Introduction
Many
systems describe simple harmonic motion (SHM),
or can be modelled to approximate SHM, but all physical systems are
damped to
some extent. As an
example, take a
free-swinging pendulum in air. It
is released
from its original position, but, as observed by Galileo, it never
reaches its
original height again, though it comes close after every period. In fact, the angle it
describes each time is
slightly lower than the previous swing.
Over time, this can be seen to decay until the pendulum is
only swinging
imperceptibly, or has stopped completely.
This phenomena is known as damping. My aim is to investigate
the effect of
damping on a system known to describe SHM.
Secondary data
I
have recovered secondary data from other sources,
and, in general, the time/displacement graph produced follows this
trend:
This
graph shows the typical sinusoidal waveform
described by all simple harmonic systems, but with a drop in amplitude
relating
directly to an exponential decay curve.
This can be shown more clearly by the example below, where
an e-ax
type curve has been added to the graph to explain the exponential decay
of the
amplitude of oscillation:
Types of damping
The
following set of diagrams shows the three different kinds of damping
possible.
As
you can see, under damping results in definite
oscillations before decaying completely, while critical damping is the
amount
of damping necessary to stop the motion fully within one oscillation. A good example of this
would be a piston
suspended by a spring. If
the piston
were free it would oscillate vertically, but when pressurising a liquid
by going
down it may return to the equilibrium position, but no further. If the piston was in a
system where the
liquid were pressurised even more, over damping may occur, where the
object
takes a long time even to return to the equilibrium position, or may
even not
return to it at all.
Forced oscillation
Under forced oscillation, at or near the
natural frequency or
resonance, of an object or system, the amplitude will continue to
increase
unless damped. Famous
examples include
suspension bridges swaying in the wind, or to the rhythm of marching
soldiers,
and eventually collapsing. The
following graphs show forced oscillation without constraint and the
effect of
damping on the same system.
Apparatus
I
intend to make use of the following equipment:
·
A metal spring of a given thickness, width,
material and
length.
·
A number of 100g masses to be attached to the
end of the
springs.
·
Clamps and clamp stands for holding the spring
and angle
measurer.
·
Angle measurer attached to a data logger and
computer for
measuring.
·
Accurate callipers to measure spring width and
coil
thickness.
·
Thin piece of stiff copper wire to transfer
vertical to
angular motion.
·
Card disks of different diameters to alter
damping.
This
equipment will be set up as shown below:
The
mass will be suspended on the spring from the
top of the clamp stand, and a copper wire attached to the angle
measurer will
pass through a ring of metal at the bottom of the spring. This will allow it to
slide through, as the
vertical motion will alter the ring’s distance from the angle
measurer. As this
wire will be loosely resting in the
ring the frictional effect will be negligible.
The angle measurer will be wired up to a computer through
a data
logger. This will
plot a graph of the
change of angle against time. I
can
tabulate this data and transfer the measure of angle to a measure of
vertical
displacement in the following way:
We
will know the length of the adjacent side, as it
is a constant - the distance between the spring and angle measurer and
our
measurements will give us the varying values of the angle, so we can use opp
= adj * Tan(a)
to find
the vertical displacement.
Before
I carry out this investigation, I have to do
preliminary work to ensure the extension of the spring is proportional
to the
force applied. I
used a ruler in a
clamp stand beside the spring in question, and placed different masses
on the
end. I waited until
the spring ceased
to oscillate, and then took a reading of the length.
The graphs on the following page show my results. It is clear from these
that the spring obeys
Hooke’s law at least from 6N to 12N.
I
have chosen 900g, within this range, to conduct my experiments with.
Here
is an example of the type of graph I hope to
achieve with the computer. This
is a
secondary source where a mass was suspended on a spring and immersed in
water. The shape
shows a sinusoidal
wave form for the oscillation, and an exponential decay for the
amplitude.
Method
I will set up the apparatus as shown above, and set up the angle measurer to be registering roughly 180 degrees at the equilibrium position. Then I will hold the mass at the natural length of the spring and release it. I will log the results for either one or two minutes depending on the decay time. I will repeat the experiment as shown, but with cardboard disks attached between the masses. The size of these disks I predict will have a direct effect on damping. I have chosen to use a small disk at 150mm diameter, a medium disk at 225mm diameter and a large disk at 300mm diameter. The surface areas of these disks, respectively, would then be 0.017671m2, 0.039761m2 and 0.070686m2. At the end I will transfer the results to Microsoft Excel in order to analyse them in more detail. I will then find a trend for the motion described, which I expect to be of the form y = e-ax cos(bx) where a and b are constants to be found.
