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The Damped Vibrations
applet illustrates

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several different
concepts related

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to differential equations.

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Initial conditions for
second-order equations,

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the phase plane for
autonomous equations,

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and damping conditions
for second-order

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homogeneous linear equations.

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The differential we're studying
appears at upper right.

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And the parameter
names reflect the fact

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that this equation describes
a mass-spring-dashpot system,

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with mass m, dashpot constant
b, and spring constant k.

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The right-hand side of
the equation is zero.

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There's no forcing term.

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The equation is homogeneous.

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The sliders at bottom
control the values

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of the system parameters.

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With the selected
values for m, b, and k,

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the solutions are
damped vibrations.

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I can let the solution evolve
by grabbing the time slider

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or by using the
animation key, which also

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serves to stop the animation.

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If the value of the
solution gets too small,

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I can blow the picture
up using the zoom slider

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at the upper left.

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Moving it to the right
squeezes the solutions.

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Moving it to the left
makes them larger.

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The window on the left lets
you set the initial condition

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of the system.

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There are two components
to the initial condition

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of a second-order equation.

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Position and velocity,
that is x and x dot.

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Together they form
the coordinates

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of a point on this plane.

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I can grab the point
and change both x and x

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prime simultaneously.

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The x-coordinate is
written vertically,

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because that's how it's written
on the right-hand graphing

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window.

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And you can see that they
keep pace with each other.

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The horizontal component is
x dot, the initial velocity.

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That's positive when
we're to the right,

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and becomes negative
when we're to the left.

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The left window
represents phase space.

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For any time t, the
values of x and x dot

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are the coordinates of
a point in phase space.

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And as the solution
evolves through time,

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that point follows
a path, sweeps out

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a path in phase space.

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And with these selected
choices of m, b, and k,

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the path is a spiral.

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This reflects the
fact that both x and x

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dot undergo a damped vibration.

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The system parameters can
be set using these sliders

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at lower left.

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Let's suggest the mass to be
1/2 and leave k to be one,

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and watch what happens
to the solutions

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as we vary the
damping constant b.

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When b is small, the solution
doesn't damp out as quickly,

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and when b is large, it
damps out more quickly.

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These things become much clearer
when you think about them

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in terms of the roots of the
characteristic polynomial

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of this equation.

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And we can display those roots
by clicking the roots button

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here.

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What's shown here
is the complex plane

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with the roots of the
characteristic polynomial drawn

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on it.

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Also we have a readout of the
two roots in green over here.

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So let's watch what
happens when we adjust b.

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I'm going to start
with b equal to zero.

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In this case,
there's no damping.

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The solutions are
sinusoidal, the roots

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of the characteristic
polynomial are purely imaginary,

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and the trajectory in
phase space is an ellipse.

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Both the solution and its
derivative vary sinusoidally.

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As I increase b away
from zero, I get spirals.

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And at the same time, the roots
move off the imaginary axis.

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They acquire a negative real
part, which is minus b over 2m,

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incidentally.

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And as I increase b further,
the spiral opens up,

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the damping occurs
more quickly, the roots

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move away from the
imaginary axis.

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They increase the size
of their real part.

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The damping occurs more quickly.

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And you'll notice that
they get closer and closer

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to the real axis as well.

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Their imaginary
part is decreasing.

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That means that the circular
frequency, the angular

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frequency of the
solution, is decreasing.

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The period, or pseudo-period
of the solution is decreasing.

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You can see that in the
graph, but it's much clearer

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to see what's happening
to these roots

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of the characteristic
polynomial.

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There are converging
on the real axis,

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and eventually they merge into
a double root on the real axis.

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This is critical damping.

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From the readout, we
see that the value of b

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seems to be about 1.41.

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Why don't you take a
moment and calculate

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what b is for these
values of m and k

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when critical damping occurs?

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If we continue to increase
b beyond critical damping,

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now the roots move
out on the real axis.

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There are two
exponential solutions,

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and the general solution is
a linear combination of them.

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And there's no oscillation
in any solution.