Biomedical Engineering Reference
In-Depth Information
if
F
s
is the restoring force due to the spring, it is often observed experimentally that the
force is directly proportional to the extension:
F
s
=−
k
s
(
R
−
R
e
)
(3.1)
The constant of proportionality
k
s
is called the force constant and it tells us the strength of
the spring. This law is known as
Hooke's law
and it applies to very many springs made
from many different materials. It invariably fails for large values of the extension, but is
good for small deviations from equilibrium.
Suppose that we now set the particle in motion, so that it oscillates about
R
e
. According
to Newton's second law we have
m
d
2
R
d
t
2
=−
k
s
(
R
−
R
e
)
(3.2)
This second-order differential equation has the general solution
A
sin
k
s
m
t
B
cos
k
s
m
t
R
=
R
e
+
+
(3.3)
where
A
and
B
are constants of integration. These constants have to be fixed by taking
account of the
boundary conditions
. For example, if the particle starts its motion at time
t
=
0 from
R
=
R
e
then we have
A
sin
k
s
m
0
B
cos
k
s
m
0
R
e
=
R
e
+
+
from which we deduce that
B
=
0 for this particular case. Normally we have to find
A
and
B
by a similar procedure.
The trigonometric functions sine and cosine repeat every 2π and a little manipulation
shows that the general solution of Equation (3.3) can also be written
A
sin
k
s
m
B
cos
k
s
m
t
2π
m
k
s
t
2π
m
k
s
R
=
R
e
+
+
+
+
The quantity
√
k
s
/
m
has the dimension of inverse time and it is obviously an import-
ant quantity. We therefore give it a special symbol (ω) and name (the
angular vibration
frequency
). We often write the general solution as
R
=
R
e
+
A
sin (ω
t
)
+
B
cos (ω
t
)
(3.4)
Atypical solution is shown as Figure 3.2 (for which I took
A
=
1m,
B
=
0,
m
=
1 kg and
k
s
=
1Nm
−
1
). Such motions are called
simple harmonic
. At any given time, the displace-
ment of the particle from its equilibrium position may be nonzero, but it should be clear
from Figure 3.2 that the average value of the displacement
R
-
R
e
is zero. As noted earlier,
it is usual to denote average values by angle brackets <...>and so we write
R
−
R
e
=
0
