damped  harmonic  oscillator   An oscillator containing frictional forces that damps its motion. The equation of motion of the system is written by modifying that of a simple harmonic oscillator* by incorporating a damping term. Since generally damping is proportional to the velocity the equation of motion of a damped oscillator is,

[d²x /dt²] + (1/t ) [dx /dt] + w ² x = 0

where  x  is the displacement of the particle from the mean position. The second term  is  due to damping. The constant, t is called the relaxation time of the oscillator, and (1/2t ) , is called the damping constant. w _o  is  the  angular frequency of the oscillator without damping. (See  simple  harmonic  oscillator). Three cases arise depending on the relative values of (1/2t ) and w _o.

1.  When  (1/2t )   >  w _o,  the  system  does not oscillate. (overdamped).

2. When (1/2t ) = w _o, the system is passing from non oscillatory to oscillatory (critically damped).

3.  When  (1/2t )  < w _o, the system will oscillate (under damped) with  angular frequency, w = [w _o² - (1/2t )²] ^1/2 . The amplitude of the oscillator decreases with time,

a(t) = a_o e ^{-t/2 t}

The energy of the oscillator is given by,

E(t) = E_oe^-t/t