Lecture 8: Differential Equations II

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Lecture 8 Differential Equations II

• Lecture 8
• Second order differential equations
• Damped harmonic motion
• Forced harmonic motion
• Problem Set 8
• Finish 5.00pm

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First order differential equations

• A differential equation is an equation which
  contains one or more derivatives of some quantity
  (y)
• The order of the differential equation is the
  order of the highest derivative which occurs in
  the equation.
• a, b and c are constant coefficients

First order
Second order
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Examples in physics

• We have already met a special case of this
  equation
• This describes simple harmonic motion
• Note that if we have the additional term
• Then if yf1(x) and yf2(x) are solutions then
• Is also a solution ? This is the principle of
  superposition This principle also applied to SHM.

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Damped Harmonic Motion

• A damped harmonic oscillator is one whose
  equation of motion is
• Where is the ?0 angular frequency without damping
  (SHM) and ? is the damping term.
• Damping arises from frictional forces of some
  kind in an oscillator.
• The negative sign is because the force opposes
  the motion.

Spring constant
eg in a spring
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Damped Harmonic Motion

• Resistance in an electrical circuit
• In differential form
• The differences from SHM arise because energy is
  lost due to resistive heating.

R
V
L
C
So for this example
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Damped Harmonic Motion

• Solution
• Try a solution of the form
• This must be true for all the times t, so

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Damped Harmonic Motion

• Our original equation is linear and homogenous
  and so the principle of superposition applies
  the general solution is
• Where A and B are constants that can be derived
  from the initial conditions.
• If we define
• Then the solution is

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Damped Harmonic Motion

• There are 3 different cases
• Then q is real and positive. There are no
  oscillations because the energy loss is so great
  that the system cannot complete one cycle.
• The actual motion depends on the initial
  conditions (eg is x is the displacement of a
  particle then the constants A and B are
  determined from the initial position and
  displacement).

(i) Heavy damping (dead beat)
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Damped Harmonic Motion
x
Initial velocity
t
Up to here Determined by initial conditions

• In an electrical circuit
• for no oscillations

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Damped Harmonic Motion

• Light damping
• In this case q is imaginary.
• Let
• So we get damped oscillations of angular
  frequency (smaller than 0)
• If x represents a physically observable quantity
  it must be real.
• Choose A and B to be complex numbers in order to
  make x real

Oscillations described by sin() Term and decay
exponentially
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Damped Harmonic Motion

• This is damped harmonic motion
• In the electrical case
• Oscillations for

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Damped Harmonic Motion

• (iii) ie q0 Critical damping
• This corresponds to an oscillation of infinite
  period or zero frequency.
• In this case, the most general solution is

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Damped Harmonic Motion
x
t

• At large times for fixed value of ?0 the value
  corresponds to the most rapid decay possible
  without overshooting and oscillations.
• Useful in needle indicators.

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Energy Loss

• In a mechanical oscillator
• then
• The energy is quite a complicated function of t.

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Energy Loss

• For the case of light damping
• Can neglect the first term in dx/dt
• The energy decays exponentially with a decay
  constant that is twice that of the amplitude.
• Describe decay by Quality factor, Q

So
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Forced Harmonic Motion

• This occurs when the harmonic oscillator is acted
  on by a sinusoidally varying external force
  driving force.
• The equation becomes
• An example would be an electrical circuit

R
L
C
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Forced Harmonic Motion

• Lets consider a situation with no damping
• Try a solution of the form
  with Agt0
• As the time dependence must cancel out we either
  have
• or

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Forced Harmonic Motion

• 3 different cases
• The requirement Agt0 means that ?0
  and so the displacement x(t) is in phase with the
  driving force.
• Then so the displacement
  is 180o out of phase with the driving force
• This condition is called resonance.
• The amplitude A is infinite the driving force
  had supplied an infinite amount of energy which
  would take an infinite amount of time if it was
  supplied at a finite rate.

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Forced Harmonic Motion

• For this special case there is a different
  solution which describes the situation after a
  finite time.
• This function describes the response of the
  system to the driving force

Amplitude grows with time
Displacement lags by ?/2
A
A0
?0
?
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Forced Harmonic Motion

• With damping (effects of energy dissipation)
• The solution has two parts
• x Forced vibrations Response
• DHM Remains
• (steady state)
• decays as

1
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Forced Harmonic Motion

• To find a solution use the method of complex
  exponentials.
• Introduce a new variable y which obeys
• (y has no physical significance, it just helps
  the maths)
• Solve this equation and take the real part z to
  find x.

2
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Forced Harmonic Motion

• Try a solution
• Then
• This is a solution provided that
• Put this back into eqn 3, write

3
complex number
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Forced Harmonic Motion

• This gives
• Which can be written in the form

where
and
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Forced Harmonic Motion
A
Amplitude
A0
?max
?
?
Phase
?0
?
ie at
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Forced Harmonic Motion

• These are smoothed out versions of the
  idealised curves for no damping.
• Maximum amplitude when
• Amax is not infinite because energy is
  dissipated.
• At resonance
• Amplification factor

for light damping
A
High Q factor ? high and narrow peak at the
resonant frequency
?
?0
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Summary Lecture 8 Diff Eqns II

• Lecture 8
• Second order differential equations
• Damped harmonic motion
• Forced harmonic motion
• Problem Set 8
• Finish 5.00pm