Accelerated Rotation, = constant:

= 0 +t

=0

+0t +

1 2

t

2

2 = 02 + 2 ( -0 )

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Vector Mechanics for Engineers: Dynamics

Sample Problem 5.1

SOLUTION:

? Due to the action of the cable, the tangential velocity and acceleration of D are equal to the velocity and acceleration of C. Calculate the initial angular velocity and acceleration.

Cable C has a constant acceleration of 9 in/s2 and an initial velocity of 12 in/s, both directed to the right.

Determine (a) the number of revolutions of the pulley in 2 s, (b) the velocity and change in position of the load B after 2 s, and (c) the acceleration of the point D on the rim of the inner pulley at t = 0.

? Apply the relations for uniformly accelerated rotation to determine the velocity and angular position of the pulley after 2 s.

? Evaluate the initial tangential and normal acceleration components of D.

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Vector Mechanics for Engineers: Dynamics

Sample Problem 5.1

SOLUTION:

? The tangential velocity and acceleration of D are equal to the

velocity and acceleration of C.

(vrD )0 = (vrC )0 = 12in. s

(vD )0 = r0

0

=

(vD )0

r

= 12 3

=

4 rad

s

(arD )t = arC = 9in. s (aD )t = r

= (aD )t = 9 = 3rad s2

r3

? Apply the relations for uniformly accelerated rotation to

determine velocity and angular position of pulley after 2 s.

( ) = 0 + t = 4rad s + 3rad s2 (2 s) = 10 rad s

( )

= 0t

+

1 2

t

2

=

(4 rad

s)(2 s)+

1 2

3 rad

s2

(2 s)2

= 14 rad

N

=

(14

rad)

1 rev 2 rad

=

number

of

revs

vB = r = (5 in.)(10 rad s)

yB = r = (5 in.)(14 rad)

N = 2.23rev

vrB = 50in. s yB = 70 in.

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Vector Mechanics for Engineers: Dynamics

Sample Problem 5.1

? Evaluate the initial tangential and normal acceleration components of D.

(arD )t = arC = 9in. s (aD )n = rD02 = (3 in.)(4 rad s)2 = 48in s2 (arD )t = 9in. s2 (arD )n = 48in. s2

Magnitude and direction of the total acceleration,

aD = (aD )t2 + (aD )2n

= 92 + 482

aD = 48.8in. s2

tan

=

(aD )n (aD )t

= 48 9

= 79.4?

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Vector Mechanics for Engineers: Dynamics

General Plane Motion

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? 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

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Vector Mechanics for Engineers: Dynamics

Absolute and Relative Velocity in Plane Motion

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? Any plane motion can be replaced by a translation of an

arbitrary reference point A and a simultaneous rotation

about A.

vrB = vrA + vrB A

vrB

A

=

r k

?

rrB

A

vrB

=

vrA

+

r k

?

rrB

A

vB A = r

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Vector Mechanics for Engineers: Dynamics

Absolute and Relative Velocity in Plane Motion

? Assuming that the velocity vA of end A is known, wish to determine the velocity vB of end B and the angular velocity in terms of vA, l, and .

? The direction of vB and vB/A are known. Complete the velocity diagram.

vB = tan vA vB = vA tan

vA vB A

=

vA l

=

cos

= vA l cos

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Vector Mechanics for Engineers: Dynamics

Absolute and Relative Velocity in Plane Motion

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? Selecting point B as the reference point and solving for the velocity vA of end A and the angular velocity 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 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.

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Vector Mechanics for Engineers: Dynamics

Sample Problem 15.2

SOLUTION:

? The displacement of the gear center in one revolution is equal to the outer circumference. Relate the translational and angular displacements. Differentiate to relate the translational and angular velocities.

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.

? The velocity for any point P on the gear may be written as vrP = vrA + vrP A = vrA + kr ? rrP A

Evaluate the velocities of points B and D.

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