FORCED OSCILLATIONS

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FORCED OSCILLATIONS AND RESONANCE
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Let us consider mass-spring system with resistive
force and applied force Fmcos?t
-bv
ma
-kx
Fm cos?t
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Here a mechanical oscillator of mass m, force
constant k and resistance b is being driven by an
alternating force Fm cos?t
Equation of motion is
The complete solution for x consists of two terms
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• Phase difference between displacement and
  applied force is(Fp/2)
• Displacement lags applied force by (Fp/2)

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• -p/2 phase comes from term i
• -F phase comes due to Zm ,a complex variable
• Amplitude of displacement A(?) is Fm/?Zm,
  which is a function of ? (driving force
  frequency)

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• Phase difference between velocity and force is F.
  Velocity lags force by angle F.If F 0 then
  velocity and force are in phase.
• Phase difference between velocity and
  displacement is p/2. Velocity is always ahead of
  displacement by angle p/2.

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Impedance of Mechanical Oscillator Zm
The amplitude of velocity is Fm /Zm
It is defined as force required to produce unit
velocity in the oscillator.
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F depends upon ? driving force frequency
Phase difference between F (t) and x (t) is
(Fp/2) and between F (t) and velocity is F
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Variation of phase difference F with ?
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Phase diff F between v and F versus ?''
v lags F
Phase diff F between v and F
b increasing
o
?
?''
v and F In phase
v leads F
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Variation of (Fp/2) versus ?'' i.e. Phase diff
between x and F versus ?''
X lags F
Phase angle ?
b increasing
Phase diff -(Fp/2) between x and F
x and F in phase
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Displacement x (t) versus ?''
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Amplitude becomes max when freq of force comes
near to the freq of oscillator. This is called
Displacement Resonance
This frequency is called resonance frequency ?r
for displacement
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Animations for SHM

• Driven SHM

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Variation of amplitude of velocity versus ?''
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velocity resonance occurs at the natural freq ?
of the oscillator
v
Increasing b
?
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• All mechanical structures such as buildings,
  bridges and airplanes-have one or more natural
  frequencies of oscillation and if the structure
  is driven at a frequency near to the natural
  frequency, the structure begins to oscillate
  uncontrollably and can have disastrous
  consequences.

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POWER SUPPLIED TO OSCILLATOR BY THE DRIVING FORCE
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In steady state the amplitude and phase of a
forced oscillator adjust themselves so that the
average power supplied by the driving force
just equals that being dissipated by the
frictional force
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P(t) Instantaneous Power F(t) Instantaneous
Driving Force v(t) Instantaneous velocity
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Variation of Pav with ? Absorption Resonance
Curve
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Sharpness of peak depends upon the value of
damping constant b
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Pav
Pmax
Pmax 2
?
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BANDWIDTH- Freq range in which power is above
the half of the maximum power
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THE Q VALUE AS AN AMPLIFICATION FACTOR
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The displacement at low frequencies is amplified
by a factor of Q at displacement resonance
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Amax
Q5
Q4
Q3
Displacement
Q2
A0
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• P 10- An object of mass 0.2 kg is hang from a
  spring whose spring constant is 80 N/m. The body
  is subject to a resistive force given by bv,
  where v is its velocity and b4N-m-1 sec.
• Set up the differential equation of motion for
  free oscillations of the systems, and find the
  time period.
• The object is subjected to a sinusoidal driving
  force given by F (t) Fm sin?t, where Fm 2N
  and ? 30 sec-1 . In the steady state. What is
  the amplitude of the forced oscillations?

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Solun 10
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Solun 10(b)
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• P 11- Consider a damped oscillator with m0.2
  kg, b4 N-m-1 sec and k 80 N/m. Suppose that it
  is driven by a force FFm cos ?t, where Fm 2N
  and ? 30 sec-1.
• What are the values of A and d of the steady
  state response described by xA cos(?t-d) ?
• How much energy is dissipated against the
  resistive force in one cycle?
• What is the mean power input?

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Solun 11
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Solun 11
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P12. An object of mass 2 kg hangs from a spring
of negligible mass. The spring is extended by
2.5 cm when the object is attached. The top end
of the spring is oscillated up and down in SHM
with amplitude of 1 mm. The Q of the system is
15. (a) What is ? for the system? (b)
What is the amplitude of forced oscillation
at ? ?? (c) What is the mean power
input to maintain the forced oscillation
at a frequency 2 greater than ??
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Solun 12
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P13- The figure shows Pav versus ? for a mass
on a spring with damping.
(a) What is the numerical value of Q? (b) If the
driving force is removed, the energy decreases
according to the equation E E0 e -? t. What is
the value of ??
Pmax
Solun

Pmax /2
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• P14- The natural frequency of oscillation of a
  mechanical system is ?1. When this system is
  driven by a force F (t) Fm cos ? t (where
  ?is
• variable), it has a power resonance curve whose
  angular frequency width, at half-maximum power,
• is ?1/5.
• At what angular frequency does maximum Pav occur?

(b) The system consists of a mass m on a spring
of spring constant k. Find the value of constant
b in terms of m and k.
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Solun 14
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• P15- The graph shows the Pav absorbed by an
  oscillator when driven by a force of constant
  magnitude but variable frequency ?.
• At exact resonance, how much work per cycle is
  being done against the resistive force? (Period
  2p/?)
• At exact resonance, what is the total energy E0
  of the oscillator?
• If the driving force is turned off, how many
  seconds does it take before the energy decreases
  to a value E E0 e-1?

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Solun 15
Pmax
10 W
Pmax /2 5 W
0.995X106 106 1.005X106
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TWO-BODY OSCILLATIONS
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Length of the spring at any time x1 x2 Change
in length of the spring x (x1-x2)-L Force
exerted on each particle by spring kx
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P 17.27
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