Driven Oscillations

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Driven Oscillations

• Driven Oscillators
• Equation of motion
• Algebraic solution
• Complex algebra solution
• Phase and amplitude of particular solution
• Algebraic
• Complex phase methods
• Argand diagrams
• Resonance phenomena
• Position of amplitude resonance
• Phase behavior of x(t) with respect to driving
  force.
• Transient solution
• Why transients are necessary?
• Typical transient behavior

• Q as the FWHM of the kinetic energy resonance
• Average kinetic energy of a driven oscillator
  over one cycle
• Resonance approximation
• Definition of FWHM

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What will we do in this chapter?
We discuss sinusoidally driven oscillators with
velocity dependent damping. The equation of
motion is an inhomogeneous differential equation.
We show that the solution consists of a transient
solution (which looks like a free oscillation
solution) and a particular solution which depends
on the driving force and frequency. We present an
algebraic solution (text) and a complex algebra
solution which I believe is better and introduces
many useful complex techniques. We next discuss
the amplitude and phase shift between the x
motion and driving force as one passes through
the resonant frequency which is close to the
natural frequency.
We show that the relative phase between the force
and x(t) passes from 0 to 1800 as one sweeps
through frequencies and 900 at the natural
frequency of wo2 k/m. We also show that the
width of resonance is proportional to 1/Q where Q
( for Quality) is a dimensionless variable
proportional to wo divided by the damping
coefficient. We conclude by discussing the role
of transient solutions and by discussing the
resonance behavior of the average kinetic energy
as a function of driving frequency.
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Driven Oscillations
We need the transient piece to match the initial
velocity and position, but it always dies out if
there is damping. As a practical matter , it
often suffices to know the particular solution.
We will review the way that the text does this
and show an alternative method using complex
variable representations.
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Algebraic Method
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Contrast this with a complex solution
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Complex solution continued
Extracting the phase and amplitude
We get the same result but I believe the
technique is much easier and gives more insight.
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Phase and amplitude of xp complex method
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Polar forms
Draw the D complex number in an Argand diagram
where we plot the real part on the horizontal
axis and the imaginary part on the vertical axis.
The modulus is the length and the phase is the
polar angle in such a diagram .
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Resonance
Q is a dimensionless variable which gives the
quality of the resonance
The amplitude resonates at the slightly different
value of wR in a way that depends on Q or b. It
is remarkable that the x(t) response is out of
phase with the driver force. At low w, x(t) and
F(t) are in phase. At w w0 ,they are out of
phase by 900 . At high w, they are out of phase
by 1800!
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Resonance phenomena
We plot the amplitude and phase versus the
driving frequency for 3 different Q values in the
upper two plots. The larger the Q value -- the
sharper the resonance both in phase and
amplitude. The lower the plot shows the ratio of
the resonant and natural frequency as a
function of Q. For high quality resonances, the
amplitude resonance occurs very nearly at the
frequency of un-driven oscillations.
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Whats the deal with transients?
Transient solutions are necessary since as shown
above the particular solution is too specific.
All aspects of xp are specified in terms of the
driving frequency and driving force. Yet we
still need some way of specifying arbitrary x(0)
and v(0). We thus need to add a function w/ two
coefficients.
You can get some pretty complicated motions when
driving frequency is very different from the
natural frequency. But the sum (upper) makes
sense when you look at the 2 components (lower).
A and ? give us the freedom to match x(0) and v(0)
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The ubiquitous Q
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More on Q
Here is a plot to illustrate the FWHM concept on
the average kinetic energy of the driven
oscillator. I have selected a damping coefficient
to create a fairly broad Q4 oscillator. Even so
the FWHM is close to 1/Q
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How can we show this?
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Eloss (continued)
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Electrical analogs
There are many important analogs to the motion
of the driven, damped harmonic oscillator. They
occur in fluid mechanics, electrodynamics,
quantum mechanics, etc and often appear in more
practical problems than that of the mechanical
oscillator. We will work out the electrical
analogy in depth. We begin with a review of some
basic elements of AC circuit theory that you
presumably learned in physics 112.
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Voltage drop across a capacitor
A very crude analogy is that voltage is analogous
to pressure, and charge is analogous to fluid
mass. A capacitor stores charge like a a tank of
a certain capacity stores water. Since the
pressure difference between the top and bottom of
the tank is proportional to the height of the
water, a large capacity (area) tank
develops little pressure drop when storing a
certain amount of fluid. The electrical analogy
is a capacitor with a large capacitance C
requires a small voltage drop to store a given
charge Q.
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Voltage drop across a resistor
A crude analogy for the voltage drop when current
flow through a resistor is the pressure drop
required to flow a viscous fluid through a narrow
pipe.
The physics behind Ohms law , a very simple and
highly intuitive result, is actually fairly
advanced and extremely important! In a perfect
crystalline metal, an electron can flow from
lattice site to lattice site unimpeded as is the
case in a superconductor. This extremely
unintuitive conclusion follows from a quantum
mechanical treatment of electrons subjected to a
perfect lattice of potentials due to the
regularly spaced positive ion cores in the metal.
The energy loss which requires a potential drop
or electrical force to allow flow follows through
through inelastic collisions of the electrons
with imperfections in the lattice site pattern.
This collision energy is transferred through
quantum vibrations called phonons. The density
of such phonons is roughly proportional to the
temperature of the metal and hence ordinary ohmic
resistance is roughly proportional to
temperature. Of course there are conditions such
that the resistance vanishes completely when
certain materials are cooled below a critical
temperature and the metal turns superconducting !
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The voltage drop across an inductor
I know of no simple fluid analogy to an inductor,
so we will begin by discussing the real physics
behind it. A solenoid is often used as an
inductor. In a solenoid, one generates a
magnetic field proportional to the current
flowing through the coil.
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The series RLC circuit
The series RCL circuit forms a perfect analogy
to the driven mechanical oscillator. An easy way
to get the relevant differential equation in the
charge is to set the total voltage drop around
the loop along the indicated path to zero.
Otherwise the voltage at a point is ill-defined
This Kirchoff law is very similar in spirit to
the work energy theorem.
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Electrical/Mechanical analogies

• Q is analogous to x
• L is analogous to m
• R is analogous to b
• E is analogous to F

The analogies follow directly by comparing the
two differential equations
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Energy analogs
Hence in the mechanical oscillator the energy
oscillates between kinetic and potential energy.
In the RLC circuit, the energy oscillates between
electrical energy ( or the energy stored in the
capacitor) and magnetic energy ( or the energy
stored in the inductor). When driven at the
natural frequency (w w0) equal average amounts
of kinetic and potential energy appears for the
mechanical driven harmonic oscillator. When
driven at the natural frequency, equal average
amounts of electric and magnetic energy appear
for the RLC circuit.