Seely F.B. Analytical Mechanics for Engineers
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PLANE
MOTION 281
gives
to
each
point
the
same
linear
velocity
and acceleration
that
the
base
point
has
at the
instant.
In
applying
the
equations,
V
P
=
v
p
-+>
V
Q
and
a
P
=
a
p
-f
a
Qj
Q
Q
to
the motion
of
any
two
points
in
a
rigid
body
having plane
motion,
the
point
Q
represents
the base
point.
However,
as
stated
above,
the base
point
may
be
any point
in
the
body. Further,
the
angular
velocity
(and
acceleration)
of the
body
is the
same
with
reference to
all
base
points
in
the
body.
Since
any
two
points
in
a
rigid body
remain a
fixed distance
apart,
there can be
no relative
velocity
of
one
point
toward
the
other,
that
is,
the
relative
velocity
of
either
point
with
respect
to the other is
perpendicular
to
the
line
joining
the
two
points.
This is
not
true,
however,
of the relative
acceleration,
since
it is
made
up
of
a
tangential
and a normal
component,
the
normal
com-
ponent
having
a
direction
along
the line
joining
the two
points.
'The
principles
discussed in this
article
are involved
in the
solution
of
the
following problem.
ILLUSTRATIVE
PROBLEM
334. A
50
h.p. engine
has
a
cylinder
10
in.
in
diameter
and
a
stroke
of
10
in.
The
engine
runs
at a constant
speed
of co
=
300
r.p.m.
The ratio
of the
length
of the
connecting
rod to
that
of
the
crank is 5.
Find
the
velocity
and
the
acceleration
of the
crosshead
when
the
crank
angle
is
30.
FIG.
318.
Solution.
Let
P be
the
crosshead
and
Q
the crank
pin.
Then
in
Fig.
318
PQ represents
25 in.
and
OQ
or
r
represents
5
in.,
according
to
the
scale
used. The
velocity
of
P
may
be found from the
equation,
282
.
MOTION OF
RIGID
BODIES
There are
six elements
involved
in the
equation
three
magnitudes
and
three
directions,
four of
which
must be found
before the
equation
can
be
used
to
determine
the
other
two.
The
direction of the
velocity
of
Q
is
per-
pendicular
to
OQ
and
its
magnitude is,
300X27r
The direction of
V
P
is
perpendicular
to
the line
joining
P
and
Q
and its
mag-
Q
nitude is unknown.
The direction of
VP
is horizontal. Hence
by
laying
off
VQ
to a convenient
scale
in the
proper
direction from
P,
and
by drawing
a line
from
the end of
VQ,
perpendicular
to the
connecting
rod,
until it intersects a
horizontal
line
through
P,
the
magnitudes
of
VP
and
vp
are determined
by
the
Q
intersection.
By
scaling
off the
values of
VP
and
vp,
the
following
results
are
Q
found
V
P
=
11.
4
ft./sec.
and
VP
=
7. 8
ft./sec.
Q
Thinking
of the
plane
motion of the
connecting
rod as
a combination
of
a
rotation and a
translation,
VP
is the
velocity
which
P
is
given by
the
rotation
Q
of the
rod about
Q
and
VQ
is
the
velocity given
to
P
by
the
translation
of
the
rod. The
two velocities
produce
the resultant
velocity
V
P
.
The
acceleration of
P is
given by
the
equation,
ap=ap
-f^
aQ.
Q
For
convenience,
ap
will be
replaced by
its
tangential
and normal
components.
Q
Hence,
'OP ia
P
\
+>
/ap\
\
Q/
t
\Q/n
Eight
elements are involved
in the
equation
four
magnitudes
and
four
directions,
six of which must
be
found before
the
graphical
construction
representing
the
equation
can
be
completed.
aQ
is directed from
Q
toward
and its
magnitude is,
/300X27r\
*
5
.,
,,
/
a
=
co
2
r=(
^
J
Xj^
=
412
ft./sec.
2
a
P
is
known
in
direction, being parallel
to PO
(horizontal),
since
VP
changes
in
magnitude only.
/a
P
\
is
directed from
P
toward
Q
and its
\
Q)
n
magnitude
is,
PLANE
MOTION
283
|P\]
\
Ql
t
has
a
direction
perpendicular
to
PQ.
The
two unknown elements
are,
therefore,
the
magnitudes
of
ap
and
/ a
P
\
.
\
Q/t
In
Fig.
319,
starting
at
P,
the
vectors
are laid
off to
a convenient scale.
Thus
CLQ
and
/p\
(both
of which are
completely
known)
are
drawn
and
\Q/n
then,
fron
the end
of ia
P
\
,
a
line
is
drawn that
represents
the direction
of
\ Q/
n
(5),
FIG.
319.
,
that
is,
it
is
drawn
perpendicular
to
PQ.
The
intersection of this
line
with
the
line
PO
(which
represents
the
direction of
a
P
)
determines
the
lengths
of the vectors that
represent
la
P
\
and
a
P
.
By
scaling
off
the
\
Q/t
values,
the
following
results
are
obtained,
200
ft./sec.
2
and 404 ft.
/sec.
2
.
Ql
t
PROBLEMS
335.
A
wheel
4
ft.
in
diameter
rolls,
without
slipping,
on
a
horizontal
track
(Fig.
320).
The
velocity
of
its
center,
at
a
given
instant,
is
4
ft./sec.
to the
right
and
the acceleration
of
the
center is
6
ft./sec.
2
in the
same,
direction. Find
the
velocities
and
the
accelerations
of
the
points
A
and
B.
336.
Find,
by
the
graphical
method,
the
velocity
and
the
acceleration of
the
point
C of the
four-link
mechanism de-
scribed
in Problem
337
(Fig.
322).
Use
the
following
scales:
lin.
=
lft.;
lin.
=
2
ft./sec.;
and
1 in.
=
10 ft.
/sec.
2
FIG. 320
Ans.
v
c
=
1.54
ft./sec.;
a
c
=55.5
ft./sec.
2
284 MOTION
OF
RIGID BODIES
135. Instantaneous
Center. It
was
shown
in
the
previous
article that a
plane
motion of a
rigid
body
may
be
considered,
at
any
instant,
as a
rotation,
about an axis
through
any
point
in
the
body,
combined
with a
translation.
However, by choosing
a
par-
ticular
axis
in the
body
(or
its
extension),
the motion of the
body,
at
any
instant,
becomes
one
of
rotation
only,
that
is,
no
transla-
tion need be combined
with
the rotation to
produce
the actual
motion of the
body.
This axis
is called the instantaneous
axis
of
rotation
or.
the
instantaneous axis
of
zero
velocity.
Its intersection
with the
plane
of motion
is
called the
instantaneous
center
of
rotation.
To show a method of
locating
the
instantaneous
center,
let
A
and B
be
any
two
points
in the
plane
of motion
of
a
rigid body
having plane
motion
(Fig. 321).
Let
V
A
denote
the
velocity
of
A,
and
V
B
,
the
velocity
of
B. From
A
draw
a line
perpendicular
to
V
A
and
from B
draw
a line
perpendicular
to
V
B
,
the two
lines
intersecting
in 0.
Any point
in
the
body
on the line
OA,
if
not at
rest,
must have a
velocity
perpendicular
to
OA,
since
any
two
points
in the
body
FIG.
321.
cannot move
toward or
away
from
each
other
if the
body
is
rigid.
Likewise, any
point
on
the
line
BO,
if
not
at
rest,
must
move
perpendicular
to
BO.
Hence,
the
point
of
intersection
of AO
and
BO
must be
at
rest
since
it
cannot,
at the same
instant,
move
in
two
directions.
Therefore,
the
body,
if not
at
rest,
must
rotate
about
the
point
at
the
instant.
If,
then,
A and
are
two
points
in
the
rigid body
having plane
motion,
A
being
any
point,
and
the
instantaneous
center,
the
equation,
reduces
to,
in
which
co
is the
angular
velocity
of the
body.
Further,
since
co
is the
same
for
all
points
in the
body,
it follows
that the
velocities
of
any
two
points
A and
B
vary
as
their
distances
INSTANTANEOUS CENTER
285
from
the
instantaneous
center.
Or,
expressed
in
equational
form,
V
B
~
u-OB OB'
It
should be
noted
that
the
instantaneous center
is
the
center of
zero
velocity
and
not of zero acceleration.
In
the case of the
motion
of
rotation
of
a
rigid
body
about a fixed
axis,
which is a
special
case
of
plane
motion,
the axis of
rotation
is
also the instan-
taneous
axis
and
it has zero
acceleration
as
well
as
zero
velocity
since
it
is at rest.
In
the
general
case of
plane motion, however,
the
instantaneous
center
changes
its
position
in
the
body
and also
in
space.
Although
the
velocity
of the
point
in the
body, coinciding
with
the
instantaneous
center
at a
given instant,
is
zero,
the
velocity
is
changing through
its zero
value and hence the
point
has
an
acceleration. Therefore
in the
equation,
a
A
=
a
A
+>
a
o,
~o
in which
A
is
any point
and is
the
instantaneous
center,
ao
is
not zero.
Hence the actual
acceleration,
at
any
instant,
of
a
point
in a
rigid body
having
plane
motion
cannot be
found
by
considering
the
body
to be
rotating
about a
fixed
axis
through
the instantaneous
center,
as
may
be done
in
determining
the
velocity
of
any point
of the
body.
ILLUSTRATIVE
PROBLEM
337. The four-link
mechanism,
ABCD,
shown in
Fig.
322
has the
following
dimensions:
A
=
6in.;
C
=
3ft.;
DC
=
2
ft.;
AD
=
4
ft.;
=
45. Find
the
instantaneous
center
for
the link
BC.
If
the crank
AB rotates at a
constant
angular
velocity
w
=
10
rad./sec.,
find
the
angular
velocities,
co2,
of
the link
DC
and,
ojs,
of
0)
s>
the link BC. Also
find
the
linear
veloc-
ity,
VH,
of
H,
the
midpoint
of
BC.
Solution.
The
instantaneous
center
for
the link BC is
0,
the
point
of
inter-
section of the lines AB
and
DC
extended.
By scaling
off the
lengths
of
OB,
OC,
and
OH,
the
following
values are
found:
OB
=
3.
1ft.
OC
=
0.96ft. O#
=
1.72ft.
286
MOTION
OF
RIGID BODIES
The
velocity
of
B
is,
v
B
=
w
X
AB
=
10
X
A
=
5
ft./sec.
Therefore
the
angular velocity
of
B,
considered
as a
point
on
BC,
is,
"3
=
^
=
3^
=
1.61
rad./sec.
and the
angular
velocities
of all
points
on BC are the
same. The
linear
velocity
of C
is,
therefore,
y
c
=
w
3
XOC
=
1.61X0.
96
=
1.54
ft./sec.
And,
v
a
=
wXOH^.
61
XI.
72=2. 77
ft./sec.
Therefore,
vc
1
.
54
W2
=
=
~~~
rad./sec.
PROBLEMS
338.
What
is the
magnitude
and
the
direction of the
velocity
of the
point
E
(Fig.
322)
on
link
DC
midway
between
D
and C.
Ans.
##
=
0.77
ft./sec.
perpendicular
to DC.
339.
Consider
the link BC
(Fig.
322)
to
be
the
triangular piece
BOG and
let
P be
a
point
on the
triangular
piece
midway
from
to
H. What is the
magnitude
and the direction of
the
velocity
of
the
point
P?
340.
Locate
the
instantaneous
center
of
the
connecting
rod
of the
steam-
engine
mechanism
described in Problem 334. Find the
angular
velocity
of
the
connecting
rod,
the linear
velocity
of the
crosshead,
and the
linear
velocity
of a
point
on the
connecting
rod
midway
from the
crosshead
to
the
crank-pin.
Ans.
0)2
=
5.47
rad./sec.;
vp
=
7.70
ft./sec.;
yjj/
=
9.12
ft./sec.
341. Find
the
instantaneous
center of the
wheel
described in
Problem
335.
Find
also
the
angular velocity
of
the
wheel
and
the
magnitudes
and
directions
of the
velocities
of the
points
B, C,
and
D.
PART
III. KINETICS
CHAPTER
IX
FORCE,
MASS,
AND
ACCELERATION
1.
PRELIMINARY
CONSIDERATIONS.
KINETICS OF
A
PARTICLE
136. Introduction. Kinetics is that branch of mechanics
which treats
of the
laws
in
accordance
with which the motion
of
physical
bodies takes
place.
A
change
in
the state of motion of a
body
always
occurs
when
an
unbalanced force
system
acts on the
body,
the unbalanced
part (resultant)
of
the force
system being
the cause of the
change
in
the motion. Common
experience
teaches
that the
change
of motion of
the
body,
is influenced both
by
the characteristics
(Art.
5)
of the
resultant
of
the forces
acting
on the
body
and
by
the nature of the
body
itself. For
example,
different force
systems
acting,
in
turn,
on
the same
body
do not
produce
the same
change
of motion of the
body. Further,
the
same
force
system applied
to
different
bodies,
even
if
they
are of the same
size
and
form,
does
not
produce
the
same
change
in
the motion of all of
the bodies.
Although experience
suggests
that relations exist
between
the force
system acting
on
the
body,
the
properties
of the
body,
and the
change
in
the motion of the
body,
it
required
the
work
of
many
eminent
men and a
period
of several
centuries
before definite
and
complete
fundamental
relations
between these
three factors
were
finally
established.
These relations
were formulated
by
Sir Isaac
Newton
(1642-1727)
and
are known as
Newton's laws
of
motion.
Newton's fundamental
laws,
however,
apply only
to the
motion of a
particle
under the action
of a
single force,
whereas
in
engineering
practice
the motion of a
body
(system
of
par-
ticles)
under
the action
of a
system
of forces
must be con-
sidered. The
force
system acting
on the
body may produce
any
287
288
FORCE, MASS,
AND
ACCELERATION
type
of motion.
Thus,
in
some cases a
motion of
translation
is
given
to the
body,
in
other cases the
body
is
given
a
motion
of
rotation,
and
in
still other
cases
the
force
system
produces
a
plane
motion of the
body.
These
three
types
of
motion,
only,
are
con-
sidered
in this
chapter.
137. The General Kinetics Problem. In
each
type
of motion
however,
the
general
character of the kinetics
problem
is
the same
:
A
physical body
is acted
on
by
a force
system
that
has a
resultant,
which causes
a
change
in
the motion
of
the
body.
In
each
prob-
lem
it
is
required
to
deduce, by
the
use of
Newton's
laws,
the
equations
expressing
the
definite relations
between
(1)
the result-
ant of the
force
system,
(2)
the
kinetic
properties
of
the
body,
and
(3)
the
change
of
motion
of the
body,
so that
the motion of
a
given body produced by
a
given
force
system
may
be
determined,
or the
force
system
required
to
produce
a
given
motion
may
be
found. The
equations
which
express
these relations
are
called
the
equations
of
motion for
the
body.
These three
elements or factors
which are involved
in
the
equa-
tions of motion
of
bodies
may
be
considered
briefly
before
stating
Newton's
laws and before
deriving
the
equations
which
express
the
relations between these
three
factors.
Change
of
motion is
measured
in
terms of
distance, time,
velocity,
and
acceleration,
relations
between
which,
for various
types
of
motion,
have
already
been
considered
fully
in
Chapters
VII and VIII. The
charac-
teristics
of the
resultant
of a force
system
have also been
consid-
ered
in
Chapter
II,
and need
only
be
reviewed
briefly
at this
point
(see
next
article)
to
show
their
connection with the
general
problem
in
kinetics.
The
nature
or
property
of
the
body
which enables it
to
have
an
influence
in
determining
its own
motion,
however,
needs
to
be discussed at
greater
length (see
Art.
139).
138.
Characteristics
of a
Force
System.
The
only part
of
a
force
system
that influences
the motion
of a
body
is the unbal-
anced
part
(resultant)
of the force
system.
The
forces
which
produce
the
i
ypes
of
motion
considered
in
this
chapter
are
coplanar
forces
or
may
be so
regarded.
1
The resultant
of
a
coplanar
force
1
Although
in most kinetics
problems
here
considered,
the
motions are
pro-
duced
by coplanar
force
systems,
the
system
of forces
may
be
non-coplanar.
It will
be
evident,
however,
after
discussing
the
equations
of
motions for the
three
types
of motion here
considered,
that
non-coplanar
forces
that
pro-
duce
these motions
may
be
replaced by
equivalent
coplanar
force
systems.
INERTIA
AND
MASS
289
system
is either a
force or
a
couple
(Art.
28) ,
although
the
resultant,
when
a
force,
may,
for
convenience,
be considered as another
force
and
a
couple (Art. 18).
If
the
resultant
is
a
force,
the
character-
istics of the resultant
which
influence the motion
of the
body
on
which the
force
system
acts are
(1)
its
magnitude, (2)
the
position
of its action line
in
the
body,
and
(3)
its sense
(Art.
5). If,
how-
ever,
the resultant is
a
couple,
the characteristics of the
resultant
which
influence the motion of the
body
on
which the force
system
acts are
(1)
the moment
of
the
couple, (2)
the
sense,
and
(3)
the
aspect
or direction of the
plane
of the
couple (Art.
16).
These
statements
appeal directly
to one's
experience
and
may
be
verified
by simple
experiments
which the reader
may readily perform.
Since the
equations
of motion of a
body
must
take account
of
all the
factors
influencing
the
motion,
there must
be
a
sufficient
number of
equations
to determine the
influence
of all of
the
charac-
teristics of
the resultant of the force
system
as
mentioned
above.
This
may
be
done,
for the
types
of motion
considered
in
this
chapter,
by
means
of three
equations.
These three
equations
will
contain
the
algebraic
sum of the
x-components
of
the
forces
acting
on
the
body,
the
algebraic
sum of the
^/-components,
and the
alge-
braic sum of
the moments of
the forces about some axis
in
the
body.
For,
if
the resultant of
the force
system
acting
on
the
body
is
a
force,
the
algebraic
sums of the
x-
and
^-components
of the forces
are needed to
take account of the influence of
the
magnitude
of
the resultant
force and of the direction of
its
action
line,
and
the
algebraic
sum
of
the moments of the
forces
is
needed
to
measure
the
influence of the
position
of the action
line of the
resultant
force.
And,
if
the resultant of the
force
system
is a
couple,
the motion
of the
body
is one of rotation
as
will
be
seen later
(Art.
147),
in
which
case,
the
algebraic
sum
of
the
moments of the
forces
is
needed to measure
the influence of
the
moment
and
sense
of
the
couple.
The
aspect
of the
couple
is
not
involved
in
the
equations
of motion since the
plane
of the
couple
always
agrees
with
that
of
the
plane
of
motion
of the
body.
139.
Inertia and
Mass.
The
property
of
a
body
by
virtue of
which
it
offers resistance to
any change
in
its
motion,
and
thereby
makes
the
body
itself
a factor
in
determining
the
motion which
unbalanced
forces
impress upon
it,
is
called
inertia.
All
physi-
cal
bodies
are
inert or
possess
inertia,
but different
bodies
possess
different
amounts of inertia.
That
is,
all
bodies
influence their
290
FORCE, MASS,
AND
ACCELERATION
own motion
according
to
the same
law
but
not to
the
same
degree.
1
Bodies
possess
many
otner
properties
such as
hardness,
grav-
itation,
elasticity, strength,
etc. In
taking
into account
the
influence
which
any
one of these
properties
has
on the
behavior
of the
body,
under the conditions
which
make the
property
a
factor
in
the
problem,
a
quantity
(value
of
the
property)
is
found
which measures the
particular
property;
that
is,
which indicates
its relation
to the
other
influencing
factors
and
which,
if
possible,
has
a constant
value
for
the
body
or for the
material of
the
body.
Thus,
when
the
property
of
strength
is
involved
in
a
problem,
the
constant
values
called
"
elastic limit
"
and
"
ultimate
strength
"
are
frequently
used
as a
measure of the
strength
of
the material.
Similarly,
the
constant value
called
"
modulus of
elasticity
"
is used as a measure of the
property
of
elasticity
or
stiffness
of
a
material.
In
the kinetics
problem,
that
is,
in
a
problem
in
which a
change
in
the motion of a
body occurs,
the
property
of
inertia
of
the
body
is involved.
In
taking
account of the
influence which the
prop-
erty
of
inertia
has
on the motion of
the
body,
a
quantity
called the
mass of the
body
is used. The two
essential
characteristics
of
mass are its
direct relation
to
the
other factors
(forces
and accel-
eration)
which are
involved
in
the kinetics
problem
and its con-
stant value. Thus the mass of a
body
is a constant
of
the
body
which measures
a
kinetic
property (inertia)
of
the
body.
The
term mass
then,
as used
in
mechanics,
denotes the amount
of
inertia
possessed
by
a
body. Or,
the
mass
of
a
body may
be
defined
as the
constant of the
body
which
measures the influence
which
the
body
exerts
in
determining
its
own
motion,
when acted
upon by
an
unbalanced force
system.
The amount of
resistance
offered
by
a
body
to
a
change
in
its motion
(that
is,
the amount of
inertia
possessed
by
a
body),
must be determined
experimentally.
An
experiment
for
determining
the mass
may
be outlined
as
follows
:
1
The
way
in which bodies resist motion is
analogous
to
the
manner in
which
elastic
materials,
such
as steel or
timber,
resist
being
stretched.
Each
material,
within
limits,
resists
according
to the same
law
(stretch
is
propor-
tional
to
stress),
but
some materials resist to a
greater degree,
that
is,
some
materials
are stiffer
than
others,
which means
that
some
materials
require
a
greater
force to
produce
a
given
stretch than do other
materials, just
as some
bodies
require
a
greater
force to overcome their inertia at a definite rate than
do
other
bodies.