Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
Подождите немного. Документ загружается.
124
Mechanical
shock
Figure
5.2.
Velocity
change
of
a
half-sine
The
displacement
is
calculated
by a
second integration;
we
will take
for
initial
conditions
t = 0, x =
0,asitis
practically always
the
case
in
these problems. This
yields
5.2.2.2.
Impulse mode
Since
V
i
= 0
To
further
the
study
of
this movement x(t),
it is
preferable
to
particularize
the
test
conditions.
Two
cases arise;
the
velocity
v
i
being able
to be:
-
either zero before
the
beginning
of the
shock:
the
object subjected
to the
shock,
initially
at
rest,
comes under
the
effect
of the
impulse
a
velocity
v
f
= AV;
- or
arbitrary nonzero:
the
specimen
has a
velocity which varies during
the
shock
duration
T from a
value
V
i
to a
value
v
f
for t = T; it is
then said that there
is
impact
NOTE:
This
refers
mostly
to
shocks obtained
on
shock machines.
This
classification
can be
open
to
confusion
insofar
as the
shocks
can be
carried
out on
exciters with
a
pre-shock
and/or
post-shock which communicates
to the
carriage
(table,
fixture
and
test
item)
a
velocity
before
the
application
of
the
shock
itself
(we
will
see in
Chapter
7 the
need
for a
pre-shock
and/or
a
post-shock
to
cancel
the
table velocity
at the end
of
movement).
Kinematics
of
simple
shocks
125
The
velocity increases without changing sign
from 0 to v
f
. In the
interval
(0, T),
the
displacement
is
thus
at a
maximum
for t = T:
It
is the
area under
the
curve v(t)
in (0, T).
Equations
[5.4],
[5.10]
and
[5.11]
describe
the
three curves x(t), v(t)
and
x(t)
in
this interval.
126
Mechanical shock
Table
5.1.
Kinematics
of
a
half-sine
shock generated
by an
impulse
5.2.2.3.
Impact mode
General
case
The
initial
velocity
V
i
is
arbitrary, zero here.
The
body subjected
to the
shock
arrives
on the
target with
the
velocity v
i
,touches
the
target (which
has a
programmer intended
to
shape
the
acceleration x(t) according
to a
half-sine)
between time
t = 0 and t = T.
Several
cases
can
arise.
At
time
t = t at the end of
the
shock,
the
velocity
v
f
can be:
-
either
zero
(no
rebound);
Acceleration
Velocity
Displacement
Maximum
at
Maximum
at
Zero slope
at
Inflection
point
at
Maximum
at
Zero slope
at
and
equal
to
for
Inflection
point
at
Kinematics
of
simple shocks
127
-or
arbitrary,
different from
zero.
It is
said there
is
rebound with velocity
V
R
(=
v
f
).
We
suppose that
the
movement
is
carried
out
along only
one
axis,
the
velocity having
a
different
direction
from the
velocity
of
impact.
The
velocity change
is
equal,
in
absolute terms,
to AV = V
R
- V
i
. The
most
general
case
is
where
V
R
is
arbitrary:
with
coefficient
of
restitution).
The
velocity change
AV,
equal
to
makes
it
possible
to
calculate
V
i
:
i.e.,
in
algebraic value,
and by
definition
The
velocity v(t), given
by
[5.6],
is
thus written:
i.e.,
since
QT = n:
and
the
displacement:
To
facilitate
the
study,
we
will consider some particular
cases
where
the
rebound
velocity
is
zero, where
it is
equal (and opposite)
to the
impact velocity
and finally
V
i
where
it is
equal
to -—.
2
128
Mechanical shock
Impact
without rebound
The
rebound velocity
is
zero
(a = 0). The
mobile arrives
on the
target with
velocity
V
i
at
time
t = 0,
undergoes
the
shock x(t)
for
time
i and
stops
at t = T.
and
The
maximum displacement
x
m
throughout
the
shock takes place here also
for
t = t
since v(t) passes
from -v
4
to 0
continuously, without
a
change
in
sign.
Moreover,
since
V
R
= 0, x
remains
equal
to x
m
for t > t.
This value
of
x(t)
is
equal
to the
area under
the
curve v(t) delimited
by the
curve (between
0 and T) and the two
axes
of
coordinates.
Kinematics
of
simple
shocks
129
Table
5.2.
Kinematics
of
a
half-sine
shock carried
out by
impact without rebound
Velocity
of
rebound
equal
and
opposite
to the
velocity
of
impact
(perfect
rebound)
After
impact,
the
specimen
sets
out
again
in the
opposite direction
with
a
velocity
equal
to the
initial velocity
(a = 1 and V
R
=
-V
i
).
It
then becomes
and
Acceleration
Velocity
Displacement^
Maximum
at
Zero
at
Maximum
at
Zero slope when
and
Zero slope
at
and
equal
Inflection
point
at
Inflection
point
at
130
Mechanical shock
The
velocity varies
from Vj to V
R
= -Vj
when
t
varies
from 0 to T. Let us
take
again
the
general expressions [5.18]
and
[5.19]
for
v(t)
and
x(t)
and set a = 1:
and,
since
x = 0
with
t = 0 by
assumption
Table
5.3.
Kinematics
of
a
half-sine
shock carried
out by
impact with
perfect
rebound
Acceleration
Velocity
Displacement
Kinematics
of
simple
shocks
131
dx
The
displacement
is
maximum
for t = t
m
corresponding
to — = 0, so
that
dt
T
If
K = 0, t
m
=
—.
The
maximum displacement
x
m
thus
has as a
value
2
In
the
case
of a
perfect rebound
(V
R
=
-V
i
),
the
amplitude
x
m
of the
displacement
is
smaller
by a
factor
n
than
if V
R
= 0. It is
pointed
out
that
the
amplitude
x
m
is
none other than
the
area
ranging between
the
curve v(t)
and the
T
two
axes
of
coordinates,
in the
time interval
(0, —):
2
Velocity
of
rebound equal
and
opposed
to
half
of
the
impact velocity
1
In
this
case,
a =
—.
The
mobile arrives
at the
programmer with
a
velocity
v
i5
2
meets
it at
time
t = 0,
undergoes
the
impact
for the
length
of
time
T,
rebounds
and
v
i
sets
out
again
in the
opposite direction with
a
velocity
V
R
= -—:
2
132
Mechanical shock
Let
us set a =
—
in the
general expressions
[5.18]
and
[5.19]
of
v(t)
and
x(t);
it
2
then
becomes
and
The
maximum displacement takes place when v(t)
= 0,
i.e. when
t = t
m
such
that
We
will take,
in (0, T),
yielding
This value
of
lies
between
the two
values
and
The
hatched area under
the
curve v(t)
is
equal
to x
m
.
Kinematics
of
simple shocks
133
Table
5.4. Kinematics
of
a
half-sine
shock
caused
by
impact
with
50%
rebound
velocity
Summary
chart
—
remarks
on the
general
case
of
an
arbitrary
rebound
velocity
All
these results
are
brought together
in
Table 5.5,
One can
note that:
- the
maximum displacements required
in the
case
of
impulse
and the
case
of
impact
without rebound
are
equal;
-the maximum displacement
in the
case
of a
100% rebound velocity
is
smaller
by
a
factor
TC;
the
energy spent
by the
corresponding shock machine
will
thus
be
smaller [WHI].
Acceleration
Velocity
Displacement
Maximum
at
Zero
at
Maximum
at
Zero slope
when
Zero slope
at
equal
to
in
in
and
to
Inflection
point
at
and