Oscillations

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Oscillations

• SHM review
• Analogy with simple pendulum
• SHM using differential equations
• Auxiliary Equation
• Complex solutions
• Forcing a real solution
• The damped harmonic oscillator
• Equation of motion
• Auxiliary equation
• Three damping cases
• Under damping
• General solution
• Over damping
• General solution
• Solution in terms of initial conditions

• Critical Damping
• Break down of auxiliary equation method and how
  to fix it
• General solution
• Solution in terms of initial conditions
• Over damping as ideal damping
• Phase diagrams
• Un-damped phase diagram
• Obtaining phase equations directly
• The under-damped logarithmic spiral
• Critical damping example
• Harmonic oscillations in two dimensions
• Lissajous figures

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What will we do in this chapter?
This is the first of several lectures on the the
harmonic oscillator. We begin by reviewing our
previous solution for SHM and use similar
techniques to solve for a simple pendulum. We
next solve the SHM using the auxiliary equation
technique from linear differential equation
theory. This allows us to extend our treatment
to the case of a damped harmonic oscillator with
a damping force proportional to drag. Three
damping cases are considered under damped , over
damped, and critically damped. The critically
damped case -- besides being very practical --
brings a new wrinkle to the auxiliary equation
technique.
We discuss the phase diagram which is a plot of
trajectories in a phase space consisting of p and
x. Several methods for computing these
trajectories are discussed and the under damped,
un- damped, and critically damped examples are
drawn. We conclude with a brief discussion of
harmonic motion in two dimensions and Lissajou
figures.
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Simple harmonic motion
We already are familiar with this problem applied
to the mass spring problem. We write the force
law and potential and the solution which we
obtained using energy conservation.
We also show that the simple pendulum put in
small oscillation has an analogous potential and
kinetic energy expression and hence will have the
same formal solution as the spring-mass system.
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SHM solution by DE methods
One way around this problem is to insist that x
is real by demanding x x
This really means that rather than having
arbitrary B1 and B2 we really only have one
arbitrary number say B1. Fortunately B1 is
complex and thus has an arbitrary amplitude and
phase. We need two arbitrary real quantities in
order to match the initial position and phase.
Complex numbers snuck in our solution! Indeed
B1,2 should be taken as complex numbers as well.
But x(t) must be real?
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DE methods (continued)
As another example we could include a linear drag
force acting on the mass along with the spring.
If we had chosen B1-i a exp(-id) we would have
obtained an equally valid solution x 2a sin(wt
- d).
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Damping cases
Under damping
Solutions will depend quite a bit on relationship
between b and w.
The under damped case will have complex auxiliary
roots and will have oscillatory behavior. The
over damped case will have real roots and thus
have a pure exponential time evolution. The
critically damped case, with a single root, has
some non-intuitive aspects to its solution.
The under damped oscillator has two constants
(phase and amplitude) to match initial position
and velocity. The solution dies away while
oscillating but with a frequency other than w
(k/m)1/2.
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An under-damped case
Here is a plot of x(t) for the under-damped case
for three different choices of phase. We have
chosen the damping coefficient b to be 1/4 of the
effective frequency w1 so you can see a few
wiggles before the oscillation fades out.
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Over damped case
In this case both the B1 and B2 coefficients must
be real so that x is real. The two terms are both
there to allow us to match the initial position
and velocity boundary conditions. Since w2 lt b
both l1 b w2 and l2 b - w2 are both
positive and thus both terms correspond to
exponential decay. We can re-arrange the blue
form to match initial conditions.
The initial condition form is easy to confirm by
expanding to 1st order in t.
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Critical damping
It is interesting to note that the critically
damped case actually dies away faster than the
over damped case. Consider the case where v00.
The critically damped case will fall off
according to exp(-b t) The over damped case will
have a exp-(bw2) t piece which dies off
faster than the critically damped case. But it
will also contain a exp-(b-w2) t piece which
dies off slower than the critically damped case.
For many applications vibration abatement,
shocks, screen door dampers, one strives for
critical damping.
To get a real x, B3 must be real. But this cant
possibly be right! With only 1 real coefficient
we cant match the initial position and velocity!
Appendix C of MT says in the case of
degenerate roots, the full solution is of the
following form which can be confirmed by direct
substitution.
In terms of initial conditions
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Example of Critical Damping
We use solutions with x01 and v00 and consider
the case of critical damping and two cases of
over damping. The critical damped case dies out
much faster and the over damped case dies out
more slowly as the over damping ratio increases.
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Phase Diagram
It is fashionable to view the motion of
mechanical systems as trajectories in a phase
space which is a plot of p (or v) versus x.
Oscillations are a perfect example of such plots.
A first phase diagram is for an un-damped
oscillator. Since we have a purely position
dependent one dimensional force we know that
energy is conserved.
In Newtonian mechanics, we specify the initial
conditions as x(0) and p(0) (or v(0)). This
corresponds to one point in the phase diagram
which specifies the initial energy or the phase
space ellipse. We note that with this choice of
coordinates, we have a clockwise phase space
orbit. This is because of the negative sign in
the equation of motion
This curve is an ellipse with an area monotonic
in energy.
Essentially this means that p must decrease when
x is positive and must increase when x is
negative. The particular phase space orbit will
depend on the nature of the forces but in general
we know that no two phase space orbits can
intersect. Otherwise several motions would be
possible for the same set of initial conditions.
Without damping the particle will execute endless
elliptical orbits of fixed energy in this phase
space
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Phase Diagrams (continued)
One can construct the phase diagram directly from
x(t) and its derivative or (in the absence of
dissipation) from the conservation of energy.
Often the phase diagram can be obtained through a
clever separation of variables
We show the phase diagram for an under damped
oscillation. Again we have a clockwise motion but
this time the energy continuously decreases with
time and eventually disappears. This sort of
curve is called a logarithmic spiral. We could in
principle obtain this directly from out solution.
The text shows a way of getting the spiral by a
variable transformation.
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Critically damped phase diagram
It is very straight-forward to draw the phase
diagram once one has x(t) and its derivative. We
plot plot x and v for 50 times for 3 sets of
initial conditions. You can easily visualize the
motion including the maximum displacement and
velocity and retrograde motion
Whoops I messed it up. How did I know?
Whats wrong with this picture?
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Harmonic motion in two directions
Lissajous figures are often a great way to
measure a phase difference on an oscilloscope.
Lissajous figure is a plot of y versus x as t is
varied. To the right is a Lissajous figure for
the case Ax Ay and wx wy and dx0 for 4
different dy phases. With these conventions, the
figure maximally opens up into a circle when dy
0.
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Lissajou figure with unequal frequency
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A cool trig identity
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Lissajous figures with different frequencies
To the right is are Lissajous figures for the
case Ax Ay and wy 3wx and dx0 for 3
different dy phases. The figure maximally opens
up when dy 90.
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Lissajous with irrational frequency ratios
To the right are a Lissajous figure for the case
Ax Ay , and dx dy 0 with the indicated
frequency ratios When the frequency ratio is not
a simple fraction (or irrational) the path does
not close and finally fills up the whole
screen. How would this figure look if
?