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Lecture No. 2:
Damped Oscillations, Driven
Oscillations and Resonance
October 15th 2014
Dr. H. SAIBI
Damped Oscillations
• Damped system: Pendulum or spring stops oscillating because the mechanical
energy is dissipated by frictional energy.
• Overdamped: Damping is very large (speed approaches zero as the object
approaches the equilibrium position).
• Underdamped: Damping is very small (system oscillates with a amplitude that
decreases slowly with time) Example: child on a playground swing when a
parent stops providing a push each cycle.
• Critically damped: Motion with the minimum damping for nonoscillatory
motion.
Damped Oscillations
• Underdamped motion: The damping force exerted on an oscillator such as the
one shown in Fig. 1 can be expressed by:
(b is a constant) Eq.1
• The energy is proportional to the square of the amplitude, and the square of the
Eq.2
amplitude decreases exponentially with increasing time:
Definition-Time constant
(A: amplitude, A0 is the amplitude at t=0, and  in the decay time or time constant). The time constant is the
time for the energy to change by a factor of e-1.
Eq.3
• From Newton’s second law, the motion of a damped system is:
with rearrangement:
(Differential equation for a damped oscillator). Eq.4
Eq.5 where
• The solution for the underdamped case is:
Ao is the initial amplitude, the frequency ’ is related to o (frequency with no
damping) by:
Eq.6.
• For a mass on a spring
. For weak damping:
and ’ is nearly
equal to o. The dashed curves in Fig. correspond to x=A and x=-A, where A is
Eq.7 . By squaring both sides of this equation and
given by:
comparing the results with Eq. 2, we have:
Eq.8
• If the damping constant b is gradually increased, the angular frequency ’
decreases until it becomes zero at the critical value:
Eq.9
Damped Oscillations
Fig. 1. (a) A damped oscillator suspended in a viscous liquid. The motion of the cylinder
is damped by drag forces. (b) Damped oscillation curve (W.H. Freeman and Company, 2008)
Damped Oscillations
•
•
•
•
When b is greater that or equal to bc, the system does not oscillate.
If b>bc, the system is overdamped.
The smaller b is, the more rapidly the object returns to equilibrium.
If b=bc, the system is said to be critically damped and the object returns to
equilibrium (without oscillation) very rapidly.
• Fig shows plots of the displcement versus time of a critically damped and an
overdamped oscillator. We often use critical damping when we want a system to
avoid oscillations and yet return to equilibrium quickly.
Fig. 2. Plots of displacement versus time
for a critically damped and an overdamped
oscillator, each released from rest (W.H.
Freeman and Company 2008)
Damped Oscillations
• Because the energy of an oscillator is proportional to the square of its amplitude,
the energy of an underdamped oscillator (averaged over a cycle) also decreases
exponentially with time:
Eq.10
where:
Eq.11 and
Eq.12
• A damped oscillator is often described by its Q factor (for quality factor):
Definition-Q factor
 The Q factor is dimensionless.
Eq.13
• We can relate Q to the fractional energy loss per cycle. Differentiating Eq. 10
gives:
Eq.14
• If the damping is weak so that the energy loss per cycle is a small fraction of the
energy E, we can replace dE by E and dt by the period T. Then E/E in one
cycle (one period) is given by:
so:
Eq.15
Physical interpretation of
Q for weak damping
• Q is thus inversely proportional to the fractional energy loss per cycle.
Damped Oscillations
• You can estimate  and Q for various oscillating systems. Tap a crystal water
glass and see how long it rings. The longer it rings, the greater the value of 
and Q and the lower the damping. Glass beakers from the laboratory may also
have a high Q. Try tapping a plastic cup. How does the damping compare to
that of the glass beaker? . In terms of Q, the exact frequency of an
underdamped oscillator is:
Eq.16
Because b is quite small (and Q is quite large) for a weakly damped oscillator,
we see that ’ is nearly equal to 0. We can understand much of the behavior
of a weakly damped oscillator by considering its energy. The power dissipated
by the damping force equals the instantaneous rate of change of the total
mechanical energy:
Eq.17
• For a weakly damped oscillator with linear damping, the total mechanical
energy decreases with time. The average kinetic energy per cycle equals half
the total energy:
Eq.18
Damped Oscillations
• If we substitute (2)av=E/m for 2 in Eq. 17, we have:
Eq.19
Rearranging Eq.19 gives:
which upon integration gives:
which is Eq.10.
Eq.20
Eq.21
Driven Oscillations and Resonance
Fig. 4. An object on a vertical
spring can be driven by
moving the support up and
down (W.H. Freeman and Company,
2008)
Fig. 5. By damping the swing,
the young woman is
transferring her internal
energy into the mechanical
energy of the oscillator (Eye
Wire/Getty from (W.H. Freeman and
Company, 2008).
Driven Oscillations and Resonance
• Resonance Width For Weak Damping:
Resonance Width for Weak Damping
Eq.22
Fig. 6. Resonance for an oscillator (W.H. Freeman and Company, 2008)
Mathematical Treatment of Resonance
• We can treat a driven oscillator mathematically by assuming that, in addition to
the restoring force for a damping force, the oscillator is subject to an external
driving force that varies harmonically with time:
Eq.23
where Fo and  are the amplitude and angular frequency of the driving force. This frequency is
generally not related to the natural angular frequency of the system o.
Newton’s second law applied to an object that has a mass m attached to a spring
that has a force constant k and subject to a damping force –bx and an external
force fo cos t gives:
Eq.24
where we have used ax=d2x/dt2. Substituting m o2 for k (=SQRT (k/m)) and
rearranging gives:
Eq.25
(Differential Equation for a driven oscillator)
• We now discuss the general solution of the Eq.25 qualitatively. It consists of two
parts, the transient solution and the steady-state solution. The transient part of
the solution is identical to that for a damped oscillator given in Eq.5. The
constants in this part of the solution depend on the initial conditions. Over time,
this part of the solution becomes negligible because of the exponential decrease
of the amplitude. We are then left with the steady-state solution, which can be
written as:
(Position for a driven oscillator) Eq.26
Mathematical Treatment of Resonance
• Where the angular frequency  is the same as that of the driving force. The
amplitude A is given by
(amplitude for a driven oscillator)
Eq.27
and the phase constant  is given by
+
Eq.28 (phase constant for a
driven oscillator)
• Comparing Eqs.23&26, we can see that the displacement and the driving force
oscillate with the same frequency, but they differ in phase by . When the
driving frequency  approaches zero,  approaches zero, as we can seen from
Eq.28.
• At resonance,  = o and  =90o, and when  is much greater than o, 
approaches 180o. The phase of a driven oscillator always lags behind the phase
of the driving force. The negative sign in Eq.26 ensures that  is always
positive (rather than always negative).
Mathematical Treatment of Resonance
• The velocity of the object in the steady state is obtained by differentiating x
with respect to t:
Eq.29
• At resonance, =π/2, and the velocity is in phase with the driving force:
Eq.30
• Thus at resonance, the object is always moving in the direction of the driving
force, as would by expected for maximum power input. The velocity amplitude
A is maximum at: