1. Damped harmonic oscillator
Real oscillators are always damped. The damped oscillator shown in the figure consists of a mass m, a spring of constant k and a vane submarged in a liquid. The liquid exerts a damping force which in many cases is proportional to the velocity (with opposite sign):
b – damping constant (22)
In this case the equation of motion can be written as
(23)
After rearrangement we have
(24)
Figure from HRW,2
Introducing the substitutions: one gets
(25)
The solution of (25) for a small damping is:
(26)
where

2. Damped harmonic oscillator, cont.
Solution (26) can be regarded as a cosine function with a time dependent amplitude Time t = τ, after which the amplitude decreases e1/2 times is called the average lifetime of oscillations or the time of relaxation.
The angular frequency ω of the damped oscillator is less than that of undamped oscillator ωo. For the small damping, i.e. for ωo>> β, solution (26) can be approximated by
(27)
The amplitude for the damped oscillator decreases exponentially with time.

29. Damped harmonic oscillator, cont.
Energy losses for the damped oscillator
The amplitude for the damped oscillator decreases exponentially with time.
For the oscillator with a small damping one obtains for the average energy:
(10.28)
The average power of losses is:
(10.29)
Therefore the average power of losses is related to the average energy as
(10.29a)

30. Damped harmonic oscillator, cont.
For the oscillating system with damping one introduces the dimentionless factor, called quality factor Q, defined as follows:
(10.30)
For low damping one obtains from the last equation substituting for power of losses from (10.29a)
(10.31)
Applying the last result to the electrical oscillator L,R,C, and introducing the analogous quantities
one obtains for the quality factor
(10.32)
Examples of quality factors Q
resonance radio circuit several hundreds
violin string 103
microwave resonator 104
excited atom 107

31. 10.7. The driven oscillator with damping
The damped oscillator responds to a periodic driving force. In this case on the right side of eq.(10.24) we introduce the driving force F(t)
(10.33)
Moving support
Substituting
and one obtains from (10.33)
(10.34)
Let the driving force be
Then the solution of eq.(10.34) takes the form
(10.35)
where the amplitude x0 is given by
(10.35a)
and the phase angle φ:
(10.35b)
The system is driven by a moving support that oscillates at an arbitrary angular frequency ω. The natural frequency of a freely oscillating system is ω0.

32. The driven oscillator with damping, cont.
Analysis of solution (10.35).
driving frequency much lower than the natural frequency ω0 (ω << ω0)
In this case:
resonance – the condition at which
the amplitude of a displacement (or velocity, or power) of oscillations is maximum.
For ω = ω0 the amplitude x0 is not generally maximum:
when is minimum.
Amplitude of the driven oscillations as a function of the driving force frequency for varying damping
(b1<b2<b3). Smaller damping gives taller and narrower resonance peak.
Thus the condition for minimum is:
what yields:

33. The driven oscillator with damping, cont.
The velocity of the driven damped oscillations can be calculated by differentiation
of eq.( 10.35):
(10.36)
The amplitude of a velocity is then given by:
From (10.37) it follows that the amplitude of
a velocity is maximum exactly for ω = ω0 .
From the analogy between oscillating mechanical
system and electrical circuit:
one obtains that the maximum
of electrical current amplitude
(resonance) is for ω = ω0.
Amplitude of a current in a resonance electrical circuit for a varying supply voltage frequency.