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We have so far studied what is called the simple harmonic oscillation,
in which the only net force exerted on the object is a linear restoring force,
such as the force from the spring. We know
that force is conservative which means the total mechanical energy of the system
is conserved. We studied that before. In reality, though
when you consider any simple harmonic system such as
you know, a pendulum. A pendulum is not going to swing forever.
Right? If you pull it to the side and let go of it, it's gonna swing
for a number of cycles. And the number will get smaller and smaller
eventually it's gonna stop. The same thing happens to
a block attached to a spring and after
several seconds of oscillations, it will slow down and eventually stop. So, what's going on here?
Because well, in the real world, there's always some resistance force.
Friction, that is, that will take energy away
it will take mechanical energy away from the oscillation system.
and that energy force is turned into heat and sound and other things.
So, we must, if we want to consider real life situations
add resistance force
and the result in oscillation is no longer simple harmonic oscillation
rather it's called damped harmonic oscillation or simply damped oscillation.
there are many different forms of resistance force
for example if you consider a sound an object
going through a flute such as air and water
one way to express the resistance force
is something like a constant times speed squared
the speed squared is
is a quadratic form
and that will add a lot of complications to our
our differential equation. In fact this could make our differential equation unlinear
so another deal is that we're gonna deal with
the simplest type of resistance force as far as solving equations is concerned,
and that that type of force is called linear resistance force.
In other words, you do get a resistance force, okay?
fr stands for resistance. It is negative,
meaning its opposite to the direction of motion.
And it is linearly
proportional to velocity, negative
V. With the force of
a constant in front of it, which we call B. Again
this is not the only time of resistance force
but as you will see, it will solve the equation. Since F is directly proportional to
v to the first power. It is a linear resistance force. That makes our calculation a lot
simplier.
So with that new force added in let's
look at what happens in x direction. What is the net force
in x direction? Well of course I have negative kx,
that is the quickest a real spring or effective spring can be, anyhow
that is the linear restoring force responsible for the oscillation, but on top of that you
also need
to add this term, the resistance force, minus
bv. Okay, v, really is just dx/dt
Right? So that's the net force.
What is that equal to? That equals mass times accleration, so mass times acceleration
which is the second derivative. Okay, my question is, with this new force
added, how does that change the oscillation?
We can look at this problem from a math point of view and a physics point of view.
From physics point of view,
you know when you add this force.
This force
always resists motion, right? Because you see, it is always
against the velocity. So when the object moves forward,
this force is backward and when object goes backward, this force goes forward. It always
tends to slow down the motion and therefore with this
additional force, you're not going to move as fast as before.
You're going to lose kinetic energy and that kinetic energy lost is not turned into
potential energy such as
in that case, this term. Rather,
it is turned into heat and sound. Therefore, you're gonna lose
mechanical energy overall and we know that mechanical energy of
a simple harmonic oscillator at a certain moment
when its amplitude is A is always 1/2ka^2, right?
And as the mechanical energy is lowered over time by this resistance force,
the amplitude has to decrease as well.
So if I compare our new oscillation
with this damping force with the previous
simple harmonic oscillation. I do in a plot,
here is T and here is X. Okay,
without the damping force, you know the amplitude is
fixed because the energy is also fixed.
It's always fixed like this. The amplitude does not change over
time. So what happens when this new term is added? We know
the energy is going decrease over time and therefore
you have situation like this, right?
You have reduced amplitude
over time. In other words, the amplitude dampers down like this.
So that's the first thing we have to consider. Amplitude is no longer a constant,
rather
it decreases over time, that's one thing. Another thing is
what happened to period of the oscillation?
In other words, does this term slow down or speed up
the oscillation? Well of course the answer is obvious. This term always resist motion, therefore it
tends to slow down the motion therefore you can imagine
when you add this resistance force, the oscillator is going to take
more time to complete the cycle
because it's always opposed by this addition of force. Therefore, the period tends to get longer,
when the frequency tends to get lower. So again, these are the two
new things from physical consideration, when you add this new term.
First of all, the amplitude would decay and secondly, the
period would get longer or the frequency would be lower.
That's from physics consideration. Mathematically,
let's look at this equation in detail. You have
x to the first power, here is dx/dt to the first power, here is the second derivative of x
respect to time. There is no such thing as
x squared, for example, or sine x, cosine x, whatever. So
every term is proportional to x to the first power.
That's good news because that means this is
linear differential equation, which is rather simple to solve, mathematically.
Okay, the idea is this, you have x,
you have dx/dt, and you have the second derivative, right?
When you move this term to the other side, you will find that these three terms will add up
to zero. Now what does that tell us? That tell us that
you are looking for a solution, x, as a function of time.
X as the function of time, such that
this function itself
and its first derivative, and its second derivative, should assume the same
function or form, in which it should have the same dependency as time.
That is way, those terms, when added together, can cancel out.
If this depends on e to the power of t, and then
the next term depends on sine of t, they will never be able to cancel each other out
for all time, right? Therefore, you're looking for the same function
of dependency.
Now the question is, what kind of function behaves
in such a way that when I take the derivative, the second derivative, it
goes back to itself? There are two functions: one
is exponential function, the other is
sine/ cosine function. Now we know that without determining sine/cosine function.
So what happens with this term?
Without this term, the solution of simple harmonic motion is
A
cosine omega
t plus initial phase angle, pi
initial.
With this new term, how should I adjust this?
First of all, the amplitude is no longer a constant, rather
it decreases over time. Now, do you expect it
to decrease linearly over time? Well,
if it decreases linearly over time, then it's going to stop, right?
There is a shock cut off. At some point, it completely stops.
Does that really happen like that? In reality,
as the amplitude decreases, so does the speed, right?
This term, the resistance force, is proportional to
the speed. What does that tell you? It tells you initially when you go very fast,
with a large amplitude, this term, the resistance force is large, which
means it's going to large a lot of energy away per cycle.
Gradually, when you slow down,
the amplitude is so small, and the speed is so small that
the resistance force automatically decreases because it's proportional to the speed.
Therefore, energy is still taken away from the system, but at a much
slower rate, which means the amplitude is not going to decay linearly, rather
it's going to decay, but it's going to keep going for a long time. In other words,
I can expect the amplitude to decay like that.
The amplitude itself, would decay like that.
So this is A as a function of time, and this is t.
Something like that, not a straight line. That's more reasonable, right? Now tell me,
what do you think this curve looks like? How about exponential curve?
Mathematically, there is a good reason because
when you take the exponential function, it goes back to itself, right?
And that's what we want, right? We want first derivative, second derivative to have the same function or
form
as x itself. How about I do this, I say e to the power
negative, some sort of constant,
c times t. Now, why a negative
sign? Because we know it decays instead of going up exponentially
over time. This is what we call our trial solution.
We have a constant, c to be determined and we have this new omega
to be determined. By the way, this omega is not the same omega as we had before.
It is not the square root of k over m.
Rather, we'd expect it to be a little lower because
again, we have a resistant force that slows down the motion.
At the next step,
all we need to do is take this and plug it into the equation.
x is substituted by that and you take dx/dt.
You second derivative. It's not hard to do, but it's a little tedious because
after all, it's a product of exponential and cosine function.
You take a derivative, one derivative, you get two terms, right?
And you take two derivatives, you get more terms coming out, but it's doable.
I'm going to leave our the derivation mathematically.
It's a little bit tedious, that's all. But when you plug it in
and you compare the two sides and make them equal,
you find that this indeed, solves the equation.
This trial solution, indeed, solves the equation
with two conditions. Condition number one, what is c equal to?
C must equal to b
over 2m. That's one thing.
So that's the decay constant, c. Another thing, what is omega equal to?
Omega must equal to
the square root of k/m minus something.
What is that something? B/2m squared.
If c equals
this, omega equals that, that turns out
this solution, indeed, solves the equation. You don't have to take my word for it. You can
plug c like this, and omega
in, and you just take one derivative, two derivatives
and see if the two sides match. They definitely will.
So here is our solution: A e to the
power b/2mt cosine omega
t plus pi initial.
Let's look at the physics
of c and omega. Now first of all, look at omega.
Omega indeed, is less than the square root of
k/m. That would be the angular frequency of
free oscillation, without any damping.
We know that with damping, you're going to have a lower value than that because
damping force slows it down.
I don't mind if the damping slows down the motion a bit.
What I'm really worried about is
that we're looking at a square root here, right? But there's a negative sign.
It is possible, mathematically, for the second term to
exceed the first term. Anything wrong with that? Well,
then you wouldn't have a real value for omega anymore.
You'd have a negative value under the square root.
That is a complex number, and that's not what we're looking for.
So what does that tell us? That tells us that if you want to sustain
oscillation,
despite the presence of friction, resistance,
you
want to make sure that omega remains a real number.
K/m must be greater than
the next term, which is b/2m squared.
If that is not satisfied,
then omega is an imaginary number and
you're not going to have an oscillation at all.
So, we have three possibilities. When this
first term is greater than the second term,
equal to the second term or even less than the second term,
and then we're going to have three different physical situations.
Situation number one:
k/m is greater than
b/2m squared. So you see, you have damped constant b but
it's not very large. It is small enough, such that k/m is
greater than that. What that tell us is that omega
is real. There is
real oscillation. So in this case,
the damping
is there but is not very strong. We say it's
under damping. That's the term we use to describe under
damping.
You have an oscillation, even though the amplitude does decrease over time.
That's the first situation. Now the second situation, is
when k/m equals the second term.
Let's write that down as well.
You have k/m
equals
b/2m squared,
so the damping constant, b, is large enough that these two terms are equal.
What do you get as a result? You get omega equals zero.
When omega is zero, you don't
have the cosine term anymore.
Omega is equal to zero, this cosine function has no time dependency, so it's just a constant.
It's some constant that you can bring into constant, A.
All you really have is just the decay function, you know
the oscillation
does not exist anymore. It's going to look like this.
No ups and down. This type of damping
is large enough so that it kills all the oscillation, but if b is
slightly smaller than that, you are going to have oscillation.
This b is barely large enough, to kill the oscillation,
we call it critical damping.
Okay, now the third case: what if
k/m is less than
b/2m squared?
Okay, so in this case, omega is not real.
There is no oscillation, okay?
So what you have here is
just x and t, it just goes down like this, monotonically.
It does not oscillate at all, and what do we call that? We call that over damping.
Over damping verses under damping of course. There is too much damping going on,
that no oscillation can sustain. Okay, so you have three possible cases:
under damping, critical damping and over damping.
If you want an oscillation, of course you b to be small enough so we can
have
under damping. But for critical, and even over damping,
these are not totally useless.
Can you give me an example of critical or even over damping? Well
here's a possibility: imagine opening a door. A lot of these doors,
such as the door to our classroom, they're spring loaded, right?
The reason why they're spring loaded is because
when you open the door, it's going to close by itself.
You don't have to close the door behind you, it's going to close by itself.
But do you want this door, after you enter the room,
do you want it to swing back and forth many times?
You don't, right? You just want it to swing nicely
and close itself. So you think of this door as an
oscillator. Do you want the oscillator to
be under damping, critical damping or over damping? Well, you probably want
critical or over damping, right? So what you do is add
enough resistance into the spring, so when the spring
pulls the door back, there is no oscillation, it just closes.
Another example would be
the needle of a balance.
So you have a balance like this, right? This is where
the needle will settle. But the thing is, if the balance is very fine, chances are, it's going to
swing to this side and swing to that side. It's annoying, it's
going to take a long time before it stops. So
if you add some damping mechanism, in particular, you can use
magnetics to introduce a magnetic damping force.
That can hopefully bring this needle into
critical or even over damping, so that
it can go nicely
and settle immediately.
So these are
the three possibilities. Next, let's look at
the value of c.
Amplitude is a function of time.
A is just a constant, e to the power of negative
c, which is b/2m times t.
The decay constant here is
b/2m.
First of all, you can verify easily that b/2m has
the right dimension. What is the dimension of b?
Well you know that the force is given by negative bv.
So b has the unit
of, well actually, let's look at it this way, you know that kx,
the restoring force.
Then you have b, right? bdx/dt
that's the damping force. These two obviously have the same
dimension, right? Because they're both forces.
That tells you what the dimension of b is.
What is the dimension of b?
These two xs cancel, so
the dimension of b would be the dimension
of k times the dimension of t, right?
So k is newton per meter, in
other words, kilograms per seconds squared,
that's k. You multiply this
by the dimension of t, that's seconds.
That will give you the SI unit of
b. So what's the SI unit of b? That will be kilograms per seconds.
Then you divide by m, m is mass,
so kilograms is gone.
So what is the unit of b/m?
That would be 1/second. You see, and now it will be canceled with t,
which is in seconds. Indeed, this is a dimensionless number, so the units
work out really well.
Secondly, take a look at this decay
constant here. This is telling us
that the greater the p value, the faster it decays, right?
And that makes perfect sense, after all, the reason
why you have a decay and amplitude is
because of the presence of this damping constant, b, right?
Without the damping force, if b is equal to zero, there's no damping. Of course
you don't have this dimension factor, so obviously
the damping is faster when you a greater value, b.
It also makes sense that you have a m in the denominator here, what does that tell me?
It tells me that for the same damping force,
if the object is massive, then the decay
is going to be slower. That makes sense because if you try to slow down a
very massive object from
oscillation, it's not going to be very effective, right?
If the object is not very massive then you use the same amount of force
of resistance, you can slow it down pretty quickly. Therefore,
the more massive the object is, the slower the decay, that makes sense.
This is how we express
the rate at which the decay occurs. So we've learned that in the real world, there's
always a decay going on, so that if you want to maintain the oscillation,
if you want to make the oscillation, going and going,
you somehow must feed energy into it, to replace the energy that's lost through the decay.
This is the restoring force, and this is the decay force.
This is the force due to the presence of friction.
In order to sustain the oscillation, you must
feed energy into it. So you must use a new term to drive the oscillation.
So we're looking at a driven, or force oscillation.
There are different forces you can use to drive it.
If you're looking for a simple harmonic expression for
the final oscillation. Then it's clear that the force
that we want here wouldn't be something really funny like a square
root or whatever. We want it to be just a simple harmonic variation
as well. Let's say the amplitude of this force
is f knot, and
there is also a cosine function of time.
So let's say, it's f knot times
a sine function of time,
sine omega t. Now
this omega can be anything I want. This is how
the force varies over time, and
it is not necessarily equal to the omega
of the free oscillation.
It's going to have a square root of k/m, right? That is not necessarily equal to this
omega. This omega is determined by the driver.
So I grab this spring, and I shake it, okay?
It depends on how fast I shake it, it has nothing to do
with how the spring is going to oscillate by itself. So
that's the force, and that is equals mass times acceleration.
So this time, I have
one more term here. The solution here is a little bit more complicated.
As a matter of fact, it consists of two terms. There is a transit term
and that is, that term is going to go away
over time. It only happens for a brief moment
when you initiate the oscillation.
And then it's followed by a steady term, which does not change over time,
which goes on and on over time.
Now, you can understand why there's a transiting term. Imagine there's a spring, and
there's a mass, nothing is going out at first.
What you do is grab that mass, and start shaking it. Of course
the system is startled by that.
It doesn't know what's going on.
It doesn't know how much force to pull it down with, it doesn't know how fast
to oscillate, right?
So after a few cycles,
you want to pull me down with this amplitude of force and
with this angular frequency. It learns to react.
There is a learning process, which may take a few cycles,
that is what we call a transit solution. We're not going to be bothered by that.
We'll go straight into the next phase, called steady
state solution.
In a steady state solution,
the amplitude of the oscillation no longer changes over time.
Let's see physically, why that's the case. Okay,
because remember, the function of this term is to feed energy
into the system. Now, initially
suppose it starts at rest.
So you take the force and you grab it and shake it, right? So initially b is pretty small.
Of course when
you add this force, it's going to stop oscillating, so v will increase.
As v increases, this resistant force also increases.
Now, this force feeds energy into the system, this
force takes energy away from the system. Initially
because the motion is relatively slow, you feed more energy into it,
and less energy leaves the system. As a result,
there is a net increase in the energy of the system, so the amplitude shoots up.
Gradually,
after the amplitude increases over time, v on average, gets greater,
so the resistant force also increases and then the energy
taken away by this resistant force also increases.
So you have two competing sources,
this one feeds energy into the system, this one takes energy away from the system. At
some point, the average speed reaches
such a value, that the energy
you feed into the system per cycle equals
to the energy that's taken away by the resistant per cycle.
So what does that tell you? That tells me that at that point,
that stage is reached, there is no
longer a net change, right?
Which means the amplitude of the system
must be a constant. That is what we call a steady state solution.
So in the steady state solution, I have
a fixed amplitude oscillation, t and x.
But
please do not confuse this with
a simple harmonic motion. This is not a simple harmonic motion, this is a force
harmonic oscillation. First of all, the
frequency is not equal to k/m necessarily,
rather it is equal to whatever driver it wants. You grab it and shake
it that fast, it oscillates that fast. If you grab it and shake it slowly,
it has to do it slowly. The system has no choice
as to what omega value it must oscillate with.
That is dependent on the driving force.
That doesn't mean the system likes to be driven like that.
You know, the system can have its own reaction, right? It has its own reaction.
Imagine a child on a swing.
You put a child on a swing like this, right? You know how you do that, right?
When a child moves towards you like this, you don't push him back.
Rather you wait for him to come over and give him a push,
and the child goes forward with greater amplitude, then he comes back,
push him again, goes even higher, comes back, push him even higher.
Now if the child swings like this,
by himself,
you don't want to grab his seat and shake it like this.
Doing that, isn't going to be very effective in terms of
increasing amplitude of the swing. So that tells you that
the system truly has no choice as to what frequency you want it to oscillate.
But it does have a choice to whether it likes the frequency or
not. By liking the frequency,
what I mean is that you are using a driving frequency,
which is close to
the natural frequency of the oscillation. The system oscillates like that naturally.
You are pushing it also like that, so
pretty much every moment that you are pushing it
in the right direction.
Essentially, you are always doing positive work to the system.
You are not resisting the motion, you don't go out of sync with the system, so when
the system goes this way, you try to push it like that, you would be doing negative work.
When you are applying
a driving force with the same frequency as
the square root of k/m, which is by the way called the natural frequency.
Let's call this omega knot just to tell it apart.
Here's the natural frequency. If the driving frequency is close to the
natural frequency, then what happens is
that the system moves in sync with the driving force.
So the driving force is always pushing the system in the correct direction, giving
it positive work. And that dramatically increases
the amplitude of the oscillation so that
if we say this is the situation where the driving force is either
too fast or too slow, it doesn't give you a very large
amplitude, right? And then when the driving force is just right
in frequency, it can give you a very large amplitude like this.
So what you have here is a simple harmonic variation
as a function of time. So it's like sine omega
t plus
pi. But there is an amplitude. This
amplitude does not change over time, because remember
this force and that force, they do the same amount of work over
one cycle, positive and negative, so the amplitude does not
change. But it does depend on the driving frequency, okay?
If this driving frequency is close
to the natural frequency, then A would be large because you are doing positive work, remember?
If omega is too fast or too slow compared to omega knot then
the oscillation will not have a very large amplitude. So this is
the
representation of the so called steady state solution.
Now you can find what that A is. All you have
to do is plug this into this expression.
Of course, you have to take two derivatives here.
You have to take one derivative here so that you can match it.
We're not going to be bothered with that, that's just mathematics but
it's a little tedious but certainly doable.
What we want to know, is the physics. And that is
what is the shape of A omega? We know
that if we plot A as
a function of omega, there is one function that stands out and that's called
natural frequency of the system. If
omega is close to omega knot, the oscillation will look very large
amplitude. When omega is too large
or too small, the response of the system will be weak, in other words, you get
a smaller amplitude. So you can expect something like this.
So
you see, around that
natural frequency, if the driving frequency matches the natural frequency, you have the greatest amplitude.
When the frequency is too large, too small,
you can have small amplitude. This phenomenon
is called resonance.
And this is called the resonance peak.
You can draw two possibilities. Let's say you have
a curve like this.
In other words, there is a resonance curve, but it's not pronounced at all. It
is shallow. They look at a different situation like that.
Look at that, that has the same resonance frequency as the first one but the peak is very
pronounced. Why is there a difference?
Well that depends on a mathematical detail of omega, which
we're not going to get into. But physically
we can understand why there's a difference.
If you look at these three forces, now this force
is simply responsible for providing the natural frequency, k, right?
This force is the driving force. Its
frequency will determine whether the system
have a large oscillation or a small oscillation. What about that force?
See, that is the resistant force. The resistant force is
always opposite to the direction of motion. It always slows things down. Does
it care about whether the frequency is too high or too low?
No it doesn't. That term cares about
whether the frequency is too high or too low.
So the resistant force is
insensitive to frequency, okay? When b is small, then
of course we're not going to worry about it too much so that you have a
very pronounced resonance peak like this. After all
this force, this term has a omega in it, when omega matches omega knot
there's a very large oscillation force. But when b is very large, then
the system is dominated by the resistant force. You don't really care that much about
the next term, which is the driving force.
So that the curve would not be very
sensitive to frequency, so you'll get a flat curve like that.
It is clear that in many situations, we want
a very sharp response like that. Now in this situation, we say the system
is highly selective. It favors this frequency,
very pronouncedly, and rejects almost all other frequencies.
Can you think of an application of a situation like that?
Well, this is a very discriminating curve.
If favors only this curve and rejects everything else.
Can you think of a situation where we want
the system to be highly selective? Well,
why not? How about
radio tuner? Or television tuner?
Our air is filled with all kinds of radio waves and all kinds of frequencies, right?
How do you tune your radio so that you select only
this particular frequency and reject all other frequencies because you don't
want to listen to two radio stations at the same time, do you?
How do you do that? It's simple. What you do is you tune the radio so that
you adjust the natural frequency of the receiving circuitry.
You are actually adjusting omega knot. Now if
omega knot matches one
frequency of one particular radio station, then
the signal from that radio station, unless a large response in
your receiver, and neighboring stations with slightly different frequencies will
be rejected. So you see, when you
buy a radio or a television set, you want a tuner that is highly selective,
so that it can pick up the whatever frequency or whatever channel
you want to listen to or watch. You don't want to buy something like that,
as a tuner because you can listen to a whole bunch of stations at the same time because you see
the response is not all that different from one another.
That is the so called the resonance, and resonance is
a wide spread application in all areas of physics.
It is sometimes a good thing, and some times a bad thing like in the case of
radio tuner, resonance is good. We want
the resonance to be very pronounce. But resonance can also be very disastrous.
For example, if you have a bridge,
the bridge, being a mechanical structure,
has its own resonance frequency. It may not have single frequency because the structure
is complicated, you
can have several natural frequencies. What does that mean? It means
that if a
natural force grabs that bridge and shakes it,
with that particular frequency, which is close to the natural frequency, that
can cause the bridge to shake violently because
there's a resonance, right? So when you design
a bridge, you want to find a way to
calculate these natural frequencies. And you want these natural frequencies to be as far
away from
the natural oscillation that can occur around the bridge. For example,
wind. Wind can blow
on and off, it's like a force rocking
the bridge, right? But if the frequency
due to the blowing of the wind, gets
close to one of the natural frequencies of the bridge itself, then
that can cause a large oscillation in the bridge that can even
collapse the bridge. I'm not just saying that as a theoretical possibility, I mean this
has actually happened. In the text book, you see
a famous example of the 1940 turbulent winds
that set up a torsional vibration in the Tacoma bridge in
Washington. This is page
474, and that caused that bridge to collapse.
So when they designed that bridge, one of the
natural frequencies happened to be around the frequency at which
the wind rocks the bridge. So that was a structurally bad design.
So you want to make sure that when you design a bridge, you don't
design a bridge to have a frequency that's close to
this range.