BASIC ACOUSTICS 4

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BASIC ACOUSTICS (4)

• Mechanical Resonance

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For a force oscillations, the general solution
The solution of equation is The sum of two parts
a transient term a
steady-state term
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For the case of a sinusoidal driving force
f(t)Fmcos(?t) applied to the oscillator at some
initial time, the solution of (1-3) is the sum of
two parts a transient term containing two
arbitrary constants and a steady-state term which
depends of F and ? but does not contain any
arbitrary constants. The transient term is
obtained by setting F equal to zero. The
arbitrary constants are determined by applying
the initial conditions to the total solution.
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After a sufficient time interval, The damping
term makes this portion of the solution
negligible. Leaving only the steady state term
whose angular frequency ? is that of the driving
force
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Zm is called the complex mechanical impedance ,Rm
is called the mechanical resistance Xm is
called the mechanical reactance
The mechanical impedance
Has magnitude
And phase angle
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ENERGY RELATION
The instantaneous power ,supplied to the system
in the steady state is equal the product of the
instantaneous driving force and the resulting
instantaneous speed.
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ENERGY RELATION
Substituting the appropriate real expressions for
force and speed
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In most situations the average power is more
significance than the instantaneous power.
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Mechanical Resonance

• In the steady state, the displacement is equal to
  

Speed gives
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If the speed as given by above is plotted as a
function of the frequency of a driving force , a
curve is obtained as follow
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The (angular) frequency of mechanical resonance
is defined as that at which the mechanical
reactance Xm vanishes, this is the frequency at
which a driving force will supply maximum power
to the oscillator.It was also found to be the
frequency of free oscillation of a similar
undamped oscillator.
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• At this frequency the mechanical impedance has
  its minimum value of ZmRm
• It is also the frequency of maximum speed
  amplitude.

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• Note that This frequency does not give the
  maximum displacement amplitude.

If the average power supplied to the system as
given by
Therefore
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• When

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based on equation ,a curve is obtained
The curve has a maximum value of F2/2Rm at the
resonance frequency and falls at lower and
higher frequencies.
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Here Qm is the quality factor of the system
The sharpness of the peak of the power curve is
primarily determined by Qm. If it is large, the
curve falls off very rapidly-a sharp resonance.
If ,on the other hand, it is small, the curve
falls off more slowly and the system has a broad
resonance.
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for
We obtain
Where f1and f2 are the two frequencies
,respectively above and below resonance
frequency for which the average power has dropped
to one-half its resonance value.
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The displacement amplitude
The frequency that makes displacement a maximum
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Displacement frequency curve is shown as follow
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TRANSIENT RESPONSE OF AN OSCILLATOR

• We will consider the effect of superimposing the
  transient response on the steady-state condition.
• The complete general solution is

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As a special case let us assume that
Substitute into above equation to obtain
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The effect of the transient is apparent in the
left-hand portion of the curve, but near the
right-hand end the transient has been so damped
that the final steady state is nearly reached.
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HomeworkP273 1-5 and 1-12

• 1-5 The diaphragm of a loud speaker weighs 1 g,
  and the displacement of its driving rod 1mm from
  equilibrium requires a force of 106dynes. The
  frictional force opposing motion is proportional
  to the diaphragms velocity and is 300 dynes when
  the velocity is 1 cm per sec. If it is assumed
  that the diaphragm moves like a simple
  oscillator, what will be its natural frequency,
  and what its modulus of decay?
• The driving rod is driven by a force of
  100,000coswt dynes. Plot a curve of the real and
  imaginary parts of the mechanical impedance of
  the diaphragm as function of frequency, from f0
  to f1000Hz.

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Equivalent electrical circuit for a simple
oscillator