1. Lecture No. 2: Damped Oscillations, Driven Oscillations and Resonance
October 15th 2014
Dr. H. SAIBI

2. 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.

3. 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)
The energy is proportional to the square of the amplitude, and the square of the amplitude decreases exponentially with increasing time:
(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.
From Newton’s second law, the motion of a damped system is: with rearrangement: (Differential equation for a damped oscillator).
The solution for the underdamped case is: where Ao is the initial amplitude, the frequency ’ is related to o (frequency with no damping) by: .
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 given by: By squaring both sides of this equation and comparing the results with Eq. 2, we have:
If the damping constant b is gradually increased, the angular frequency ’ decreases until it becomes zero at the critical value:
Eq.1
Eq.2
Definition-Time constant
Eq.3
Eq.4
Eq.5
Eq.6
Eq.7
Eq.8
Eq.9

4. 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)

5. 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)

6. 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:
where: and
A damped oscillator is often described by its Q factor (for quality factor):
The Q factor is dimensionless.
We can relate Q to the fractional energy loss per cycle. Differentiating Eq. 10 gives:
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:
Q is thus inversely proportional to the fractional energy loss per cycle.
Eq.10
Eq.11
Eq.12
Eq.13
Definition-Q factor
Eq.14
Physical interpretation of Q for weak damping
Eq.15

7. 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:
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:
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.16
Eq.17
Eq.18

8. Damped Oscillations
If we substitute (2)av=E/m for 2 in Eq. 17, we have:
Rearranging Eq.19 gives:
which upon integration gives:
which is Eq.10.
Eq.19
Eq.20
Eq.21

9. 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).

10. 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)

11. 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:
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:
where we have used ax=d2x/dt2. Substituting m o2 for k (=SQRT (k/m)) and rearranging gives:
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:
Eq.23
Eq.24
Eq.25
(Differential Equation for a driven oscillator)
(Position for a driven oscillator)
Eq.26

12. 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)
and the phase constant  is given by
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).
Eq.27 +
Eq.28
(phase constant for a driven oscillator)

13. Mathematical Treatment of Resonance
The velocity of the object in the steady state is obtained by differentiating x with respect to t:
At resonance, =π/2, and the velocity is in phase with the driving force:
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:
Eq.29
Eq.30