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we have so far studied what is called simple harmonic oscillation
in which the only net force exerted on the object is a linear restoring force
uh such as the force from a spring. We know
that force is conservative which means the total mechanical energy of the system
is conserved, and we studied that before. In reality, though,
when you consider any simple harmonic system such as
you know a pendulum, the pendulum is not going to swing forever
if you pull it to the side and let go of it, it's going to swing
for a number of cycles and the amplitude will get smaller and smaller
eventually it's going to stop. The same thing happens to
a block attached to a spring, and after
several cycles of oscillation, 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, take mechanical energy away from the oscillation system,
and that energy, of course, is turned into heat and sound and other things
so we must, if we want to consider a real-life situation,
add a resistance force
and the resultant 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 solid object
going through a fluid, such as air or water,
one way to express the resistance force
is something like a constant times speed squared.
The speed squared
is a quadratic form
and that will add a lot of complications to
our differential equation. In fact, it's going to make our differential equation nonlinear,
so we're not going to deal with that. We are going to deal with the
simplest type of resistance force as far as the solving equation is concerned,
and that type of force is called linear resistance force.
In other words, you do get a resistance force
f sub r, "r" standing for resistance. It is negative,
meaning it's opposite to the direction of motion,
and it is linearly
proportional to to velocity, negative
V, with, of course,
constant in front of it, which we call "b"
Again, this is not the only type of resistance force,
but as you will see in solving the equation since "f" is directly proportional to
"v" to the first power is a linear resistance force. That makes our calculation a lot
simpler.
So, with that new force added in
let's look at what happens in the x-direction. What is the net force
in the x-direction? Well of course I have negative kx
that could be a real spring or an effective spring. Anyhow, that
is the linear restoring force responsible for the oscillation, but on top of that, we also need
to add this term, the resistance force minus bv.
Okay, V is really just dx/dt,
as you know, right? So that's the net force.
What is that equal to? That equals mass times acceleration, so mass times acceleration
which is the 2nd derivative. With this new force
added, how does it change the oscillation?
We can look at this problem from a math point of view and a physics point of view.
From the physics point of view,
you know when you add this force
This force always resists motion because, you see, it is always
against the velocity, so when the object moves forward,
this force is backward, and when the object moves backward, this goes forward, it always
tends to slow down the motion and therefore, with this
additional force are you not gonna move as fast as before
you are gonna lose kinetic energy, and that kinetic energy lost is not turned
into potential energy such as
in the case of this term, rather, it is
turned into heat and sound, and so forth, therefore, you're gonna lose
mechanical energy overall, and we know the mechanical energy
of a simple harmonic oscillator at a certain moment
when its amplitude A is always 1/2 k*a^2
and as the mechanical energy is lowered over time by this resistance force,
we know the amplitude has to decrease as well.
So if I compare our new
oscillation with this damping force with the previous
simple harmonic oscillation by doing a plot,
here is T and here is x,
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 with this new term added? Well, we know the
energy is gonna decrease over time and therefore
you're gonna have a situation like this.
you have reduced amplitude
over time. In other words, the amplitude tapers down like this,
so that's the first thing that we have to consider. The amplitude is no longer a constant,
but rather, it decreases over time, that's one thing. Another thing is
what happened to the period of the oscillation?
In other words, does this term slow down or speed up
the oscillation? Of course the answer is obvious; this term always resists motion, therefore
it tends to slow down the motion, therefore you can imagine
when you add this resistance force the oscillator is gonna take
more time than before to complete one cycle
because it's always opposed by this additional force and therefore the period tends to get longer
or the frequency tends to get lower. So, again, these are the two
new things from the physical consideration when you add this new term,
and that is, first of all, the amplitude will decay, and secondly,
the period will get longer or the frequency will be
lower. That's from the physics consideration.
Mathematically, let's look at this equation in detail. You have here is
x to the first power, here is dx/dt to the first power, here is the second derivative
of x with respect to time. There is no such thing as
x^2, for example, or sin x, cosine, whatever.
So every term is proportional to x to the first power,
so that's good news because that means this is
a linear differential equation, which is rather simple to solve mathematically.
ok the idea is this: you have x,
you have dx/dt, and you have the second derivative,
and when you move this term to the other side, you'll find that these three terms
will add up to 0. What does that tell us? It tells us
that you are looking for a solution x, as a function of time, such that
this function itself
and its first derivative and its second derivative should assume the same
functional form. In other words, it should have the same dependency on time
that is why this term, this term, this term, if you add them together, they can cancel out.
If this depends on e to the power t
and the next term depends on sin t, you know they're never gonna be able to cancel each other out
for all times. So therefore you are looking for the same functional dependency.
So now the question is, what kind of function behaves
in such a way that when I take the derivative, I take the second derivative,
it goes back to itself? Well, the only thing we can think of
actually, two things we can think of. One is an exponential function
the other would be sine/cosine function. Now, we know that without determining the sine/cosine function,
and that's what happened with mister okay
without his term look like this I the solution somehow my social you're
familiar with now
he I and then cosine
dnt plus initial phase angle
financial I'm wynonna right now
with this new tire how should I justice
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rather decreases home to now do you expect
decreased merely all the time
well decrease immediately all the time then it's gonna stop crime
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are nasty amplitude decreases so to speak I
abs you see this term the resistance forces
positives wanna tell you details initially will be fast
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which means you can take ma energy away for SAC okay
but gradually housing slowdown somehow gonna
the everything is so small and speed is a small flat these
resistance force automatically decreases because board mislead
devore energies to taken away from system but at
much smaller rate which means the company's not too keen riney
robber is good UK by is gonna keep on going going for a long time
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let's look at the physics
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there's always a *** ain't going on so that if you want to maintain the
oscillation
you wanna meet opposition only and going and going
you must somehow feed energy into it to replace the energy has lost decay
ok this system wrist when you force and this dickey for sake
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in order to sustain our solution you must
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if you're looking for us simple harmonic expression
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and his %ah patterns in Galway
overtime it only happens for brief moment when
be initiated the ostrich
and that's followed by steady turn which does not change over time
which which which which calls on on how
now you can understand why did a transient imagine there's a spring
and there's a mass right nothing's gonna first
would you do is you grab mass used force okie he's touching
a horse you know the system is startled by
tried to happen and you what's going on I could haha
you know how how much force one point how is he doesn't know how fast you want
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back is with a transient solution not gonna be mobbed by
be woodall Street into next phase Andrews was class steady
state solution okay
steady-state
steady state solutions in a steady state solution
d amplitude of the oscillation no longer changes
overtime let's eat mathematical one has the keys
be because remember a function this term is two feet
energy into the system now initial
suppose he starts at rest oscillation start
crest CT casualty prodigy initial these pretty small
you agree he's pretty small okay and a horse
when you add this force I it's just a hassle easily will increase
has be- increases this raises for schools
increases now %uh this force feed energy into the system
this 46 energy away France's initially
because motion is not relatively small are you feed more energy
into it and last energy visa system as a result
nap during nap increase in energy persistently empty
shooter increased up gradually
at the MPD increased all the time v on average his crater
the reasons for supposing increases and then the energy
taken away by this reasons forced also decreased
and so you have to compete sources
this one feat energy to system this 22g way from the system
at some point the speed reaches
the average speed which is such a pal that the energy
you feat inconsistent per cycle equals
to energy has taken away find resistance prozac
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longer any met change energy of the system
i'd which is the amplitude system
must be constant that he's steady states
so in a steady state solution I have just
in a fixed amplitude oscillations
T and axe
please do not confuse this we use
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harmonic passage first the Akkadian
frequency he's not equal to scored a camera and necessarily
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she could have passed is costly to past and grumpy she could slow
it has to do slowly okay the system has no choice
as to what oh my god damn musasa leaves
act is dependent on drops
what that doesn't mean the system ike's to be dry
you know the system can have its own reaction I had some reaction
imagine you know a choc ice
you put a shot past you don't you hung dude
on a charge moves 20 like this you push back
rocker you wait for him to come over and you push
and chuckles forward with great aptitude he comes back
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pushing higher now each house we lack is
I him South my kids my kids
you don't want to pry his seat and shake a leg is because you know
between Seoul is not gonna be very effective in terms of
increase in after to the bustling so that tells you
the system truly has no choice as to what frequency you want us lead
hot don't have a choice whether it lacks the speakeasy
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driving frequency which he's close
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natural frequency
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the system moves in seek withdrawn first
dramas always push the system in the correct direction
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its if we see this is a situation where driving forces
even too fast or too slow to give you a very large
am pretty right and when a driving force is just right
I in frequency you can give it a very large number do you like these
see like this I
so what you have here is a sin is this is a simple harmonic variation
at a function time so it's like sign
on my kitty plus
far but there is a amputee
this amputee has not changed over time because remember
this force a nap that was the do the same amount work
over one cycle positive iniki's the amplitude
does not change but it does deep and hard rock britain's
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as a function home thirst my africans it stands out an ***
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and this is called resins peak ok
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occur like this
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hacked
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pronounced why is there a difference
well that depends on a mathematical details of be only
we're not which may not get by physically
we can understand why there's a difference %uh
if you look at these three fourths ok out this force
simply is responsible for providing the africans
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its frequency on will determine whether this is my
largely since March wanna force
see that ease the reasons the resistance forces
always opposite to the direction of motion how slow things down
doesn't care about whether to frequencies to how to how
no it doesn't actor cares about
when it freezes too hard to convince africans soon
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insensitive to freaks okay when B is small
then of course when I don't worry about it too much soon and you happy
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this force destroying have homemade in it monica mattos
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then the system is barred by the resistance force you you can't have much
about
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so mad at the truth will not be here would not be beary
sensitive to frequencies and get a flock
okay %uh it is clear happy many situations we want
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the reproduce rejects almost home africans
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the system to be highly selective
well why not how how I
revealed soon okay martel engine you know
our various fields all have dreams all have frequencies
I how do you to you read his slack
only this particular frequency okay reject of africans
because you don't listen to to reduce the student Andrea
how do you do that simple which you do is you too real
soon at you just natural frequency after receiving circuitry
of course is natural hostage the idea is
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rejected you see so we knew bach
review on TV set you wanna tuner that is highly selective
so that you can pick up whatever frequency hundred-channel
you wanna listen to watch ok you know I'm
as a junior because then you listen to how much is decent hockey see
response is not active home Sol
patties to support residence and residents
widespread applications all areas of Physics
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cases review tuner presence is good we long
dresses to be very pronounced other draft class can also cause disastrous
ok for attack %ah you happy bridge
you have a bridge the bridge be and mechanical structure
has gone rat as frequencies it may not have a single frequency become
structures hockey
you can have several natural freaks without me
it means that if evil force GB if
actual was crap stockbridge Sheikh Syed
was that particular freaked which is close to the natural ricans
that's because the bridge to sheik party
because residence right so when you design
a bridge you wanna find a way to
talcum predict these natural frequencies you want the statue frigates to be far
away
from natural oscillation current bridge
for example weekend okay in a week ball
%ah on off on off is like an *** rocking
my was rocking the bridge but Eve the frequency
due to due to the blowing in the wind
gets close to a natural frequencies for the bridge itself
back and cost your march ha solution in the bridge second even
class the bridge and I'm not just seen at human capacity
discuss action happened in tax
you see a famous exact okay 1940 term
weeds asap torsional vibration Tacoma Narrows Bridge
in wash %ah this is not page
for 474 and back cause the bridge to collapse
when he designed a bridge be your wanna
natural frequencies happen to be around frequency
which twin in rocks to you bridge happen once the statue impact is %uh
saw you wanna make sure average you don't
is on a bridge to have a frequency data as close to
to his ranch