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29 Οκτ 2013 (πριν από 4 χρόνια και 9 μέρες)

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1

All figures taken from
Vector Mechanics for Engineers: Dynamics
, Beer
and Johnston, 2004

ENGR 214

Chapter 17

Plane Motion of Rigid Bodies:

Energy & Momentum Methods

2

Principle of Work and Energy for a Rigid Body

initial and final total kinetic energy of rigid body

Work done:

For a couple:

If M is constant:

total work of external forces acting rigid body

3

Kinetic Energy of a Rigid Body in Plane Motion

Plane motion

combination of translation & rotation

For non
-
centroidal rotation:

G

G

v
G

4

Principle of Work and Energy: System of Rigid Bodies

Can be applied to each body separately or to the system as a
whole

sum of kinetic energies of all bodies in system

work of all external forces acting on system

Useful in problems involving several bodies connected
together by pins, inextensible chords, etc. because internal
forces do no work &

reduces to the work of external
forces only.

5

Conservation of Energy

Example: rod is released with zero velocity from horizontal
position. Determine angular velocity after rod has rotated

.

0.5 l

6

7

Power

Power = rate at which work is done

For a rotating body:

8

Sample Problem 17.1

Knowing

that

the

bearing

friction

is

equivalent

to

a

couple

of

magnitude

60

lb

ft,

determine

the

velocity

of

the

block

after

it

has

moved

4

ft

downward
.

A

240
-
lb

block

is

suspended

from

an

inextensible

cable

which

is

wrapped

around

a

drum

of

1
.
25
-
ft

radius

attached

to

a

flywheel
.

The

drum

and

flywheel

have

a

combined

moment

of

inertia

I
=
10
.
5

lb

ft

s
2
.

At

the

instant

shown,

the

velocity

of

the

block

is

6

ft/s

downward
.

9

Sample Problem 17.1

with

with

10

Sample Problem 17.2

The system is at rest when a moment M=6 Nm is applied to gear
B
.

Neglecting friction,
a
) determine the number of revolutions of gear
B

before its angular velocity reaches 600 rpm, and
b
) tangential force
exerted by gear
B

on gear
A
.

11

But

Sample Problem 17.2

12

Sample Problem 17.2

For gear A:

13

Sample Problem 17.3

A sphere, cylinder, and hoop, each having the same mass and radius,
are released from rest on an incline. Determine the velocity of each
body after it has rolled through a distance corresponding to a change of
elevation
h
.

14

Sample Problem 17.3

Friction force in rolling does no work

15

Sample Problem 17.3


Each of the bodies has a different centroidal
moment of inertia,

NOTE:


For a frictionless block sliding through the
same distance,

16

Can also be solved using
conservation of energy:

17

Sample Problem 17.4, SI units

A 13.608
-
kg slender rod pivots about the point
O
. The other
end is pressed against a spring (
k

= 315.212 kN/m) until the
spring is compressed 25.4 mm and the rod is in a horizontal
position. If the rod is released from this position, determine its
angular velocity and the reactions at the pivot as the rod passes
through a vertical position.

1.524 m

0.305 m

18

Sample Problem 17.4

19

Sample Problem 17.4

To get pin reactions:

W

20

Sample Problem 17.5

Each of the two slender rods has a
mass of 6 kg. The system is released
from rest with
b

= 60
o
.

Determine
a
) the angular velocity of
rod
AB

when
b

= 20
o
, and
b
) the
velocity of the point
D

at the same
instant.

21

Sample Problem 17.5


Evaluate the initial and final potential energy.

SOLUTION:


Consider a system consisting of the two rods. With
the conservative weight force,

22

Sample Problem 17.5


Since is perpendicular to
AB

and is horizontal,
the instantaneous center of rotation for rod
BD

is
C
.


and applying the law of cosines to
CDE
,
EC

= 0.522 m


Express the final kinetic energy of the system in terms
of the angular velocities of the rods.


Consider the velocity of point
B


For the final kinetic energy,

23

Sample Problem 17.5


Solve the energy equation for the angular velocity,
then evaluate the velocity of the point
D.

24

Principle of Impulse and Momentum

For a rigid body in general plane motion:

Can be split into 2 components (x and y)

25

Principle of Impulse and Momentum

For non
-
centroidal rotation

26

Conservation of Angular Momentum


When the sum of the angular impulses pass through
O,
the
linear momentum may not be conserved, yet the angular
momentum about
O

is conserved,


Two additional equations may be written by summing
x

and
y

components of momenta and may be used to determine
two unknown linear impulses, such as the impulses of the
reaction components at a fixed point.


When no external force acts on a rigid body or a system of rigid
bodies, the system of momenta at
t
1

is equipollent to the system
at
t
2
. The total linear momentum and angular momentum about
any point are conserved,

27

Sample Problem 17.6

The system is at rest when a moment M=6 Nm is applied to gear
B
.
Neglecting friction,
a
) determine the time required for the angular
velocity of gear B to reach 600 rpm, and
b
) tangential force exerted by
gear
B

on gear
A
.

28

Sample Problem 17.6

Gear A:

Gear B:

Solving yields:

29

Sample Problem 17.7

A uniform sphere of mass
m

and radius
r

is projected along a rough
horizontal surface with a linear velocity and no angular velocity. The
coefficient of kinetic friction is

Determine
a
) the time
t
2

at which the sphere will start rolling without
sliding and
b
) the linear and angular velocities of the sphere at time
t
2
.

30

Sample Problem 17.7

Along x:

Rotation about G:

Solving:

rolling

31

Sample Problem 17.8

Two solid spheres (radius = 3 in.,

W

= 2 lb) are mounted on a spinning
horizontal rod (





= 6 rad/sec) as shown. The balls are
held together by a string which is
suddenly cut. Determine
a
) angular
velocity of the rod after the balls have
moved to
A’

and
B’
, and
b
) the energy
lost due to the plastic impact of the
spheres and stops.

SOLUTION:


Observing that none of the external
forces produce a moment about the
y

axis, the angular momentum is
conserved.


Equate the initial and final angular
momenta. Solve for the final angular
velocity.


The energy lost due to the plastic impact
is equal to the change in kinetic energy
of the system.

32

Sample Problem 17.8

Sys Momenta
1

+
Sys Ext Imp
1
-
2
=
Sys Momenta
2

SOLUTION:


Observing that none of the
external forces produce a
moment about the
y

axis, the
angular momentum is
conserved.


Equate the initial and final
angular momenta. Solve for
the final angular velocity.

33

Sample Problem 17.8


The energy lost due to the
plastic impact is equal to the
change in kinetic energy of the
system.

34

Eccentric Impact

Period of deformation

Period of restitution

As for particles:

Same relation applies for rigid bodies

Note: velocities are along line of impact

35

Eccentric Impact

If one or both of the colliding bodies rotates about a fixed
point
O
, an
impulsive reaction

will develop

36

Sample Problem 17.9

A 0.05
-
lb bullet is fired with a horizontal velocity of 1500 ft/s into the side
of a 20
-
lb square panel which is initially at rest. Determine
a
) the angular
velocity of the panel immediately after the bullet becomes embedded
and
b
) the impulsive reaction at
A
, assuming that the bullet becomes
embedded in 0.0006 s.

37

Sample Problem 17.9

For entire system:

Impulse & momentum:

x

components:

y

components:

Moments about
A
:

but

Solving:

Then:

b

38

Sample Problem 17.10

A 2
-
kg sphere with an initial velocity of 5 m/s strikes the lower end of an 8
-
kg
rod
AB
. The rod is hinged at
A

and initially at rest. The coefficient of
restitution between the rod and sphere is 0.8.

Determine the angular velocity of the rod and the velocity of the sphere
immediately after impact.

39

Sample Problem 17.10

Moments about
A
:

Impulse & momentum:

where

Solving:

+

40

Sample Problem 17.11

A square package of mass
m

moves down conveyor belt
A

with constant
velocity. At the end of the conveyor, the corner of the package strikes a
rigid support at
B
. The impact is
perfectly plastic
.

Derive an expression for the minimum velocity of conveyor belt
A

for
which the package will rotate about
B

and reach conveyor belt
C
.

41

Sample Problem 17.11


Apply principle of impulse and momentum at impact (just before & just after
impact)

Moments about
B
:

42

Sample Problem 17.11


Apply principle of conservation of energy (just after
impact until maximum height)

(solving for the minimum

2
)