

Where that which goes out, comes back...
Extending our notion of what makes up the motion of a particle
or system of particles, we should now consider that kind of
motion where the object dithers about in the neighborhood of a
point in the space of our reference frame. Such motion is called
"vibration" or "vibratory motion".
In fact we have already
introduced motion of this sort with the spring and block thing we
used to illustrate some of the ideas about work and energy. We
will use that same mechanism to begin the study of vibration. In
the Simple Harmonic Oscillator
display the position and velocity of the block are plotted vs.
time with time running down the page.
You may have noticed the label on the previous display was
"Simple Harmonic Oscillator". This terminology comes
from the form of the position vs. time plot. An oscillator in
which the restoring force is proportional to the displacement
vibrates with a particularly simple function of position vs. time.
The function is
x = A*cos(w*t +
f) .
The sine and cosine functions are called harmonic functions since
they may be used to create any waveform, however complicated. The
"simple" terminology just means that a single term of
the sine or cosine function fully describes the motion. We will
abbreviate the simple harmonic oscillator, SHO.
Many real dynamical systems behave very much like the SHO as
long as the size of the vibratory motion is small compared to the
size of the system. For that reason we will go into some
additional detail on the SHO. The behavior of the block fits the
basic description of vibration set forth above. It dithers about
in the vicinity of the zero reference point. Because sooner or
later I will confuse you by beginning to refer to vibrations as
oscillations, let's agree at this point that these two terms
are interchangeable. With that out of the way we can go on to
discuss the other terms related to vibrations.
The maximum displacement from the neutral position in
vibration is called the "amplitude" of the vibration.
In the SHO modeled here that is .25 meters. The "state"
of the system at any instant is defined by the position and
velocity of all its moving parts. The state of the block in our
example is constantly changing but because the motion is
"periodic" the states are repeated in equal intervals
of time. The time it takes for the moving block to return to the
same state is called the "period" of the oscillation.
The number of times the system revisits the same state in a unit
of time is called the "frequency" of the oscillations.
From these definitions you can see that frequency and period are
reciprocals of each other, Period=1/Frequency.
Remember there are two ways of looking at the spring and
block. One is based on the Hooke's law nature of the spring
such that the force it exerts on the block is proportional to the
block's displacement from zero. The other is to ignore the
spring and just say that the potential energy of the system, for
whatever reason is higher at other locations than it is at zero.
It is a fundamental principle in dynamical systems, and all of
nature for that matter, that systems tend to seek their least
energy configuration. Mother Nature as it turns out is lazy. We
will work for a bit with the first point of view but keep the
other in mind.
The force from the spring on the block is f=k*x, where k is the
spring constant and x is the displacement. Note that we have
dropped the vector notation since we are working here in 1
dimension. Then according to Newton's second law,
m*x'' = f = k*x .
Here we have used the x'' notation to represent the
acceleration, indicating that acceleration is the rate of change
of the rate of change of position with respect to time, in other
words, the second derivative of position with respect to time.
The mass of the block is represented by m. The equation
m*x''=k*x may be rewritten as
x'' = k/m*x .



Whatever the function of
time which is a solution to the differential equation above, its
second derivative must be proportional to the negative of the
function itself. If your memory is extremely good and you were
paying strict attention, we have already found a function which
meets that criteria, way back in the Rates
of Change section of the course. Look at the Second Derivative of a Sinusoidal display
as a reminder.
Notice that the sine function and its second derivative are
negatives of one another. The first derivative of the sine is
recognizable as a cosine function. The third derivative of the
sine, being the second derivative of the first derivative, would
be a negative cosine function meaning that the cosine also fits
the requirements for being a solution to the original
differential equation of motion. In fact any function of the form
x = A*sin(w*t +
f) or x = A*cos(w*t + f) ,
can be shown to be a solution to the equation of motion for the
simple harmonic oscillator.
The
equation of motion we developed from Newton's law involves
derivatives, in this case the second derivative of position with
respect to time. To "solve" that equation means to come
up with a function where x is the dependent variable and time is
the independent variable, x=f(t), such that when we plug f(t)
into the equation of motion in place of x, the equation holds
true. The reason for doing this is that we want to predict the
future and if we get x as a function of time we can do that just
by plugging a future time into the function an calculating a
future x.
The branch of mathematics that solves equations like our
equation of motion is called "differential equations".
Do not let the name turn you off. In fact there is only one way
to solve differential equations. You guess at a solution and
check to see if it works. There are some fancy guessing
techniques (see the image below) that we do not need to study right now. We can guess
just based on the curve of position vs. time displayed by the
model.

Let's take the function
x = A*cos(w*t +
f)
as a tentative solution to the equation of motion,
x'' = k/m*x .
We could show by repeating the development which led up to the
Second Derivative of the Sinusoidal display, that the first
derivative of A*cos(w*t + f) is w*A*sin(w*t +
f) and the second derivative of
A*cos(w*t +
f) is w^{2}*A*cos(w*t + f).
Substituting for x, the expression A*cos(w*t + f) and for
x'', w^{2}*A*cos(w*t + f) we get
w^{2}*A*cos(w*t + f) = k/m *
A*cos(w*t +
f) .
If we choose the constant w such
that w^{2}=k/m then the
equation of motion is satisfied, indicating that our tentative
solution is in fact a solution as long as we choose w = (k/m)^{0.5}.
The constants A and f remained
undetermined in our analysis of A*cos(w*t + f) as a
possible solution to the equation of motion. This means that any
choice whatsoever for A and f still
satisfy the equation so that a large variety of motions is
possible for the oscillator. The parameter
w is common to all allowed motions but A and f may differ among them. the parameters A
and f we will see are determined by
how the motion is started, the so called initial conditions.
Now we are going to reason our way through
the meaning of the parameters A, w and
f. This is something that gets easier
with experience. You have to have some intuition about the effect
of parameters on functions, which only comes with looking at lots
of functions.
Since the range of the cosine function is 1 to 1, it is clear
that the symbol A is the amplitude of the SHO motion, the peak
value that x may have either positive or negative is A. The
parameter w is called the angular
frequency. If the time t in
x = A*cos(w*t +
f)
is increased by 2*p / w, the function becomes
x = A*cos(w*(t +
2*p / w) +
f) .
This expands to
x = A*cos(w*t +
2*p + f)
.
Adding 2*p to an angle just brings it
back to its starting point so 2*p /
w must be the period of the motion.
T = 2*p / w ,
where T is the period. But since
w^{2}=k/m we have w =
(k/m)^{0.5}, so
T = 2*p *
(m/k)^{0.5} .
The oscillatory period T is determined only by the mass and the
force constant k. It is independent of amplitude.
The frequency, f, is the number of complete oscillations per
unit time,
f = 1/T = w /
2*p = 1/(2*
p) * (k/m)^{0.5} .
From this we see that
f = w / 2*p or w = 2*p * f .
The angular frequency w then differs
from the frequency f by a factor of 2*p. It has dimensions of reciprocal time, the
same as angular velocity, radians/second.
Use the SHO display to calculate the
ratio of m/k used in that model. Determine the period from the
plot. Then use
T = 2*p *
(m/k)^{0.5}
to get the required ratio. You should find that m/k is about .5,
remembering that picking of the period using the cursor is an
approximation.
What about this constant, f, we have
been carrying along through all this discussion? That quantity is
called the phase angle of the motion. As you can see from its
position in the function
x = A*cos(w*t +
f) ,
it plays the role of offsetting time from zero. If for example
f = p/2
then the function
x = A*cos(w*t +
f) = A*cos(w*t  p/2) =
A*sin(w*t)
so that displacement x is zero at t=0. If
f=0 then x is A at t=0. Other initial conditions
correspond to other phase angles. The amplitude A and the phase
angle f are determined by the initial
position and velocity of the particle.



Before exploring the effect of varying initial position and
velocity, we should generalize our SHO model. Strictly speaking,
the spring is not a necessary part of our system. In the SHO case
the potential energy as a function of x was 1/2*k*x^{2}
as we discovered in the section on Potential
Energy and Fields , since that was the work done on the
spring. Review the Work on Spring
display to see the shape of the pe(x) curve. Let the display run
long enough to fill in the curve.
Now if any mechanism, be it spring, electricity or whatever,
creates a condition in space such that the potential energy as a
function of displacement is proportional to the square of the
displacement, an object placed in that space will behave like an
SHO. To illustrate that concept run the
Potential Energy SHO display. Notice that as you start the
object in different positions with different velocities that the
amplitude of the motion and the time when the object passes
through the zero position take on different values.
To farther clarify the relationship between initial conditions
and the parameters A and f we will
dispense with the physical representation of the object as we did
in the free body diagram display, and just show the position of
the object as a function of time. Run the
Initial Conditions display.

Next we will explore the relation
between simple harmonic motion (SHM) and uniform circular motion.
Consider a circle in the (x,y) plane, centered at the origin, of
radius 5 meters with a point on the circle moving around at
constant angular speed, w, expressed
in radians per second. Next suppose that the initial position of
the point on the circle was at angle f
from the xaxis. The projection of that point on the xaxis at
any time t would be at
x = 5*cos(w*t +
f) .
This expression for x is exactly that we derived for the motion
of the SHO. Run the Circular to SHM
display to see these relationships.
From this we see that the parameters
w and f have a geometric
interpretation. The
angular frequency w is the angular
velocity of the circular motion related to the linear simple
harmonic motion. The phase
angle f is the starting angle of
the related circular motion. We might also note that the
amplitude A of the simple harmonic motion is the radius of the
related circular motion.
If we had taken the projection of the circulating point onto the
Yaxis instead of the x, we would have found the motion to be
x = 5*sin(w*t +
f) .
This is another example of simple harmonic motion that differs
only in phase angle from the x motion. If we replace f by fp/2, cos(w*t +
fp/2) may
be replaced with sin(w*t + f) since sin and cosine functions are
identical except for a p/2 phase
difference.



Now if we turn our thinking inside
out, we can synthesize uniform circular motion from the
combination of two simple harmonic motions. Two simple harmonic
motions along perpendicular lines, of equal amplitude and angular
frequency, which differ in phase by
p/2 radians combine to produce the circular motion we
described. Run the SHM to Circular
Motion display for an illustration. In this display we see a
green oscillator and a red oscillator making a cyan circle.

In many dynamical systems, motion that appears to be quite
complex can be understood as a combination of simple harmonic
motions. In the case where the frequencies are the same we have
x = A_{x}*sin(w*t +
f_{x}) and y = A_{y}*sin(w*t +
f_{y}) .
Suppose the phase angles f_{x}
and f_{y} were also equal.
Then dividing the y equation by the x equation we get,
y = (A_{y} / A_{x}) * x
,
which is the equation of a straight line passing through the
origin with slope A_{y} / A_{x}.
If the phase angles are different the resulting motion will be
elliptical. In the special case where the amplitudes are equal
and the phase angles differ by p/2,
the motion is circular, as we have seen. Run the Combined SHM display to experiment with
various combinations of parameters.
All of this emphasis on simple harmonic motion is appropriate
under the heading of vibration. Many of the vibrations
encountered in dynamical systems can be very closely approximated
by combinations of SHM. Next we will look at a specific vibrating
dynamical system. We will study the vibration of atoms in a
solid.
In the section of the course on translation when we were
developing the notion of center of mass, we talked about the
constraint that in a solid the distance between atoms is fixed.
It is true that the average distance between atoms is fixed but
the atoms do exhibit some vibratory motion about this fixed
position. We know from what we have learned here that in
vibration, the object stays in the vicinity of a point and that
point for an atom is the position where the potential energy is a
minimum. Remember that potential energy is the energy it takes to
change a system's configuration.



Imagine a solid object made up of atoms. These atoms have
parts that have an electrical charge on them. It is the nature
of electric charges to effect the space in which they are located
so that other charged particles in that space experience a change
in potential energy with a change in position. Therefore, work is
involved in moving charged particles around in the space. We may
think of the space as having a potential energy field in it. If
we pick an atom in the solid to examine, we find that the space
in which it exists has a potential energy function pe(r)
where the potential energy is a function of the atom's
position r. All the other atoms in the vicinity create
that condition of space.
It turns out that the net effect of all this electric stuff is
very similar to having each atom connected to all its immediate
neighbors by a little spring. Any displacement of an atom from
its equilibrium position would compress some springs and stretch
others, building up potential energy. Then as we saw in the
discussion on Potential Energy and
Fields , the potential energy gets converted to kinetic
energy, which goes, back to potential and so on and so forth. To
fully model this behavior in three dimensions for a significant
number of atoms would bring my computer (and probably yours) to
its knees. To give you some sense for how this works, I will put
together a nineatom model in only one dimension. Run the Atomic Solid display to see this in
action.
The force between any pair of atoms is k*dx where k is an
effective spring constant and dx is the change in distance to its
nearest neighbor. We ignore any forces due to displacement
relative to more distant neighbors. From Newton's third law
we know that this force will be negative on the one atom and
positive on the other one. The forces are transmitted from atom
to atom through the mutual effect of the disturbance in the
potential energy field caused when an atom is out of its
equilibrium position.

Actually the restriction that we work in one dimension is not
as severe as you might think. In a solid, the arrangement of
atoms can be looked at as a series of strings. If we grab an atom
and shake it, the motion of that atom may be resolved into
components along the directions of all the strings of which it is
a member. So the model we show is fairly accurate for one of the
components of any atom's motion.
Notice that energy was imparted to the system but the system
as a whole did not move except by the amount that the center of
mass was affected by the initial displacement of the left atom.
If we were looking at the lump of stuff composed of these nine
atoms as a particle in a larger system, energy in that system
would appear to not be conserved. Basically energy may seem to be
lost from systems made up of bulk matter because some of it is
transferred into vibratory energy of the matter itself. The
amount of energy stored in the vibrations of the atoms making up
an object is called heat energy, and the average speed of the
vibrating atoms is proportional to the temperature of the
object.
You also might note that if you keep your eye on the rightmost
atom when you initially click the Action button, you can see that
it takes some time for the disturbance to be felt at the far end
of piece of matter. The velocity with which the disturbance
propagates down the string of atoms is the velocity of sound in
the material. It is evidently dependent on the mass of the atoms
and the stiffness of the bond between them since those are the
only two parameters that have been set in this model. By halting
the action with the Cut button, you freeze the positions and
velocities at that instant. Then clicking Action again adds an
additional shot of energy. That way you can build up a lot of
agitation in the atoms. The Reset button restores the atoms to
their initial position and zero velocity.
Next we will extend the idea of vibration to include the
motion of planets in their orbits about the sun. To do that we
need to examine the idea of gravity which we will handle in the
next section.
Are there any questions?


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